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THEORETICAL AND EXPERIMENTAL STUDY OF WIND—AIDED
FLAME SPREAD OVER VAPORIZING SOLIDS

By

SANJAY AGRAWAL

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1992

g.) u,
o / n)?“ 3.x / >\

ABSTRACT

THEORETICAL AND EXPERIMENTAL STUDY OF WIND-AIDED
FLAME SPREAD OVER VAPORIZING SOLIDS

By

SAN J AY AGRAWAL

This study presents a theoretical and experimental investigation of wind-aided ﬂame
spread over vaporizing solids. Effects of free stream velocity and oxygen mass frac-
tion, solid-phase unsteadiness, blowing and surface radiative heat loss on pyrolysis and
ﬂame front speed (location) are investigated. Investigations of the thermal degradation
of vaporizing solids and the stability of wind-aided ﬂame spread across a ceiling are

also performed.

Pyrolysis and ﬂame from location and production rates of major chemical species
were measured as functions of time for ﬂames spreading over PMMA samples for
various coﬂow speeds and oxygen mass fractions. The surface heat ﬂux distribution in
the burning zone is determined from burned samples using a regression model. Mass
loss rate and in-depth temperature measurements were obtained during pyrolysis of

PMMA samples in nitrogen environments for various heat ﬂuxes.

Explicit expressions are deduced for the pyrolysis front speed, ﬂame tip location,
surface regression rate, mass loss rate and surface heat ﬂux under the ﬂame. Numeri-
cal results are correlated to provide expressions for the surface heat ﬂux in the preheat

zone and the excess pyrolyzate.

The pyrolysis front speed varies linearly with both free stream velocity and oxy-

gen mass fraction, comparing favorably with experiments. The solid is always

transient during the ﬂame spread process, as conﬁrmed by degradation experiments.
Inclusion of surface radiative heat loss is necessary to produce a phenomena observed
during the experiments, ﬂame spread termination before the end of the sample. Using
burned samples and a model it was determined that the net surface heat ﬂux in the
burning zone decreases with the streamwise distance, and increases with the free stream
velocity. The inﬂuence of the gas-phase reaction rate and solid-phase cooling is lim-
ited to the ﬁrst 3 cm of the sample. This, coupled with surface radiative heat losses
and solid-phase unsteadiness are the main reasons for the shorter experimental ﬂame
overhang.

An inviscid stability analysis suggests that in the ceiling conﬁguration the ﬂame is

stable because of forced convection. For small Froude numbers it is necessary to

include ﬁnite reaction rates in the analysis.

Copyright by

SANJAY AGRAWAL
1992

To my family

ACKNOWLEDGEMENTS

I am grateful to my advisor Dr. Indrek Wichman, without whose support and gui-
dance this completed work would not have been possible. I am especially grateful to
him for providing ﬁnancial support, encouragement and the means to attend numerous

professional meetings through the course of my doctoral program.

Dr. Arvind Atreya has also been instrumental in helping me achieve this goal.
He provided great moral and professional support. I am especially grateful to him for
his help in the experimental portion of my research and for letting me use his lab.

The long nightly discussions with him will be fondly remembered.

I am thankful to Drs. John Lloyd, Manoochehr Koochesfahani and Milan Mik-
lavcic, who formed the rest of my committee, for carefully reading my work and giv-
ing useful suggestions. The use of G-splines, suggested by Dr. Miklavcic is greatly
appreciated. I am thankful to my colleagues and friends, especially Dr. Kamel Mekki,

for his help in conducting ﬂame spread experiments.

I dedicate this thesis to my family: my parents - who have been my strongest
source of inspiration kept me constantly motivated and provided invaluable support
through regular long-distance phone-calls and letters; my sister and brother-in-law,
brother and Iother family members for their abiding conﬁdence in my abilities; and my

wife - Madhu, for her constant love, understanding, companionship and support.

Funding for this research was provided by U.S Department of Agriculture under
contract numbers 86-FSTY-9-90192 and 90-37291-5768. Equipment was funded by
NSF under the grant number CBT-8415423.

vi

TABLE OF CONTENTS

List of Tables .............................................................................................................
List of Figures ............................................................................................................
Nomenclature ..............................................................................................................

Chapter I: INTRODUCTION ....................................................................................

Synopsis ..........................................................................................................
1.1. Introduction .............................................................................................
1.2. Wind-Aided Flame Spread .....................................................................
1.2.1. Statement of the Problem ........................................................
1.2.2. Literature Review .....................................................................
1.2.3. Theoretical Framework ............................................................
1.3. Objectives of the Present Work .............................................................
1.4. Organization of the Thesis .....................................................................

Chapter 2: A PRELIMINARY MODEL OF FLAME SPREAD .............................

Synopsis ..........................................................................................................

2.1. Introduction .......... '. ..................................................................................

2.2. Problem Formulation .........................................

2.3. Solution ...................................................................................................

2.3.1. Pyrolysis Zone (0 < g < ép) ...................................................

2.3.2. Preheat Zone (ép < 5 < 1) ......................................................

2.4. Results .....................................................................................................

2.4.1. Pyrolysis Front Speed ..............................................................

2.4.2. Flame Front Speed ...................................................................

2.4.3. Flame Location ........................................................................

2.5. Discussion ...............................................................................................

2.6. Conclusions .............................................................................................
Appendix 2.A: Green’s Function Solution of Gas-Phase Preheat Zone

Equation .................................................................................

vii

xi
xii
xvi

15

15
16
18
27
31
33
35
37
37
38
39
49

51

Appendix 23: Laplace Transform Solution of Gas-Phase Preheat

Zone Equation ....................................................................... 53

Appendix 2.C: Green’s Function Solution of Solid—Phase Preheat
Zone Equation ....................................................................... 54
Appendix 2.D: Evaluation of the Integral I (A ,B) ........................................ 55
Chapter 3: HEAT FLUX DISTRIBUTIONS ............................................................ 57
Synopsis .......................................................................................................... 57
3.1. Introduction ............................................................................................. 58
3.2. The Model ............................................................................................... 59
3.3. Solutions for Surface Heat Flux and Surface Temperature .................. 65
3.3.1. Heat Flux ....................... 65
3.3.2. SurfaceTemperature ................................................................ 70
3.4. Solutions for Normalized Fluxes and Overall Fluxes ........................... 72
3.4.1. Excess Pyrolyzate .................................................................... 81
3.5. Conclusions ............................................................................................. 86
Chapter 4: AN EXPERIMENTAL INVESTIGATION ............................................ 90
Synopsis .......................................................................................................... 90
4.1. Introduction ............................................................................................. 91
4.2. Experimental Setup ................................................................................. 92
4.2.1. Combustion Wind Tunnel ....................................................... 92
4.2.1.1. Inlet Section .............................................................. 92
4.2.1.2. Turbulence Manipulation Section ............................ 94
4.2.1.3. Test Section ............................................................... 95
4.2.1.4. Exhaust Section ......................................................... 96
4.2.2. Gas Analysis Equipment ......................................................... 96
4.2.2.1. Total Hydrocarbon Analyzer .................................... 98
4.2.2.2. Water Analyzer ......................................................... 98
4.2.2.3. CO-C02 Analyzer .................................................... 98
4.2.2.4. Oxygen Analyzer ...................................................... 98
4.2.3. Data Acquisition ...................................................................... 99
4.2.4. Sample Preparation .................................................................. 99
4.2.5. Experimental Procedure ........................................................... 99
4.3. Results ..................................................................................................... 99
4.3.1. Temperature Measurements ..................................................... 101
4.3.2. Flame Spread Rates ............... , .................................................. 103
4.3.3. Species Production Rates ....................................................... .. 108

viii

4.4. Discussion ..................................................................................... 111

4.5. Conclusions ............................................................................................. 112

Chapter 5: THERMAL DEGRADATION OF A VAPORIZING SOLID .............. 114

Synopsis .......................................................................................................... 1 14

5.1. Introduction ............................................................................................. 114

5.2. Goveming Equations .............................................................................. 117

5.3. Solution ................................................................................................... 120

5.3.1. Preheating Stage ....................................................................... 121

5.3.2. Vaporizing Stage ...................................................................... 121

5.3.2.1. In-depth Temperatures Proﬁles at the

Vaporization Temperature ....................................... 123

5.3.2.2. Transient Mass Loss Rate ........................................ 123

5.4. Experimental Setup .................................................................... . ............ 125

5.4.1. Effect of Sample Thickness and In-depth Absorption 131

5.5. Comparison with the Literature .............................................................. 133
Chapter 6: FLAME SPREAD OVER AN UNSTEADILY VAPORIZING

SOLID ...................................................................................................... 137

Synopsis .......................................................................................................... 137

6.1. Introduction ............................................................................................. 138

6.2. Model Problem ........................................................................................ 139

6.3. Solution ................................................................................................... 143

6.3.1. Surface Heat Flux under the Flame ........... 143

6.3.2. Mass Flux in the Pyrolysis Zone ............................................ 146

6.3.3. Flame Front Location .............................................................. 147

6.4. Results and Discussion ........................................................................... 148

6.4.1. Effect of Blowing .................................................................... 151

6.4.2. Effect of Radiant Losses from the Surface ............................. 155

6.4.3. Inﬂuence of Unsteady Pyrolysis .............................................. 156

6.6. Conclusions ............................................................................................. 158

Appendix 6.A: Blowing Correction ............................................................... 159

Appendix 68: Turbulent Wind-Aided Flame Spread .................................. 162

Chapter 7: SURFACE HEAT FLUX IN THE BURNING ZONE .......................... 164'

Synopsis .......................................................................................................... 164

ix

7.1. IntroduCtion ............................................................................................. 165

7.2. Net Surface Heat Flux in the Burning Zone ......................................... 167

7.3. Regressed Surface Location ................................................................... 148

7.4. Pyrolysis Experiments: Determination of so ......................................... 170

7.5. Flame Spread Experiments: Determination of to .................................. 171

7.6. Results and Discussion ........................................................................... 173

7.6. Conclusions ............................................................................................. 176
Chapter 8: STABILITY ANALYSIS ........................................................................ 177
Synopsis .......................................................................................................... 177

8.1. Introduction ............................................................................................. 178

8.2. Steady Flame Spread .............................................................................. 181

8.2.1. Basic-State Velocity Proﬁle ..................................................... 185

8.2.2. Basic-State Temperature Proﬁle .............................................. 185

8.3. Stability Analysis .................................................................................... 193

8.4. Results and Discussion ........................................................................... 195

8.5. Conclusions ............................................................................................. 205
Appendix 8.A: Reaction Zone Solution for Temperature ............................ 208
Appendix 83: Asymptotic Approximation for T, as n—mf ....................... 210
Appendix 8.C: Analysis of the Disturbance Equation .................................. 211
Chapter 9: CONCLUSIONS ........................................ , ............................................. 214
Synopsis .......................................................................................................... 214

9.1. Conclusions ............................................................................................. 215

9.2. Suggestions for Future Work ................................................................. 220

LIST OF REFERENCES ........................................................................................... 221

LIST OF TABLES

Table 3.1 Ratios of the heat released by the ﬂame, or received by the
surface in each zone, to the total heat released, as a function of
Y0... Here tire is the excess pyrolyzate, "'2’, is the total mass
lost from the solid, 9" and gm are the total ﬂame heat release
in the pyrolysis zone and the preheat region, respectively, and
g-py, Em: and E” are the total surface heat ﬂux in the pyrolysis
zone, the under-ﬂame preheat zone and the preﬂame preheat
zone, respectively. The quantity 3707 is the total heat released

by the ﬂame. ........................................................................................ 85
Table 6.1 Property values ..................................................................................... 154
Table 8.1 Table of parameters used. Parameters a -b and QYO .Jv

are obtained from Eq. (8.A.2) and (8.17) respeCtively ....................... 192
Table 8.2 Values of the nondimensional activation energy [3 used

for the different B numbers ................................................................. 192

xi

Figure 1.1

Figure 1.2
Figure 1.3
Figure 2.1

Figure 2.2

Figure 2.3
Figure 2.4

Figure 2.5

Figure 2.6

Figure 2.7
Figure 2.8

Figure 3.1
Figure 3.2

Figure 3.3a

Figure 3.3b

LIST OF FIGURES

Modes of ﬂame spread .........................................................................
Schematic diagram of wind-aided ﬂame spread ..................................
Thesis ﬂow chart ....................................
Schematic diagram of wind-aided ﬂame spread; Pyrolysis zone

extends till xp (1 ) and ﬂame reattaches at xf(t ). Superscript *

signiﬁes a dimensional quantity ...........................................................
Equations and boundary conditions in the pyrolysis zone ..................
Equations and boundary conditions in the preheat zone ....................

Variation of mass ﬂux parameter (M) and surface fuel mass
fraction (Yps) as a function of QYO .JL ..............................................

Variation of pyrolysis front speed (5p) and ﬂame front speed
(ﬁf) as a function of free stream oxygen mass fraction (Y0 -) ..........

Spread rates versus mass ﬂux parameter. Here é is the non-
dimensional pyrolysis front speed, Q is the nondimensional

ﬂame front speed, HP is the approximate (zero-M assumption)
nondimensional pyrolysis front speed and if is the approximate
(zero-M ) nondimensional ﬂame front 5 eed. The reason the

curves do not coincide is that E and f are evaluated for

M = 0 whereas in the present eory the minimum value of M

is 0.0636 ................................................................................................

Ratio of spread rates (ép/ E”) versus mass ﬂux parameter M ...........

Flame locations at different nondimensional time instants. The x
and z coordinates are nondimensional Howarth coordinates ..............

Model conﬁguration ..............................................................................
Plot of h(§,§p ), given by Eq. (3.3) et seq, versus é. Observe
that h is piecewise continuous at if ...................................................

Plot of the normalized nondimensional heat ﬂux gT(§) versus 5,
obtained from the solution of Eq, (3.4) using G-splines. The
stepsize A: is increased by the factor 10 at 3Q. The maximum

of the curve represents the location 9 ...............................................

Plot of the nondimensional heat ﬂux, g :00; /2)gT/‘5c_ , versus
é. Here g (Q) is always less than g (ép), even though g (Q) is
always greater than gﬁﬁp) ..................................................................

xii

3
6
13

19
23
24

4o

41

43
44

46
6O

64

68

69

Figure 3.4

Figure 3.5a

Figure 3.5b

Figure 3.6a
Figure 3.6b

Figure 3.7

Figure 4.1
Figure 4.2
Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6
Figure 4.7

Figure 4.8

Figure 4.9

Figure 5.1
Figure 5.2

Plot of the reduced temperature I: , given by Eq. (3.10), versus
E. The ﬁrst kink (from left) in the graph (for Y0..=1) represents
the location of the ﬂame front. The second and third kinks
(numerical error) occur because of step size changes that make
the computation faster. This is also seen in Fig. 3.3a for Y0..=1

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

Heat ﬂux correlation for the under-ﬂame preheat zone, showing
the rise of the heat ﬂux as the downstream ﬂame tip (reattach-
ment point) is approached ....................................................................

Heat ﬂux correlation for the pre-ﬂame preheat zone. Note how
rapidly the decay occurs .......................................................................

Heat ﬂux of Figure 3. 3a for ﬁxed x (xo=1) versus time t .................

Surface temperature of Figure 3.4 for ﬁxed x (x0 =1) versus
time t.
Plot of the ratio gig/g};- given by Eq. (3.13). Also shown is
the graph of 5/90! (dotted line), for comparison. In this ﬁgure
the parameters are for PMMA with a wind speed of 1 m/s . In
our model, because of the Oseen ﬂow assumption, the wind
speed appears as a non-dimensional quantity, and hence, will
not change the above ratio.

Schematic of the experimental apparatus ............................................
Schematic of the gas analysis equipment ............................................

Schematic of wind-aided ﬂame spread in the tunnel over a
ceiling-mounted sample of PMMA ......................................................

Measured surface temperatures and their rate of change with

time for PMMA during ﬂame spread (U .. = 0.9 m/s ,

Y0... = 0.331). Here surface temperatures are plotted for

different streamwise locations on the sample ......................................
Flame front location as a function of time. Note the slight
acceleration for xf 2.40 cm. Also note the negligible error
[(O(mm)], encountered in determination of ﬂame front location
using VCR ............................................................................................
Pyrolysis-front location as a function of time .....................................

Dependence of the pyrolysis and ﬂame front speeds on the free
stream velocity. ......................................................................................

Dependence of pyrolysis and ﬂame front speeds on the free
stream oxygen mass fraction (blank symbols represent Up and
crossed symbols represent U f) ............................................................

Dependence of the species production rates on pyrolysis front-
location (U, = 0.9 m/s; 1 - Y0, = 0.23; 2 — Y0“ = 0.43;

3 — Y0... = 1.0) ......................................................................................
Schematic of the solid phase model ....................................................
Surface temperature and total mass loss as a function of time

for PMMA. Heat ﬂux = 2.1 W/cm2

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

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

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

xiii

73

74
77

93
97

100

102

104
105

106

107

109
118

Figure 5.3
Figure 5.4

Figure 5.5
Figure 5.6
Figure 5.7
Figure 5.8
Figure 5.9

Figure 5.10
Figure 5.11
Figure 5.12

Figure 5.13

Figure 6.1
Figure 6.2

Figure 6.3
Figure 6.4

Figure 6.5

Figure 6.6

Figure 7.1
Figure 7.2
Figure 7.3
Figure 7.4
Figure 7.5

Figure 8.1a
Figure 8.1b

Figure 8.2

Figure 8.3a

Surface temperatures as a function of time, during preheating of
the solid .................................................................................................

In-depth temperatures as a function of time, during preheating
of the solid ............................................................................................

Mass loss rate as a function of time .................................................... '

Varying external heat ﬂux ....................................................................

- Mass loss rate as a function of time under varying heat ﬂux ............

Effect of sample thickness on mass loss rate ......................................
Schematic of the apparatus ..................................................................

In-depth temperatures as a function of time, during vaporization
of the solid under q =3. 4W /cm

Mass loss rate as a function of time. Comparison of integral
model with the data of Vovelle et a1. (1987) ......................................

Mass loss rate as a function of time. Comparison of numerical
model with the data of Vovelle et al. (1984b) ....................................

Mass loss rate as a function of time. Comparison of integral
model with the data of Vovelle et al. (1984a) ....................................

Schematic of the model ........................................................................

Dependence of total oxygen depletion (total fuel production
x v) on the pyrolysis front location. Symbols represent experi-
mental results. 1—Y0”=0.233; Z-Yo..=0-43; 3—Yo,,=0.61 ..................

Blowing correction ...............................................................................

Comparison between the convective heat ﬂux and the net heat
ﬂux ........................................................................................................

Dependence of oxygen depletion rate on the pyrolysis

front location (Parametric dependence). Yo ..=O.233. Symbols
represent experimental results. Solid curve represents the model
predictions. Calculated Curves (1a) to (1e) are for different
conditions as explained in Secs. 6.4.1-643 ........................................

Effect of blowing correction with transient mass ﬂux on
total oxygen depletion rate. Solid curve represents the model
predictions. Calculated curves (1a), (1b) and (1f) are for

different blowing corrections as explained in Appendix 6.A .............
Schematic of the gas-phase model .......................................................
Schematic of the solid-phase model ....................................................
Regressed surface location as a function of time ................................
Net surface heat ﬂux in the burning zone ...........................................
Net surface heat ﬂux near the leading edge ........................................
Physical conﬁguration of the wind-aided ﬂame spread ......................
Schematic diagram of the model .........................................................

Basic- state temperature and velocity proﬁles. Note that
T—proﬁle plotted here is for the inﬁnite-reaction-rate .........................

Inner zone (reaction region) temperature proﬁles, .8 =0.275.
Curves 1-6 represent proﬁles for different [3’s ....................................

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

xiv

122

124
126
127
128
129
130

134
135
136
140
150
152

153

154

161
166
169
172

174

175
179
166

186

189

Figure 8.3b
Figure 8.3c

Figure 8.4

Figure 8.5a

Figure 8.5b

Figure 8.5c

Figure 8.6a

Figure 8.6b

Figure 8.6c

Figure 8.7

Figure 9.1

Inner zone (reaction region) temperature proﬁles, B =1.125.

Curves 1-6 represent proﬁles for different B’s .................................... 190
Inner zone (reaction region) temperature proﬁles, B =1.6.

Curves 1-6 represent proﬁles for different B’s .................................... 191

Froude number F r0 as a function of free stream velocity
and streamwise distance ....................................................................... 196

Stability curve; non—dimensional frequency versus Froude
number, B =0.275. Curves 1-6 are for different inner tempera—
ture proﬁles of Fig. 8.3a ....................................................................... 199

Stability curve; non-dimensional frequency versus Froude
number, B =1.125. Curves 1-8 are for different inner tempera-
ture proﬁles of Fig. 8.3b ...................................................................... 200

Stability curve; non-dimensional frequency versus Froude
number, B =1.6. Curves 1-6 are for different inner temperature
proﬁles of Fig. 8.3c .............................................................................. 201

Frequency curve; neutral frequency versus streamwise distance,
B =0.275. Curves 1-6 are for different inner temperatru'e
proﬁles of Fig. 8.3a .............................................................................. 202

Frequency curve; neutral frequency versus streamwise distance,
B =1.125. Curves 1-8 are for different inner temperature
proﬁles of Fig. 8.3b .............................................................................. 203

Frequency curve; neutral frequency versus streamwise distance,

B =1.6. Curves 1-6 are for different inner temperature proﬁles

of Fig. 8.3c ............................................................................................ 204
Variation of frequency (0)0) with —Im(ao ). Curves plotted

are for temperature proﬁle #5 for F r0 equal to 2 and 4;

B =1 . 125. .............................................................................................. 206

Variation of kpCp as a function of temperature ................................ 219

XV

NOMENCLATURE

CHAPTERS 2-3

 

mmmgwmenmowb

3“)
3?
v

N «sax g < q: H”2JI>UIQ$§5-5?F‘ml

Pre-exponential factor

Mass transfer number

Speciﬁc heat

Speciﬁc heat ratio (gas to solid)

Coefﬁcient matrix (Eq. 3.5)

Mass diffusion coefﬁcient

Activation energy

Heat ﬂux to the interface

Normalized heat ﬂux

Total heat ﬂux (Eq. 3.13)

Total heat released by ﬂame

Transformed heat ﬂux deﬁned before Eq. (3.4)
Function deﬁned as h (x )= ‘11—: xexp (x 2)erfc (-x)
Variable deﬁned in Eq. (3.4)

Latent heat of vaporization; reference lengths in gas and solid
Lewis number, Le =0t/D

Mass ﬂux from solid to gas (dimensional)

Mass ﬂux parameter

Prandtl number -

Heat of combustion per unit mass of fuel vapor
Thermal responsivity ratio (gas to solid); reaction term
Universal gas constant

Time

Temperature

Streamwise velocity; variable deﬁned after Eq. (3.11)
Streamwise velocity (free stream, pyrolysis front, ﬂame front)
Transverse velocity

Transverse mass ﬂux

Streamwise coordinate

Transverse coordinate

Mass fraction

Mass based transverse coordinate

xvi

O
-t
8
=-

d a eNDW<TZ Pawns:

Thermal diffusivity

Coupling function (gas); 7' — 1 (solid)

Modiﬁed transverse coordinate in solid

Similarity variable in gas and solid, see Eqs. (2.12)
Thermal conductivity

Dynamic viscosity

Kinematic viscosity; mass based stoichiometric coefﬁcient
Similarity variable , é = x/t

' Density

Fuel mass fraction in pyrolysis vapor
Variable deﬁned after Eq. (2.21)

is»

ondimensional time

Subscripts

f
fl
F
g
o

P
PF
PY
PR
S
UF

Flame reattachment point; ﬂame
Flame height
Fuel vapor
Gas

Oxygen
Pyrolysis front
Preﬂame
Pyrolysis
Preheat

Solid; surface
Underﬂame
Free stream

Superseripts

*

Dimensional quantity

CHAPTERS 4-7

 

\
\

335gh~=m nm>
§- E“

Variable deﬁned after Eq. (5.7)
Mass transfer number, Eq. (6.19)
Speciﬁc heat

Heat of combustion per unit mass of fuel vapor
Blowing correction

Convective heat transfer coefﬁcient
Thermal conductivity

Heat of gasiﬁcation

Lewis number

Mass loss rate; Mass ﬂux
Nondimensional mass loss

Prandtl number

xvii

Heat ﬂux

Reynolds number

Regresged surface location

Time, t=t-tv deﬁned after Eq. (5.6)
Temperature, T=T -T,,

Streamwise velocity

Streamwise velocity (free stream, pyrolysis front, ﬂame front)
Transverse velocity

Sample width

Streamwise coordinate

Transverse coordinate

Mass ﬁaction

0

~<~<><§<q=~1-m7e-o.‘

Greek

(1 Thermal diffusivity
[3 Variable deﬁned in Eq. (6.10)
5 .81 Boundary layer thickness: hydrodynamic, thermal
e Emissivity
n y ' /5'

v Kinematic viscosity; mass based stoichiometric coefﬁcient
p Density

f Flame, ﬂame location
g Gas

0 Oxygen
p Pyrolysis front, in pyrolysis zone, pyrolyzing fuel
3 Solid- gas interface; in solid (for properties)

v At vaporizing conditions

oo Free stream

Superscripts

* Non-dimensional quantity

CHAPTER 8

 

Mass diffusion coefﬁcient, Damkohler number
Activation energy

f (n); f'=u . f”=u’

Froude number

a constant used in Eqs. (8.12) and (8.13) and in deﬁnition of g
A Pre-exponential factor

b constant used in Eqs. (8.12) and (8.13) and in deﬁnition of 9
B Mass transfer number

0 cola

C Speciﬁc heat

D

E

f

F

W

xviii

C) N‘<‘<>*€<Q='~i"m|m©t~°o
'l
8
w

(Dome-€13 <1: >-‘.:3.mm-<'cag

Acceleration due to gravity
Reference length in gas

Heat of combustion per unit mass of fuel vapor
Universal gas constant

Reaction term

Time

Temperature

Streamwise velocity

Free stream velocity

Transverse velocity

Transverse mass ﬂux
Streamwise coordinate
Transverse coordinate

Mass fraction

Mass based transverse coordinate

Wave number

Nondimensional activation energy
Coupling function

Disturbance amplitude

Modiﬁed inner variable for 11
Similarity variable in gas

Thermal conductivity

Dynamic viscosity

Kinematic viscosity; mass based stoichiometric coefﬁcient
Density

Stream function

Frequency

Vorticity

Disturbance stream function

Modiﬁed inner variable for temperature

Subscripts

*8 $9100: “:1“.

Flame; inﬂection point

Fuel vapor

Gas

Oxygen; modiﬁed variable for a, B and Fr
Solid; surface

Solid surface (wall)

Free stream

Inﬂection point

Superscripts

A

*

Modiﬁed variable Eq. (8.26)
Disturbance variable Eq. (8.26)
Dimensional quantity

xix

CHAPTER 1
INTRODUCTION

SYNOPSIS

 

In this chapter the problem is introduced and the relevant literature is reviewed.

The objectives and organization of the thesis are discussed.

1.1. INTRODUCTION

The present study involves investigation of wind-aided ﬂame spread across a
vaporizing solid in the ceiling. conﬁguration. A typical practical scenario is a corridor
ceiling ﬁre.

The threat of uncontrolled ﬁre has been a formidable problem to mankind for
centuries, yet a complete understanding of the factors affecting its growth has not been
achieved. The investigation into ﬂame spread over solids is motivated by the need to
prevent and control ﬁre growth in 5 buildings and other conﬁned spaces. Spread of the
ﬂame results from the complex interaction of transport processes in the gas and the
condensed phases, the vaporization of the fuel and the chemical reaction of the fuel

vapors with the gaseous oxidizer.

The spread can occur in horizontal, vertical or any other ﬂow conﬁguration
(Fernandez-Pello and Mao, 1981); can be assisted or opposed by the natural convec-
tion (wall ﬁres) or forced convection (ceiling or ﬂoor ﬁres) or both (Fig. 1.1). Natural
convection is generated by the density stratiﬁcation in the vicinity of the ﬂame.

Forced convection occurs because of the wind (corridor ﬂow).

In ﬂame spread processes where the oxidizing gas ﬂows in the direction of ﬂame
propagation, the environment is said to "favor" the propagation of the ﬂame. As a
result the spread process is usually more rapid than other modes (and consequently
more hazardous) and has been proven to be extremely difﬁcult to understand and quan-
tify (Fernandez-Pello and I-Iirano, 1983; Fernandez-Pello, 1984). This kind of spread
can be further classiﬁed as buoyancy-controlled upward ﬂame spread (wall ﬁres) or
wind-aided ﬂame spread (ceiling or ﬂoor ﬁres). The interest that this mode of ﬂame
propagation represents for the ﬁre-safety ﬁeld has motivated the development of the
ASTM E-84 (Anonymous, 1977) tunnel test of ﬂammability. This test, which simu-
lates wind-aided ﬂame spread across a ceiling (Fig. 1.1) in a corridor, uses a forced

ﬂow of hot oxidizing gas to enhance the rate of ﬂame spread, thus subjecting the

 

 

N N
—+ <— —> -—>
—> —>
—> —>
—> —>
Opposed Convection (ceiling mode) Forced Convection (ceiling mode)
_> —>
__..> —>
_’ —>
—> <— —> —>
.x\ R
Opposed Convecrion (ﬂoor mode) Forced Covection (ﬂoor mode)
N , i \\‘
Natural Convection Natural Convection
OPPOSED FLOW FLAME SPREAD WIND-AIDED FLAME SPREAD

Figure 1.1 Modes of ﬂame spread

ﬂammability properties of the material to stringent conditions. The ten minute test
principally involves simple observation of ﬂame spread rate and how far ﬂame con-

verge (Parker, 1977).

The ﬁrst attempt to develop a comprehensive model of wind-aided ﬂame spread
was made by Carrier et al. (1980). In fact this model was developed with the aim of
describing the ASTM E—84 tunnel test. The complicated analytical techniques needed
to reduce the problem to a manageable computational form sufﬁce to demonstrate the
difﬁculty of the problem. On the experimental side there were few studies because of
the relative hazard.» This prevented the experimental study for long time (Femandez—
Pello and Hirano, 1983; Femandez-Pello, 1984). Largely because of unavailability of
experimental information, the development of theoretical models for ﬂame spread in
forced concurrent ﬂow started only in the last decade (Femandez-Pello, 1979;
Annamalai and Sibulkin, 1979b; Carrier et al., 1980, 1983, DiBlasi et al., 1987, 1988a.
1988b). We note, however, that the development of theoretical models for buoyancy-
controlled upward spread began in early seventies (Markstein and deRis, 1972, 1973;
Orloff et al., 1974; Sibulkin and Kim, 1977; Fernandez-Pello 1978; Annamalai and
Sibulkin 1979b; Fernandez-Pello and Quintiere, 1982; Hasemi, 1986; Saito et al., 1986;
Kulkarni and Fisher, 1988). The theoretical model development for opposed ﬂow
ﬂame spread began even before this (deRis, 1969; Sirignano, 1974, Fernandez-Pello
and Williams, 1975; Wichman and Williams, 1983). Because of the objectives of the
present study and limited literature, the remainder of this section is conﬁned to wind-
aided ﬂame spread. Several reviews exist which discuss various other modes and
scales of ﬂame (or ﬁre) spread (Friedman 1968; Magee and McAlevy 1971; Sirignano,
1972; Williams, 1976; Hirano and Akita, 1979; Emmons, 1980; Femandez-Pello and
Hirano, 1983; Fernandez-Pello, 1984).

1.2. WIND-AIDED FLAME SPREAD

1.2.1. STATEMENT OF THE PROBLEM

For the ﬂame to spread the solid ahead of the ﬂame should pyrolyze to provide
fuel to react with the oxidizer. The hot still-reacting post-combustion gases move
ahead of the pyrolysis zone and enhance the rate of heat transfer to the virgin solid.
Heat transfer is also augmented by the radiation from the ﬂame. This results in a tem-
perature rise at the surface. ‘The fuel produced in the pyrolysis zone is not completely
consumed in the bunting zone. The excess pyrolyzate move ahead of the pyrolysis
front (though still reacting with the oxidizer) and extends the diffusion ﬂame. Thus
we have a ﬂame front (ﬂame tip, xf) moving ahead, followed by a pyrolysis front (xp)
(Fig. 1.2). Trailing both of them is a burnout front which signiﬁes unavailability of
the fuel; this could occur because of charting or total fuel consumption (vaporizing or
non-charting solid). For the typical downstream fuel element the series of events is
initial heating, pyrolysis, ignition, ﬂame spread and extinction.

A systematic study is performed for a ﬁre that is spreading in this mode over a
continuous solid fuel slab. Speciﬁc objectives of the present work are discussed later

in Sec. 1.3.

1.2.2. LITERATURE REVIEW

All of the theoretical studies of wind-aided ﬂame spread assume the ﬂow is lam-
inar (Carrier et al., 1980, 1983; Fernandez-Pello, 1979; DiBlasi et al., 1987, 1988a,
1988b). Most of the models (except DiBlasi et al., 1987, 1988a, 1988b) can be
classiﬁed as thermal models. In these models the rate of ﬂame spread is determined
by the rate at which the surface temperature of the combustible is raised to a
. prescribed value (vaporization temperature), through heat transfer from the ﬂame and

external heat sources. The combustible is assumed to evaporate as soon as it reaches

PMMA sample ‘

 

 

 

 

Gravity l

 

 

 

Figure 1.2 Schematic diagram of wind-aided ﬂame spread

this prescribed temperature. In all of these models a one step inﬁnitely fast gas-phase
reaction (va+v00—> products) is assumed.“ DiBlasi et al., (1987, 1988a, 1988b)
assumed a zeroth order Arrhenius type solid fuel pyrolysis rate. They also took into
account the ﬁnite-rate gas-phase chemical kinetics (one-step second-order irreversible
Arrhenius reaction). It was found that effect of ﬁnite-rate chemistry is limited to the
ﬂame tip and the upstream leading edge, and is prominent for reduced oxygen concen-
trations (e.g. the extinction limit). They also found that for wind-aided ﬂame spread
streamwise diffusion is negligible, and the gas and solid-phases can be accurately
represented by boundary layer equations. The experimental study of Loh and
Fernandez-Pello (1984) shows the unimportance of ﬁnite-rate chemical kinetics. The
possibility of developing a theoretical model of wind-aided ﬂame spread that considers
only heat transfer processes represents a great simpliﬁcation; it is not clear that this
will always be a good approximation.

The momentum equation introduces strong nonlinearities into the problem. Gen-
erally, in all theoretical models developed, the energy equation is decoupled from the
momentum equation by assuming a uniform ﬂow velocity (Oseen approximation) (Car-
rier et al., 1980, 1983) or a parabolic channel-ﬂow proﬁle (given by Hagen-Poiseuille
formula) (DiBlasi et al., 1987, 1988a, 1988b). Carrier et al. (1983) employed the
boundary-layer approximation, but their results were not much different from those
obtained by using the Oseen approximation. We observe here that there have been no
systematic studies to assess the validity and limitations of the Oseen approximation.
Despite its relatively common use, its origin remain unexplored: recent work in

micro-gravity (rig) research is changing this, however.

The ﬂame length depends on the total amount of fuel produced. In all of the
models (except Carrier et al., 1980) the solid phase in the pyrolysis zone is assumed
steady. Although DiBlasi et al. (1987) assumed the solid as transient, but their

assumption of mass loss occurring at the surface with zeroth order Arrhenius-type

dependence on surface temperature, made the .mass ﬂux constant since during vapori-

zation the surface temperature is relatively constant (Vovelle et al., 1987).
Most of the theoretical models predict spread rates that are in qualitative agree-

ment with the experimental observations. Based on their experimental results Loh and

Fernandez-Pello (1984) produced an expression for ﬂame spread (pyrolysis front
speed):

 

2
_ pg cps Atr Tf'Tv
Up _ U” pscsls [TV-Too (1.1)

The experiments were done on thick PMMA sheets in the ﬂoor conﬁguration. Equa-
tion (1.1) shows that for thick non-charring materials, ﬂame spread is linearly depen-
dent on ﬂow velocity, proportional to the square power of the oxygen mass fraction

and steady.

' Theoretical models of wind-aided ﬂame spread over vaporizing solids predict a
linear dependence of spread rate on free stream gas velocity (Fernandez-Pello, 1979;

Annamalai and Sibulkin, 1979b; Carrier et al., 1980, 1983).

For non-charting materials the ﬂame spread is steady whereas for charting materi-
als it accelerates before reaching steady state (DiBlasi et al., 1987, 1988b; Loh and
Fernandez-Pello, 1984; Saito et al., 1986). In all of the studies ﬂame (soot) radiation
and surface heat losses (reradiaton) are neglected. Carrier et al. (1980) included ﬂame

radiation in a very approximate form. The emissivity was assumed constant.

Most of the models developed consider the material as non-charting because of
difﬁculty in analyzing solid-phase processes. Non-charring solids vaporize completely
without leaving any residue (char), e. g. plastics. Charring makes problem inherently
transient. Carrier et al., (1983) treated char and virgin material as solids with different

densities. The volatiles in the virgin material vaporize at constant temperature and

char erodes at some different constant temperature. Solutions are provided only for
steady-state case, although the formulation is transient. DiBlasi et al. (1988b)
developed a theoretical model of the spread of a ﬂame over thin charting combustible
solids. Tire charting of the solid is modeled by treating the solid as a two-component
material; the combustible component that gasiﬁes, and the char that remains occupying
the initial solid volume. In the'gas phase, the combustion reaction is modeled with a
one-step irreversible second order Arrhenius chemical reaction. The results of the
analysis suggest that simpliﬁed model of Saito et al. (1986) could predict certain

aspects of the ﬂame spread process without much error.

In conclusion, based on .the available literature the present understanding of

wind-aided ﬂame spread can be expressed as follows:

"Wind-aided ﬂame spread can be treated as the spread of a pyrolysis front which
varies linearly ,with the free stream velocity and as square of the free stream oxygen

mass fraction, and is steady for non-charring materials"

1.2.3. THEORETICAL FRAMEWORK

A simple model is presented in this section for wind-aided ﬂame spread that gives
us the essential parametric dependencies. Let us assume that the solid is only heated
under the ﬂame, i.e, before the ﬂame tip (xf) the surface temperature of the solid is
ambient (Tn). This assumption is used by (Sibulkin and Kim, 1977; Fernandez-Pello,
1979; Quintiere, 1981; Loh and Fernandez-Pello, 1984). If the gas is assumed steady
and the ﬂame radiation and surface heat losses are neglected, then the net heat ﬂux
into the solid is just the convective heat ﬂux qczm. By further assuming that the solid

is thick (semi-inﬁnite) the problem is reduced to the solution of the following equation:

dl' d2T

pscs‘gt' ="- As _dy2 , (1.2)

10

and the boundary conditions are

dT
-13 [_] =qconv ’

T (°°,t )=T..

Here p, c and 7t. are the density, speciﬁc heat and thermal conductivity respectively.
Subscripts s , g and co represent conditions in the solid, gas and the ambient, respec-
tively. The onset of the pyrolysis front is characterized by the surface reaching its
vaporization temperature, Tv . The time required 'to do so, At , can be found easily
from Eq. (1.2). Also the pyrolysis front in this time has traveled from xp to x,, thus
Up =(xf -Jtp )/At. The location xf has faced 4,2,, for the time interval At. Thus one

obtains (Sibulkin and Kim, 1977; Quintiere, 1981 ; Annamalai and Sibulkin, 1979b)

U __4_ deliﬂxfﬂxf-xp]
* p n Mean-T42

 

(1.3)

In the boundary layer the convective heat ﬂux goes as x'l’z. Now if we assume
(1,2,, (x )=(pg cm, 1.8 UJPn)la(Tf -T,,) (Loh and Fernandez-Pello, 1984), then Eq. (1.3)

becomes

7. T -T 2
Upszpgcpg g( f V) 1 x" (1.4)
ps cs As (Tv —Too)2 xf

If xf >> xp one obtains Eq. (1.1). The ﬂame length xf primarily depends on the total
fuel available, which depends on the heat ﬂux distribution in the pyrolysis zone and

11

the solids’ degradation behavior. Thus, the important parameters to study are the heat

ﬂux distribution, Up , Jcp , xf , and the solid degradation behavior.

Equation (1.3) is only valid when there is no preheating of the solid upstream of
xf. In the next chapter we analyze this in greater detail and formulate the problem
with preheating extending all the way to U mt. Note that locations x > U m,t are undis-

turbed.

1.3. OBJECTIVES OF THE PRESENT WORK

A systematic study is undertaken in order to determine the major variables (fac-
tors) that affect wind-aided ﬂame spread. Other than the usual environmental variables
like free-stream velocity and oxygen mass fraction, the effects of unsteadiness of the
behavior of the solid, heat ﬂux near the leading edge, surface heat loss and stability of
the ﬂame on wind-aided ﬂame spread phenomena are also examined. In doing so,
explicit expressions for the pyrolysis front speed, estimates for the realistic ﬂame from
location and transient mass loss rate are sought. No particular emphasis is placed on
the solution approach, i.e., whether experimental or theoretical. This is a debatable
philosophical point that has spawned libraries full of weighty expostulations by many
wise savants. I would say: "The difﬁculty of the problem dictates that the method of
solution vary according to the particular question being examined; it is not possible to
force this complex problem to yield to a certain solution method, or even to a single

restricted class of solution methods".

The goal is to develop a foundation for the wind-aided ﬂame spread problem that
will allow major tasks, like modeling of turbulent wind-aided ﬂame spread over char-

ring solids, to be successfully undertaken.

12

1.4. ORGANIZATION OF THE THESIS

This thesis is organized into 7 chapters other than Introduction and Conclusion.

It is best understood in the context of Fig. 1.3. Chapters 2, 4 and 6 deals with the

theoretical description and experimental determination of pyrolysis front movement and

ﬂame front speeds and locations. Chapters 3 and 7 deal with the determination of the

surface heat ﬂuxes theoretically and experimentally, respectively. Chapter 5 deals

exclusively with the solid phase. The transient effects in the solid are analyzed

theoretically and experimentally. Chapter 8 deals exclusively with the gas-phase. Here

the stability of the ﬂame is analyzed and characteristic pulsation frequency of the ﬂame

is determined. The speciﬁc objectives and the main conclusions of each chapter are:

1.

Based on the need, present understanding of the problem, a theoretical model
which assumes a steady pyrolysis zone is developed. This results in an explicit
expression for pyrolysis ﬁ'ont speed. The expression suggests linear variation with
free stream velocity. It also suggests linear variation with free stream oxygen
mass fraction, disagreeing with the experimental results of Loh and Femandez-

Pello (1984).

Analysis of model 1 is extended to obtain heat ﬂux distributions in the pyrolysis
region and preheat region. The numerical results for surface heat ﬂux in the
pyrolysis zone and in the preheat zone can be reduced to expressions which
depends only on ﬂame front speed and pyrolysis front speed. An expression for
excess pyrolyzate is also obtained. The results indicate that underﬂame preheat-

ing is the major contributor towards preheating of the solid.

An experimental investigation is performed in order to understand the reasons for
the disagreement between our model and experimental results, and also to under-
stand the controlling mechanisms better. The experimental results suggest a
linear dependence of pyrolysis front speed on free stream velocity. It was found

that for higher oxygen concentrations there is substantial ﬂame radiation; this is

l4

reﬂected from the walls of the combustion tunnel and strikes the sample surface,
increasing the ﬂame spread rate. For cases with no reﬂection the spread rate
agrees with the expression obtained previously. For cases with some reﬂection it
agrees with the previous experimental results (Loh and Fernandez-Pello, 1984).
Moreover it was found that solid is always transient during the experiments, sug-
gesting a much shorter ﬂame tip (which primarily depends on amount of fuel)

than predicted by the assumption of a steady solid.

A simple one-dimensional model for a vaporizing solid is developed in order to
determine its transient response. An explicit expression is obtained which gives
the uansient mass loss as a function of time and the applied heat ﬂux. The
expression agrees favorably with the concurrently-performed experiments and

numerical results.

A model is developed for wind-aided ﬂame spread over a transient solid. The
model predicts a realistic ﬂame tip location as a function of the pyrolysis front
location (speed) and time for both laminar and turbulent ﬂows. The predictions
agree with the experiments. They also. suggest that surface heat losses cannot be

neglected.

Surface heat ﬂuxes in the burning zone are obtained from the burned PMMA
sample by use of a simple model. The results suggest that the net surface heat
ﬂux in the burning zone increases with the free stream and decreases with the

streamwise distance, though it does not vary as x"1’2.

In order to check for the stability of the ﬂame, a linear inviscid parallel stability
analysis is performed. The results indicate that in the ceiling conﬁguration the
ﬂame is stable because of forced convection (i.e, the Froude number is high). It
also suggests that for lower Froude numbers the stability of the problem could be

lost to inﬁnite-reaction-rate kinetics.

l4

reﬂected from the walls of the combustion tunnel and strikes the satnple surface,
increasing the ﬂame spread rate. For cases with no reﬂection the spread rate
agrees with the expression obtained previously. For cases with some reﬂection it
agrees with the previous experimental results (Loh and Fernandez-Pello, 1984).
Moreover it was found that solid is always transient during the experiments, sug-
gesting a much shorter ﬂame tip (which primarily depends on amount of fuel)

than predicted by the assumption of a steady solid.

A simple one-dimensional model for a vaporizing solid is developed in order to
determine its transient response. An explicit expression is obtained which gives
the transient mass loss as a function of time and the applied heat ﬂux. The
expression agrees. favorably with the concurrently-perforrned experiments and

numerical results.

A model is developed for wind-aided ﬂame spread over a transient solid. The
model predicts a realistic ﬂame tip location as a function of the pyrolysis front
location (speed) and time for both laminar and turbulent ﬂows. The predictions
agree with the experiments. They also. suggest that surface heat losses cannot be

neglected.

Surface heat ﬂuxes in the burning zone are obtained from the burned PMMA
sample by use of a simple model. The results suggest that the net surface heat
ﬂux in the burning zone increases with the free stream and decreases with the

streamwise distance, though it does not vary as x‘m.

In order to check for the stability of the ﬂame, a linear inviscid parallel stability
analysis is performed. The results indicate that in the ceiling conﬁguration the
ﬂame is stable because of forced convection (i.e, the Froude number is high). It
also suggests that for lower Froude numbers the stability of the problem could be

lost to inﬁnite-reaction-rate kinetics.

CHAPTER 2
A PRELIMINARY MODEL OF FLAME SPREAD

SYNOPSIS

 

This chapter theoretically analyzes ﬂame spread over thick non-charting (vaporiz-
ing) solid fuels in concurrent oxidizer ﬂows. Emphasis is placed on making rationally
justiﬁable physical approximations that produce useful results. Exact solutions for the
pyrolysis and ﬂame front speeds are obtained. These quantities are both linearly
dependent on the free stream velocity. The pyrolysis front speed is also linearly

dependent on the free stream oxygen mass fraction.

15

16

2.1. INTRODUCTION

Wind-aided ﬂame spread is a hazardous process that is difﬁcult to quantify both
in the wildland and in controlled small-scale laboratory experiments. The wildland ﬁre
poses an exceedingly formidable problem whose complications we do not presently
wish to examine. This chapter focuses instead on the simpler small-scale problem.
Even for this case, however, the problem is quite complex, since ﬂame spread usually
occurs under turbulent conditions, buoyancy and radiation often play major roles in the
heat transfer mechanism, the solid fuel may char or display other complex behavior,

and the detailed gas-phase chemical processes are unknown.

These difﬁculties have, generally speaking, produced two general classes of
wind-aided ﬂame spread models, (A) the simpliﬁed ones that derive practical correla-
tion formulae and (B), the comprehensive numerical ones that describe ﬂame spread in
terms of tables and graphs. Before describing some of their differences we emphasize
ﬁrst their similarities: they always assume laminar ﬂow; they usually do not include
radiation; they usually do not describe charrin g or other complicated solid behavior. If

they include any of the latter two it is an approximate or simpliﬁed way.

Models of class (A) have their origin in a sequence of phenomenological assump-
tions that the authors do not attempt to rigorously justify. They are based more on
practical experience and qualitative order-of-magnitude estimates than on deductive
reasoning ﬂoor the ﬁrst principles. Examples are a steady-state gas phase (Fernandez-
Pello, 1979), a constant difference between ﬂame and pyrolysis zone lengths (Quin-
tiere, 1981), and a thermal balance at the pyrolyzing surface between only the incom-
ing gas phase heat ﬂux and the energy for vaporization (i.e., neglect conduction into
the solid) (Fernandez-Pello, 1979), etc. Nevertheless, they produce useful formulae for
practical engineering data correlations while providing insight into the many complexi-
ties of this problem. Happily enough, these assumptions can in fact be justiﬁed by the

rigorous analysis of the governing equations.

17

Our theory, however, is not general enough to discuss the validity of various
recent models of class (A) that have undertaken a rather novel approach (Saito et al.,
1986; Kulkarni and Fisher, 1988). As their starting point they take the ﬂame spread
formula of a highly simpliﬁed qualitative analysis (Quintiere, 1981). They then later
artiﬁcially reintroduce physical complexity by substituting formulae describing the
external radiant and piloted heat ﬂuxes (Saito et al., 1986), the mass ﬂux distribution
from charting solids (Kulkarni and Fisher, 1988), etc. We may, with conﬁdence,

describe these as purely phenomenological models of wind-aided ﬂame spread.

By contrast, models of class (B) attempt a comprehensive description of wind—
aided ﬂame spread. These analyses employ the governing conservation equations, usu-
ally without imposing too many restrictive modeling assumptions. Thus, they retain
streamwise and transverse diffusion (DiBlasi et al, 1987, 1988a, 1988b), radiation
(Carrier et al., 1980, 1983), charring (DiBlasi et al., 1988b; Carrier et al., 1983),
solid-phase pyrolysis kinetics (DiBlasi et al., 1987, 1988a, 1988b), and ﬁnite-rate gas-
phase chemistry (DiBlasi et al., 1987, 1988a, 1988b). Their drawbacks are (1) they are
difﬁcult to understand and interpret because results are presented in graphical or tabu-
lar form and (ii) they do not produce deﬁnite and usable data correlations since depen-
dencies on certain expected dominant parameters (e.g., the free-stream oxygen concen-
tration) are usually all that computational time and cost constraints allow them to
determine. Thus the inﬂuences of more routine parameters (such as the thermal
responsivity ratio) usually go unstudied even though they form an important part of a

successful correlation.

Present analysis attempts to follow a middle course between those of types (A)
and (B). We do this by on the one hand analyzing the conservation equations them-
selves, not a set of postulated phenomenological relationships, and on the other hand
by making enough justiﬁable approximations So that resorting to a numerical solution

becomes unnecessary. In short, we will use analysis to extract as much worthwhile

18

information about this physical problem as possible. We wish to simultaneously
deepen our physical understanding and improve our theoretical predictions.

The model problem we study closely resembles that of Carrier, et al. (1980); in
fact, it is identical except we introduce one additional modeling assumption that is
based on their numerical results and on the analysis of a related, simpliﬁed model
problem (Wichman and Baum, 1988). It is this assumption that makes the difference

between numerical and analytical solution.

2.2. PROBLEM FORMULATION

 

The gas is initially quiescent with ambient temperature T; and oxygen mass frac-
tion Y5... The solid fuel also has initial temperature TL. At time t' = 0" the gas is
swept into motion with free stream velocity U 2, in the direction parallel to the solid
surface while the fuel is simultaneously ignited at the leading edge. A thermally
undisturbed region, a ﬂame front, and a pyrolysis zone each propagate downstream
thereafter with velocities U 1,, U; and U; respectively.

We adopt the idealized two dimensional geometry of Figure 2.1. The coordinate
system is attached to the leading edge, past which a uniform ﬂow of gas into the
region x' > 0, y' > 0 above the solid begins at t" = 0. The solid occupies the region
x’ > 0, y,’ < 0, where the subscript s here denotes the solid (it may also designate
surface). At any instant the pyrolysis zone extends to xp“ (t'). The ﬂame reattaches

at xf " (t' ), as will be discussed later.

It is now necessary to state some of our major assumptions for simplifying the
governing equations (Williams, 1985). Concerning diffusion, we assume here that
high rates of , streamwise convection produce much larger gradients in the direction per-
pendicular to the free stream than along it, allowing the boundary-layer assumption to
be invoked, thereby enabling streamwise diffusion of mass, momentum, energy and

species to be neglected.

19

_ Pyrolysis zone Preheat zone Dead zone

 

 

 

  

Ix.

"P

 

C
Q
C

 

X
4!
C2
8
'9

Solid

 

 

 

Figure 2.1 Schematic diagram of wind-aided ﬂame spread. Pyrolysis zone extends till
- x‘, ”(t ) and ﬂame reattaches at xf(t ). Superscript * signiﬁes a dimen-
sronal quantity

20

The gas-phase chemical reaction is assumed to occur through the simple one-step
irreversible reaction F + v0 —) products , where v is the mass-based stoichiometric
coefﬁcient. A Schvab-Zeldovich analysis will be performed for the spreading diffusion
ﬂame with absolutely no regard for ﬁnite-rate chemical processes, whether near the

inception point or anywhere downstream.

In this study gas-phase radiation is assumed negligible. This is a reasonable
approximation for small laboratory scale experiments for some polymeric materials
such as polymethylmethacrylate (PMMA), although recent experiments with wood
have shown that a minimum external radiant ﬂux level is necessary for spread to occur
(Saito et al., 1987). Consequently, attention is focused on simple vaporizing solids
such as PMMA; the inﬂuences of charring for cellulosic materials and external radia-

tion under large-scale conditions are left for future consideration.

Finally the convective terms in the transport equations are simpliﬁed by assuming
that the gas ﬂows over the surface with uniform and constant velocity U f. This
assumption is entirely consistent with the ﬁrst two because diffusion ﬂames with
inﬁnitely fast chemistry can logically be expected to locate themselves near the exter-
nal ﬁ'ee stream far from the surface, where the convective supply of oxidizer is
greatest and the inﬂuences of ﬁnite-rate chemistry, ﬂame quenching (and therefore

streamwise diffusion) are least.

Tire equations for conservation of species and energy in the gas and the equation

for conservation of energy in the solid therefore become

 

 

 

 

 

 

 

 

eDYi‘ a e ani. e
p , - , pD , =—v,—R ,i=0,F, (2.1)
Dt 3y 3y
.DT‘ a . .31" Q‘ .
- D = R , 2.2
p Dt' 31'") 8f] 6,? ( )

 

and

21

 

 

 

 

 

,‘ - j . =0, (2.3)
3 Dr cs 32
where
D“E 8‘+U; a. +ve a“
Dr 3: 8x
and

D: a +ni*(x‘,:‘) a

Dr’ ar‘ 9; Byi'

 

 

 

are the transient-convective operators in the gas and solid and

R‘ =A‘p'2Y0’YF’exp(-E’/R'*T*)

is the reaction term. Here m'(x‘,t*) = p' (x*,0,t*)v‘(x*,0,t') is the mass ﬂux from
the solid to the gas, v; = 1 and v0 = v. The following additional minor simplifying
assumptions have been made in deducing Eqs. (2.1)-(2.3): equal binary diffusion
coefﬁcients, constant mixture speciﬁc heat, unity Lewis number (Le =0t/D ), low Mach

number, and. negligible solid-phase regression rate.

The boundary and initial conditions for Eqs. (2.l)-(2.3) are most easily discussed
with reference to Fig. 2.1, where it is shown that there are three primary zones, (i) a
pyrolysis zone, 0 < x’ < x;(t' ), where the solid is ejecting volatile combustible into
the gas, (ii) a preheat zone, x;(t') < x‘ < U Lt' , where the fuel surface is being
heated prior to pyrolysis, and (iii) a dead zone, x‘ > U Lt. , where there is no
inﬂuence of the upstream ﬂow. The term "dead zone" is understood by observing that
the wavelike transient-convective operator of Eqs. (2.1) and (2.2) limits streamwise
signals (originating at the leading edge) to the region 0 S x’ S U it" because stream-

wise communication by diffusion has been eliminated by the boundary layer

22

approximation.

The initial, free stream and upstream boundary conditions are assumed identical
(see Fig. 2.1). Thus. the ambient temperature is T L , the ambient oxidizer concentra- -
tion is Y5... , and the fuel concentration is zero, Y; = 0. The ambient temperature of
the solid is assumed identical to that of the gas. In the pyrolysis zone we assume that
the solid pyrolyzes at the constant vaporization temperature T3, a conventional and
well-justiﬁed approximation. In chapter 5 we will discuss more on this approximation.
When 7" < T: no pyrolysis can occur. Furthermore the solid fuel regression rate, as
already mentioned, is neglected, as is the depletion of fuel (semi-inﬁnite solid). The
boundary conditions for the temperature and fuel concentration ﬁelds are shown in
Figs. (2.2) and (2.3), where L‘ is the endothermic speciﬁc heat of pyrolyzation and x
is the mass fraction of pure fuel in the mass ﬂux from the surface. In the preheat zone
the mass ﬂux from the solid vanishes and the surface temperature is therefore always
less than T5. The ﬂame reattaches at x' = x; because the assumption of inﬁnitely fast
gas-phase reaction requires all fuel entering the gas to react - mixing with the oxidizer
stream is impossible. This condition is obviously never satisﬁed in practice but experi-
ence in the laboratory shows that under some conditions (e.g., ﬂame spread in a ceiling
conﬁguration, where buoyancy pushes the ﬂame closer to the surface near the tip) it is
reasonable. In the preheat zone the heat ﬂux from the ﬂame to the surface must equal
that into the Solid since there is no vaporization. Also, the gas and solid surface tem-
peratures are identical, the normal gradient of Y I; vanishes under the ﬂame and the
normal gradient of Y5 vanishes ahead of the ﬂame. At the reattachment point we have

Y5 = Y I; = 0. All of these conditions are also shown in Figs. (2.2) and (2.3).

The governing equations (2.1)-(2.3) and the initial and boundary conditions of

Figs. 2.2 and 2.3 are now put into simpler form. First we introduce the time-

3
dependent Howarth transformation, with the mass coordinate z I = I p' dy " and the
o

 

=1

 

 

 

 

 

5

OBY Y0”
5v ywﬁkrt—v'ﬁ] . Q

23

PYROLYSIS ZONE

 

 

at) am 1 alt-1
= .— + w— - ——
ox dz Pr 3:2

an

L

as,_

'52"

113%
R 8;,

+Per ,

PRE-I-IEAT
‘- ZONE -—

 

Bs=0

 

[35 = [33:

LV- Prkﬁ .—

C 3C.

3255 __
BC? ‘

 

Bs=T-1

Bs=0

Figure 2.2 Equations and boundary conditions in the pyrolysis zone

24

PRE—HEAT ZONE

 

 

‘ Z Br=Br=0
BFBYP
L=§(—-+--a:(l---1—820
at dx Fr 322
PYROLYSIS DEAD
ZONE L{sy}= 0 : ZONE
Br B E
[31‘5” :
a a , a a
[31°an %=0. (27%; [37%. Taurus)
KP tss=ess g=t :
212-2111. - Bs=°
a: at} _

55:0

 

C.

Figure 2.3 Equations and boundary conditions in the preheat zone

25

- y' 1"
modiﬁed transVerse mass ﬂux w’ = p'v’ + I-aﬁdy‘ + Uijgerdy‘. This, along
0 ‘ o x

with the assumption p'zD * = constant, eliminates the density from the convection and

diffusion terms. The equations and boundary conditions are then nondimensionalized

by deﬁning
p=p’/p..'. D =D'ID..'. L =L'/cp..‘,T..'. Q =Q‘lc ..'T..'.
t =2'U,‘/Lg‘, r =T“/T..“, u =u’/U..*, v =v’/U..*, w =w’/p..’U..’.

x =x*/Lg‘, z =z'/p.,’Lg’, z, =y_.*/L,*. Y0 = Y5. Yp =Y;-.

where the subscript co denotes free stream conditions, L; = v: / U ”I is the reference
length in the gas and L; = a;/U:, is the reference length in the solid
(as. = As'lpfcs‘). Note that the nondimensional streamwise velocity u is unity.

The Schvab—Zeldovich variables

. YOTYo»
1) Br=——'YF’

v
_ Y-Y,,
iilﬂr=TQ1+ ovo,

(2.4)

 

are then used to eliminate the reaction terms in two of the gas phase equations. We
do not need to consider the third nonlinear equation (for T, say) because inﬁnite-rate

chemistry has been assumed. In the solid we deﬁne, for notational consistency,

%=R-L ' as

Thus, in the gas we obtain

26

93.; 93.; ab.- 1 325,.

a: + dx +w dz Pr 322

 

=0, i=Y,T, (2.6)

where Pr is the Prandtl number. Note that in this transformed coordinate system, the
continuity is satisﬁed by Bw/Bz=0 since u=1. In the solid we deﬁne a new transverse

coordinate,

t

z: _ ys

x/L;/L,’ [war afag’xu 1,] ’

 

 

so that Eq. (2.3) reduces to

2
ﬁ+ —~IPrR 3‘35 J36“ 4:0, (2.7)

at BC. at;,2

 

where C = CI, ,.;/c is the speciﬁc heat ratio and R= p“, cpﬂAngﬁ ';A is the thermal
responsivity (Wichman and Williams, 1983).

Tire initial and boundary conditions for Eqs. (2.6) and (2.7) are

BT=BY=BS=0att=0,x=0,andz=oo,C,=-oo. (2.8)

At the interface we have

 

aBr _ You
0<x<x 2i)—a—— wPr[By+T+x],
ti)Q§5—B:- =\/’;—'-a—ES +Pr Lw,
Y; (2.9)
iii)B1=T Q 1-——v—,

iv)BS=Tv_1°

27

a
xp<x<t: i) ﬁ =0,

 

dz
ii) QBET— = \jE 35s (2.10)
Oz R at” '

The boundary value problem described by Eqs. (2.6) to (2.10) is now solved after
one more important assumption has been made. Without this, the solution must be

found numerically (Carrier et al., 1980).

2.3. SOLUTION

The nature of the solutions to Eqs. (2.6) and (2.7) can be deduced by an integral
analysis of a set of simpliﬁed model equations (Wichman and Baum, 1988) and by a
rigorous mathematical analysis of equations describing the impulsively-started ﬂow of
an incompressive ﬂuid Over a semi-inﬁnite plate. Instead of entering into a detailed
discussion of these results, however, we note the following two facts. First, the
difﬁculties encountered with equations such as (2.6) describing an impulsively-started
event (here the impulsive initiation of ﬂame spread at x = 0 , t = 0) under the
boundary-layer approximation are well known. The analogous semi-inﬁnite ﬂat plate
problem, for example, is one of the problems of ﬂuid mechanics, having been attacked
for 40 years with few concrete results. Second, and more importantly, it turns out
then these mathematical complexities are not an obstacle to a satisfactory physical
understanding of the problem, which can be obtained by answering the question; what
does an observer stationed at a ﬁxed downstream position x from the leading edge see

after the start of the process?

To explain this let us consider the pure ﬂuid mechanical problem just mentioned,
which corresponds to startup ﬂow in a low-speed wind tunnel. Since signals from the
leading edge propagate streamwise through the ﬂuid with maximum speed U z, (the

free stream velocity) there can be no inﬂuence of the leading edge at location 1’ until

28

the time t' =x*/U:,. Thus, for 0 < t‘ < x*/U:. the ﬂow must be of Rayleigh type
, meaning that the observer sees a ﬂow ﬁeld identical to one over an inﬁnite ﬂat plate
with no leading edge. This corresponds to the dead zone discussed in the previous
section. The other extreme of behavior occurs in the limit t‘ >> x' IU; , when the
initial transients have disappeared and a steady ﬂow has been established in the tunnel.
Since we have invoked the boundary layer approximation, this steady ﬂow must be of
the Blasius type , meaning that for t’ >> x‘lU; the observer sees a steady Blasius

boundary layer whose leading edge is at x = 0.

In sum, for x‘lULt’ << 1 the ﬂow is of Blasius boundary layer type, steady but
dependent upon x, while for 1 < x'IULt' < no it is a Rayleigh boundary layer, tran-
sient but independent of x . In between there must be a transition of some sort which,

however, is of no concern to us at present.

What is important is the approximation that this reasoning suggests: in the pyro-
lysis zone near the leading edge (see Figs. 2.1-2.3) the unsteady terms in the equa-
tions may be discarded, whereas in the preheat zone, where there is no pyrolysis,
they must be retained because the surface temperature in this zone is a function of
time and space (T3’ET;(x* ,t‘ )). Consequently we are led to postulate a steady

state in the pyrolysis zone.

Once this has been done Eqs. (2.6)-(2.10) can be solved exactly after substituting

11, 0<x <xp(t),

«E

0 , x >xp(t) , (2'11)

W:

. w—ﬁ—r .
for the transverse convective mass ﬂux. Here M = w*\/x IplrraU, In terms of the
dimensional quantities. It is well known that for constant velocity streaming over a ﬂat

plate this is the only surface blowing function that allows self-similar solutions of Eqs.

29

(2.6)—(2.10). This distribution is consistent with the boundary layer approximation
when we assume a direct proportionality between the gas-phase heat ﬂux to the surface
and the ensuing mass ﬂux. Note that w = M N}- for all z. This approximation
(w=M/‘1)? ) is quite reasonable for many materials, since it implies the heat ﬂux (which
turns out to have the boundary layer form, 8T /82 (x ,0) ~Q0 N; ) is directly propor-

tional to the mass ﬂux.

For analytical solution, Eqs. (2.6) and (2.7) may be written in a simpler and more

convenient form. Substitute Eqs. (2.11) for w , then introduce the similarity variables

 

§=xh,

n =(‘1P_r/2)z/‘/; (gas), (2.12)

n. = (VPrR /C)€.N§ (solid) ,
to ﬁnd

313: BB.- 325i .

4§(1-E.)-a—§-+2(M‘1P_r—n -5n—-—an-2-=O,I=Y,T, (2.13)

in the gas and
2 2 a 2
_§C £+M8BS_aBs "O (2.14)

Fri? 8:. an, 3113'

in the solid. Note that these equations can be used to justify the steady-state approxi-
mation in the pyrolysis zone, since for small é (i.e., x/t << 1 , ﬁp ~ 0(10‘3)) all
terms containing the time t vanish. This assumption may be questionable in the solid
for some values of C and R; however, the assumption of constant surface heat ﬂux,‘
temperature (T v) and Constant blowing velocity M (all in the E, — n coordinate sys-
tem), leads to a steady-state solution as the only bounded solution of Eq. (2.14). In
other words, under these conditions an unsteady bounded solution of Eq. (2.14) in the

pyrolysis zone does not exist.

30

The initial and boundary conditions of Eqs. (2.8) become B1 = By = B5 = 0 at

g = 0 and at n = co , ns = —oo , while the interface conditions reduce to

 

 

 

 

 

 

 

Pyrolysis zone (0 < g < S, 2:
Y
i) %=2WM(By+—9vi+x),
.. 3131 _ 2W7 35s L
.11) w— QC an: +25M‘jﬁ,
__ Tv ‘1 You. (2'15)
iii) B13- — Q - -V— ,
iV) B33 = Tv -1
Preheat zonejg‘IL < t; < 1):
an , (216)
ii) ESL = 2‘17”? 313s °
3n QC ans '

Observe that M = 0 in the preheat zone, lip < l; < 1.

Equations (2.13)-(2.16) are now solved separately in the pyrolysis and preheat
zones (in the dead zone the solutions are simply By = B1 = B5 = 0). Because of the
step change in the mass ﬂux at Q the solutions for BY and BT will have continuous
values but discontinuous derivatives of all orders. Thus, the gas phase solutions are
"patched" at 5,, , not matched in the asymptotic sense (Van Dyke, 1975). Note ﬁnally .
that the solutions for any dependent variable (such as By , BT, BS , or the pyrolysis
front speed EP) will depend on t linearly through the similarity transformation of Eqs.
(2.12). Therefore, acceleratory ﬂame spread is impossible in this self-similar version

of the ﬂame spread problem.

31

The mathematical solutions of Eqs. (2.13)-(2.16) [or (2.6)-(2.10)] are now written

in the pyrolysis and preheat zones.

2.3.1. PYROLYSIS ZONE (0 < 5 < ép):

We note that in this region the gas and solid phase problems are mathematically
uncoupled. By now putting a(-)/a§ = o Eqs (2.13) and (2.14) reduce to simple ordi-
nary differential equations that are easily solved subject to the far-ﬁeld boundary con-
ditions [Eqs. (2.8)] and, at the interface, Eqs. (2.15.i) and (2.15.iii) (for the gas) and
Eq. (2.15.iv) (for the solid). Thus,

erfc( 11 -M‘1P_r )

 

 

 

0 BY = BYS erfc( -M\/P7)
.. ___ erfc( 11 - ME )
"”31 B” erfc(—MVP?) ’ (2'17)
17035 = 1335 CXP(MT13 ),
where B15 and B55 are given by Eqs. (2.15.iii) and (2.15.iv) respectively, and
Bys- [ v fx]l+h(M\/ITr-) ,h(¢)- “(be erfc(-¢). (2-18)

One can now use Eqs. (2.17) and (2.18) to derive expressions for the gas-phase
surface fuel mass fraction Yps and the mass ﬂux eigenvalue M. Since

Bys = -Y,.-S - YOJV one ﬁnds, from Eq. (2.18),

_ xh(M~17>7)-Y0../v
F3" 1+h(M~fP—r) ’

 

(2.19)

32

which shows that a minimum non-zero mass ﬂux is necessary to produce a non-
negative value of Yrs ; it is found by solving the transcendental equation
h( M \lP_r ) = Y0 m,/(vx). The requirement of a minimum positive M states a simple
physical fact: ﬂame spread is not possible unless the surface is pyrolyzing at a rate
sufﬁcient to sustain it. The eigenvalue M is found by substituting Eqs. (2.17.ii) and

(2.17.iii) into the surface energy balance of Eq. (2.15.ii). This gives

QY eo/v - (TV _1)
h(M\/P7 )-_- L0+(T -1)/c as, (2.20)

 

where B is the mass transfer number deﬁned by Spalding. Note that B must satisfy

the condition B > Y0 _,/(vx); without this ﬂame spread is not possible.

Equation (2.20) suggest two limiting cases, one neglecting (T, — 1)/C, the other
neglecting the vaporization enthalpy L . Both will obviously overpredict M . In the
ﬁrst case the solid-phase temperature gradient term in Eq. (2.15.ii) is discarded. This
is reasonable when Pr, M, and C are of order unity, Q is large, and L/Q is of order
unity, and suggests that the energy balance occurs between the incoming gas phase
heat ﬂux and surface vaporization enthalpy, as for ﬂame spread over a thin fuel. This
procedure was utilized by Fernandez-Pello (1979). On the other hand, the second case
has zip—r/QC ~ 0(1), Pr ~ 0(1) and ML/Q << 0 (1). These conditions are satisﬁed
when the vaporization enthalpy ﬂux ML is much smaller than the heat ﬂux Q from the
ﬂame. Since so little energy is required to vaporize the fuel most of the incident heat
ﬂux is conducted into the solid. Our result is of course the composite of these two

cases.

33

2.3.2. PREHEAT ZONE (5,], < E, < 1):

Before deriving the solutions in this zone we note that even though the governing
equations in the gas change discontinuously across §= §P we can still apply Eqs.
(2.17.i) and (2.17.ii) as "initial" conditions at g, for the solutions to By and B1 in
§> tip. This is achieved by writing n = (Way/45;? in Eqs. (2.17.i-ii), and it
means that By and Br will in fact be continuous across the line i = 54,, but that the
streamwise derivatives of all orders will be discontinuous. The further requirement of

continuous heat ﬂux into the solid at g (i.e., lim 3 Ian = lim 8 Ian) allows us to
” H; B1 t—rc '37 A

derive the pyrolysis zone spread rate formula (Sec. 2.4]: this condition is necessary
because the - heat ﬂux from the ﬂame to the surface cannot have a step change at any
value of §. What this requirement produces, unfortunately, is a solid phase preheat
zone solution that cannot be patched to the pyrolysis zone solution at Zip. Although
this result is somewhat unsatisfactory it is reasonable (given the assumption of a steady
pyrolysis zone) because a step change in the mass ﬂux across the pyrolysis front, cou-
pled with a continuous normal gas phase heat ﬂux there, requires a step change in the
heat ﬂux entering the solid [compare boundary conditions (2.15.ii) and (2.16.ii)]. This
in turn produces the discontinuity in B; at é = 5p . We do not expect this to produce
any difﬁculties, however, since the solid phase solution only enters the problem
through the boundary conditions. As long as they are consistent with the gas-phase
processes the resulting predictions will be accurate enough.

To solve for By we put w = 0 in Eq. (2.6) and apply the initial condition,
By (x ,z ,0) = 0, the far-ﬁeld boundary condition, By (x ,oo,t) = 0, the interface condition,
313,, (x ,0,t )/82 = 0, and the patching condition obtained by evaluating the upstream
solution given by Eq. (2.17.i) at xp = épt. The solution is found by using Green’s

functions after a suitable change of variable (Appendix 2.A):

34

= _ - S + u _ .
By (§,11) erfc (—M m {1? 0e cosh2su erfc [ms M VPr ]ds , (2 21)

 

1 1
where u = 71150 - €11)“: - QB; and (0 = [(53 — €p)l(§p(1 ‘ 5.0)]?

i The procedure for ﬁnding B1 is slightly more complicated because the boundary
condition at the interface is inhomogeneous. But since the equation and boundary con-
ditions are linear we use superposition and write B1 = B11 + B12, where the initial and
boundary conditions for B11 are identical to those for BY (witir BYS replaced by B13)
and for B12 we impose the conditions B12(x,z,0) = B12(x,oo,t) = BT(xp,z,t) = 0 and
8Br2(x,0,t ya: = g (x ,t), where g (x ,t) is the unknown heat ﬂux distribution from the
ﬂame. The solution is (see Appendix 2.A for B11, Appendix 2.3 for B12)

 

 

314%“) = WﬁlﬁJ‘T—fé e‘“ + “2)cosh23u erfc[0)s — MWT’FMs
o
‘11 g: g (S) 25
_ - T .. —n (1 —s)
24; 1(1—s)m-,——.r)“""[ e—s it“ (2'22)

where u and to are deﬁned after Eq. (2.21) and 327(5) = aBT(§,0)/an = 2g (x ,t NW? .

Finally for the solid phase we put w = 0 in Eq. (2.7) and apply the conditions
BS (x ,C, ,0) = B5 (x ,oo,t) = B; (x =t,§_,,t) = 0 and 8B; (x ,0,t )IBC, = g (x ,t ). The solution
(Appendix 2.C) is

 

 

1
_ rig 83(5) .. 413:5]

35

where gs (§) = BBS (gm/ans, ”(13 = Cs/Z‘I; = [C l2‘lPrR 111,. For future reference we

note here that g (x ,t) and g“ (x ,t) are related to one another [see Eq. (21011)] by

T.
Q g = is . (2.24)

while g7 (é) and gs (§) obey the relationship (2.16.ii), giving

_ __1_
gr - Q ﬁgs - (2.25)

We are now able to deduce formulas for the pyrolysis front and ﬂame front

spread rates, the ﬂame location, the heat ﬂux to the interface, etc.

2.4. RESULTS

We begin by deriving a complicated integral equation giving the heat ﬂux to the
surface in terms of the various thermophysical parameters. This is done by requiring
that the temperature across the interface be continuous. The following relationships
are obtained at the surface of the preheat zone under and in front of the ﬂame, respec-

tively;

 

 

,0 Y
i) was): as: ’- 3°“. a, sisif.
.. . BS (QC) (2'26)
u)Br(§.0)= Q was). t, sisr.

By substituting for By, B1 and BS and gs from Eqs. (2.21), (2.22), (2.23) and (2.25)
we ﬁnd

36

 

 

 

§ 1 i
_ ‘11 i 81(3) ill;- 8T(S)
”“9 ‘ 2t; 10 -s))/'s—\/E——sds +m2ﬁls-WJE—JEds’ (2'27)
where
Yo...
v +BTSI(§);§p <§<§f
”(’9 = (Bu— Bys)1(§);§f < i <1 , (228)
with

_ _2_” .32 erfc((ns -M1/ﬁ)
Hg)— if e erfc(-MW) d5

0

 

(2.29)

being the integral of Eqs. (2.21) and (2.22) evaluated at 1] = 0. Equation (2.27) is a
type of Fredholm-Volterra integral equation of the ﬁrst kind for the unknown heat ﬂux
distribution gT (é), which is related to the physical heat ﬂux [see Eq. (2.22) et. seq.) by
g = \[P—rgT/Z‘J; . We expect g to decrease in the streamwise direction, eventually to
zero at g = 1. This decrease however, may not be monotonic since local increases in
gT can produce local relative maxima of g. We may expect a relative maximum to
appear only at the ﬂame reattachment point, there being no physical reason for ﬁnding
one anywhere else. In any case, Eq. (2.27) is exceedingly difﬁcult to solve, because
very little is known about integral equations of this kind (Linz, 1989). An asymptotic
analysis (coordinate expansion) about the points x = Q, and i = 1 suggests that the
second term on the right-hand side of Eq. (2.27) produces near Q the growth,
g7. 8: (2(Tv - 1)/Q WHEN—17:5), while the ﬁrst term is responsible for the decay near
(5: 1, g,- = 1 -— 5,. Whether or not the growth in gT is strong enough to produce a
local maximum in g is still an open question that we do not wish to explore here.
Chapter 3 deals with the solution of Eq. (2.27). It will sufﬁce for our purposes to

evaluate Eq. (2.27) at certain choice points where the solution for g1 (é) is easy.

37

2.4.1. PYROLYSIS FRONT SPEED

One such point is Q, , where the left-hand side of Eq. (2.27) becomes
H (ip) = (T, - 1)/Q , so that it reduces to a simple Abel integral equation, whose solu-

tion is

 

(§>-—2- g” T"1 (230)
87' p -‘/7—t l_§p QVR- . .

But from Eq. (2.17.ii) we also have
= __2- M ' 2 31
37%”) EB" erfc(-MVP?) ’ ( ' )

so that the requirement of a continuous heat ﬂux into the surface gives, after equating

Eqs. (2.30) and (2.31) and reintroducing physical variables,

 

 

2
e a e e e e 2
U3, - U; p313; T: - T; erfc (-M‘1Pr ) '

Here the ﬂame temperature is deﬁned as T; = T; + Q‘ Y5,/VC;.,. Note the similar-
ity between this result and the Loh and Fernandez-Pello (1984) formula, which
becomes even more striking when we consider small M , and observe that

U;/(U:. - U;)=U;/U:, under most conditions.

2.4.2. FLAME FRONT SPEED

This is found by using the fact that there is no fuel or oxygen at the ﬂame reat-
tachment point. From Eq. (2.4) we have, by deﬁnition, By (§f ,0)/B” =
(Y0 ,Jv)/(Y0,,/v + YFS), which we equate to Eq. (2.21,) evaluated at g = Q, 11 = 0, to

38

ﬁnd

 

 

Y0, = _2_°J:8_s,erfc(mfs - MW)dS (2 33)
Y0..+VYps~ ‘56 0 erfc(-MW) ’ .

where (of = (0(§f ). In the general case this integral can be evaluated in terms of an
inﬁnite series (Appendix 2.D) so that the root éf can be determined by using a stan-
dard bisection method. For the case of low mass ﬂux, however, it can be evaluated in

closed form to give, after some working, the very useful formula

2 1: Y5../VY;S
2 1+ 13¢ng

 

U f" = U;cosec (2.34)

Hence U; is always greater than U;, as expected. In the special case Y5,/vY,§S << 1,

which is reasonable for low oxygen concentrations, this reduces further to

,_ 4UP , , _2
U f z 1:2 (You/VYps) . (2.35)

 

2.4.3. FLAME LOCATION
The ﬂame position in the pyrolysis zone is found by putting Y0 = Y; = 0 in Eq.
(2.17.i) and in the preheat zone by applying the same conditions to Eq. (2.21). It may

be compared with experimental ﬂame shapes if and when they are ever measured.

39

2.5. DISCUSSION

 

' In this section the predictions of the previous formulas are examined in detail by
thorough calculation of a numerical example and by comparing the ﬂame spread for-
mulas to others already in the literature. For the numerical example it turns out that
the single most important parameter is the ratio QYmJL = Q ' Y5../L' , which
increases with Q * and the oxidizer mass fraction and decreases with the vaporization
enthalpy. Therefore we study its effect by ﬁxing the solid phase parameters for PMMA
(Wichman, 1983; Anonymous, 1988) (p; = 1.19 g/cm3, c; = 0.5 cal/gK,
1.; = 6.4x10'4 cal/cm s K, T: = 643 K, x =1, L" = 380 cal/g PMMA) and by
ﬁxing the ambient gas phase parameters for air at T 3 (p3, = 5.36x10‘4 g/cm3, ‘
(2;, = 0.25 cal/gK, A; = 1.2x10‘4 cal/cm s K, T; = 295 K, Y5, = 0.233,
Pr = 0.7, v = 1.92). For the present we choose U 1 = 90 cm Is and put
Q’ = 4500 cal lg fuel. These parameter choices give R = 4.22x10‘5, C = 0.5,
i Q = 61, L = 5.15, Q/L = 11.84, and B = 0.83 [from Eq. (2.20)]. The minimum mass
ﬂux from the formula h( M ‘(P—r ) = You/(Vx) is M = 0.0636.

Shown in Fig. 2.4 is a plot of M and the surface fuel mass fraction Yps versus
QYO ,JL. Note that in Eq. (2.20) the quantities Q and Y0, are ﬁxed and L is varied
through large and small values. As L/QYO” -) 0, Eq. (2.20) gives
h(Mt/P_r- ) ~ QCYOJV(T,, - 1), which produces the horizontal asymptote. The limiting
value of h (M W ) likewise produces, from Eq. (2.19), the asymptote for Yps.

Figure 2.5 shows a plot of the nondimensional pyrolysis and ﬂame front speeds
versus the free-stream oxygen mass fraction. We observe that the pyrolysis front speed
varies linearly with the oxygen mass fraction, not with its square as predicted by Loh
and Fernandez-Pello formula (1984). In our formula the increase in oxygen concentra-
tion increases the fuel mass ﬂux M which in turn decreases the factor
exp(-—M2Pr )lerfc (—M1/P7 ), apparently as Y 07.0.5 . Consequently the effect of the oxy-

gen is to change this quantity into a mass ﬂux correction factor of the type that

40

 

 

 

 

 

 

 

0-0 I ' l ' i ' 1
0 10 20 3D

YoooQ/ L

Figure 2.4 Variation of mass ﬂux parameter (M) and surface fuel mass fractiOn (Yps)

as a function of QYO ,JL

41

 

SPREAD RATES
P
O
8
l

1.1.rmlml\.

 

 

0.0 0:2 ' 0.4

Figure 2.5 Variation of pyrolysis front speed (
function of free stream oxygen mass

 

 

0.0 ' 0.8 ' 1.0

and ﬂame ﬁ'ont speed (ﬁf) as a

§ )
ITBCIIOII (Y0 g.)

42

frequently appears in physical problems with combined heat and mass transfer (Atreya
and Wichman, 1989; Bird et al., 1960). Although our analysis assumes an Oseen ﬂow
it still appears because the mass ﬂux distribution is of the boundary-layer type
(Emmons, 1956).

The pyrolysis front speed is found to decrease linearly with M (Fig. 2.6). This
occurs because the volatile mass ﬂux into the gas can be increased in only two ways,
by increasing either M or ép. In the former case the fuel entering the gas increases
tirroughout the pyrolysis region whereas in the latter an increment of mass ﬂux is
added in the interval from 5,, to Q, + Ag? . Thus as M becomes larger the incremental
fuel vapor produced by changing Q, can actually be negative, i.e., 5,, can decrease;
correspondingly, Fig. 2.6 and Eq. (2.32) show that as Q, increases M must decrease.
Consequently it is not unusual that Eq. (2.32) should predict a smaller pyrolysis zone
spread. From Fig. 2.6 we also see that the ﬂame front speed initially increases linearly
but then levels off and decreases with further increase of M . The reasons for this are
clear from the preceding discussion, with the assistance of Eq. (2.34) [or (2.35)] and
Fig. 2.4. As QYOJ L increases Fig. 2.4 shows that M and YFW both increase.
Therefore U; decreases (as already discussed) and by either of Eqs. (2.34) or (2.35)
the factor multiplying U; increases. This dominates the decrease of U; so that U ; irri-
tially increases. But, since Y W increases to its asymptotic maximum faster than M
(see Fig. 2.4), the multiplicative factors of Eqs. (2.34) and (2.35) level off while U;
continues its decrease. . Thus, for larger values of M the ﬂame front velocity U ; even-

tually decreases.

Figure 2.7 shows that although 0; and U; both ﬁnally decrease with M, their
ratio U ; IU; increases linearly. This result may be useful for experimental correlations
since it is much easier to measure U; (by surface temperature) and the total mass ﬂux
(by species concentrations) than the ﬂame speed, which requires fast photography,

image processing or other sophisticated techniques.

43

 

0.010 r I m 1 . 1 ' i ' l ' i ' l '

0.0084

0.0031

1

0.0041

SPREAD RATES

 

 

 

 

 

 

,°-° 0.1 0.2 ' 0.3 0.4 ' 0.5 ' 0.6 ' 017 0.0

Y00°

Figure 2.6 Spread rates versus mass ﬂux parameter. HereE is the non- dimensional
pyrolysis front speed, Ef is the nondimensionalp ﬂame front speed, %ms
the approximate (zero-M assumption) nondimensional pyrolysis
speed and E, is the approximate (zero-M) nondimensional ﬂame front
speed. The freason the curves do not coincide is that E and Eare
evaluated for M= 0 whereas in the present theory the minimum v no of
M is 0.0636

 

 

 

 

30 . I a I ' f ' I ' I ' I r I

20% i ‘
m5"
\..
w

10% "‘

O ' I ' I ' I ' I ' I ' I .' I '
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
M

Figure 2.7 Ratio of spread rates (Ep/ Ef) versus mass ﬂux parameter M

45

A note of caution, however; the higher values of M in Figs. 2.6 and 2.7 are very
near to the maximum permissible blowing rates for which boundary layer analyses of
this kind are valid. For if 2 0.62 the analysis of Emmons (1956) predicts ﬂame
blowoff. We note from Fig. 2.4 that the maximum value of M = 0.7. Since our

M W is Emmons’ 117, this produces a maximum value of M = 0.74.

Also shown in Fig. 2.6 are the predictions of the Loh and Fernandez-Pello for-
mula (1984), EI, and Eq. (2.34), Ef, which are valid for small M only. Both produce
upper bounds for the respective spread rates. We note that the Loh and Fernandez-
Pello limit of Eq. (2.32) is justiﬁable when M ~ 0 (10*), which by Fig. 2.4 is possi-
ble only when Q IL ~ 0 (1). For PMMA burning in air, however, we have
Q/L ~ 0 (10), which gives M = 0.37 and thereby invalidates the Loh and Femandez—
Pello’s (1984) formula by introducing a non-negligible correction factor, [exp(-M2Pr )/
erfc(-M11173]2 = 0.44.

For illustrative purposes the ﬂame location was calculated according to Sec. 2.4.3.

A plot of ﬂame height versus streamwise position is shown in Fig. 2.8.

We can compare our pyrolysis front formula, Eq. (2.32), to the recent predictions
of Carrier, et. al. (1990) for their solution of practically the same problem (they do not
invoke the Oseen approximation). Using Eqs. (2.18) and (2.32) with
M417; 13 = 0.508 In (1+3 )IB ”’54 , ose $20 (Carrier et al., 1990) gives

 

 

 

 

2
U. . :0.” * CD 2
+7 =1r(0.508)2 9”} f, , L y + f ”“150?” , (2.36)
U, - Up psAsc, cp..(Tv —T,,) cp, B '

which is exactly the same (with a different constant multiplicative factor) as the result

of Carrier et. al. (1990).

46

 

 

 

 

X

0.4 r I I I ' I I ' r ' I
-— 1.80.5
. -- t=
'- t=10
0.3“ ..........
[I \\\\
I \
N 0.24 I, \‘\
II ‘ \\
u 'I \\
'i \\
0.1~ L ‘t -
i' l
r‘ \
1‘ \
l. i ‘1 ‘
i ‘r
0-0 I ' 'I ' I ‘ I ' I r I ' I ' I ' I L '
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Figure 2.8 Flame locations at different nondimensional time instants. The x and z coor-
dinates are nondimensional Howarth coordinates

47

The other of our two formulas that can be compared with previous results is Eq.
(2.34) [or Eq. (2.35)] for the ﬂame front velocity. We ﬁrst rewrite this in a more fam-
iliar form by subtracting U P. from both sides and using some simple trigonometric

identities to ﬁnd

Y5 .. IVY;S
1 + Y5 ,, Ivy},

 

(1;: (U; - 0;) tanzd) ; <12 = 12‘- (2.37)

If on the left-hand side we put 0; = dx;/dt' and on the right-hand side we write
U; =xf'lt‘ and U; = xglt' and then deﬁne r' = tiltan2<i> we obtain the well-known

spread rate formula of Quintiere (1981),

 

dx‘ x'-x
_i:_= f . P , (2.38)
dt ‘t

which has been extensively used to predict spread rates and to correlate experimental
data (see, e.g., Quintiere et al., 1986). It is here, however, that our analysis and theirs
diverge. Whereas we have solved a particular problem and derived a speciﬁc formula
relating U f' to the known pyrolysis front speed [from Eq. (2.32)] and therefore the
relevant thermophysical parameters [for example, Eq. (2.35) gives U ; ~ Yﬁsz and
U; ~ 5...], they on the other hand have used Eq. (2.38) as the starting point from
which to derive (hopefully) more general results . This is done by postulating vari-
ous functional forms for, e.g., the ﬂame location x;(t') in terms of the heat ﬂux to the
surface and the mass ﬂux from the surface (Saito et al., 1986; Kulkarni and Fisher,
1988) and the time t" (t‘ = (1t/4)p;c;A;[(T: - T; )/q,: 12 is commonly used [Quin-

tiere, 1981]). Recent studies have attempted to model the local heat ﬂux at the surface

48

in terms of fourth-order polynomials in the time t' to simulate the mass ﬂux distribu-
tion for more complex materials such as wood (Kulkarni and Fisher, 1988). Our result
[Eq. (2.37)] bears no relation to these formulations although it does justify the use of
Eq. (2.38). Whether or not this simple result can then be legitimately extended to
describe more general cases can only be answered by examining them in the same

detail as here.

An important omission from our analysis is ﬁnite-rate chemical kinetics.
Although this should not inﬂuence the pyrolysis front speed calculation it will doubt-
less affect the ﬂame speed calculation. The ﬂame will never completely reattach at the
surface, even if Y5... = 1. On the contrary it will break open when the Damkohler
number, D = A "' Y5_exp[—E " IR ‘ T;]/(p;U; 2N3), obtained by nondimensionalizing
Eqs. (2.1) and (2.2), is less than some extinction value Dan. The value of Em at
which this occurs is found by ﬁrst calculating Tf'm from the formula for D , then put- '
ting Y0 = 0 in Eq. (2.22) and determining the point where T; has decreased to this
value. Clearly, Em must be greater than Ep and less than Ef . This ﬁnite-rate
analysis, although complete in principle, requires knowledge of g7 (E), which in turn
requires the solution of Eq. (2.27); we will contend with this difﬁcult problem in the
next chapter. A better sense of the unreality introduced by the assumption of inﬁnite-
rate chemistry can be obtained by rcexamining the steps leading to Eq. (2.33). From
Eq. (2.4), BY(E,0) = — Yoalv — Yps (E) for E less than Ef; equating this to Eq. (2.21)
evaluated at 11 = 0 yields

 

 

Yo... + VYFs(§) 2 °‘ _ 2ng(t0(E)5 - MW)
= __ s 2.
You + vrps 45);" erfc(—MW ) d3 ’ ( 39)

from which Eq. (2.33) follows directly by putting Y ps (Ef) = 0 and (0(Ef) = (of. Now

the maximum value for both sides is unity; it occurs when E: E, , where

49

Yrs (E) = Y ,3 and (0(Ep) = 0. The minimum possible value of the left-hand side
occurs when Y rs (é) = 0. This generally requires large (0(E), since the right-hand side
is small only when this is true; note that for large 0) the right-hand side is 0((0'1).
Consequently if one deﬁnes the location at which Y R,- (E) = 0 as the "ﬂame reattache-
ment" point, it follows that 00(E) must be "large", giving Ef = Ep(l + (0}); even for
(of ~ 3, Ef is already ten times greater than Ep. If, however, the ﬂame terminates
somewhere in the gas because of ﬁnite-rate chemistry the fuel mass fraction beneath it,
at the surface (Yps (E, )) can be substantially larger than zero. Then the left side of Eq.
(2.39) is much closer to unity and (0(E) must be small, giving :, = g, (1 + (0}); for
(of < 1 the ﬂame tip and the pyrolysis ﬁ'ont may approach one another very closely
indeed. The only remaining requirement in this ﬁnite-rate case is that Y [:5 (E) must
vanish at or before E = 1, i.e., all of the fuel liberated by pyrolysis must eventually be
consumed; unlike the previous (inﬁnite-rate) case this need not occur at a reattachment
surface ﬂame. In sum, ﬁnite-rate chemistry effects could produce a ﬂame front velo-
city only slightly larger than the pyrolysis front velocity. The distinction between

ﬂame front and pyrolysis front would then become almost irrelevant.

Another omission that is generally much more important than ﬁnite-rate chemical
kinetics is gas-phase radiation. It is well known that ﬂame spread is not possible over
materials such as wood without a substantial background external radiant ﬂux level
(Saito et al., 1987). We also know from this and previous theoretical analyses (Carrier
et al., 1983) that it is necessary for acceleratory spread over simple materials of the
kind analyzed here. Nevertheless the proper means for including radiation are not

clear and further study beyond the preliminary scope of this work is warranted.

2.6. CONCLUSIONS

 

Analytical expressions have been derived in this study for the pyrolysis front and
ﬂame front speeds. This was made possible by the simplifying assumption of Sec. 2.3,

50

namely, a steady state in the pyrolysis zone. The formulation in temrs of the simi-
larity variables (E11) makes these spread rates constant. Thus, physical quantities that
cannot be reduced to self-similar form in these variables, such as the radiant ﬂux from
the ﬂame for example, are necessary to produce acceleratory spread. The pyrolysis
front speed is found to depend linearly on the free stream velocity and the ambient
oxygen mass fraction, while the ﬂame front speed is proportional to the inverse of the
oxygen mass fraction. The ﬂame front to pyrolysis front velocity ratio varies linearly

with the mass ﬂux.

We have shown that Loh and Femandez-Pello’s formula for the pyrolysis front
speed can only be used in the limit of small mass ﬂux; the discussion of Sec. 2.5 sug-
geSts this assumption is inappropriate even for PMMA under ordinary burning condi-
tions. We have also shown that the well-known spread rate correlation of Quintiere
[Eq. (2.38)] has a basis in this more detailed theory of ours. It is not clear, however,
whether it is legitimate to extend this simple result to cases of far greater generality as
some authors have done. A more comprehensive theory of the kind developed here,
perhaps including radiation, perhaps charring materials, etc., will be necessary to

answer this question.

In next chapter the numerical solution of Eq. (2.27) is obtained, along with the

surface heat ﬂux distribution. An expression for the excess pyrolyzate is also obtained.

51

APPENDIX 2.A: Green’s Function Solution of Gas-Phase Preheat Zone Equation

Here the equations for By and B1 are of the form B, + BJr = (l/Pr)B,,, B = 0
(initial condition), B = B2 = O (in the far ﬁeld, 2 = co) and B, = 0 [at the surface, see
e.g., Eq. (2.10.i)]. In addition, to patch the preheat zone solution for B to that in the
pyrolysis zone we. apply Eqs. (2.17.i,ii) at 11p = (JP—r/2)z NE; , i.e., B = F (Ep)
.=. Bserfcmp — MVP?)/erfc(-M\/Fr-) at xp = Ept.

To solve we deﬁne the new variables x = (x - Ept)/(1 - Ep) and p. =52 - t
(Baum, 1980). Since E= x/t s l we have 1.1 s. 0, so that from the deﬁnition of It the
range of t is now lul s t S T, where T is some arbitrary upper bound. The coordi-
nate x is now eliminated from the equations and boundary conditions, which reduce to
B, =(1/Pr)Bu, Bz =0 at 2 =0 and z =oo and B=F(11p), where
11,, = (Way/«127W, since it = 0 gives t = ‘11 = lul. We therefore write
B = B(z,t;|1rl) since u is only a parameter, so that the patching condition, written for-
mally, becomes B(z,|ul;|u|) = F[(\(P7/2)z/\IEP_W]. The solution in terms of Green’s
functions is now straightforward. Deﬁne G (2 ,t IC,1:) as the solution to the equation
G1 + (l/Pr)G§§ = 5(C — z)5(1: - t) subject to G = 0 when 1: = T, G; = 0 at C = 0 and
G = G); =0 at C: co, whereby

G(z,t| 12,0 = (2.A.1)

V
a

_ ‘11—); —Pr (g—z )2 —Pr (§+z )2
— 40 4) + e 4U '4’) , t _
2‘1 1t(t -1:)

and therefore

Ben; ii”) = —jG<z.z| his) Basin; '9') dc . (2.A.2)
0

52

It is easy to now manipulate Eq. (2.A.2) into the form

Wink-7% e'(""‘)2cosh(2su) F ((9.?) ds. (2.A.3)
0

where u and (0 are given after Eq. (2.21).

53

APPENDIX 2.B: Laplace Transform Solution of Gas-th Preheat Zone

Equation

The equations, initials and boundary conditions are identical to those of APPEN-
DIX 2.A except the patching condition is now homogeneous (B = 0 at x = Ept) and
the interface condition is inhomogeneous (B, = g (x ,t) at z = 0). In this case it is
simpler to work in E11) coordinates from the outset; the equationand boundary condi-
tions become [see A Eqs. (2.13) and (2.16)] 4E(1 - E)B§ = 211Bn + B“,
B(E,oo) = B(Ep ,n) = 0, We introduce the simultaneous change of independent and
dependent variables, u = ln[E(l - Ep )lEp(l - E)], B = exp(—112/2)f (u ,11), followed by
11 = 0N2. . The 7 equations and boundary conditions simplify to
foo - 2f, — (oz/4 + 1/2)f = 0, f (ac/~12) = 0, f (u ,..) = 0 and
f c,(u,0) = (Ila/5hr (u ), where hT (u) 5 gr (E). By now taking Laplace transforms in u
we obtain the parabolic cylinder equation, f w -- (02/4 + (2s + l/2))f = 0, whose

solution subject to the boundary conditions is (Abramowitz and Stegan, 1972)

 

+1 .. " “i
f (3.0/15)=2;1n- hT(s )2 21"(s+-%)D_(23+1) (o') s (11(3) 1(3) (2,3,1) ~

I- 2‘15

We now substitute Eq. (2.13.1) into B: exp(—62/4)f(s,o/\i2) and take inverse
transforms “with the assistance of the convolution integral to ﬁnd (Roberts and Kauf-

man, 1966)

-_r]2§( l-s )

t
Jute gr(s)e j“
Mar) 2 f1; 1 ( 1-s)~ET§—_s) ds. (2.13.2)

 

54

APPENDIX 2.C: Green’s Function Solution of Solid-Phase Preheat Zone Equa-

tion

We solve Eq. (2.7) with w = 0. For convenience we deﬁne E, = — {3, so that
0 s E S +oo. The boundary conditions are speciﬁed before Eq. (2.23), except that
BB3 (x ,0,t)/8E, = - g (x ,t). The associated Green’s function satisﬁes
G,+G§§=5(t;-E,)5(t-r), with G =0 at 1:=T, G =G§=0 at t;=0 and
G = G; = 0 at c: .., Thus, G(§,.r I at) is given by Eq. (AD with Pr =1 and 2
replaced by Es , and

1
Ba (Cam = — IG (6, .z I 1.01;; (mm . (2.c.1)

By now substituting E, = - C3 = 4511,. and g = g5 /2\1; into Eq. (2.C.1) we obtain
immediately Eq. (2.23).

55

APPENDIX 2.D: Evaluation of The Inteng I (A ,8)

104,3) =1)-1’ erfc (Ax - B) dx. Since I(A.0) = ﬁzz —( lNr'c)tan-1A we
0

differentiate I with respect to B holding A constant. This gives

 

 

 

'01 2". 2 2 2A3 32 1 -le(l+A2) AB
—=— ex —(1+A )x x+ dx= __._e er 0
BB «JED p[ [ 1+A2 1+A2 VI+A2 f \/1+A2
C
a .x2
=— e erfc(Ax)dx ,
880

where C :8 N1,+A2. Therefore an equivalent form for I is

C
_ l, l_ —l -x2
1- 4J2 tan 4+4} erfc(Ax)dx. (213.1)

We now deﬁne J (A ,C ) as the integral on the right-hand side of Eq. (2.D.1), which we
now evaluate by ﬁrst differentiating with respect to A , holding C ﬁxed. This yields,

after some working,

tan'lA
I = —l- £(1+erf C) — 2tan‘l A +e‘C2 I e“C2““29d0 . (2.D.2)
ii 2 0

It remains only to evaluate the integral in Eq.(2.D-2). Although it may be possible to
do this by contour integration, we found it easier to expand the exponential in a

power series, assume interchangeability of integration and summation, then to evaluate

56

the simpler integral coefﬁcients of an inﬁnite series. Thus,

tan“A a _ n 2" tan“A
j e-<Cm9>’de=2(—1'§— j tan2”0d0
0 n=0 "- 0

no (_1)nc?.n n—l (_l)mA2(n-m)-l

= ——-—'——-2

 

+(—1)"tan-1A (2.D.3)

Substitution of Eq. (2.D.3) into Eq. (2.D.2) gives the desired solution.

CHAPTER 3

HEAT FLUX DISTRIBUTIONS

SYNOPSIS

In this chapter the heat ﬂux distributions in idealized wind-aided ﬂame spread are
determined. The surface heat ﬂux and the surface temperature can be found by numer-
ically solving an integral equation. Algebraic expressions independent of the free
stream velocity and the oxygen mass fraction are obtained for the surface heat ﬂuxes.
The ratio of total surface heat ﬂux in the under-ﬂame zone to that in the pre-ﬂame
zone is found as a function of the pyrolysis and ﬂame-front speeds. In addition, the
fractions of the total ﬂame heat release that appear as heat ﬂuxes to the pyrolysis
region and to the preheating region are determined. An analytical expression is

obtained for the excess pyrolyzate.

57

58

3.1. INTRODUCTION

 

Tire self-sustaining feedback mechanism for wind-aided ﬂame spread is well
known: the heat ﬂux from the ﬂame must be sufﬁcient to produce a spread rate
sufﬁcient to produce a mass ﬂux sufﬁcient to support the heat ﬂux from the ﬂame, etc.
In order to theoretically determine the rate of wind-aided ﬂame spread over solid fuels
it is therefore necessary to know something about the heat ﬂux to the fuel surface, and
also about the relationship between the heat ﬂux (or the surface temperature) and the
fuel mass ﬂux. For the various engineering analyses that use integral methods to solve
highly simpliﬁed forms of the conservation equations, the heat ﬂux along the entire
surface must be speciﬁed before solving for the spread rate. In theories employing
more complicated forms of the conservation equations it is necessary, for analytical
solution, to specify the heat ﬂux only in certain regions, such as downstream of the
advancing ﬂame (Fernandez-Pello, 1979), or in (and slightly downstream of) the pyro-
lysis zone (Carrier et al., 1990), or only in the pyrolysis zone (Wichman and Agrawal,
1991; Chapter 2); in Carrier et al. (1990) and Wichman and Agrawal (1991) (Chapter
2) the heat and mass ﬂuxes in the pyrolysis zone are directly proportional to one
another (they have the same functional dependence). For the numerical studies using
still more complicated equations (Carrier et al., 1980; DiBlasi et al., 1987), no condi-
tions are needed on the heat ﬂux; it is determined along with the ﬂame spread rate dur-
ing the course of the solution, although some restrictions on the mass ﬂux may Still be

necessary (DiBlasi et al., 1987).

In this chapter we will examine the heat ﬂux and surface temperature distributions
arising in our idealized model of wind-aided ﬂame spread. The previous chapter was
devoted primarily to deriving formulas for the pyrolysis and ﬂame-front speeds. For
practical reasons, obtaining ﬂame spread formulas is certainly a desirable goal. In
order to better understand ﬂame spread, however, it becomes necessary to examine the

heat ﬂux distributions in the physical plane, for it is the heat ﬂuxes that determine the

59

rate at which ﬂame spreading actually occurs. In other words, the ﬂame heat release
and the ensuing heat ﬂuxes to the fuel surface are the principal causes of ﬂame
spread. For this reason it is crucial that the heat ﬂux distributions be determined for
our (or any other) model. But there are also other important reasons. The knowledge
of the heat ﬂux distributions may result in the development of revised, perhaps more
useful, interpretations of wind-aided ﬂame spread. Knowledge of heat ﬂux distribu-
tions also forces us to recognize the weaknesses and limitations in our current models,
that newer and better models might follow. It is of no ultimate use, therefore, to cloak
a model’s intrinsic defects by emphasizing only its most favorable and most visible

features.

In section 3.2 we discuss the model and the surface heat-ﬂux formulas that can be
derived ﬁ-om it. In section 3.3 the heat ﬂuxes reaching the preheated surface are cal-
culated, as well as the preheat zone surface temperatures. In section 3.4 we ﬁrst dis-
cuss the normalized heat ﬂuxes (the heat ﬂuxes found in sec. 3.3 depend upon Yo_;
they are normalized to factor out Y0 ,.) and then develop expressions for the fractions
of the total ﬂame heat release that appear as heat ﬂuxes to the pyrolysis region, to the
under ﬂame preheat region, and to the pre-ﬂame preheat region. An expression for the
excess pyrolyzate is also developed. In section 3.5 we present our conclusions and

suggest future work.

3.2. THE MODEL

 

Shown in Fig. 3.1 is the schematic illustration of the model problem. In labora-
tory experiments the ceiling conﬁguration is the easiest to examine (Mekki et al., 1990;
Chapter 4) because the adverse inﬂuences of buoyancy (which, in the ﬂoor
conﬁguration, lift the ﬂame away from the fuel surface) are minimized; in this case the
ﬂame much more closely resembles the model conﬁguration shown in Fig. 3.1, where

buoyancy and ﬁnite-rate chemistry are neglected. Note, however, that in the ceiling

 

 

 

 

 

 

 

 

 

GAS (oxidizer)
I'__—” :
U” .
§—>Up §-> Uf
YROLYSIS—t §= PREHEAT ZONE = DEAD
ZONE - s g ZONE
Diﬁ’usion ﬂame
x :=U°°t

’ "p SOLID

Figure 3.1 Model conﬁguration

61

conﬁguration buoyancy may produce instability, since cold gas ﬂows between the
ﬂame and the surface (Agrawal and Wichman, 1992b; Chapter 8). Since gravity is
neglected in our model, we are simply drawing attention here to the fact that the actual
tests in the ceiling conﬁguration resemble the illustrations of Fig. 3.1 much more

closely than do tests in the ﬂoor conﬁguration.

After ignition, the tip of the surface-pyrolysis zone propagates downstream with
velocity U;, while the ﬂame tip propagates downstream with a higher velocity U I? ,
burning the "excess" (Pagrri and Shih, 1976) fuel species released from the pyrolyzing
surface and not already burnt there. The wind speed, U L, is greater than both U; and
U f . As shown in Fig. la, the ﬂame is embedded inside a thermal boundary layer.
Tire thermal boundary layer is slightly thicker than the velocity boundary layer, since

the Prandtl number for gases is generally less than unity.

In the model problem shown in Fig. 3.1 there are four distinct zones to be con-
sidered. The ﬁrst is the dead zone, x'ZULt' , where there is no inﬂuence of any of
the processes occurring in the region 09:" <U:.t*. This zone is the consequence of
our use of the boundary-layer approximation, whereby the streamwise diffusion of
mass, energy, and momentum are neglected. The second is the pre-ﬂame preheat zone,
x;<x" <U:.t* , which lies ahead of the ﬂame tip. There are no fuel volatiles here,
these having all been burned at the diffusion ﬂame in 0<x'<x;. The third zone is the
under-ﬂame preheat zone, x;<x" or}, between the pyrolysis front and the ﬂarrre tip.
The fourth is the pyrolysis zone. We assume that vigorous pyrolysis occurs only after
the surface has been heated to the "vaporization temperature", T5. Thus, our Study is
restricted to "thermally simple" materials that "vaporize" only once T: has been

reached.

Consequently, the rate of ﬂame spread (either U I: or U f’ ) is determined by the

rate at which the solid fuel can be heated to T: .

62

The important assumptions in our model are the following: (1) Inﬁnite-rate chem-
istry, which requires that the spreading diffusion ﬂame reattach at the ﬂame tip, x}.
(2) A uniform (Oseen) gas ﬂow, which decouples the momentum equation from the
species and energy equations. With the constant-pressure assumption the solution of
the streamwise momentum equation is simply u‘ = U L (3) A steady state in the
self-similar pyrolysis zone. This requires a solid phase whose adjustment to any
changes in gas-phase conditions is instantaneous. A corollary of this assumption is
that the fuel mass ﬂux is assumed proportional to the incident ﬂame heat ﬂux, regard-
less of the magnitude of the local gradients of temperature in the solid phase. Also
neglected is radiation, bath from ﬂame and surface. It will be discussed to some
extent in Secs. 3.4 and 3.5. Assumption (1) produces the ﬁctitious but convenient
quantity, x}, which we will show enables us to obtain algebraic expressions for the
heat ﬂuxes. Assumption (2), though retained by us (Wichman and Agrawal, 1991;
Chapter 2), was discarded in the recent theoretical analysis of Carrier et al.‘ (1990).

Nevertheless, their ﬂame-spread formula,

2

 

U. _ kU' ch;-l.: L“ CI ln(1+B) 2
P "' °° at t a e a a + a __0.54 , (3.1)
. p, c, A, cp..(T,,—T.,) 0,... B

is identical to ours (Wichman and Agrawal, 1991; Chapter 2) except for the magnitude
of the constant factor k; we ﬁnd k=rr(0.508)2=k,, while they obtain k=k0/4. Inclusion
of the viscous boundary layer changes only the effective free stream velocity depen-
dence (kU.,) (Carrier et al., 1990). Thus, although it is clear that the viscous ﬂow
analysis is more realistic we do not believe the inclusion of viscous-ﬂow proﬁles con-
tributes very signiﬁcantly to our physical (or, for that matter, our theoretical) under-
standing of the ﬂame spread mechanism; obtaining a more precise estimate for k in

Eq. (3.1) does not seem to warrant the effort. For this reason we also do not believe

63

the use of the viscous ﬂow proﬁle will dramatically alter the heat ﬂux distributions we
calculate in the following section, especially because we are interested primarily in
ratios of heat ﬂuxes in various zones. Assumption (3), however, is another matter. It
was discussed in some detail in Wichman and Agrawal (1991) (Chapter 2), and we
note that it is a limitation on our analysis that can be removed only by a more detailed
examination of the pyrolysis zone equations (Chapters 5 and 6). As will be discussed
in Secs. 3.4 and 3.5, it is perhaps the main reason for the disagreement between theory

and experiment.

Under assumptions (l)-(3) we are able to deduce Eq. (3.1) for the pyrolysis front
speed (also see Eq. (2.47)) and an expression for 0} (Eq. (2.48)). Once these ﬂame
speeds are known, the heat ﬂux to the preheat zone (both pre-ﬂame and under-ﬂame)

can be calculated (Wichman and Agrawal, 1991; Chapter 2) from the integral equation

§ 1
_ 81(8 ) 81(5 )
h (§.§p )-‘jl—-\El mdf + ‘17?- ﬁlmds , (3.2)

where Mag, ) is given by Eq. (2.28). Note that h(§,§p)=2«IEH(&,). In Fig. 3.2, the
funcrion h(E,Ep) is plotted against E for E, <E<1 for various values of Y0“; the solid-
phase parameter values are for PMMA (Wichman and Agrawal, 1991; Chapter 2). We
observe that h (EEP) is continuous but that all of its derivatives are discontinuous at
5;.

Equation (3.2) is a mixed Fredholm-Volterra integral equation of the ﬁrst kind,
whose unknown is the normalized heat ﬂux, gT(E). It is related to the physical heat
ﬂux through the relationship g mgr/2155'. The numerical solution for g,- (E) is

presented in the following section.

 

 

 

--- . o...,=1
-- 0.31

-- 0.432
-- 0.333
- - 0.331

-- 0.233

 

 

 

Figure 3.2 Plot of h(E,EP ), given by Eq. (3.3) et seq, versus E. Observe that h is

piecewise continuous at Ef

65

3.3. SOLUTIONS FOR SURFACE HEAT FLUX and SURFACE
TEMPERATURE

 

3.3.1. HEAT FLUX

Into Eq. (3.2) we put G(s )=gT(s )/s3’2(1—s ), then discretize the interval from E,

to 1 into N not necessarily identical steps, with E1=Ep and EN+1=1. Thus, we obtain

— iém

. < ., I”
Hens-E FT%)j =E SG—L-S)dr+RT-7:(1:)G(‘)ds. (3.3)
i=1 Ji=j ‘j-ZE?

 

Regardless of how the integrals over the subintervals [E5 ,Em] are evaluated, we ﬁnd

[é](N+l)x(N+l) [GI(N+1) = [H ](N+1) t (3:4)

as the linear system of equations to be solved for the solution vector [G]. Here [G ] is
the matrix of coefﬁcients that is obtained after approximating the integrals in Eq. (3.3),
and [If ] is the known input vector, or forcing function, obtained by discretizing the
left-hand side of Eq. (3.2). Note from Eq. (3.2) that because of the singularity at E,
the ﬁrst and second integrals in Eq. (3.3) must be evaluated analytically in their

respective intervals on either side of E j.

We determined that the most appropriate way to solve Eq. (3.3) was by using G-
splinesl , also referred to as "linear" splines. Thus, the function G in any interval

[Quin-+1] is approximated as

 

’ 1 Higher-order splines cannOt be used because they require continuity of derivatives. This
will smootlren the g7 proﬁle and destroy the anticipated piecewise continuity of g1 at Ef (see
Fig. 3. 3).

66

 

G (§)-G, é‘gi (3 5)
——G= .

”1‘6 §i+1"§i .

By substituting Eq. (3.5) into Eq. (3.3) we obtain integrals over [gig-+1] that can be

evaluated analytically. The result is

H(§]’§1)-ZGR (§I’§l+1§j)_EGR (gl’gl-lgj)

_r\/T
l-ﬁjj

where

n-

20.- lea—ea -:.-.1.1—:,->- >2 éRU-énl-é: 1.1-e) (3.6)

= i=j+l

 

 

 

“ _ 3/2 3/2
Rta.b.x)=2(bx_ zﬂx-Z- -\/x -b 2(2): _b)(x a) -(x-a)
-a b—a

 

 

2 (x-a )5’2—(x -—b )5”
5 b-a '

 

(3.7)

The solution vector G is now obtained by using Gauss elimination to solve the linear
system (343-111

The accuracy of this numerical scheme was tested by using (3' (§)=§”', with
092156. The function I?(§,§P) was then evaluated from Eq. (3.3). By using this
mag), 6 (g) was calculated from Eqs. (3.5)-(3.7). For m=0 and m=1, (7’ was com-
pletely recovered, as expected (linear splines reproduce linear functions exactly); for
larger values of m the error norm was found to be given approximately by

HG (§)-5(§)II“SC (§)/(N -—1)2. Thus as N ->oo the error becomes vanishingly small.

67

Unfortunately the actual problem [H (§,§p) given by h(§,§p)/\IT:E;, as seen from
Eq. (3.2) et seq.] is n0t so well behaved, see Fig. 3.2. There are two distinct regions,
one where 2‘, ~ 0.(§f)=0(10-2), the other where g ~ 0 ( 1). By using several stepsizes,
. ranging from A: ~ 10‘4 to Ag ~ 10‘2, it was found that the solutions for
(32, . . . 13m changed very little (<5%), but that (31 varied somewhat (10%). This is
to be anticipated because pivoting in Gaussian elimination piles the greatest error on
the last element calculated, which is GI. To check this effect, 62, (i3 and (34 were
used to extrapolate values for GI that were compared to analytically calculated values

at the pyrolysis front using (Wichman and Agrawal, 1991)

 

61:6 (gp )=

 

 

3T 1 2 Eap [TV-1]

= — — . (3.8)
gnu—é,» gnu—e.) VT: l-ép QV'R'

Agreement was found to be within 5%.

Figure 3.3a shows the normalized ﬂux gT (g) for various values of Y0... It is clear
that ﬁgT increases uniformly with Y0... Also, the maximum of gy(§) occurs at the
ﬂame reattachment location, éf, where g7 (é) is piecewise continuous. This is caused
by additional heat reaching the surface because of ﬂame reattachment. The function
gT (é) drops sharply after ﬁf. The abrupt changes of slope in the curves for §=3§f
occur because of the change of grid size; for §S3§f we used A§=O.1§f, while for
§>3§f we used A§=0.025. Figure 3.3b shows a plot of the actual (nondimensional)
heat ﬂux, g=~lP—rgT/2‘E ; we have put t=1, so that §=x. We have also used2 Pr=0.7.
We note by comparison to Fig. 3.3a that although g is piecewise continuous at Q

there is no conspicuous local maximum of the kind seen for gr.

 

2 This value of Prandtl number is chosen because the characteristic length in the gas-phase
is still the viscous diffusion length, V/U... even though an Oseen velocity profile is assumed
(see, e.g., Carrier, et al., 1980). ‘

68

 

 

 

 

 

 

Figure 3.3a Plot of the normalized nondimensional heat ﬂux gr (é) versus é, obtained
from the solution of Eq, (3.4) using G-splines. The stepsize A§ is
increased by the factor 10 at 39. The maximum of the curve represents
the location Ef.

69

 

 

 

 

 

 

0.06 0.03 ' 0.10

Figure 3.3b Plot of the nondimensional heat ﬂux, g =(‘1757 /2)g7~ N; , versus é. Here

g(§f) is always less than g(§p), even though gT(§f) is always greater
than 87' (gp)

70

3.3.2. SURFACE TEMPERATURE

A normalized nondimensional surface temperature in the prehea't zone is calcu-

lated using (Wichman and Agrawal, 1991; Chapter 2)

l

 

. Tho-T; (1.5 )6 (s)
T 5 =45 ————ds . 3.9
,(§) QWE/(zﬁcg...) ! \ls—z; (. )

Note that the surface temperature function in Eq. (3.9) is actually the second integral
on the right-hand side of Eq. (3.2). The proﬁles of T} are plotted in Fig. 3.4 for vari-
ous Y a -. We observe ﬁrst that the surface temperature is uniformly higher for larger
Y0... and second that it is always smooth and above ambient at E, . The same abrupt
changes of slope occur for T} as g7 (or g) because of the switch to a coarser grid

downstream of éf.

 

 

 

71

H
N
.1

 

— Yo..=1
-- 0.31
--- 0.432 .
- - 0.332

- 0.331

—- 0.232 _

 

 

NORMALIZED SURFACE TEMP. (TS)

 

 

 

F1gure 3.4 Plot of the reduced temperature 7:, given by Eq. (3.10), versus g. The ﬁrst
kink (from left) in the graph (for Y 0 ”=1) represents the location of the
ﬂame front. The second and third kinks (numerical error) occur because

of step size changes that make the computation faster. This is also seen in
Fig. 3.3a for Yo..=1

72

3.4. SOLUTIONS FOR NORMALIZED FLUXES and OVERALL FLUXES

 

It is of interest to determine whether the many gr (6) proﬁles of Fig. 3.33 can be
arranged to [fall on a single curve, thereby eliminating the explicit appearance of the
oxidizer mass fraction from these graphs. We found that this was indeed possible,
although separate' correlations were necessary in the under-ﬂame preheat zone
(£1, <§<§f) and in the pre-ﬂame preheat zone ( ﬁf <§<1 ). In the region Q <§<§f the
g1 (é) curves collapsed when gr VE/gyp Vi: was plotted against the modiﬁed coordinate
(ﬁ—épméf ﬁp). The resulting single curve, shown in Fig. 3.53, can be accurately
represented (i4 % accuracy) by the function

 

 

gT<§WE =1+5[ H,

, n=0.6. (3.10)
81‘, (Q NE; gf ‘E-tp ]

In the region é, <§<1 the curves collapse when g1 ﬁ/grlﬁf was plotted as a function of
Uéf. The resulting curve shown in Fig 3.5b, is accurately correlated (13 %) by the

function3

 

8T(§)§ {if 13 . (3.11)

81,(§f)§f - é

As a consequence of Eqs. (3.10) and (3.11) we can calculate g7 (é) at any streamwise
position, 5, in the preheat zone for any oxidizer mass fraction Y a ,,, once the nondi-
mensional pyrolysis and ﬂame front speeds (Q, and g, ), and the normalized» nondi-
mensional heat ﬂux at the pyrolysis front, gT (ﬁp ), have all been calculated from Eqs.

(3.1),(2.48) and (3.8), respectively. We will return to this later, when we compare the

 

3 Note that Eq. (3.11) shows algebraic decay, not exponential (as generally assumed).

73

 

 

 

 

l I
0.0 0.2 0.4 0.3 . 013 1.0

E-S, / 6:5,

Figure 3.53 Heat ﬂux correlation forathe under-ﬂame preheat zone, showing the rise of
the heat ﬂux as the downstream ﬂame tip (reattachment point) is
approached

74

 

 

 

 

 

 

 

1.0 l ' I V l l l l I I I I I
0.84 d
u}; 0.34 _
III
m -t
\ ..
“1‘-
no 0.4T -
0.2+ -
1 I l I I I I I —:I=I¥ I I I I
O 4 8 ~ 12

Figure 3.5b Heat ﬂux correlation for the pre-ﬂame preheat zone. Note how rapidly the
decay occurs

75

ﬁ-dependence of the physical (g (§)) and normalized (gT (§)) nondimensional heat

ﬂuxes.

At present it is instructive to make comparisons of our results to those of Carrier
et a1. (1990), based on the under-ﬂame correlation given by Eq. (3.10). In physical
terms, Carrier et al. (1990) postulate that the preheat zone ﬂux, g =8T/ay , is weakened
downstream of the pyrolysis front, 1,, , by a factor f“2 because of vertical dilution and
by an additional factor of Jc’l’2 because of the functional dependence of army on the
self-similar coordinate, ‘n=y/\/J_t- . This means that gx (or gT‘E for the normalized
ﬂux) should be constant in the preheat zone. From Fig. 3.5a we ﬁnd that this is
indeed the case, at least when ﬁ-ép is small. As é-ép increases, however, this corre-
lation function will remain at unity while ours rises to a value of 6 at éf . This rise is
caused by impending ﬂame reattachment, which bends the ﬂame back toward the sur-
face downstream of the pyrolysis front. Although strict reattachment is impossible, the
"excess pyrolyzate" calculations of Pagni and Shih (1976) show that the ﬂame
approaches the surface because the fuel vapors are all on the surface side. The com-
peting effect (at least in the ﬂoor-spread case) is buoyancy, which lifts the weakening
ﬂame away from the surface. In ceiling ﬂow, buoyancy pushes the ﬂame closer to the

surface.

Impending ﬂame reattachment, along with the correlation given by Eq. (3.10), can
also be used to explain the difference between Figs. 3.3a and 3.3b. In Fig. 3.3a we
plot gT(§) versus i (or g7 (x) vs x when t=1) for which Eq. (3.10) gives
8T,(§f)/ST,(§p) =6VFP7EF, indicating that .between ﬁp and Q the gT(§) curve must
rise. For Fig. 3.3b, however, we recall that g=(‘IF;/2)gT/‘1; , so that when t=1 we
ﬁnd the ratio g f (x, )lgp (xp )=6xp le , which is smaller than unity, demonstrating that
between xp and xf the g (x) curve must decrease. Note that g(x) is still only piece-

wise continuous at xf , where it has a small local maximum.

76

We now observe that the plot of the heat ﬂux versus é, as given in Figs. 3.3a, is
somewhat vague, since coordinate §=x It is only a similarity variable. In order to
obtain a clearer picture of the heat ﬂux variation it helps to ﬁx either time, and vary
only x (as in Fig. 3.3b), or to ﬁx the location and vary only I. When 1: is ﬁxed, say
x=xo, and t is allowed to vary, the dead zone passes at t=xo, and the ﬂame reattach-
ment and pyrolysis fronts pass at t=x0/§f and taco/é], , respectively. Since Q, and Q
are quite small, this has the effect of compressing the pre-ﬂame preheating zone and
lengthening both the under-ﬂame preheating zone and the pyrolysis zone. This is
shown in ﬁgs. 3.6a, for the heat ﬂux g7 of Fig. 3.33, and 3.6b, for the surface tem-
perature of Fig. 3.4; we have used x0¥1 for these graphs. In Fig. 3.6b the region C-A
represents the pre-ﬂame preheating zone and A-B represents the under-ﬂame preheat-
ing zone. Comparison to experimentally- measured temperature proﬁles for PMMA
(Loh and Fernandez-Pello, 1984; Mekki et al., 1990; Mekki, 1991) shows that the
predicted under-ﬂame preheating zone exists over a considerably larger time interval.
The reason for this discrepancy (Mekki et al., 1990; see Chapters 4 and 6) is that the
theoretical analysis employs a mass ﬂux w=M ‘5 that has attained its steady state
(maximum) value whereas in the experiments this maximum value is asymptotically
approached only for large times. Consequently more fuel volatiles are released to the
gas in the theory, producing more excess pyrolyzate and thus a longer ﬂame. This will

be discussed in the Conclusion section in greater detail.

For ﬁxed x=xo and variable t it is not difﬁcult to calculate the total heat ﬂuxes to

the pre-ﬂame and under-ﬂame surfaces. We have

X,/§, x0];
871’)? = I gdt ; .901? = J- gdt . (3.12)
I, X,/§f

respectively, where g (rte/04W /2)gT(x0/t)/‘/.? . The separate values of g are of no

77

 

 

 

 

 

 

 

 

I 6 I r r T 1 I I
_ You-=1
" 0.51
"" 0.432
' ' 0.332
‘- 0.331
‘" 0.232
\
\
.-,~..~
~ . ‘\ “~
4 N‘-:‘?~:to\
-—-—_._;'_’:"s= ------------------------- ‘
Méi’ ...................
' I ' ' ' l V V I l r I I I I
400 800 1200 1300

TIME

Figure 3.63 Heat ﬂux of Figure 3.3a for ﬁxed x (x0=1) versus time t

‘1_

 

SURFACE TEMERATURE (K)

78

 

 

 

-- YO..=0.232
- - 0.331 -

 

 

 

- 0.333
-- 0.432 1
' T I l — 1.0
a . . . . . . , . . a
400 800 1200 1600 2000
TIME

Figure 3.6b Surface temperature of Figure 3.4 for ﬁxed x (to =1) versus time t

79

great interest to us -- the ratio is. To obtain this, we use Eq. (3.11) for g” and Eq.
(3.10) with n=0.5 for g'UF; using n=0.5 instead of n=0.6 allows us to evaluate the
integral in Eq. (3.12.ii) exactly while introducing. an error of less than 2% into [we

when evaluated with n=0.6. By neglecting Q5 and using gT,\/E}T/gr,‘/E; = 6, we ﬁnd

fur _ 5 (6-1)
E: — '5'[03I2 —1 “PS-037] I (3°13)

where 0' = Q I§p. As a —-) 1 the ratio vanishes, since the under-ﬂame region has
disappeared. As 0 -> oo we ﬁnd g'UF/g'pp=503’2/9, showing that the under-ﬂame heat
ﬂux completely dominates the surface preheating. In this case the pre-ﬂame preheating

may be discarded.

A plot of Eq. (3.13) is given in Fig. 3.7, which also provides the second abscissa,
Yo... (non-linear); the data for the 0 vs. Y0... relationship is taken from Fig. 3.3 of
Wichman and Agrawal (1991) (Chapter 2). We see that o for PMMA is large enough
over the entire range of Y0, for the large-o approximation, g’UF lg'pp~ Sam/9, to be
quite accurate. Consequently, our model states that for laboratory-scale wind-aided
ﬂame spread over PMMA (in the ceiling conﬁguration) the fuel surface preheating

occurs almost exclusively under the advancing ﬂame.

80

 

 

 

 

 

30 I I ' I I [ I
20"- .
Eh
Q.
DD
\ . 4
Ch
not)
10" -
0 . r ' l ' I '
0.0 4.0 8.0 12.0

0'

Figure 3.7 Plot of the ratio g’Up/g‘PF given by Eq. (3.13). Also shown is the graph of
5/90‘”2 (dotted line), for comparison. In this ﬁgure the parameters are for
PMMA with a wind speed of 1 m/s. In our model, because of the Oseen
ﬂow assumption, the wind speed appears as a non-dimensional quantity,
and hence, will not change the above ratio.

81

3.4.1. EXCESS PYROLYZATE

Pagni and Shih (1976) have suggested that excess pyrolyzate is responsible for
ﬁre growth. Excess pyrolyzate burns far downstream from its source and preheats the
virgin solid. Depending upon free stream conditions (speed and oxygen concentration)
both convection and radiation heat transfer to the surface could contribute signiﬁcantly,
or could be neglected individually, depending on the situation. An increase in wind
speed increases the convection heat transfer to the surface by decreasing the ﬂame
height. It also extends the preheat region. Radiation from the ﬂame depends on ﬂame
temperature and emissivity, which primarily depend on free stream oxygen mass frac-
tion, and fuel type. Part of the ﬂame radiation is lost to. the environment, whereas the
rest reaches the surface. For compartment ﬁres all of the ﬂame radiation reaches one
wall or another. For a high-convection system in air the ﬂame radiation is negligible
(Mekki et al., 1990), but it becomes much more important for higher oxygen concen-
trations. Even for low oxygen concentrations it becomes important under quiescent
conditions (e.g. microgravity); hence, it is obviously important to include both radia-
tion and convection (with boundary layer terms, i.e., relaxing the Oseen-ﬂow assump-
tion) for a complete analysis. Solution of the resultant governing system of equations,
if not impossible, is extremely difﬁcult. One way to retain the inﬂuences of radiation
in an approximate form is to include its effects only in the region where preheating of

the solid is a maximum.

To ﬁnd the region we ﬁrst determine the excess pyrolyzate. This is the fuel that

could not burn in the pyrolysis region. It is given by (Pagni and Shih, 1976)

Y}:
nae = j(p*u*r;),; dx“ , (3.14)
0

where y}, is the location of the ﬂame at the pyrolysis front xp‘. The ﬂame lies along

82

the locus 11:1] f, (where Y;=Y;=O) which is given implicitly by (Wichman and
Agrawal, 1991; Chapter 2),

Byserfcmﬂ ”MVP—r) _ £1

erfc (-M\/7’7 ) v

 

Converting Eq. (3.14) to similarity variables (Wichman and Agrawal, 1991; Chapter .

 

 

2), we obtain
11]:
. Inmwn
e 0 Y3
= =1+ ——ex [Mp7 2— 34W 2], 3.15
"3p M ‘17,? B P ( ) (11 fl ) ( )
x;
where nip is the total mass loss from the solid, mp=wa'dx*.
0

We now perform an energy analysis of the idealized wind-aided ﬂame spread
problem to determine the relative importance of each zone (pyrolysis, under—ﬂame
preheat and preﬂame-preheat). The total heat lost from the ﬂame in the pyrolysis zone
(gem) is the total heat ﬂux conducted away from the ﬂame in the transverse direction
(both above and below the ﬂame) since there is no streamwise conduction because of

the boundary-layer approximation. Thus,

1‘;
gm, .—_- j [1* £5] dx‘ . (3.16)
0 y TI}:

Note that g with a hat (g‘ ) represents the total heat loss from the ﬂame and g with an

overbar (g’) represents the total heat received by the solid surface. Now, the total heat

83

produced by the fuel (g‘mT) is the product of the heat of combustion and the total

mass loss ﬁ'om the gasifying solid. Hence,

x,
81m“ = Qt'hp =Qwa'dx‘ =§Py +§PR . (3-17)
0

where gm is the total heat lost from the ﬂame in the preheat zone. Converting Eq.

(3.16) to similarity variables we obtain (Wichman and Agrawal, 1991; Chapter 2),

 

 

 

g" = GILT; ” (318)
am 2MP? Q " dn “" ’ '
where
' exPl—(Tl ‘M W r )2]
Jr I - 2L f’ (in. (3.19)

"E 11,, ' agar“ erfc (-M\/1-D; )

The above equation includes heat losses from the ﬂame to the regions above and
below it. From Eqs. (3.18) and (3.19), and using
B = ﬁM‘lP—rerfd-M‘JF; )exp(M2Pr), one obtains gPRE/gm (same as Eq. (3.15)).
Still unknown are the ﬁ'actions of the total ﬂame heat release that appear as heat ﬂuxes
to solid surface in the pyrolysis zone (g‘py/gTOT), in the under-ﬂame preheat zone

(gap @701). and in the preﬂame preheat zone (gnu/gm;- ). The ratio g'py/g‘mT is

found by using Eq. (3.18) and evaluating ﬁ- at n=0. Also, only the heat loss below

the ﬂame is considered (heat ﬂux reaching the surface). Thus,

84

g-PY = I315
—B .

 

g (3.20)
870T

From Eqs. (3.13), (3.17) and (3.20), using gTOT=§py+§UF+§PF, we obtain fup/irm
arid EFF/8101'-

These results are tabulated in Table 3.1 as a function of free-stream oxygen mass
fraction. The dependent variable (Y0 ..) is in column one. Column 2 represents the
excess pyrolyzate (fraction of total mass loss). It decreases with increase in Y0 ,,. This
suggests that the ﬂame length decreases as the oxygen mass fraction increases. For
higher oxygen mass ﬁ'actions the fuel does not have to travel far ahead in order to ﬁnd
oxidizer. For air we found the excess pyrolyzate to be 0.586. Pagni and Shih (1976)
found it to be 0.57. Since their analysis used the boundary-layer momentum equa-
tions, this result is yet another indication that the Oseen ﬂow approximation is indeed
a good approximation for the wind-aided ﬂame spread problem. Column 3 presents the
ratio of the ﬂame heat release in the pyrolysis zone to that in the preheat zone. For
lower oxygen mass fractions the ﬂame heat release in the preheat zone is larger. For
100% oxygen ﬂow, both zones are equally important. Column 4 to 6 presents the
fractions of the total ﬂame heat release that appears as surface heat ﬂuxes to the pyro-
lysis region, the under-ﬂame preheat region and the preﬂame preheat region, respec-
tively. It clearly indicates that the under-ﬂame preheat zone is the most important as

far as preheating is concerned.

Since the ﬂame is ﬂat and relatively close to the surface, it could be assumed
that the inﬂuences of radiation are limited to the under-ﬂame zones. The streamwise

radiation ﬂux can probably be neglected~ (Beier, 1982).

85

Table 3.1 Ratios of the heat released by the ﬂame, or received by the surface in each
zone, to the total heat released, as a function of Y0... Here the is the excess
pyrolyzate, 711,, is the total mass lost from the solid, g" and gm are the
total ﬂame heat release in the pyrolysis zone and the preheat region, respec-
tively, and g'py, gm. and g'pp are the total surface heat ﬂux in the pyrolysis
zone, the under-ﬂame preheat zone and the preﬂame preheat zone, respec-
tively. The quantity gm, is the total heat released by the ﬂame.

 

 

You ﬁle/"3p . gray/8m fry/ﬁrm Eur/8701 in: @101
0.232 0.586 0.706 0.123 0.824 0.053
0.331 0.578 0.730 0.123 0.810 0.067
0.383 0.570 0.754 0.123 0.802 0.075
0.432 0.565 0.770 0.123 0.794 0.083
0.610 0.542 0.845 0.123 0.765 0.112
1.000 0.499 1.004 0.123 0.711 0.166

 

86

Finally we examine the connection between our heat-ﬂux distribution (Fig. 3.3)
and certain previously - assumed distributions. A nondimensional plot of the heat ﬂux
at a ﬁxed streamwise location versus time for our model is shown in Fig. 3.63. Quali-
tatively it compares well with the assumed distribution of Annamalai and Sibulkin
(1979a), there being only two minor differences. The ﬁrst is that their heat ﬂux jumps
from zero to a constant under-ﬂame value; we have a non-zero, and in many cases
non-negligible, preheat zone heat ﬂux, depending upon the value of a. Second, they
assume that at xp the heat ﬂux drops to a new constant value, a phenomenon explained
as heat-ﬂux blockage by volatile fuel vapors. This process has not been explicitly
included in our model, although it appears implicitly in the mass ﬂux factor containing
the B-number (see Eq. (3.1)). The heat ﬂux distribution of Fig. 3.63 also compares
well qualitatively with that of Quintiere et al. (1986), although they study acceleratory
upward ﬂame spread. Nevertheless, they ﬁnd that pre-ﬂame preheating is negligibly
small, which corresponds to the limit a -9 co in Eq. (3.13). It is also encouraging to
observe that their surface heat-ﬂux measurements correlate decently when plotted
against x/xf, which is the same as E,/§f for ﬁxed t. Although they did not use our
ﬂux function [given by ng/gTIxf = gx 3’2/gfxf3’2 (for ﬁxed t)], they did extended their
correlation up past the pre-ﬂame zone (x Ixf >1), and also extended it downward, at
least for PMMA, to 1:09 :0.03. We observe that a limited downward extension of the

abscissa of our correlation from Fig. 3.5b to Fig. 3.53 is also possible, because

(é—ép )/(§f -§p) = §I§f when §>§p; note, however, that the ordinate has changed from
gvgféf to 815/ng3

3.5. CONCLUSIONS

 

In this chapter the heat ﬂux distributions occurring in idealized wind-aided ﬂame
spread have been examined. By numerically solving an integral equation [Eq. (3.3)]

we were able to determine normalized (gr (5,» and physical (g (§)) heat ﬂuxes, as well

87

as the surface temperature, 123(5). We then eliminated the Y 0,, -dependence from the
ﬂuxes, producing the analytical correlations of Figs. 3.53,b (see Eqs. (3.10) and
(3.11)). The pre-ﬂame surface ﬂux was found to decay algebraically, not exponentially
(see Eq. (3.11)). The correlations were then used to derive expressions for the total
heat ﬂuxes to the pre-ﬂame and under-ﬂame regions; their ratio is given by Eq. (3.13),
which shows that as o = g, lép —) co the heat ﬂux from the ﬂame to the surface occurs
primarily in the under-ﬂame preheat zone. It was found that the excess pyrolyzate is
nearly 50% of the total, and that it decreases with increase in the free stream oxygen
mass fraction. An energy analysis suggests that most of the preheating occurs in the
under-ﬂame preheat zone and that nearly 80% of the total heat produced by the ﬂame
falls onto the sample in this zone. This fraction also decreases with increase in the

free stream oxygen mass fraction.

These results are in direct contrast to the experimental measurements (Mekki et
al., 1990; Chapter 4), which show that the ﬂame tip and the pyrolysis front are closer
than predicted, and that most of the preheating of the surface occurs in front of the
ﬂame tip, in the pre-ﬂame preheat zone. There are several reasons for this contrast,
one of which has already been brieﬂy discussed, namely the difference between the
theoretical mass ﬂux proﬁle in the pyrolysis zone and the "actual" (experimentally
measured) proﬁle. The latter never becomes as large as the former, so that much less
pyrolyzate is released for gas-phase combustion in the experiments. This produces
shorter ﬂames, less under-ﬂame preheating, and closer ﬂame and pyrolysis fronts. It
suggests that putting w=M\5t- is an idealization that is achieved for PMMA only for
very large burning times [which are beyond the capacity of current measurement
equipment to study (Mekki, 1991)]. Consequently, if we deﬁne an "x‘m-solid fuel" as
one that quickly establishes the self-similar w=M ‘1; pyrolysis-zone mass-ﬂux distribu-
tion we can safely conclude that PMMA is not an "ideal x‘m-solid fuel". This does

not mean that no such material exists, or that materials do not exist that approximate

88

better this dependence; it only means that PMMA, the material chosen for the particu-
lar set of experiments of Mekki et al.(l990) (Chapter 4), does not show this ideal
behavior. This observation may lead to signiﬁcant engineering insight when we ask
the following question - what characteristic features of PMMA (and similar solid fuels)
produce the discrepancies between the theoretical and practical results? Once these
features are known and understood one may design materials that maximize the non-
ideality; one may produce materials that exhibit slow spread rates, short ﬂame
overhang, low volatile release rates, etc. This is a desirable ﬁre-safety design goal. It
may be necessary, for other applications (not Fire Safety) to design materials that
optimize the spread rate; in this case one would try to exploit as many features of the
"ideal x’m-solid fuel" as possible. Another reason for the discrepancy is ﬁnite-rate
chemistry. In the actual case there is never ﬂame reattachment; the ﬂame always
extinguishes before it once again touches the surface. As discussed in Wichman and
Agrawal (1991) (Chapter 2), under the correct conditions this can bring ﬂame tip and
pyrolysis front into rather close proximity. Consequently what was once an under-
ﬂame preheat ﬂux now becomes a pre-ﬂame preheat ﬂux, thereby upsetting the balance
given in Table 3.1. Finally, discrepancies also arise because our discussion of preheat-
ing is limited to ﬂame heat ﬂuxes since "hot gas wash" and radiation have been
ignored in both this and our previous analysis (Wichman and Agrawal, 1991; Chapter
2). The simplest way to treat "hot gas wash" (Carrier, et al., 1990) is to require that
the incoming ﬂow (across the leading edge at x=0) have a uniform initial temperatme
T,- greater than ambient; for large vertical distances from the surface the temperature of
the gas in the region 0 < é <1 will be T,- > T a» where T“ is the "ambient" or "room"
temperature.4 Thus, depending on how much greater T,- is than T”, the surface can be
signiﬁcantly preheated. In the Steiner—tunnel apparatus this preheating is important.

 

4 Our analysis considers the simplest case, T,- = T”.

89

For this reason, the pioneering study of Carrier, et al. (1980) (whose motivation was to
develop a detailed theoretical model of the Steiner test) has an extended discussion of

"hot gas wash" preheating effect.

Despite these limitations we conclude by making an observation that nearly all of
the heat transfer to the sample occurs in front of the pyrolysis zone. Our model sug—
gests 88%, of which the under-ﬂame surface receives the far greater fraction, between
71% and 82% of the total. Since for high oxygen concentrations actual laboratory-
scale ﬂames are quite luminous it seems appropriate in future work to focus attention

on the inﬂuences of soot and .ﬂame radiation on wind-aided ﬂame spread.

In next chapter detailed experiments are performed in order to determine the
pyrolysis front speed and the ﬂame front speed. The solid behavior is analyzed through

species measurement.

CHAPTER 4

AN EXPERIMENTAL INVESTIGATION

SYNOPSIS

This chapter presents the results of 3 detailed experimental investigation on lam-
inar forced ﬂow wind-aided ﬂame spread over PMMA in the ceiling conﬁguration.
The speed of propagation of the pyrolysis front and ﬂame front and the production
rates of major chemical species are measured as a function of time. The objectives are
to study the dependence of the ﬂame and pyrolysis ﬁ'ont speeds on the free stream
velocity and the oxygen mass fraction and to infer the local fuel pyrolysis rates from

the measured production rates of major chemical species.

90

91

4.1. INTRODUCTION

 

This work presents a detailed experimental investigation of laboratory-scale lam-
inar wind-aided ﬂame spread along a ceiling-mounted PMMA slab. The gas ﬂow
along the slab is forced and its speed and composition are controlled.

Apart from the ASTM E-84 tunnel test (Anonymous, 1987), most experimental
investigations of wind-aided ﬂame spread are for the buoyancy-controlled upward
ﬂame spread (Quintiere et al., 1986; Saito et al., 1986; Kulkarni and Fisher, 1988).
However, in these investigations, independent control of environmental variables such
as the ambient gas speed. gas composition and external radiation was not possible.
This capability has proved very helpful in identifying the controlling mechanisms of
wind-opposed ﬂame spread (Fernandez-Pello and Hirano, 1983). For wind-aided ﬂame
spread, there is only one such detailed experimental study on PMMA (Loh and
Fernandez-Pello, 1984).

Several theoretical models of wind-aided ﬂame spread over non-charting solids
have been presented in the literature (Fernandez-Pello, 1979; Annamalai and Sibulkin,
1979b; Carrier et al., 1980; DiBlasi et 31, 1987). Recent models for non-charting
materials (Chapter 2, Carrier et al, 1990) have sparked a considerable controversy
since they seemingly disagree with the detailed experimental study of Loh and
Fernandez-Pello (1984). The models suggest that the speed of the pyrolysis front
varies linearly with the free stream oxygen mass fraction, whereas the experiments
suggest a quadratic dependence. This clearly points out the need for further experi-
mental investigation to determine which of the many assumptions made during the
development of the models or possible experimental errors are responsible for the
discrepancy. Thus, the objectives of this work are: (i) To provide a physical under-
standing that will serve as the basis for the development or reﬁnement of theoretical
models (Chapter 6); (ii) To provide additional measurements for non-charting materials
like PMMA.

92

4.2. EXPERIMENTAL SETUP

 

A schematic of the experimental apparatus is shown in Fig. 4.1. It consists of
three main parts: (i) A small-scale combustion wind tunnel which is capable of produc-
ing a desired external radiant ﬂux on the sample surface along with a ﬂat-plate boun-
dary layer ﬂow at a given free-stream velocity and composition. (ii) A set of continu-
ous gas analyzers for measuring the depletion of oxygen and production of C0 2, C0 ,
H 20 and total unburned hydrocarbons. (iii) Data acquisition equipment to collect,
store and process the data. This apparatus was designed by Dr. Arvind Atreya and

was built under his guidance by his previous students.

4.2.1. COMBUSTION WIND TUNNEL

The combustion wind tunnel is the heart of this apparatus. It consists of three
main sections, namely inlet, test and exhaust sections. Gases are supplied to the inlet
section by two sources: (i) pressure-regulated laboratory air supply, and (ii) high pres-
sure gas cylinders (02 or N 2). The gas composition and speed are controlled and
metered by critical (sonic) nozzles. The resultant mixture enters the tunnel test section

through a large settling chamber and a set of ﬂow conditioners.

4.2.1.1. Inlet Section

The desired air ﬂow and composition results from the mixture of air and oxygen
(or nitrogen) after ﬂowing through sonic oriﬁces, as illustrated in Fig. 4.1. The air
supplied by the building compressor ﬁrst enters a large tank in order to dampen the
mechanical oscillations. The air then ﬂows through a sonic oriﬁce, of known diameter,
at a certain upstream pressure. This pressure should be higher than the critical sonic
pressure, whereas the downstream pressure is nearly atmospheric. However, oxygen
and nitrogen are supplied by pressurized gas cylinders to a large tank before ﬂowing

through another sonic oriﬁce. When the nozzle diameter, upstream pressure and ﬂow

93

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

RADIANT HEATERS

 

 

 

 

 

Moewples
Heat Flux Gages

  

 

 

 

 

 

 

 

 

DWI-Observation Window
PAoProbe Acme:
TM-Wenee Mampulators
HL-Hemd Line
SN-Sontc Neale

 

 

 

Figure 4.1 Schematic of the experimental apparatus

94

density and temperature are known, the mass ﬂow rate can be computed as (Holman,

1984)

 

. 28c Y 2 7'1
= e 4.
m A” P‘VRTI y+1[y+l] ( 1)

where

77': = mass ﬂow rate (lbm/sec)

AN = area of the nozzle (in 2)

P,- = inlet static pressure (psia)

gc = gravitational conversion factor (32.2 lbm.ft/lb.sec 2)
R = gas constant (ftlbf/lme)

T,- = inlet temperature (R)

7 = ratio of speciﬁc heats of the gas (op/cw)

The sonic nozzles were calibrated using a tracer-gas technique, where a known
ﬂow rate of methane is introduced downstream of the oriﬁce, and the mole fraction of
the methane-air mixture was measured. The discharge coefﬁcient CD of each nozzle
was carefully determined from the ﬂow measurements and was found to be 0.97
(Abu-Zaid, 1988). The mixture ﬂows through a 20 foot long, 3 inch inside diameter
tube before entering the settling chamber through 8 pipes, which branch out of the
manifold at equal angles.

4.2.1.2. Turbulence Manipulation Section

The ﬂow exits the settling chamber with large-scale and small-scale eddies.

Hence, the ﬂow is ﬁrst conditioned in the turbulence manipulation section before

95

running through the test section. Combinations of honeycombs, glass beads and ﬁne
screens were tested for a Blasius type boundary layer ﬂow in the test section. The
honeycombs are used to damp the large scale eddies and the ﬁne screens are used to
dissipate the small-scale eddies. A two-inch space full of glass beads provides an inlet
uniform ﬂow. The high velocity near the walls of the settling chamber was decreased
using a screen frame. Final results for velocity distributions were very close to the

Blasius boundary-layer over 3 ﬂat plate. Details are described in Mekki (1991).

4.2.1.3. Test Section

The usable portion of the tunnel test section is 0.81 m long and 0.153 m wide.
The test sample (.76 m long, .076 m wide and .0127 m thick) is placed horizontally
along the tunnel ceiling while the bottom of the. tunnel is hinged at the inlet, making
the tunnel depth adjustable. The tunnel depth at the inlet is 0.1 m and that at the
exhaust end can be increased to 0.13 m . This provides a maximum 30% increase in
the cross-sectional area at the exhaust end to compensate for acceleration of the gas
core because of boundary layer growth and gas expansion due to heat release. The
damper on the exhaust fan and the exit tunnel depth were adjusted to provide atmos-
pheric pressure in the tunnel test section to within l><10"4 torr. This was necessary to
prevent gas leakage in or out of the tunnel for chemical measurements and also to
maintain a nearly constant free stream velocity. A maximum 10% increase in the
free-stream velocity at the exit was observed. To further reduce the effect of variation
in the free-stream gas velocity, data for only the ﬁrsr 0.5 m were used. Measurements
of the velocity proﬁle inside the tunnel showed that the ﬂow was nearly laminar.
Also, the RMS level of turbulent ﬂuctuations was found to be less than 1% of the

free-stream velocity.

External radiation on the sample surface can be provided from below by an

electrically-operated radiant heater with a water-cooled shutter (not used in the present

96

study). The hinged bottom portion of the tunnel contains a 0.153 m wide and 0.76 m
long infrared optical glass window which transmits about 70% of the infrared radiation
to the sample surface. The heaters were arranged such that the radiation measured at
the sample surface was uniform to within :1: 3% over the entire length. The entire tun-

nel test section was maintained between 315 K and 335 K by cooling water.

4.2.1.4. Exhaust Section

The stratiﬁed products of combustion are mixed in the electrically-heated exhaust
section of the tunnel in order to obtain a representative gas sample for transient chemi-
cal analysis. The chemical measurements, along with the measured mass ﬂow rate
inside the tunnel, yield the production and destruction rates of chemical species. These
data were then reduced in terms of the mass production rate of the species at the
instant they were produced. A detailed description of the gas analysis equipment
along with the time lag and response-time corrections is given elsewhere (Atreya,

1983).

4.2.2. GAS ANALYSIS EQUIPMENT

For the species concentration measurement, a constant ﬂow rate of a representa-
tive sample from the tunnel was supplied to the gas analyzers by a metal bellow
vacuum pump. To reduce errors due to condensation, the lines running to the total
hydrocarbon [THC] and H 20 analyzers were heated by either electrical tape or hot
water as shown in Figure 4.2. The gas was then passed through a cold-trap at -—5° C
and dried before passing through the 02 and the C0 -C02 analyzers.

Prior to every experiment, the gas analyzers were ﬁrst zeroed with nitrogen gas,
then adjusted to the proper reading for the known concentration of every specie of the

calibration gas.

97

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

      
 
   
  

Dew Point
Meter

Heated line

CO-CO
2

  
    

Analyzer

Flow
meter

Figure 4.2 Schematic of the gas analysis equipment

98

4.2.2.1. Total Hydrocarbon Analyzer

The total hydrocarbons were measured by 3 Flame Ionization Detector (FID) gas
chromatograph GC —3BF , which used a 40%1-1 2—60%He mixture for fuel. The FID

had a very good response time and a time constant of 1.0 second.

4.2.2.2. Water Analyzer

The water vapor concentration history was measured by a condensation Dew-
Point hygrometer (General Eastern 1200APS ). The dew-point temperature was meas-
ured by optically detecting the condensation on a temperature-controlled mirror sur-

face. The instrument had an accuracy of i0.2°C and a time constant of 1.0 second.

4.2.2.3. CO-C02 Analyzer

An infrared IR 702 nondispersive dual gas analyzer was used for C0 and C02
concentration measurements. The ﬂow was dried before entering the analyzer. The
meter had a good accuracy of :l:I% of full scale (CO : 0-3%, 0-12%; C02 : 0-6%, 0—

20%) and a time constant of 2.6 seconds.

4.2.2.4. Oxygen Analyzer

The oxygen concentration was measured by a Beckman 0M -11 polarographic
analyzer. This meter also required dry ﬂow. The meter had an accuracy of i0.l% and

a time constant of 1.5 seconds.

More details regarding experimental setup are described in Abu-Zaid (1988), Nur-
bakhsh (1989) and Mekki (1991).

99

4.2.3. DATA ACQUISITION

The analog signals from the thermocouples, gas analyzers and the radiometer
were fed to a data acquisition/control unit (HP 3497A). The data were then acquired
by an HP 486 personal computer using LabWindows software (National Instruments).
The handshaking commands and data transfer between the computer and the HP data
acquisition unit were assured by the use a GPIB card with a standard IEEE 488 inter-
face cable. The data were taken at 12 readings per second with 0.1% accuracy.

4.2.4. SAMPLE PREPARATION

The samples used for these experiments were obtained from white sheets of
PMMA. These were instrumented with nine thermocouples on the surface 0.05 m
apart (Chromel-Alumel; 76 um diameter). The ﬁrst 0.15 m of the sample was not

. instrumented to avoid errors due to ignition.

4.2.5. EXPERIMENTAL PROCEDURE

Once all the desired conditions were set, the sample was ignited with a small
methane porous-metal burner placed at the tunnel inlet with its face parallel to the
sample surface as shown in Fig. 4.3. The fuel ﬂow rate to the igniter was controlled
such that the ﬂame overhang on the sample surface was about 0.02 m . The data were
collected by the computer-controlled data acquisition system and the ﬂame spread pro-

cess was recorded by a video camera.

4.3. RESULTS

After ignition, the physical process that occurs inside the tunnel is schematically
Shown in Fig. 4.3. Here, the fuel vapors generated in the pyrolysis region, A which

extends from x = 0 to x = x are burned in the diffusion ﬂame, which extends from

p!

x = 0 to x = xf, with xf > x The hot combustion products that ﬂow downstream

p0

100

X

 

 

 

 

PMMA sample ‘ p : gniter
l f ‘ Xf Flame : U <——
Gravity 4—
Water cooled plate

Figure 4.3 Schematic of wind-aided ﬂame spread in thetunnel over 3 ceiling-mounted
sample of PMMA

101

of xf , and the ﬂame extension (xf — xp ), help to convectively and radiatively heat the
pristine solid to a temperature (TV) at which it begins to vigorously pyrolyze and con-
tribute fuel to the ﬂame. Thus, the ﬂame spread process consists of the spread of the
pyrolysis front. Clearly, the rate of ﬂame spread will depend on how fast the surface
temperature of the solid is raised to its pyrolysis temperature. This, in its turn,
depends on the radiative and convective heat ﬂux from the ﬂame. The convective heat
ﬂux is proportional to the gas phase temperature gradient at the surface, which depends
on both the boundary layer thickness and the ﬂame temperature. Thus, the two most
important variables that affect this mode of ﬂame spread are the wind speed, which
controls the boundary layer thickness, and the oxygen concentration, which controls
the ﬂame temperature. The results of a systematic study of these two variables are

described below.

4.3.1. TEMPERATURE MEASUREMENTS

Figure 4.4 shows the measured surface temperatures for PMMA as the ﬂame pro—
pagates along the ceiling-mounted sample. These temperatures are typical of all our
measurements. Measurements for PMMA show a plateau at about 643 K, which is

taken as the melting or vaporization temperature.

Figure 4.4 also shows the rate of change of surface temperature for PMMA. From
visual observations it was found that the peak rate of change in the surface temperature
after ignition occurs at the instant the ﬂame tip arrives at the thermocouple location for
both wood and PMMA. By writing a surface energy balance it can also be demon-
strated that this peak corresponds to a sharp increase in the incident heat ﬂux which is
caused by the ﬂame tip arrival. Such peaks are shown in Fig. 4.4. Thus, the ﬂame tip
location, xf (Fig. 4.3), may be determined as a function of time from the measured
temperature proﬁles by calculating the maximum value of dis ldt . Results of such cal-

culations show excellent agreement with 1:, determined from video records (Fig. 4.5).

102

 

 

 

 

 

1000 20
A 800- '10
M
v
a 600-1 '12
D
E u
E 400-1 '8
E13 .
[-0 .
200- -4
0 . r .‘ , . . . o
0 100 200 300 400 500 600 700 800 900 1000

TIME (see)

Figure 4.4 Measured surface temperatures and their rate of change for PMMA during
ﬂame spread (U... = 0.9 m/s, Y0, = 0.331). Here surface temperatures are
plotted for different streamwise locations on the sample

dT/dt (K/s)

103

The pyrolysis front location xp is found by deﬁning a constant surface temperature T,
at which the solid begins to vigorously pyrolyze and contribute fuel to the ﬂame. For
PMMA this is the "vaporization" temperature (643 K). Unfortunately, at these tem-
peratures the thermocouples often detach from the surface due to melting. 'Ihus, opera-
tor judgement is required to determine xp . To eliminate this difﬁculty, A: between xf
and xp was consistently determined by (T, - Txl)/(d1'sldt)mu. An example of this

method of determining xf and xp is illustrated in Fig. 4.4.

It is also important to note that the surface temperatures at xf are much higher
than ambient. They range from 533 K to 573 K for PMMA. This implies that most
of the temperature rise has occurred in the preheat zone ahead of the ﬂame. This
observation is in sharp contrast with the previous assumption (Fernandez-Pello, 1979;
Annamalai and Sibulkin, 1979b; Loh and Fernandez-Pello, 1984) where the surface

temperature is taken as ambient until the arrival of the ﬂame tip.

4.3.2. FLAME SPREAD RATES

Once xf and xp are determined as a function of time, the pyrolysis-front speed,
Up (deﬁned as dxp Idt), and the ﬂame-front speed, U f (deﬁned as dxf ldt), are
obtained from the slope of the least-square-ﬁt line. Data for only the ﬁrst 0.5 m were
used to minimize errors from changes in the free stream velocity (U m). In the ﬁrst
0.5 m of the sample, both xp and xf increase nearly linearly with time, whereas in the
last 0.15 771 they exhibit slight acceleration (Figs. 4.5 and 4.6). Figure 4.7 shows Up
and U, plotted against the free stream velocity; it is seen that Up and Uf increase
linearly with U ..

Figure 4.8 shows Up and U f plotted against the oxygen mass fraction (19.).
Note that several different measurements of U p and U f for PMMA (from reﬂective to
blackened water-cooled bottom aluminum plate) are presented. It is also noteworthy

that Up and Uf are nearly equal (see Figs. 4.7 and 4.8).

104

 

. l .
. o from Temp. -

- from VCR -

   
 

FLAME FRONT LOCATION. xf (cm)

 

 

 

 

v ( .
e 23 _
v 3 .33
o .9 .33 ‘
A .3 .43 -
D .9 .61 .
I o .8 1
. . . . , . . . 1 . . .
800 1200 1600 2000
TIME (sec) 1.

Figure 4.5 Flame-front location as a function of time. Note the slight acceleration for
x, 240 cm . Also note the negligible error [O(mm)], encountered in deter-
mination of ﬂame front location using VCR

105

 

60

50-1

 

 

 

 

l ' I ' O
1000 2000 3000 4000
TIME (see)

Figure 4.6 Pyrolysis-front location as a function of time

106

 

O
'6

4E-04" ~ 6 .. .
3E-04-1 -i

3E-04-i -

SPREAD RATES (m/s)

(0
‘7’
O
.5

1
1

 

 

2E-04 - t . l . j . I
0.4 0.8 1.2 1.6 2.0
' FREE STREAM VELOCITY U,° (In/s)

 

Figure 4.7 Dependence of the pyrolysis and ﬂame front speeds on the free-stream
velocity

107

 

6E-03 . I I I I I I I
)K Reﬂective Aluminum Plate
o—o Dull Aluminum Plate .
. 1:1 Blackened Aluminum /’ .
A Blackened Water-cooled /"
-- Loh at Fernandez-Pello (1984)
413-034 - - Theory Eq (2 32) )K B -

 

SPREAD RATE (m/s) PMMA

 

 

 

0.2 ' 0:4 ' 0:6 2 0:8 1:0
OXYGEN MASS FRACTION You.

Figure 4.8 Dependence of pyrolysis and ﬂame front speeds on the free-stream oxygen
mass fraction (blank symbols represent Up and crossed symbols represent

U,)

108

Figure 4.8 also shows that Up for PMMA depends upon the surface ﬁnish of the
aluminum plate directly under the burning sample. The aluminum plate was used only
for PMMA since external radiation was not required. As is evident from Fig. 4.8, the
surface condition of this plate (and to a lesser degree, the two vertical sides of the tun-
nel that contain the observation windows) considerably alters the ﬂame spread rate
because it reﬂects the ﬂame radiation back to the sample surface. This effect
magniﬁes as Y 0.. increases because the ﬂame radiation increases. Measurements show
that the ﬂame radiation at the plate surface increases linearly from zero at Y0. = 0.2 to
3.5 W/cm2 at Y0. = 1. Thus, for a reﬂectivity as small as 0.1, the reﬂected radiation
becomes substantial for higher oxygen. The spread rate measured with a reﬂective
aluminum foil for Y0... = 0.6 is 2.7 times the spread rate for blackened and water-
cooled aluminum plate and 1.6 times the spread rate for a dull aluminum plate. Thus,
the ﬂame spread rate depends upon the reﬂections inside the tunnel and the result
closest to the truth is that of the blackened and water-cooled plate. For PMMA, these
measurements show that U p ~ Y 01;? . This differs from the previous measurements for
PMMA (Loh and Fernandez-Pello, 1984; dotted line in Fig. 4.8) which show that
U, ~ 1’}... However, the present results for a dull aluminum plate agree well with
those of Loh and Fernandez-Pello (1984), indicating that reﬂection of radiation inside
the tunnel may be the reason for the disagreement between theory and experiment.
Results for a water-cooled plate agree well with the recent theoretical predictions for

PMMA (Chapter 2), which are shown as dashed line in Fig. 4.8.

43.3. SPECIES PRODUCTION RATES

Figure 4.9 shows the species production rates for PMMA plotted against the pyro-
lysis length xp. Production rates of C0 and total unburned hydrocarbons are not
presented because they are two orders of magnitude smaller than the others. The data

for small xp are omitted since the measurements are affected by the igniter ﬂame and

109

 

 

 

 

 

100.01 ' .1
300-: 1
’a‘ 1 1
\ 40.0- -
M - .
v
N .
O
'7 10 0-
X 320-5 -‘
0 '1 I
'35 40-? ‘
0: ' . -.
U)
m . 1
c0 _____
a ..........
1.01 .......... 1
03‘: 1..»2" - - 02-depl. -;
I ' I r _IC012 I I ‘
10 20 40 60 30 100 .

Pyrolysis Front Xp (cm) I

Figure 4.9 Dependence of the species production rates on pyrolysis front-location
(U... = 0.9 m/s; 1— Y0” = 0.23; 2 — Y0... = 0.43; 3 - Y0, = 1.0)

110

henCe they are subjected to a large percentage error. Also, the soot production rate
was not measured even though a substantial amount of soot was formed, especially

during experiments at high Y 0 ..-

Species data were collected even after the ﬂame had spread over the entire sam-
ple, i.e., in the boundary-layer burning zone. These data are presented only for
Y0... = 0.43 (for purposes of clarity). Vertical straight lines are obtained (see Fig. 4.9)
because 1:, does not change during this period since the pyrolysis front has already
reached the end of the sample. For PMMA the mass production rate more than dou-
bles during this period before achieving a steady-state value. The fact that the species
production rates continue to change substantially even after- the pyrolysis front has
reached the end of the sample shows that steady-state mass loss is not achieved in the

boundary layer burning zone.

For complete combustion, the production rates of C0 2 and H 20 and the deple-
tion rates of 02 are directly related to the total fuel production rate between 1: = 0 and

x = x For the local mass ﬂux to vary as x‘05, the total fuel production rate any

I"

- 1
instant must vary as xp

’2. Incompleteness of combustion will only serve to further
reduce this power. However, the data for PMMA show that this power is greater than
0.5 and closer to l; for PMMA at Y0... = 1.0 it is even greater than one. Also note
that the production rate of H 20 (rhﬂzo) follows the depletion rate of 02 (77°10 2) for all
the experiments. However, the C0 2 production rate (rhea 2) is slower. For
Y0... = 0.23, rilco2 > 75102, whereas for Y0... = 1.0, n'tco2 < ”'10: for PMMA. Crossover
between ﬁlm2 and 77102 can be seen at xp = 0.5 m for PMMA (Y0... = 0.43). This
indicates that the chemistry is changing as xp and Y0... are increased. Carbon in the

fuel is converted to unburned soot leading to lower Inc-02. This agrees with the physi-

cal observation that both larger ﬂames and ﬂames at higher Yo... are brighter.

111

4.4. DISCUSSION

 

It is interesting to note that there are several areas of agreement and disagreement
between these experimental results and the recent theoretical models developed for

non-charring materials (Chapter 2, Carrier et al., 1990).

As predicted by previous theories (Chapter 2), both Up and U f are found to vary
linearly with U .. (Fig. 4.7). However, these theories also predict that Uf is
signiﬁcantly larger than Up , which is in contrast with the experimental results (Figs.
4.7 and 4.8), which show that U, 2 Up. Likewise, xf is found to be only slightly
larger than xp (between 5 and 10%) for PMMA, regardless of the free stream velo-
city or the oxygen mass fraction. It clearly suggests that the ﬂame reattachement
point (treated as location of xf in chapter 2) can not be taken as ﬂame tip location.
This discrepancy arises because the previous theoretical models utilize the steady-state
"Emmons" (1956) solution in the boundary layer burning zone. It will be discussed in
much more detail in the next chapter, and is also the basis for that study. As dis-
cussed earlier (Fig. 4.9), the burning zone is unsteady in the solid phase during the
ﬂame spread process. This leads to lower instantaneous fuel mass production rates,
resulting in smaller-than-predicted values of xf and Uf. The case for an unsteady
burning zone is further supported by the fact that the species production rates vary

roughly as xp rather than as xp05.

As is evident from Fig. 4.8, the theoretically predicted (chapter 2) dependence of
Up on Y0... agrees well with that determined experimentally. An increase in Y0-
increases Up , primarily by increasing the heat ﬂux from the ﬂame to the as-yet-
unburned solid surface. This heat ﬂux increases because the ﬂame temperature
increases in proportion with Y0... in the absence of ﬂame radiation and incompleteness
of combustion. This heat ﬂux also decreases because of increased shielding of the fuel
surface from the ﬂame by the evolved fuel mass ﬂux (Note that the fuel mass ﬂux
increases roughly proportionally with Yo n; Fig. 4.9). As discussed earlier, the previous

112

theoretical models overestimate the evolved fuel mass ﬂux in the burning zone by
using a steady-state solution there. This results in an overestimation of the heat block-
age factor. It appears to be compensated by overestimatin g the ﬂame temperature, by

neglecting ﬂame radiation and incompleteness of combustion.

4.5. CONCLUSIONS

 

Several signiﬁcant conclusions can be derived from the experimental results
presented here: (i) The pyrolysis-front and the ﬂame-front speeds for PMMA vary
nearly linearly with the free stream oxygen mass fraction and the free stream velocity.
(ii) The pyrolysis-front and the ﬂame-front are much closer to each other than
predicted by recent theoretical models. This is also true for the pyrolysis-front and the
ﬂame-front speeds regardless of the free stream velocity and/or the oxygen mass frac-
tion, primarily because unsteady conditions exist in the burning zone. (iii) Species
measurements suggests that the pyrolysis mass ﬂux is roughly constant for PMMA,

rather than varying as x'o's.

Although these experiments bring several interesting facts to light, it is worth
remembering that the theoretical models were constructed in the absence of informa-
tion. And despite their limitations, the steady-pyrolysis-zone models comprise self-
contained and self-consistent descriptions of the spread process. They prOvided valu-
able clues for the subsequent experimental work. More importantly, as with all
"clean" models, these descriptions (Carrier et al., 1990; Wichman and Agrawal, 1991;
Chapter 2) provide an orderly basis for thinking about the more cOmplex problem.

In sum, the theoretical models for non char-ring materials predict the pyrolysis
front speed quite accurately despite the steady-state solid-phase assumption in the
burning zone. This underscores the fact that the ﬂame spread rate depends primarily

on local heating of the solid by the ﬂame tip in the adjacent preheat zone.

113

In next chapter we investigate the transient behavior of a vaporizing solid. An
expression is developed for instantaneous mass loss which together with the present

conclusions forms the basis for Chapter 6.

CHAPTER 5

THERMAL DEGRADATION OF A
VAPORIZIN G SOLID

SYNOPSIS

This chapter presents a simple model for the gasiﬁcation of a vaporizing solid
under an external heat ﬂux. An expression is obtained for the transient mass ﬂux as a
function of the net heat ﬂux. This expression was corroborated by transient pyrolysis
experiments conducted on PMMA for heat ﬂuxes ranging from 1.6 to 5 W/cm2 in

nitrogen atmospheres.

114

115

5.1. INTRODUCTION

The previous chapter has demonstrated that the thermal degradation of a solid
plays an important role in ﬁre development. During early stages of ﬁre growth, solid
pyrolysis is transient, and the ﬂame length (height) depends on the availability of fuel.
To incorporate this feature into models of wind-aided ﬂame spread (Chapter 6) or
zone-type compartment-ﬁre models (Peacock and Bukowski, 1990), simple and accu-
rate expressions for the transient mass loss rate are needed. The present work is lim-
ited to gasiﬁcation of vaporizing solids like PMMA. A proper understanding of the
rate of polymer gasiﬁcation in a ﬁre environment is essential to the mathematical pred-

iction of ﬁre growth on such materials.

Depending on heating rates (exposure), two types of thermal degradation of
PMMA can be observed (Khalturinski and Berlin, 1983) (i) isothermal bulk degrada-
tion (chemical regime) and (ii) linear pyrolysis (ablation regime). Our interest lies in
linear pyrolysis because the PMMA samples used in this study are thermally thick.
Under these conditions a narrow layer near the heated surface liquiﬁes when the sam-
ple is exposed to a radiant ﬂux. Small clusters of degradation products must escape
from the surface. Since the boiling point of the degradation products is much less than
the polymer degradation temperature and since a signiﬁcant quantity of only partially
degraded polymer is still available at the surface, the degradation products are
superheated. Consequently they nucleate and form bubbles which grow by mass
diffusion through the molten layer, mass vaporization and thermal expansion
(Kashiwagi and Ohlemiller, 1982). Furthermore, because of the higher temperatures,
the degree of superheating and bubble nucleation rate increase while surface tension
decreases as the surface is approached. Transport of bubbles from the interior of the
polymer towards the surface is also affected by the existence of viscosity gradients
(Kashiwagi et al., 1990). In general the degradation of PMMA occurs according to the

following main stages: depolymerization initiation, propagation of reaction chain and

116

termination. Depolymerization is believed to initiate at Chain ends, at random points
along the chain, or at isolated " weak links" (Lengelle, 1970). Free radicals, formed
during initiation, give rise to the so-called "unzipping", that is the propagation of the
reaction chain. Chemical processes at this stage may be characterized by the length of
the reaction chain, that is, the number of monomer units produced on average for one
initiation. Termination reactions account for the stabilization of free radicals which

occurs through the combination with an H atom.

Modeling Studies of PMMA degradation can be divided into two main categories
(3) Kinetic and (b) Thermal. Kinetic modeling can be further classiﬁed as (i) one-step
global models which employ a one-step global Arrhenius-rate reaction to account for
all chemical processes. Such an approximation has been used with the description of
physical processes to model solid-phase processes (Krishnamurthy and Williams, 1973;
Vovelle et al., 1987). (ii) Detailed degradation models where kinetic schemes,
accounting for chain initiation reactions, depropagation reactions and termination reac-
tions, have also been proposed (Kashiwagi et al., 1985, 1986; Hirata et al., 1985; Inaba
et al., 1986, 1987). Such models have never been coupled to the description of physi-

cal processes.

Even the one-step global model of Vovelle et al., 1987 is too complex to incor-
porate in the wind-aided ﬂame spread problem which requires simultaneous solution of
the gas and solid-phase problems. Vovelle’s model involves numerical solutions of
partial differential equations; the model is transient and includes a one-step Arrhenius-
type degradation reaction in the sub-surface regions that contribute to the gasiﬁcation
rate. What is needed is a thermal model which can provide an explicit expression for
the transient mass loss. The model that now exists is too simple (Tewarson and Pion,
1976), suggesting gasiﬁcation rates proportional to the net applied heat ﬂux. Recently,
Steckler et al. (1991) and Delichatsios et al. (1991) proposed a thermal model which

assumes that gasiﬁcation occurs only at the surface and at a constant "gasiﬁcation

117

temperature". Thus the surface regresses at a constant temperature. Results were

obtained by numerical solutions of ordinary differential equations.

An extensive literature exists for solid-phase modeling (V ovelle et al., 1984a,
1984b, 1987; see the references therein and those cited above) but no direct expression
is available for the transient mass loss. Thus, the objectives of this chapter are to pro-
vide an explicit expression for the transient mass loss as a function of constant and

variable applied heat ﬂuxes.

5.2. GOVERNING EQUATIONS

As shown schematically in Fig. 5.1, during heating the solid undergoes two
different heating stages: (i) The inert preheating stage, in which the surface tempera-
ture increases until it reaches the "vaporization" temperature, and (ii), the vaporization
. stage, in which the solid surface regresses at a constant temperature, liberating fuel
vapor. The assumption of a constant predeﬁned "vaporization" temperature is widely
used in solid modeling (Steckler et al., 1991; Delichatsios et al., 1991), see the experi-
mental results presented in Fig. 5.2. Plotted are the surface temperature and total mass

loss as a function of time for PMMA exposed to a radiant heat ﬂux of 2.1 W/cm 2.

The governing equation for one dimensional heat conduction during the preheat

stage is

2
£31.91, 0<y<,,,,>o-, (5.1)
dyz Otdt

subject to the initial condition, T(y ,0)=T,., and the boundary conditions

"k ['3] :‘qnet (t )=QHV _ qloss = QIN-es C(Ts4-To4o )-hs (T3 -Too)r y =09 t >03 (5'2)
y=0

118

Surface Temp
/

 

Surface Location

 

 

 

 

l 50)

Preheating i Vaporization

Time

Figure 5.1 Schematic of the solid phase model

17

119

 

 

 

 

 

0.8 I I l 1 T r I l r 400
s ----------- - - — ----..I
*3 i
\. 0.6- : a-300
E :
no . . .
v l
to : ,

8 0.44 . --200

r—o I

u :

VJ - I ..

‘5 l

E 0.2-i E 4-100

'8 .

4.) I

O . : .,
0.0- . . , . . . . I . . , . . o

0 tv 400 300 1200 1600 2000

Time (sec)

Temperature (C)

Figure 5.2 Surface temperature and total mass loss as a function of time for PMMA.

Heat ﬂux = 2.1 W/cm2

120

In the vaporizing stage, the regressing top surface moves downward at a rate ds(t )ldt.
Let tv represent the time when the surface starts regressing, i.e., when the surface

reaches TV . The governing equation is

 

d2T 1 dT
.___-.——, r< <00,t>tv, 5.3
dyz a d: S( ) y ( )
subject to the interface condition:
Jr
qm(t) + k —— =pL d3 (' ) , T(s (t ),t )=T,, (5.4)
dy y=s(t) d‘

where L is the. latent heat of gasiﬁcation. These equations were solved both numeri-
cally and analytically. FOr the analytical solution presented below, a semi-inﬁnite solid
was assumed with T(oo,t )=T.., where T g. is the "ambient temperature. However, for
the numerical solution, a thick solid, with convective and radiative heat losses ﬁ'om the

bottom surface, was employed.

5.3. SOLUTION

An extensive literature exists on the pyrolysis of "vaporizing" solids (Steckler et
al., 1991; Delichatsios et al., 1991; Vovelle et al., 1987), but explicit expression for
transient mass loss is not available. Recently Delichatsios et al. (1991) presented an ‘
integral analysis with an exponential proﬁle, similar to the present work. This, how-
ever, resulted in ODE’S and was solved numerically. Thus, a simple but fairly accu-
rate integral model is developed here. The problem is divided into two parts, preheat-

ing and vaporizing.

121

5.3.1. PREHEATING STAGE

During preheating the surface temperature is a function of the applied heat ﬂux,
hence Eq. (5.1) must be solved along with the boundary condition (5.2). A very useful
and accurate expression for the preheating time is obtained by Atreya (1983) for semi-
inﬁnite solids exposed to a constant heat ﬂux. After adjusting the multiplicative con-

stant the expression becomes:

“:30ka [ 7 g t In (ZEN-VEXMB) _ 3112330 ’ (5.5)
6 e2 2 f2 83/2 (2§m+v'6xr—~IB) Bf

where y=T,—T,., er=—(h McoT}, ), §=I-250’T.2./3, 046—43312) and
f =(d;,/8mt§yz). Note that qw is constant here.

Equation (5.5) gives Iv (time to reach T v) very accurately. In Fig. 5.3 results are
plotted for temperature as a function of time. Numerical solution of Eq. (5.1) com-
pares very favorably with Eq. (5.5). The sample thickness was taken as 0.127 cm.
The other parameters used were (Vovelle et al., 1987) k=0.185W/mzs,
Cps=2150KJIkgK , p=ll90kg /m3, T v=635K , Tg,=295K , L=l6501lgm, 8=0.92, and
h=0.4 W/mzK was chosen to give (2,3,, = 0.82W/cm2. We will explain this choice

later in the experiments section.

For varying external heat ﬂux Eq. (5.1) must be solved numerically because in

this case Eq. (5.2) becomes a function of time.

5.3.2. VAPORIZING STAGE

The integral method is used to solve Eq. (5.3). We choose

TO .t')=f;expl-a (7)0 -s (2' ))1 (5.6)

122

 

 

 

 

TV. , ,,,,
300%
tr: 2.9 ,x q"=1.err/cm2
“é
' m 2004
z I
1223 1
Pa :5
8
100
35.
0:.
:3
V3 -- NUMERICAL
0 — APPROX
l ‘ l ‘ l r I
0 100 200 300 400
TIME (see)

Figure 5.3 Surface temperatures as a function‘of time, during preheating of the solid

123

where f=r-T,,, t'=t-t, and a (0)=rj,,';,(t'=0)/k7"', (from Eq. (5.4)).

5.3.2.1. In-depth Temperature Proﬁles at the Vaporization Temperature

It is important to check that in-depth temperature proﬁles obtained by the integral
method, at the onset of vaporization (i.e. surface reaches TV, Eq. (5.6) evaluated at tv)
be very close to those obtained by the numerical solution of Eq. (5.1). Results are
presented in Fig. 5.4. The match is reasonably good. It improves as the applied heat
ﬂux decreases. Moreover the transient effects are much more prominent at lower heat

ﬂuxes.

5.3.2.2. Transient Mass Loss Rate
Integrating Eq. (5.3) from ymd‘) to y=oo gives ds/dt‘=[ti,{;,(2')-ka(t')f,]/pz.
Using this in Eq. (5.4) leads to an ODE in a (t‘):

 

7 IO"—
59: qw( )a2 - [0. + —v J03.
dt pL pL

Here we have used Eq. (5.6) to evaluate (dT/dx)s(,). For the variable net heat ﬂux,
the above equation is solved numerically to give a value of a which can be substituted
into the equation for d5 Id}- to give the transient mass loss. However, for a constant net
heat ﬂux the above equation can be solved exactly to give the transient mass loss rate

as:

 

--—_-, (5.7)

124

 

400 I I I I I I T I T If

  
 
 

U l I .
— NUMERICAL
--~ INTEGRAL ,
300-4 s.‘ ..
“'5‘13? W/sz
200- --------- ..- -
Q“.-.
.‘~~.‘.‘~ J
1 00 q \‘s k..."' -

\
\
~
\
“
\

s
‘Q

Q
s...
.Q

IN DEPTH TEMPERATURE (C)

 

 

 

 

0 I l I I I l I I I I
0.000 0.002 0.004 0.006 0.006 0.010 0.012 0.014

DISTANCE FROM THE SURFACE (m)

Figure 5.4 In-depth temperatures as a function of time, during preheating of the solid

125

where

A (t‘)—1 exp[A (t‘) ]_ -Cp.é.;37

—— —exp _
Ao-1 A0 (PkLXL +Cp, To)

The results are plotted in Fig. 5.5 for four different heat ﬂuxes along with the
corresponding experimental results. Equations (5.1) and (5.3) were also solved numer-
ically to predict the surface temperature during preheating and the surface regression
rate during solid vaporization. They are also plotted for the same heat ﬂuxes in Fig.

5.5. The agreement is good, especially for higher heat ﬂuxes.

For a varying external heat ﬂux (Fig. 5.6) the results are plotted in Fig. 5.7.
Also plotted are the numerical and experimental results. The agreement is good.
Numerical solutions where obtained by using a program developed by Nurbakhsh
(1989) for moving boundary problems.

In order to check the effect of sample thickness, the integral solution was com-
pared with the numerical solution obtained for various sample thicknesses. The results
are plotted in Fig. 5.8 for various constant external heat ﬂuxes. As the sample thick-
ness increases the numerical results approach the integral result. Depending upon the
time duration of the experiment, a correct sample thickness can be chosen from the

above graph to ensure that sample can be treated as thick.

5.4. EXPERIMENTAL SETUP

 

Experiments were performed on PMMA under various heat ﬂuxes to obtain its
degradation characteristics. These were done in a controlled nitrogen environment
cone heater apparatus shown in Fig. 5.9. Although it has been shown that mass loss
rate for PMMA is different in oxygen environments (Kashiwagi and Ohlemiller, 1982),

nitrogen was used because in actual ﬁre situations there is no oxygen between the

MASS LOSS RATE (lrg/nn2 sec)

126

 

 

 

 

 

 

 

200 I 400 ' 600 I 600 ' 10'00 '

TIME (see)

Figure 5.5 Mass loss rate as a function of time

 

12'00 ' 1400

127

 

 

 

 

 

4IIIII7IIIIIIIITI
A —HeetFlux
m d
8 1
{

3-1 ..
E __
N.

.2
ISL.
4.32- 2 ..

a!

Q)

m d
'73
5:1- 1 -
I...
0)
4.:
x q
LII-J
OIIIIEIIIIIIIIIIII
0. 200 ,400 600 800 1000 1200 1400 1800 1800

TIME (see)

Figure 5.6 Varying external heat ﬂux

128

0.0016

 

. G—O E'xptil
. A-A Exptﬁz
q — Integral
0.001%. -- Numerical

.1

0.0008-1

Mass Loss Rate (gm/cmzsec)

 

 

 

 

’ I I I I ' I I I ﬂ I '
0 200 400 600 800 1000 1200 1400 1800
TIME (see)

Figure 5.7 Mass loss rate as a function of time under varying heat ﬂux

129

 

 

 

 

 

 

 

I I 1 ' I ’1 I I
’0‘ I x . an we we 2.3 mm“ I —- i=.766m
o _ -
NW q"=5W/cm2 , _ $3111
8 0-002‘ ~--- acclaim)“
Q — Integral
8
no .t
v
m I I ’ - \ .... e
E. PI’T-ﬂ—‘F -_ ........ .. . é‘
a 0.001 ! - SW/cmz -
(I) I l
S
2
g 1.5W/cm
0.000 t . ’ . . . . . . . r
0 1000 - 2000 3000 4000 5000
TIME(sec)

Figure 5.8 Effect of sample thickness on mass loss rate

130

Cone Heater

a

 

Water cooled cover

 

/

 

 

 

 

 

Viewing
window

 

RN

l

l\\\

 

 

 

 

 

 

 

\

I.

 

N 2
Water cooled Sample , $
housing Thermocouples
~\\\‘ \ ‘V w
\ § §
s s s ow
\ § .0 straightners
\ \ \\‘
_\ \ \
——s s .
Si \ m

 

 

 

'////////////////////////////////////'

\\\\\\\\\V

[/5

7///////////

 

 

 

Load cell

Y-Traverse

’//////////////////.

Pressurized
chamber

 

Figure 5.9 Schematic of the apparatus

Sonic
. nozzle

 

131

diffusion ﬂame and the pyrolyzing surface. During operation, the top part, which
houses the cone heater, is bolted to the bottom part, which houses the sample, weigh-
ing scale and air ﬂow straighteners. The water-cooled bottom housing is connected to
a pressurized chamber which is fed by two independent metered gas lines. A data
acquisition system is used to collect, store and process the voltage outputs from the
weighing scale and the thermocouples. Once the cone heater reaches the desired heat
ﬂux, the water-cooled plate is pulled out, exposing the PMMA sample which was
10 cm x 10 cm and 1.27 cm thick. The top surface of the sample was covered by a
reﬂective aluminum foil with a circular opening (31 cm 2) in the center to eliminate
edge effects. These samples A were subjected to four different (1.6, 2.9, 3.4 and
5 W/cm 2) heat ﬂuxes. The results for the mass loss rate are plotted in Fig. 5.5. From
the steady state value of the mass ﬂux and the applied heat ﬂux it was determined that
surface heat loss (convection and radiation) was 0.82W/cm 2. (Orloff et a1. (1974)
found a value of 0.86W/cm2). The literature (Vovelle et a1, 1987) value of surface
emissivity of PMMA is 0.92. Thus, the convective heat transfer coefﬁcient
(h =0.4W/m 2K ) was calculated accordingly. For variable heat ﬂux (Fig. 5.6) the

results for mass loss are presented in Fig. 5.7

5.4.1. EFFECT OF SAMPLE THICKNESS AND IN-DEPI‘H ABSORPTION.

To check for the sample thickness effect two samples were glued using acrylite
glue making the sample twice as thick. The combined sample was then subjected to a
high heat ﬂux (3.5 W/cm 2). The mass ﬂux remained within the experimental error.
During this experiment the in-depth temperature proﬁles were also obtained. They
match nicely with the numerical results obtained from Eqs. (5.1) and (5.3) (Fig. 5.10).
In order to check for surface absorption a sample was painted black and was subjected
to low heat ﬂux (1.6 W/cm 2). The results obtained for the mass loss rate were identi-

cal with the untreated, sample. Hence, it was concluded that for the white PMMA

132

 

 

 

 

 

 

500 T e 1 I . . . , . .
— Expt.
‘ --- T6635 K
4004 TOP - - races 1:
a far-{____________________:_,_
v - x I
I
135' ' . I
m 300-1 0' "5/
D ’9'”
E . MIDDLE .,I
A?
La zoo-1 .x”
s
Er: ‘ ’/"
E-l ,/ BOTTOM
100% ,v’ /'
,/'/ /
d //" 4/
o - . , . . , . - ' . , . . . . . .
p 0 400 800 1200 1600 2000
TIME (sec)

 

Figure 5.10 In-depth temperatures as a function of time, during vaporization of the

solid under q' =3.4W/cm2

133

sample used in this study, there was negligible in-depth absorption of radiation.

5.5. COMPARISON WITH THE LITERATURE

As a further check of the transient mass loss rate model (Eq. 5.7), it was com-
pared with the data of Vovelle et. a1 (1987). These comparisons are shown in Fig.
5.11. Vovelle et al’s results are also compared with numerical predictions, as shown
in Fig. 5.12. For this case we chose L=10071 lgm as suggested by them. The
decrease in the mass loss rate is due to sample thickness. Also, the predictions show
some disagreement because of inaccuracy in predicting the back boundary condition.
The predictions for bath cases agree well with the data. For varying heat ﬂux we
compare Vovelle et al. (1984a) with the integral model. The heat ﬂux and mass loss
rates are shown in Fig. 5.13. Clearly, for all cases the predictions of the integral
model are encouraging.

MASS Loss RATE (kg/m2 sec)

134

 

0.010-1

CI

0.008“

.

0.006-1

q

0.004~

 

' I ' I ' I
H Vovelle at 0.1.0987)
-- INTGL.

--
"’-
‘—
a’
o‘
"
’

400

’
”

 

 

600 800
TIME (sec)

' 1600

----'----I

 

 

1500 ' 1400

Figure 5.11 Mass loss rate as a function of time. Comparison of integral model with
the data of Vovelle et al. (1987).

MASS Loss RATE (gm/cmzsec)

135

 

0.0016-i

1

0.00124

0.0008ﬁ

 

. , . , . .
O Vovelle ct al.(1984b) -
. -- Numerical

  
    

 

q"=3.1 W/crnz

r l

l

 

 

 

 

000000 0002.5 .
a ..
o0
- .1 ~
°°-° . 1.7 °°o. ‘
I. 0. 000° 0 o 00 ‘ O O .=. 0 0° 0 —
. 0° - .
.. 00° 0 ONOQOOOOOONOOOOOQ
861’ 0 00 °
O O 1.4 /
. r . w
1000 2000 3000 4000
TIME (sec)

Figure 5.12 Mass loss rate as a function of time. Comparison of numerical results
with the data of Vovelle et al (1984b).

136

 

 

A O ' 1 I l 3000

0 . 0 a

0 0.0016-1 0 ””2 0°° n a
an ‘ _ mm :0

a " ”" maxim o I

{ - — Hoot Flux /

a 0.0012: ./ -2000 ’3
:9 . ' . g
o . V
*3 0.00084 9
c: . ~ 8
% ~1000 F
3 03004-1

m .

m

d

a .

0.0000 I , a

 

 

 

we...

1 2 4 5
External Heat. Flux (W/cmz)

Figure 5.13 Mass loss rate as a function of time. Comparison of integral model with
the data of Vovelle et al. ( 1984a).

CHAPTER 6

FLAME SPREAD OVER AN UNSTEADILY
VAPORIZIN G SOLID

SYNOPSIS

 

This chapter presents a simple model for the ﬂame front speed during wind-aided
ﬂame spread on a thick vaporizing solid like PMMA. Model predictions compare
favorably with the experiments conducted in the ceiling conﬁguration. Effects of the

blowing, surface heat loss and the unsteadiness of the solid are analyzed.

137

138

6.1. INTRODUCTION

Wind-aided ﬂame spread is the most hazardous type of ﬂame spread in building
ﬁres, and has been the most difﬁcult to understand and quantify. An improved under-
standing of wind-aided ﬂame spread will result in better material hazard classiﬁcation
and greater ﬁre safety. For ﬁre safety, the knowledge of the burning rate as a function
of time is important because it controls the ﬂame size or the ﬂame length. During
wind-aided ﬂame spread, the burning rate depends upon the local burning rate and the
burning area (or in the 2-D ceiling conﬁguration), which changes with time. Previous
studies (Fernandez-Pello, 1979; Carrier et al., 1980, 1990; Loh and Fernandez-Pello,
1984; Wichman and Agrawal, 1991; Atreya and Mekki, 1992) of wind-aided ﬂame
spread have primarily focussed on determining the length of the burning (or pyrolyz-
ing) zone and have assumed the local pyrolysis rate to be steady. As a result, the
pyrolysis front location and speed (which primarily depends on the ease of ignition of
the as-yet-unburned material) have been extensively analyzed but predictions of the
ﬂame front speed or ﬂame length do not show close agreement with the limited experi-
mental data. This is especially true for the ceiling conﬁguration. Thus, this chapter

focuses on determining the ﬂame front speed.

Previous experimental investigation (Mekki et al., 1990; Chapter 4) on wind-aided
ﬂame spread over a vaporizing solid (PMMA) has shown that the ﬂame front speed is
essentially controlled by solid vaporization. More importantly, it was found that
though the vaporization rate was unsteady throughout the ﬂame spread process, the
spread rate was directly proportional to the free-stream velocity, implying a quasi-
steady behavior of the gas-phase thermal boundary layer. Thus the characteristic time
for the development of a steady thermal boundary layer is much shorter than the
characteristic time for the solid-phase to achieve steady state. Recent experiments

112

have shown that the convective heat transfer under the ﬂame is proportional to x‘ ,

exhibiting steady behavior of the thermal boundary layer (Atreya and Mekki, 1992).

139

Also, for most building materials the surface emissivity is nearly unity (0.92 for
PMMA) resulting in a substantial (~ 1W/cm2) radiative heat loss. This further compli-
cates the problem. Even for the laminar case, the problem is inherently transient and
necessitates the use of approximate solution methods. Thus, the objectives of this
study are: (i) To develop a simple but workable steady laminar gas phase model; (ii)
To couple it with the transient solid phase model (obtained in the previous chapter)
using appropriate boundary conditions at the surface to predict the ﬂame front location
(xf) as a function of the pyrolysis front location and time, and (iii) To provide physi-
cal understanding which could serve as a basis for material hazard evaluation or

development of more complex models.

6.2. MODEL PROBLEM

In the gas-phase energy balance, the heat supplied by the ﬂame to the solid is the
difference between the heat generated by combustion and that convected downstream
by the ﬂow. The heat convected downstream depends upon the thermal boundary
layer thickness and the temperature distribution within it, which is affected by the
ﬂame location. Estimates of the thermal boundary layer thickness show that, for typi-
cal velocities (~ lm/s), it soon becomes substantially larger than 0.1 m while experi-
ments (Mekki et al., 1990; Atreya and Mekki, 1992) show that in the ceiling
conﬁguration the ﬂame stand-off distance is only about 3 mm. Thus, the error in deter-
mining the heat convected downstream by assuming the ﬂame to lie on the surface is
expected to be small. It will result in a slight underestimate of the heat convected
downstream by the ﬂow and hence a slight over estimate of the heat supplied to the
solid. This is conservative from the ﬁre-safety perspective. Hence, to simplify the
gas-phase analysis, we assume at the outset that the ﬂame lies at the solid-gas inter-
face and has a constant temperature T f. This ﬂame is maintained by diffusion of

I oxygen to the solid-gas interface. A schematic of this problem is shown in Fig. 6.1.

140

Solid

 

 

Figure 6.1 Schematic of the model

141

A surface ﬂame was employed by Wichman and Williams (1983) for opposed-ﬂow
ﬂame spread in their analysis of spread into an Oseen ﬂow of oxidizer. In their case,
the ﬂame actually lay on the surface, because of the inﬁnite reaction rate, the high
rates of downstream convection and because the mass ﬂux boundary condition was
linearized. Their surface-ﬂame theory, in other words, was justiﬁed by, and consistent
with, the remainder of the analysis. This is not the case here because the gas ﬂow is
not Oseen (it is a boundary-layer ﬂow) and the mass ﬂux is large near the leading edge
(infact, it is inﬁnite at the origin). Hence, from the standpoint of the equations and
boundary conditions, the surface-ﬂame hypothesis is strictly not possible. The
observed closeness of the ﬂame to the surface arises because of three factors: (i) The
solid is transient, hence the mass ﬂux is small. Also, the boundary layer approxima-
tion breaks down near the leading edge, making the mass ﬂux much smaller (details in
the next chapter). The location of the ﬂame primarily depends on the stoichiometric
ratio. Since the fuel is much less than the corresponding steady-state value, the ﬂame
moves to the surface. (ii) High streamwise convection and (iii) Buoyant forces also

push ﬂame closer to the sample surface.

We take the gas-phase governing equations to be the steady isobaric system

 

 

 

 

Bu 8v
3x + By =0, (6.1)
a“ a" -2. 3_u
Pug-H” a), - a), [it By]. (6.2)
ar ar -2. a'_r

pCp UFV-a—y']-ay [k y], (6.3)
8Y0 8Y0 _ a D 8Y0 64

“ewe“Epe ‘”

 

 

142

We introduce the following non-dimensional parameters,

'

x*-_-x/xf, “Eu/u”, Ref=uwxoolvw T'=(Tf-T)/(Tf-T,,), Y;=Y0/Yo..,

y y
y e =(Ref1’2/xf )Ip/p,dy , v' =Rcf1/2(pv+uI-3%dy )/p,,u ... p2v=p3v” and Le =1,
0 0

The governing equations in the non-dimensional form are

 

 

3“, + 3", =0, (6.5)
8x 8y

i t 2 I
at Bu +V. Bu _a u (6.6)

 

 

ax‘ ay' ‘ay‘z’

+ - _1
axl v 8”. Pr ayiz

 

 

1. e 2 e
u' 3T .8T _1 8T (6.7)

 

. 3Y3 . BY; 1 321’;
—.+V —.—=— I. .
3x a); Pr 8); 2

(6.8)
The boundary conditions are:
aty’=0:

u‘=0;

T'=0 for 0 < x*<l, T‘=1 for 1 < x*< oo;

v' (x * ,0)=Re}’2m;(x ,t )Ipmu, for 0<x ' <xp*, v‘ =0 for x‘ >x;;

Yg=0 for 0<x*<1, erg/ay*=o for l<x‘<oo.

143

aty'—>oo:

Note that the transient behavior of the solid comes through mass ﬂux, mp ”(x ,t ).

6.3. SOLUTION

6.3.1. SURFACE HEAT FLUX UNDER THE FLAME
Th integral method is used to solve Eqs. (6.5)-(6.8). Integrating the continuity
equation (Eq. (6.5)) from the surface to some location Y outside the boundary layer

gives

Y

d t *
. ud , (5.9)
a! y

 

v‘=M(x*)—

where M (x’ )=v' (x' ,0) is the non-dimensional mass ﬂux from the solid surface
divided by the density. For M (x' )=0 and negligible unheated starting length, the
integral solution using cubic proﬁles is given in the literature (Kays and Crawford,
1980). An exact solution for the mass ﬂux from a steadily vaporizing solid under a
boundary layer diffusion ﬂame is also available (Emmons, 1956). It shows that the
mass ﬂux decreases as x‘m. Since this solution neglects the radiative heat loss from
the sample surface and assumes steady vaporization of the solid, it gives the maximum
mass ﬂux. Thus, M (x‘ )=0 and M (x' )=cx"’1’2 represent the minimum and maximum
limits of the mass ﬂux from the sample surface which vaporizes unsteadily during the

ﬂame spread process (Mekki et al., 1990).

144

The integral solution using cubic proﬁles gives the hydrodynamic boundary layer
thickness as: (i) 6" (x"‘)=(2bx"/a)1’2 for M(x“)=0, and (ii) 8"(x‘) = (2[b+c]x'/a)m
for M(x*)=cx*"1’2._ Here a=39/280 , b=3/2 and c is the non-dimensional blowing
constant. Also, the thermal boundary layer thickness (8;) is related - to 5' as:
5H8" =0.977Pr'1’3. Clearly, the boundary layer thickness increases because of blow-
ing represented by c. Note that egg/83;,“ = (1+c /b)“2 is a constant that depends on
the value of c . Thus, the heat ﬂux to the sample in the presence of maximum blowing
may be obtained by simply dividing the heat ﬂux for zero blowing with this constant.
However, for unsteady solid pyrolysis considered here, the above "blowing correc-
tion" is inappropriate. Instead, the local blowing correction recommended by deRis

(1979) is used (see Appendix 6.A for the derivation). This correction has the form:
5(a) = B/(eB—l), where [3 = tagger” (T, -T,)/ 4.2.... (6.10)

where qcom, is the convective heat ﬂux to the sample for zero blowing. For com- *
parison with experiments, ma; (x) is the local average mass ﬂux evolved during the

experiment (see Appendix 6.A for the effects of transient mass ﬂux),

uu,
mm (x) = I a, (x ,t)dt.
xP/UP

Here I is the length. of the sample. Formulas for mp"(x,t) and q'c’;m(x) are derived

below.

The convective heat ﬂux from the ﬂame to the surrounding is

qf_,.. = -k(dT/dy )y=0. Using the variables deﬁned previously for M = 0, we obtain:

145

1/2
x

,. R
chase)=0.332k(Tf—T..)Pr1’3 x . (6.11)

 

where the adiabatic ﬂame temperature T f is given by

 

Y0... Y0,
(AH—L) — (-—v—C,,,+ Cpg )(T,-T,,)

V
T =T + . (6.12)
f v
Cpga + YOJV)

 

Similarly, the oxidizer mass ﬂux from the free stream to the ﬂame located at the sam-

ple surface is given by

.. Rel’2
rho (x)=o.332-"—-Pr1’3 " r0... (6.13)
Cpg x

 

The convective heat ﬂux from the. ﬂame to the solid for M = 0 is obtained from the

difference between the amount generated and convected to the ambient,

rh;(x)AH

4.2m) = — 4;’_...(x) . (6.14)

Thus, the net heat ﬂux into the solid in the presence of blowing (M at 0) can now be

obtained as
41;: (I) = [qc’dnv (x )]E (B(x )) — (Ir'e'rad - (6°15)

Since the surface temperature T, is constant, an,“ is also constant. Thus, q'w(x) is

146

time-independent in the pyrolysis zone (x,-<x<xp ). This considerably simpliﬁes the
solid-phase analysis. Note that by locating the ﬂame on the surface on; is over-
estimated because some enthalpy is also convected downstream between the ﬂame and
the solid surface by the fuel. However, this is small because in our earlier work

(Mekki et al., 1990; Chapter 4) it was noted that xf is only slightly larger than xp.

6.3.2. MASS FLUX IN THE PYROLYSIS ZONE

To develop a simple solid-phase model of "vaporizing" materials like PMMA, we
note that in the pyrolysis zone, the surface temperature remains constant ("vaporiza-
tion" temperature) with the streamwise coordinate x. Also, the maximum temperature
gradient in the x direction. in the preheat zone is at least an order of magnitude smaller
than the temperature gradient in the y direction. Thus, the solid-phase heat transfer is
approximated as one—dimensional normal to the surface. In the "vaporizing" stage, the
regressing top surface moves downward at a speed ds (t )/dt which is important for fuel
production but is too small to alter the ﬂat plate geometry considered in the gas-phase
problem. Hence, the one-dimensional regressing surface model of Agrawal (1992a)

(Chapter 5) is used to predict transient mass loss rate:

 

.. x A C T
ritp(x)=p£_= 4““ ) 1 5’. , A0=—%, (6.16)
dt L A(t) L+Cp,r,

where T=T—T,,, t-=t -t,, (t, is the time it takes the solid to reach its vaporization tem-

perature) and

(A (F)-1)/<A 6—1)exp(A (WA 0) = expt—Cp. éﬁf/tpkL )(L +9.7". )1.

which determines the function A (I).

147

6.3.3. FLAME FRONT LOCATION
Assuming complete combustion, the total fuel produced in the pyrolysis zone
(1:,- < x < xp) is consumed in the ﬂame, which extends till xf (xf 2 xp ). Hence, the

ﬂame front location is determined from the solution of the following integral equation:

11(1)

1%,“ MP Im 'dex

 

(6. 17)

Here, mp is given by Eq. (6.16) and m0 is given by Eq. (6.13) and x,- is the leading
edge of the solid. Eq. (6.17) is true in general, and does not require Eq. (6.13). For

the steady case, i.e t-eoo, we have n} " = 'p'imdy = 6,;«1. +C,(T,—T,.)). Now with

P
4,;ad=0, E (13):] and xi=0, one obtains from Eqs. (6.1l)-(6.17):

 

2
x AHY V-C TV -T..
._f_-_- A ’ where B = OJ pg ( ) , (6.18)
x, 1+Yo .Jv +Cp, (T, -T,.)

For this case xf lxp is a constant. Equation (6.18) gives the maximum ﬂame length as
a function of the pyrolysis front location. Also, for ﬂame spread to occur the condi-

tion B >Y0 “Iv must be satisﬁed.

148

6.4. RESULTS AND DISCUSSION

Previous experiments and theories (Fernandez-Pello, 1979; Carrier et al., 1980,
1990; Loh and Fernandez-Pello, 1984; Mekki. et al., 1990; Wichman and Agrawal, .
1991; Atreya and Mekki, 1992) have suggested a constant pyrolysis front speed Up.
Hence, when the pyrolysis front is at xp , a location x upstream of xp has already been
exposed to a constant heat ﬂux q}; (x) for (xp —x )/Up seconds. Thus for a moving
pyrolysis front, t in Eq. (6.17) is equal to (xi, -x )/Up. Note that the ﬂame length will
continue to increase even if the pyrolysis front has reached the end of the sample
(Mekki et al., 1990), i.e. xp = I. For this case, I is equal to At+(l-x)/Up. Here At

represents the time elapsed after the pyrolysis front has reached the end (I ).

To compare the predictions of the present model with our previous experiments
(Mekki et al. 1990; Chapter 4), we brieﬂy describe the ignition method employed. This
is essential for determining the location of the leading edge of the pyrolysis front x,-
used in Eq. (6.17). The sample was ignited by a pilot ﬂame which was left on during
the experiments. The igniter was 3.25 cm long and produced a pilot ﬂame with a
3 cm overhang across the test sample. Thus, we assume that the start of the maximum
pyrolysis rate is 3 cm downstream of the sample’s leading edge (x;=3 cm) or 6.25 cm
downstream of the inception of the thermal boundary layer. Between the start of the
sample and the tip of the ﬂame the net heat ﬂux is assumed to follow a quadratic
power law. It is assumed that heat ﬂux is zero at the start of the sample and is max-
imum at the ﬂame tip with its derivative being zero there. This is justiﬁed since the
solid demands a two-dimensional solution near the leading edge. The solid region
next to the water-cooled igniter is nearly at ambient temperature, which makes the
streamwise conduction large; hence it cannot be neglected. This is discussed in greater
detail in Agrawal (1992b) (Chapter 7). Due to incompleteness of combustion, oxygen

depletion data was considered most appropriate for comparison with the model (Mekki

I,
et al., 1990; Chapter 4). For any x , the total oxygen consumed is vwx Im;(x)dx,

It

149

where w is the width of the sample (5.7 cm). The predictions of the above expression
. for PMMA burning in different oxygen concentration environments are presented in
Flg. 6.2 along with the experimental results. Given the simplicity of the model, the
agreement is quite good except for higher Y 0,, and when the ﬂame becomes large.
Under these conditions incomplete combustion and the resulting ﬂame radiation has a
signiﬁcant effect (Mekki et al., 1990; Chapter 4) which is not included in the model.
For calculations presented in Fig. 6.2, the following data were used: AH =26.56KJIgm,
v=1.92 and Cpg=l.3KJ/kg in Eq. (6.12) (Sibulkin et al., 1981). Also, k and Pr for
Eqs. (6.11) and (6.13) were calculated at the ﬁlm temperature, (T, +T,.)/2 for air

(Incropera and Dewitt, 1985); Cp was based on the integrated average shown below

. T!
C, = (140.99”: + YWCPOI, 9:1] cpdr. (6.19)

Cp~,=9-3355 - 122.560'15 + 256.3%2 — 196.080'3 cal/gm -moleK ,

6,058.9465 + 00048044615 - 42.6799-1-5 + 56.61594,

where 0=0.01T(Kelvins) (V anWylen and Sonntag, 1973). The property values for
various oxygen mass fractions are given in Table 6.1. Here Up is the pyrolysis front
speed obtained in Chapter 4. The ﬂame temperature, T, , and C” are obtained from
Eqs. (6.11) and (6.19) respectively, 46;“)? is obtained ﬁ'om Eq. (6.14).

150

 

0.4a

0.3-1

0.24

 

OXYGEN DEPLETION RATE (gm/s)

 

 

 

 

Figure 6.2 Dependence of total oxygen depletion (total fuel production x v) on the

pyrolysis front location. Symbols represent experimental results.
l-Yo..=0.233; 2—Y0..=0.43; 3—Y0..=0.61.

151

Table 6.1. Property values

 

 

Y0- U, T, k C, Pr claims/E
m/s K W/mK J/kgK W/m1~s
0.233 0.00025 2342 0.083 1000 0.7 6807
0.383 0.00066 3441 0.126 1098 0.68 14749
0.432 0.00092 3769 0.140 1114 0.67 18450 '
0.61 0.00175 4856 0.225 1152 0.61 41497

 

Figure 6.3 shows the calculated values of 8(6) for different oxygen mass frac-
tions. Fig. 6.4 shows 6,2,, and q); for different values of the oxygen mass fraction.
Clearly the net heat ﬂux into the solid is far less than qconv and also is relatively ﬂat.
Fig. 6.5 shows the calculated oxygen consumption rates for the experiment in ambient
oxygen condition (#1) with different assumptions. This experiment was chosen
because the ﬂame was blue and incompleteness of combustion and ﬂame radiation
were not signiﬁcant factor. The experimental data is plotted as open circles and the
solid curve (1) represents the model predictions, as in Fig. 6.2. The calculated curves

(1a) to (1e) are for different conditions that are described below.

6.4.1. EFFECT OF BLOWING

In Fig. 6.5, curves (1a) and (1b) are calculated exactly as curve (1), except that
the steady state rather than the average mass ﬂux was used in the blowing correction
[153(0)] for curve (1a) and no blowing correction was used for curve (1b ). For this

experiment under ambient 02 conditions, the evolved mass ﬂux is small, hence the

152

 

 

 

 

BLOWING CORRECTION E(ﬂ)

 

 

 

' Figure 6.3 Blowing correction

 

153

 

 
 
    

 

 

1.4E-l-06 . , . , , ,

, - — 9:00” .

1.2E-l-05- .... q not ..

0’2‘ ‘ 1- omar-0.233 -

E 1.0E+O5-l 2- 0,-0.432 -

3 ‘ 3- o,,,,=0.810 '
v 8.0r:+044
E'. 6.OE+04-1

E—o -(

. a ,

 

 

 

 

 

 

 

Figure 6.4 Comparison between the convective heat ﬂux and the net heat ﬂux

154

 

 

 

 

 

 

 

 

‘ m‘ 004 I I I I I I I l I I I I I
o"
-( I".
"o' a
DD o"
'0
v '0'
0"
0"
m 0 .‘ 'o' -
B I"
'l
o
d o’
o
'o
o
m It " O
I """" "
1e x «I
l '0
o '0‘
z ....... 1
,,,,,, c
" 0' O
—I ooooo '0'
H ,,,, "o' «-
E-1 t' ......
” oooo
' '0 ‘o
a o o
1' """""
o" """"""
. . 1d ......
'0 ''''' cl
” """
’0 ooooo
’I '''''
I’ 00000
. ’0 '''''''
0. 1 ~ .I ..... _
’ 0' 0'
l a '0
z 1' """""
m I " 0’"
. I - ,. 1b u--“‘"
I, 0’ i "' --'-—w . . O C . . . .
I ' 0' .00....” O O
....... one...
a '0' ”a” . O O __
1 a
o I l I I I I F

Figure 6.5 Dependence of oxygen depletion rate on the pyrolysis front location
(Parametric dependence). Y0 ..=0.233. Symbols represent experimental
results. Solid curve represents the model predictions. Calculated Curves
(1a) to (1e) are for different conditions as explained in Secs. 6.4.1-6.4.3

155

inﬂuence of blowing is small. To clearly see the effect of blowing, consider curve
(1d ), which represents steady-state purely convective burning of the solid as discussed
in previous models (Carrier et al., 1990; Wichman and Agrawal, 1991; Chapter 2).
Here, the evolved mass ﬂux is much larger than measured during the experiments.
Thus, the blowing correction is substantial. This is shown in curve (1e), which is cal-
culated for the same conditions as curve (1d) except with E ([3) = 1, i.e. no blowing.
The effect of blowing was also substantial for higher Y0... cases (Fig. 6.2) because the
evolved mass ﬂux was much larger. More generally it is seen from Fig. 6.3 that blow-

ing has a substantial inﬂuence on cases where the evolved mass ﬂux is large.

6.4.2. EFFECT OF RADIANT LOSSES FROM THE SURFACE

It is interesting to analyze the effect of heat loss from the vaporizing surface.
This is constant since T, is constant, and is primarily due to re-radiation. The curve
(1c) in Fig. 6.5 was calculated for the same conditions as curve (1), except $2,“ = 0.
Neglecting surface re-radiation signiﬁcantly increases the evolved mass ﬂux and hence

the 0 2 consumption rate.

Re-radiation also has another effect on the ﬂame spread process. It eventually
causes the pyrolysis front speed Up to become zero. For simplicity, consider the
steadily pyrolyzing case with x,-=O in Eq. (6.17). Also, to examine the eventual dis-
tance the ﬂame will propagate, xp M, the effect of blowing can be legitimately
ignored because the sample at this location is barely pyrolyzing. Thus, with mp’M
= 6,;«1. +Cp (T, -T,,)) one obtains from Eqs. (6.11)-(6.17):

2

AI-IY v-C T,—T..
] ,whereB= OJ ”A )

L+Cp, (T, -T,,)

 

 

. (6.20)

xf 1 érerad‘fx—p- v(1+B)
1 , ..
xP 2 qcoanx—

v+Yo~

156

This equation shows that xf/xp decreases as xp increases. This decrease will continue
until the not heat ﬂux into the solid at the pyrolysis front becomes zero (6,2,, =q',:,d).
This location deﬁnes the maximum pyrolysis front length (xp M) of a semi-inﬁnite
sample and is primarily controlled by surface heat loss. Here, xf /xp is one fourth of
the steady state value without re-radiation (Eq. (6.18)). This is in agreement with the
experimental results. As shown in Fig. 6.2, for the experiment under ambient 02
conditions, the pyrolysis front stopped at point B and never reached the end of the
sample (66 cm). Similar results were also obtained for experiments with 18% oxygen
and free stream velocity of 0.9 m/s . Since no sample is inﬁnitely thick, after some
time the leading edge will burn out and result in an increase in the convective heat
ﬂux at x, (since xp -x,- decreases). The pyrolysis front will then move ahead and stop

somewhere downstream where q}; =0. Clearly, for this case the ﬂame spread occurs

as fast as the burnout front navels.

6.4.3. INFLUENCE OF UNSTEADY PYROLYSIS

To examine the effect of unsteady solid pyrolysis, curve (1d ) in Fig. 6.5 was cal-
culated under exactly the same conditions as curve (1c) except the solid was assumed
to be steadily pyrolyzing and the corresponding E (B) was used. Thus, curve (1d)
represents steady state purely convective burning of the solid as assumed by previous
models (Carrier et al., 1990; Wichman and Agrawal, 1991). Two points are
noteworthy: (i) a comparison of curve (1d ) with the experimental data which includes
both re-radiation and unsteady effects shows signiﬁcant disagreement, and (ii) a com-
parison with curve (10) shows that even after eliminating the effect of re-radiation, the
unsteady effects are substantial. The fact that the sample undergoes transient pyrolysis
throughout the wind-aided ﬂame spread process is evident in Fig. 6.2. The point A
(for Y o ..=O-43) represents the end of the sample. Here, the pyrolysis front reaches the

end but the ﬂame continues to grow until a steady state is achieved. This increase in

157

the ﬂame length is represented by the vertical line commencing at A.

Now let us consider the effect of unsteadiness of the solid when E (B)=1 and

d,’,',,,,=0. From Eq. (6.16) one obtains

 

 

' " / C T" A _
. :nP (1 xp) = [1+ p: v][1_ o- ] ’ (6.21)
meme/e) L 40)

where A (t-)Efunction ($.37). Without surface heat loss 4;, goes as x’“2 (Eq. (6.15))
and t_=(xp -x )/Up, this implies that A (7)5function (x /xp). On integration of Eq. (6.21)
from zero to x/xp one obtains mp =const. xmpmmy . Thus, the total mass loss rate
from a sample at any instant is a constant fraction of its steady state value. This con-
stant fraction remains the same throughout the ﬂame spread process. It increases with
increase in wind speed and oxygen mass ﬁaction and decreases with increase in pyro-
lysis front speed.

Since the ﬂame front location is directly related to total oxygen depleted [Eq.
(6.17)], agreement between experimental and theoretical predictions of total oxygen
depletion (Fig. 6.2) will also result in agreement of the ﬂame front location. Hence,
the present model also predicts U f :Up as obtained experimentally in chapter 4. How-
ever, this still disagrees with the predictions of the steady- state model developed in
chapters 2 and 3 where U f 2UP. The main reasons are (i) steady pyrolysis as dis-
cussed in this section and (ii) ﬁnite rate chemistry. In the actual case there is never
ﬂame reattachement, the ﬂame always extinguishes before it once again touches the
surface, thus decreasing the ﬂame overhang. This is discussed in greater details in the

Conclusions of Chapter 3.

158

6.6. CONCLUSIONS

In this study a simple but realistic model for the transient ﬂame front location
during wind-aided ﬂame spread across a ceiling-mounted thick vaporizing solid has .
been developed. The model predictions are in good agreement with the experimental
results. This model is based on the experimental observation that the solid—phase is
transient while the gas-phase is quasi-steady. The experimental fact that in the ceiling
conﬁguration, the ﬂame stand-off distance is much smaller than the thermal boundary
layer thickness is used to considerably simplify the gas-phase analysis. It yields an
expression for the ﬂame heat ﬂux as a function only of the streamwise distance. This
enables including surface re-radiation and blowing, which were found to have a large
inﬂuence on the burning process. To account for transient pyrolysis, an analytical
expression is developed for transient mass loss from the sample subjected to a constant
heat ﬂux. It compares favorably with the numerical and experimental results obtained
in this study and those available in the literature. This expression is combined with
the gas-phase heat ﬂux to yield the ﬂame front location. It was found that because of
the surface heat loss, for cases where the ﬂow is laminar and ﬂame radiation is small,
the pyrolysis ﬁ'ont speed will eventually become zero for a thick solid. Also, as a
result of re-radiation, blowing and transient pyrolysis, the net heat ﬂux into the solid

1’2, as assumed by previous models. To summarize, the determina-

does not vary as x'
tion of Up does not result in a complete understanding of wind-aided ﬂame spread.
The process is inherently transient and the determination of the ﬂame front location
and speed is perhaps more important.

In next chapter surface heat ﬂux in the burning zone (pyrolysis zone) is deter-

mined from burned PMMA samples. A simple model is developed which gives the

regressed surface location as a function of applied heat ﬂux and time.

159

APPENDIX 6.A. Blowing Correction

Since the ﬂame is very close to the surface, the streamwise velocity is very small
between ﬂame and solid surface. Also the temperature gradients in the streamwise
direction are much lower than temperature gradients in the transverse direction. Hence

udl‘ldx<vJI‘/dy. Using this in Eq. (6.3) one obtains

v— = a— , (6.A.1)

For no blowing, v=i), one obtains, using T=T, at y=0 and T=Tf at y=yf,

T—T" y (6A2)
Tf-Tv yf , . .

 

where yf is the ﬂame-standoff distance. Similarly, for blowing (van), one obtains

T-Tv = exp(vy let—l)

 

_ (6.A.3)
Tf -T, exp(vyf/0t-1)
Now the "blowing correction E " is given by
' .. (v #0) , ,.
E =7c—,?""——— , where gm, = [It-‘11] , (6.A.4)
qconv (V =0) dy y=0

and from Eqs (6.A.2)-(6.A.4) one obtains

5(8) = _p_ B = yfv/a. x (6.A.5)

For v=0, 4'62", = h(Tf-T,) = k(Tf—T,)/yf (from Eq. (6.A.2). Thus h=k/y,. Using
this, v=mnlp and deﬁnition of at one obtains ,B=m"Cp /h'. Note that we have used con-
stant k and p in reducing Eq. (6.A.1) from Eq. (6.3). However the use of Howarth

transformations (Eq. (6.7)) will result in the same blowing correction.

Also note that we have used an average mass ﬂux, m", over the experiment dura-
tion (Eq. (6.10)) which results in a constant blowing correction factor (Eq. (6.A.5)) and
in a constant net surface heat ﬂux (Eq. (6.15)). However this assumption could be
A relaxed in order to obtain total oxygen depletion rates. Using the integral model for
varying external heat ﬂux obtained in the last chapter, one can iterate between Eq. _
(6.15) and transient mass loss (Sec. 5.3.2.2) to obtain values of m" and q,;. All other
calculations remain the same. The results are plotted in Fig. 6.6 for air. Curve 1f
represents the present calculations. Also plotted are the calculations previously
obtained (Fig. 6.5). The solid line represents the calculation obtained with a blowing
correction based on average mass ﬂux. Also plotted are results for zero and maximum
blowing correction. Clearly the inﬂuence of other variables like surface reradiation
and solid unsteadiness (see Figs 6.2 and Fig 6.5) is much greater. Thus blowing

correction could be based on the average mass ﬂux to save computation.

161

0.08 . ﬁ . I I I ' I ' I ' I '

 

0.06-1 ‘ ,,,,, . - ~

 

0.04%

OXYGEN DEPLETION RATE (gm/s)

 

 

 

70

 

Figure 6.6 Effect of blowing correction with transient mass ﬂux on total oxygen deple-
tion rate. Solid curve represents the model predictions. Calculated curves
(la), (1b) and (1f) are for different blowing corrections as explained in
Appendix 6.A

162

APPENDIX 6.B. Turbulent Wind-Aided Flame Spread

It is important to investigate the effect of turbulence on the ﬂame length. We
choose the same conﬁguration as shown in Fig. 6.1. The details of the formulation are
very similar to the laminar case and are not repeated. Details are presented in Kays
and Crawford (1980) and Hines and Maddox (1982). We now present the equivalent
equations for convective heat loss from the ﬂame and the oxidizer mass ﬂux into the

ﬂame.

4/5

 

- " 1/3 Rex
q,_,_, (x )=0.0292k (T, 4,)!» x , (6.B.1)
. n k m Ref,”5
m0 (x) = 0.0292-5—Pr —-x—-Y0.. . (6.3.2)
- ps

The equation for the net heat ﬂux into the solid remains same. Using Eqs. (6'14)'.
(6.17) and (6.B.l)-(6.B.2), we obtain an expression for the maximum ﬂame length
(equivalent to Eq. (6.18))

 

574
If _ 1+B
xp —[1+Y0.JV] , (6.3.3)

Clearly the ﬂame is shorter in turbulent ﬂow than in laminar ﬂow. Now, if we include
the radiative heat loss from the surface, using Eqs. (6.14)-(6.l7) and (6..B.1)-(6.B.3)

one obtains the equivalent equation to Eq. (6.20)

,, 54
31:1 4 9MP” _I_tB_ 5"
xp 5 éCvax 1’5 1+Y0 oJV

 

(6.3.4)

163

Thus, xflxp decreases as xp increases. The decrease will continue until the net heat
ﬂux into the solid at the pyrolysis front becomes zero (4,2,, =q',:,ad). This location
deﬁnes the maximum pyrolysis front length (x, m) of a semi-inﬁnite sample and is
primarily controlled by surface heat loss. Here, Up=0 and x, Ix, is 0.1337 of its max-
imum steady state value (Eq. (6.24)) Note that surface heat loss results in much
shorter ﬂame in turbulent ﬂow than in laminar. Now let us consider the effect of

unsteadiness of the solid. For simplicity we neglect surface reradiation and blowing

effect. From Eq. (6.16) one obtains

in, (x lxp)

 

 

 

 

+CP‘TVM1 A0] ' (685)
L A(t‘) ’ "

mp .steady (x lxp )

where, A (7)5function ((527). Without surface heat loss q"; goes as x‘“5 (Eq.
(6.15)) and t_=(xp —x )lUp, this implies that A (7) is not a function of (x /x,,,. Thus, the
total mass loss rate from a sample at any instant is not a constant fraction of its steady
state value as was the case in the laminar ﬂow. This makes the problem inherently

transient and necessitates to treat solid as transient during the ﬂame spread.

CHAPTER 7

SURFACE HEAT FLUX IN THE BURNING ZONE

SYNOPSIS

 

An expression is obtained which provides the net surface heat ﬂux in the burning
zone during wind-aided ﬂame spread. This expression uses the ﬁnal thickness of the
burned sample to calculate the net heat ﬂux. This expression was corroborated by
transient pyrolysis experiments conducted on PMMA for various heat ﬂuxes in N 2
atmospheres. It was found that the net surface heat ﬂux in the burning zone increases
with the increase in free stream velocity and decreases with the streamwise distance,

though not as x‘m.

164

165

7.1. INTRODUCTION

 

Wind-aided ﬂame spread is the most hazardous type of ﬂame spread in building
ﬁres, and has been the most difﬁcult to understand and quantify. An improved under-
standing of wind—aided ﬂame spread will result in better material hazard classiﬁcation
and greater ﬁre safety. Although several studies have been reported for determination
of of burning zone length (Femandez—Pello, 1979; Carrier et al., 1980, 1990; Loh and
Fernandez-Pello, 1984; Wichman and Agrawal, 1991 (Chapter 2); Atreya and Mekki,
1992) or ﬂame length (Agrawal and Atreya, 1992; Chapter 6), only one set of heat
transfer measurements have been reported (Atreya and Mekki, 1992) for the ceiling
conﬁguration (see Fig. 7.1 for the schematic). In Atreya and Mekki (1992) a detailed
experimental study is performed, and the surface heat ﬂux distribution is obtained in
the preheat zone (both preﬂame and underﬂame), but not in the burning zone, mainly
because of the experimental difﬁculties in obtaining direct measurements. Theoretical
results are presented in Agrawal and Wichman (1992) (Chapter 3) for the surface heat
ﬂux distribution in the preheat zone for the idealized wind-aided ﬂame spread model of
Wichman and Agrawal (1991) (Chapter 2). Thus, the objective of the present study is
to experimentally determine the surface heat ﬂux distribution in the burning (pyrolysis)
zone ﬁom indirect measurements of the ﬁnal thickness of the burned PMMA samples.
This study is limited to thick vaporizing solids and for the ﬂow conditions where ﬂame

radiation is negligible (lower oxygen concentrations).

In the following sections, gas-phase approximations are presented; a simple
solid-phase model is formulated under the approximations described. An integral solu-
tion for the regressed surface location is subsequently presented. Experimental results
and comparisons with the model followed by a discussion and conclusions are then

presented.

166

 

 

 

    
   

 

 

  

 

 

 

 

 

 

> Xp PMMA sample
T5 = cilia) i ‘ W i i q’mm
Flow qmmd=const qmmd(x,t)
—-> Flame
i ’ Xf
Underﬂame Preﬂame
Pyrolysis Zone Preheat Zone >

Figure 7.1 Schematic of the gas-phase model

167

7.2. NET SURFACE HEAT FLUX IN THE BURNING ZONE

In our previous experimental investigation (Mekki et al., 1990; Chapter 4) on
wind-aided ﬂame spread over a vaporizing solid (PMMA) it was found that while the
vaporization rate was unsteady throughout the ﬂame spread process, the spread rate
was directly proportional to the free-stream velocity, implying quasi-steady behavior of
the gas-phase thermal boundary layer. Thus the characteristic time for the develop-
ment of a steady thermal boundary layer is much shorter than the characteristic time
for the solid-phase to achieve a steady state. Recent experiments of Atreya and Mekki
(1992) have shown that the convective heat transfer to the surface under the ﬂame is

proportional to x’l’z, exhibiting steady behavior of the thermal boundary layer.

Under experimental conditions where the ﬂame radiation is negligible (lower oxy-
gen concentrations), the net heat ﬂux into the solid under the ﬂame in the pyrolysis
zone is time-independent, since convective heat transfer is only a function of location
and the surface reradiation is constant (the surface temperature being the constant
"vaporization" temperature). However in the underﬂ ame preheat zone the surface tem-
perature is not constant, hence the surface reradiation is a function of time and location
making the net surface heat ﬂux transient. In the preﬂame preheat zone the convective
heat ﬂux is a function of time and location (Agrawal and Wichman, 1991; Atreya and
Mekki, 1992). Thus in the burning (pyrolysis) zone, a location x0 faces a net heat ﬂux
4,; (x0) for time to during vaporization (see Fig. 7.1). Here to is the maximum (total
duration of experiment) time minus the time taken for the pyrolysis front x, to reach
x0 (to =tmu—t, ). Thus the solid has regressed a distance so at x0 in time to under the
net heat ﬂux 4,; (x0 ). These observations are used in order to develop a one-

dimensional thermal degradation model. It requires to and so and gives the value

q.n£t (x0 )-

168

7.3. REGRESSED SURFACE LOCATION

 

For developing a simple solid-phase model of "vaporizing" materials like PMMA,
we note that in the pyrolyzing zone, the surface temperature remains constant (vapori-
zation temperature) with respect to the streamwise coordinate x . Also, the maximum
temperature gradient in the x-direction in the preheat zone is at least an order of mag-
nitude smaller than the temperature gradient in the y -direction. Thus, the solid-phase
heat transfer is approximated as one-dimensional normal to the surface. Fig. 7.2
schematically shows this heating process. The solid undergoes two stages: (i) The
inert preheating stage in which the surface temperature increases until it reaches the
vaporization temperature, and (ii) The vaporization stage in which the solid surface
regresses at a ‘ constant temperature liberating fuel vapor. The model and its predic-
tions for the preheating stage are presented elsewhere (Chapter 5). In the present
chapter I, is determined experimentally (explained later in ﬂame spread experiments
section). In the vaporizing stage, the regressing top surface moves downward at a
speed ds(t )ldt. Here t, represents the time when the surface starts regressing, i.e.,
when the temperature of the top surface reaches T, . The governing energy equation

is:

(FT 1 err
—=——, t< <oo, ,, 7.1
dyz a dt S () )’ t>t ( )
subject to the interface conditions:
4;. + 435] =pti-‘ﬂ , T(s(t).t)=T. . (7.2)
dy r=r(t) d‘

Also T(oo,t )=T.. in the far ﬁeld. The above solid-phase model has been widely used

169

Surface Temp
/

Surface Location

 

 

 

 

Preheating Vaporization

<——t°—>

 

tv tmax
Time

Figure 7.2 Schematic of the solid-phase model

170

to model vaporizing solids like PMMA (Delichatsios et al., 1991; Steckler et al., 1991;
Agrawal, 1992; Chapter 5).

The integral method is used to solve Eq. (7.1) subject to Eq. (7.2). We choose
the proﬁle T(x,t-) = iexp[-a (t_)(x -s (t—))], where f=T-.T,., T=t-t, and a(0)=q,;;,/k7-‘-,
(from Eq. (7.2)). Integrating Eq. (7.1) from s(t) to no one obtains
ds/dt"=[q,,';,—ka(i')f,]/pz.. Using this in Eq. (7.2) leads to an ODE in a (7) which can

be solved exactly to give the regressed surface location:

_ :- kT‘, - - C T,
s“) = A .71 1_A(_t) ; Ao=__P"__-’ (7.3)
qnet l-AO L+CpsTv

where

(A (?)-1)/(A.—1>exp(4 (2)/A. )=cxpi—C,...q‘..'3?/(pkL )(L+c,,r', )1.

Knowledge of s and t- can produce 4,2,. This particular integral method was recently
used to predict the transient mass loss during gasiﬁcation of PMMA. The results
agreed favorably with the experiments and i with the numerical results (Agrawal, 1992;
Chapter 5).

7.4. PYROLYSIS EXPERIMENTS: DETERMINATION OF SQ

 

To conﬁrm the. predictions of Eq. (7 .1), experiments were performed on
10 cm x 10 cm and 1.27 cm thick samples of PMMA under various heat ﬂuxes.
These experiments were conducted in a water-cooled controlled-environment cone
heater apparatus in a N 2 atmosphere (Fig. 5.9). White pigment PMMA was used to

minimize in-depth absorption of external radiation. The top surface of the sample was

171

covered by a reﬂective aluminum foil with a circular opening (31 cm 2) in the center to
eliminate edge effects and a water-cooled plate was used to protect the sample during
heater transients. These samples were subjected to various heat ﬂuxes. The total mass
loss was measured as a function of time, as plotted in Fig. 7.3. Here the mass loss is
divided by the surface area and density to give the regressed surface location. The
predictions of Eq. (7.4) are also shown in Fig. 7.3. The match is favorable. The sam-
ple thickness was taken as 0.127 cm. The other parameters used were (Agrawal,
1992; Chapter 5) k=0.185W/m2s, C,,=2150KJ/kgK. p=ll90kg/m3, T,=635K,
T..=295K, L=1650 J/gm and 4,2,“ = 0.82W/cm2.

7.5. FLAME SPREAD EXPERIMENTS: DETERMINATION OF ta

 

The experimental apparatus (Fig. 4.1) used consists of a small-scale combustion
wind tunnel which is capable of providing a ﬂat plate boundary layer ﬂow at a
speciﬁed free stream velocity and chemical composition across the sample surface. A

detailed description of this apparatus is given in Chapter 4.

The test sample (0.76m long, 0.076m wide and 0.0127m thick) is placed horizon-
tally along the tunnel top. The samples used for these experiments were obtained from
white sheets of PMMA. These were instrumented with nine thermocouples on the sur-
face 0.05 m apart (Chromel-Alumel; 76 pm diameter). The ﬁrst 0.15 m of the sample
was not instrumented to avoid errors due to ignition. Once all the desired conditions
were set (ﬂow speed and composition), the sample was ignited with a small methane
porous-metal burner placed at the tunnel inlet with its face parallel to the sample sur-
face. The fuel ﬂow rate to the igniter was controlled such that the ﬂame overhang on
the sample surface was about 0.03 m . The data were collected by the computer-
controlled data acquisition system and the ﬂame spread process was recorded by a
video camera. The ﬂame is extinguished with the nitrogen before it reaches the end of

the sample. It is well known that PMMA vaporizes at a constant temperature, and this

172

 

03
+1..

J

.1;
J n

l

l

00
J_

l l

l

REGRESSED DISTANCE s (mm)

1 ‘l f
—- Experimental

 

-CD
L

 

 

Theoretical ’
, w ..
,” "=2.75 W/crnz -
I
I
I .
I
I d
I
I, u
. 2.20 ,’ ’
I ’ 4
I’ 1.82 ,’
l/ I I’ 1.52 1.32
I”” ’Igv’I’ ””””””” 1
z ', ”1"” , I . I . I j
500 1000 1500 2000
TIME (sec)

Figure 7 .3 Regressed surface location as a function of time

2500

173

fact has been extensively used in the literature to characterize the onset of pyrolysis in
PMMA (Loh and Fernandez-Pello, 1984; Mekki et al., 1990; Chapter 4). Thus, from
the measured surface temperature proﬁle at x0 , tv represents the time required by the

surface to reach the vaporization temperature. Also, to =tmu-tv.

7.6. RESULTS AND DISCUSSION

The burnt sample sheets are out along the longitudinal center line and its thick-
ness measured with a micrometer (accuracy of $0.0254 mm) at various positions along
the center line. The net regressed distance 3 is obtained by subtracting the ﬁnal thick-
ness from the initial sample thickness. Results for the net surface heat ﬂux in the
pyrolysis zone as determined from Eq. (7.4) with the use of measured so and to are
presented in Fig. 7.4. The net heat ﬂux increases with free stream velocity and
decreases with the Streamwise distance. Note that the decrease does not follow Jc’l’2
dependence. A typical Jc'l’2 proﬁle is also shown in Fig. 7.4. Some of the reasons
for this discrepancy were discussed in Chapters 4 and 6 (solid-phase unsteadiness and.

surface re-radiation).

Note that in all cases the maximum net surface heat ﬂux is around 3 em down-
stream from the leading edge. No data are presented for region 1: < 3 cm because this
region is prone to gas-phase ﬁnite reaction-rate effects and solid—phase cooling (see
Fig. 7.5 for the surface heat ﬂux near the leading edge). In this region the net heat
ﬂux decreased with the decrease in x . Other then the inﬂuence of ﬁnite gas-phase
reaction-rates (Chen and Tien, 1986; DiBlasi et al., 1987) the solid edge (touching the
igniter) was cooled because of the water-cooled i gniter. To quantify this effect numer-
ical solutions were obtained for (11‘ Id: =a(d7’I'/dx 2+d 2T ldy 2). The boundary condition
representing the water-cooled side of the solid was chosen as T (x =O,t )=T... Far from
the leading edge the solid was assumed as one dimensional (dT (x —-)oo,t)/dt = 0). The

bottom side was assumed adiabatic. The sample was subjected to a constant heat ﬂux

q"m(W/cm')

174

 

l ' I '

I

O
U
A

 

 

 

 

x (cm)

Figure 7.4 Net surface heat ﬂux in the burning zone

 

q"...(W/cm')

175

 

 

 

 

2.0 ‘. l I '

. 't O U..=2.0rn/r
1.8... t“ D U..=1.5m ﬂ
1 3 . §-- 1/\/X Dependence A U-=0.9m;ﬁ

0 ﬂ “ d
1.4; ° ° ° _'

q ‘\‘ O O O O 0 .

1.2-j O Ox‘q D O O -.:
x‘ o o a
1.04 o a ““““ a a o d

. ‘ ““““““ ' o u '-

o. - ° ~~~~~ ‘
81 o A ............... . ° .
0.6+ D A ‘ ¢ ........ “M ...... 'M;

_. A A A A A
0.41 g d
0.2- "
0.0 I ' l '

0 5 10 15

x (cm)

Figure 7.5 Net surface heat ﬂux near the leading edge

176

of the order shown in Fig. 7.4. It was found that the leading edge effects are limited
to the ﬁrst 2-3 cms of the sample.

This, coupled with surface radiative heat losses and solid-phase unsteadiness seem

to be the main reasons for the short ﬂame overhang (Chapter 4).

This technique (using burned samples) has been previously used to predict to the
regression rate of a PMMA sample burning under turbulent conditions (Zhou and
Fernandez-Pello, 1991). The regression rate was determined by dividing the regressed
distance by effective time (so/t0 ). Note that in doing so, the solid is essentially
assumed steady. Since the solid thermal processes are transient (Mekki et al., 1990;
Chapter 4; Agrawal, 1992; Chapter 5; see also Fig. 7.3), this produces an under-
estimate of the maximum regression rate and subsequently the net surface heat ﬂux.
For a thick sample, the maximum regression rate is directly proportional to the net sur-

face heat ﬂux (steady state condition).

7.7. CONCLUSIONS

 

The net surface heat ﬂux in the burning zone during wind-aided ﬂame spread is
determined using measurements of the ﬁnal sample thickness. A simple solid-phase
model is developed which gives the net surface heat ﬂux based on the regressed dis-
tance and the effective burning time. The model agrees favorably with the pyrolysis
experiments. It was found that the surface heat ﬂux increases with the increase in free
stream velocity and decreases with the streamwise distance. This study is limited to

vaporizing solids and for environmental conditions where ﬂame radiation is negligible.

CHAPTER 8

STABILITY ANALYSIS

SYNOPSIS

 

A hydrodynamic stability analysis is performed for the wind-aided ﬂame spread
problem in the ceiling conﬁguration. The effect of the reaction zone temperature
proﬁle on stability is analyzed. It is found that the Froude number is the only parame-
ter affecting the pulsation frequency of the ﬂame once the temperature and velocity
proﬁles are speciﬁed. The inﬁnite-reaction-rate ﬂame temperature proﬁle is the most
unstable: generally, the sharper the temperature proﬁle at the ﬂame, the higher is its
characteristic oscillation frequency. It is also found that the oscillation frequency

increases with the B -number.

177

178

8.1. INTRODUCTION

It is well known that a light ﬂuid beneath a heavy ﬂuid is generally in a buoy-
antly unstable condition (Taylor, 1950). This condition occurs for wind-aided ﬂame
spread across ceilings (Fig. 8.1a). Despite the apparent importance of the diffusion
ﬂame stability problem for determining conditions leading to turbulence, for altering
heating rates, etc., there are relatively few studies available in the literature. A brief

review of the literature follows.

Large jet diffusion ﬂames have been observed to ﬂicker at a frequency of about
12 Hz, which is insensitive to ﬂow rate, burner size or gas composition (Buckmaster
and Peters, 1986). This ﬂickering is apparently related to the stability properties of the
ﬂame. Since the 12 Hz frequency is largely independent of ﬂame geometry the insta-
bility mechanism is presumed to have hydrodynamic origins. The study of Kimura
(1965) for tube burner diffusion ﬂames raises a fundamental question as to the choice
of undisturbed velocity and temperature proﬁles because these change signiﬁcantly
throughout the ﬂame. He adopts the bell-shaped proﬁles measured well above the
ﬂame. Since the instability seen within the body of the ﬂame is caused by the proﬁles
at much lower heights, and these proﬁles are far from being bell shaped, doubts were
raised regarding Kimura’s (1965) choice of proﬁles by Buckmaster and Peters (1986).
Grant and Jones (1975) studied the stability of the free diffusion ﬂames for different
temperature proﬁles and suggested that the invariance of the frequency for small-scale
free diffusion ﬂames cannot be obtained from a linear stability analysis where the dis-
turbances are purely convective, i.e., where all dissipative process such as diffusion of
momentum are neglected. Although the stability of free diffusion ﬂames (Buckmaster
and Peters, 1986; Ellzey and Oran, 1990; Grant and Jones, 1975; Kimura, 1965; Puri
and Hammins, 1990; Toong et al., 1965) has been studied, there are no available stu-
dies of diffusion ﬂame stability for ﬂows with contiguous walls. The present study is

an attempt to understand the stability of such ﬂames.

179

 

 

 

 

 

SAMPLE
\\\\\\\\\\\{\\\\\\\\\\\\\\‘ ——I> Up
_> Fuel
’.
Wind Direction Thermal B.L.
Gravity

Figure 8.1a Physical conﬁguration of the wind-aided ﬂame spread

U

Thermal B.L.

-le Flame

 

 

Flow

TW ———-> X

 

Figure 8.1b Schematic diagram of the model

.180

The questions that this study addresses are: (1) when can instability occur, (2)
what are its principal causes? Accepting that the temperature (density variation) may
be the driving force (or principal cause) for the instability mechanism, we ask further; ,
(3) how signiﬁcantly does the temperature proﬁle affect the stability characteristics of
the ﬂame? We wish to determine, if possible, the relevant non-dimensional
parameter(s) on which the most unstable frequency depends and the frequency range of

instability under representative experimental conditions.

The problem we will examine is illusu'ated in Fig. 8.1a. We see that since the
density decreases from the wall to the ﬂame the region in between is potentially
unstable. Because of large density variations we perform a hydrodynamic stability
analysis of the conservation equations. Analysis of the disturbance equation (Appen-
dix 8.C) shows that an inﬂection point in the velocity proﬁle is a necemary but not
sufﬁcient condition for instability to occur and that the unstable region exists between
the wall and the ﬂame. The velocity proﬁle possess an inﬂection point because of the
blowing of fuel from the "vaporizing" surface. This condition exists only in the pyro-
lyzing zone of the ﬂame spread (Fig. 8.1a). In the preheat zone there is no blowing at
the surface; hence, instability there can only occur if pyrolysis zone is unstable. Thus,
only the pyrolysis zone of the wind-aided ﬂame spread problem will be analyzed.
Numerous temperature proﬁles are examined to identify the principal mechanisms for
the instability. This is the most important part of our analysis for it goes to the heart
. of question (3). If the instability mechanism is essentially independent of the detailed
ﬂame temperature proﬁle, a comprehensive analysis of the diffusion ﬂame su'ucture is
unnecessary: one could simply employ the relevant inﬁnite reaction rate proﬁle and
produce a generic, universally-valid stability analysis for a given class of ﬂows such as
jets, ceiling ﬁre spread, etc. If, however, the stability characteristics depend strongly
on the detailed nature of the ﬂame temperature proﬁle, the stability analysis becomes

more complex because the reaction-zone temperature proﬁle depends on the chemical

181

rate parameters therein (like the pre-exponential factor and activation energy for a
single-step reaction). We note that outside the reaction zone the inﬁnite and ﬁnite-rate
proﬁles are virtually identical. Thus, our question is whether the temperature proﬁles
in an extremely thin region of the ﬂow ﬁeld can affect the stability characteristics of
the global problem. We will demonstrate that the answer to this question is

afﬁrmative.

The governing equations for the wind-aided ﬂame spread problem are presented
in the next section. Basic state velocity and temperatm'e proﬁles are obtained
therefrom. In Sec. 8.3 the disturbance equations are identiﬁed and a single equation is
obtained that uses the steady values obtained in Sec. 8.2 as its basic state. We note
that our analysis is for an assumed inviscid parallel ﬂow. Despite the important work
done on non-parallel stability analyses for boundary layer ﬂows (Saric and N ayfeh,
1977; also see refs therein) we do not feel these additional complications presently
warrant inclusion in this preliminary investigation: our study is focused on the princi-
pal thermal causes of the instability, not its purely ﬂuid-mechanical subtleties. Results
are presented in Sec. 8.4. Appendices 8.A and 8.B deal with the inner zone solutions
of the temperature proﬁles. A classical Rayleigh-Fjortoft analysis of the disturbance
equation is conducted in Appendix 8.C.

8.2. STEADY FLAME SPREAD

 

Shown in Fig. 8.1a is the schematic illustration of the model. After ignition, the
tip of the surface-pyrolysis zone propagates downstream with velocity U I: , while the
ﬂame tip propagates downstream with a higher velocity U}, burning "excess" fuel
released from the pyrolyzing surface and not already burnt there. The pyrolysis zone
lies between x' =0 and x;=U;t' . The preheat zone extends from x; to the location
ULt‘; there is no effect of leading edge beyond this location. It is in the pyrolysis

region of the ﬂow ﬁeld that we wish to examine the ﬂame stability, for reasons already

182

mentioned in the Introduction.

In the analyses of Wichman and Agrawal (1991), Chapter 2 and Carrier et al.
(1990) for the wind-aided ﬂame spread problem, the pyrolysis zone is assumed steady,
which requires the fuel mass ﬂux to be proportional to the incident heat ﬂux. In the
gas-phase the chemical reaction is assumed to occur though the simple one-step
irreversible reaction F + v0 -> products , where v is the mass-based stoichiometric
coefﬁcient. Other simplifying assumptions are negligible streamwise diffusion (boun-
dary layer approximation), equal binary discussion coefﬁcients, constant mixture
speciﬁc heat, unity Lewis and Prandtl numbers, low Mach number and negligible
solid-phase regression rate. The ﬂame radiation is neglected, which is valid for low
ambient oxygen concentrations (Mekki et al., 1990; Chapter 4). Limitations and res-
u'ictions of the model are discussed at length in Chapter 3 (Agrawal and Wichman,
1992).

The steady boundary layer equations describing wind-aided ﬂame spread across a

ceiling in a cartesian coordinate system (see Fig. 8.1b for the schematic) are

ags‘u‘)+a(p’v‘)=0 (8.1)
ax‘ ax“ ’

mass:

in sau' a sau‘ 3P. 8 r sau'

 

 

 

 

 

 

 

 

 

 

 

 

x-mom: u . v . = t + t t )' , (8.2)
p 3:: +9 ay 8x ay “1 ay
P‘ t t it
y-mom: a , =-(p..-p )g . (8.3)
8y
. . .ar“ .aT‘ ~ 3 .ar‘ .2 . . . .ym'r
energy: p C [u ——T+v .]= . 2t .]+Ap Q YoYpe , (8.4)
” 9x 3y 3y 3y
, P , BY; , ar'l P, airM . . .
oxidizer. p u —-;-+v f = 3* p D‘ f —vAp*2Y;Y;e-5 ’R T , (8.5)
. ax ay l 8y . ay .
’ a e I ti
,,ay .ar “or
fuel: p u —f—+v f]: a. p D f —Ap 2Y0Y1t~e45mT , (8.6)
. at 3y J 8y .' 3y .

 

 

 

 

183

state: P I =p' R ' T' . (8.7)

3
By using the Howarth transformation, with the mass coordinate 2.: I p'dy' and the
o

transverse mass ﬂux w'=p' v' “,1 (ap' /8x")dy* and with zero pressure gradient
(streamwise) we can easily obtain the corresponding incompressible ﬂow boundary
layer equations (Wichman and Agrawal, 1991; Chapter 2). These equations are non-
dimensionalized by deﬁning u=u'/U:,, x=x'/L', z=z'/p;L', w=w'lp;U:.,
7:171"; and L'=v;/U;, ter/1;, Q=Q‘/C;,,T;, Y,=Y,.‘. Here v is the mass-based
stoichiometric coefﬁcient for the one- step reaction (F +v0 -—)P ); we have Vp=1 and
v0=v. Using the stream function val—27 f (n) and the similarity variable 11:2 Nix- and

u =aw/az and w=-8\|I/3x , the x-momentum equation reduces to
f + ff " = 0 . (8.8)

Here u=f' and w=(1'1f'—f )N2_x_. The boundary conditions are f '(0)=0, f '(oo)=l and
f (O)=~—f"(O)B, where B is the "8 number for mass transfer from the surface".
Details of the derivation of Eq. (8.8) and the boundary conditions can be found in
Emmons (1956). We note here that buoyancy is neglected in the derivation of Eq.
(8.8) Wind-aided ﬂame spread experiments conducted in air in the ceiling
conﬁguration have shown that for a free stream velocity as low as 0.6 m/s , the ﬂame
was stable (Mekki et al., 1990, Chapter 4), although this may not hold true for ﬂames
spreading across a ﬂoor or ﬂames spreading in much higher oxygen concentrations,
where the density variation is large. In addition to the effects of buoyancy, ﬂame radi-
ation has to be included for higher oxygen concentrations. Hence, as already stated

the present analysis is limited to ﬂame spreading across a ceiling in low oxygen

184

concentrations, where ﬂame radiation can- be neglected. In what follows a prime () as
superscript denotes the derivative of that variable with respect to n; x' = dx/dn, x" =
dzx /dn2, etc. By choosing Lewis and Prandtl numbers of unity we obtain the follow-

- ing energy and species equations:

-fT' = r"+2xQR' , . (8.9)
—fY,,' = Y,,"+2xv§ , ' (8.10)

-fY,.1 = Y£+2xE , (8.11)

where 18:03:, YFexp[-B(1-T)/(1—ot(1-T))], with B = E‘(T;-T;)/(R‘r;2) and a =
=(Tf-T;)/T;. The quantity D=A" p‘v;U;-2 exp(-E‘/R‘r;), the Damkohler
number, is the ratio of the thermal diffusion time to the chemical time.

For solution we employ the Schvab—Zeldovich coupling functions 70 =T IQ +l’0 Iv
and 7p=TlQ +10.» to obtain 73+fyé=0 and y£+f 713:0- Suppose now that u(n)=f' can
be obtained from the solution of Eq. (8.8); the solutions for 70 and 7;- can then be

obtained as linear functions of u (Emmons, 1956),

Yo=aou+bo . (8.12)

Y): mpu'i'bp . (8.13)

By then imposing the boundary conditions at 11:0 (u =0, T=Tw, Yo =0, Y F =pr) and at
11->°° (u =1. T=T,., Y0 =Y0.., Yp =0) we determine the integration constants as

do: ko—(Tw -T..)/Q, b0 =Tw IQ, a1.- =—YFW —(Tw —T,,)/Q and 12,. =Tw/Q +Ym.

185

8.2.1. BASIC-STATE VELOCITY PROFILE

In order to solve for u (1]) only one parameter is required, namely B . We used
the values B=0.275, B=1.125 and B=1.6, for which, the heat losses from the ﬂame
towards the surface and towards the free sueam are identical, see Appendix 8.A. This
preserves the symmetry of the reaction zone, which would otherwise be skewed to one
side or the. other, severely complicating the stability calculation. We employ the
Runge-Kutta-Gill method with a stepsize of 0.001 to determine u , u' and u" from Eq.-
(8.8). The results are plotted in Fig. 8.2 for B=1.125, from which it is clear that the
velocity proﬁle has an inﬂection point at 11:11. . The velocity at the inﬂection point, .
ns=l.405, is u. =0.448.

8.2.2. BASIC-STATE TEMPERATURE PROFILE

From the known solutions for u , yo and 7;, the mass fractions Y0 and Y I: can be

easily expressed as linear functions of T ,

T
Y = b0-— , 8014
a (a0u+ Q) ( )

Yp=v(apu+bF-%) . (8.15)

Equations (8.14) and (8.15) are valid throughout the physical domain (i.e., not simply
on one or the other side of the ﬂame). Thus, the reaction term if in Eq. (8.9) can be
written in terms of T; consequently Eq. (8.9) can now be numerically solved for cer-
tain values of D , B and 0t, for the boundary conditions speciﬁed previously. If we
wish to develop an asymptotic solution in the inner (reaction) zone we have recourse
to asymptotic methods (Buckmaster and Ludford, 1982; Linan, 1974; Williams, 1985).
In this case the outer zone solutions are given by the homogeneous forms of Eqs.

(8.9)-(8.11) with the condition T=Tf applied at the ﬂame, or by Eqs. (8.9)-(8.11) with

186

 

 

 

 

 

 

 

 

Figure 8.2 Basic-state temperature and velocity proﬁles.
here is for the inﬁnite-reaction-rate

Note that T-proﬁle plotted

187

5 replaced by a 8—function heat release term at the ﬂame sheet location. Since the
objective of this study is to examine the effect of the inner (reaction) zone on the sta-
bility we will eventually subject the energy equation to the high-activation energy

asymptotic analysis. The reaction zone calculations are described in Appendix 8.A.

We presently focus on the outer zone calculations for T. First we determine the
location 11f of the ﬂame. Here Y0 =Yp =0. Let us now expand T' near any n-value
using Taylor’s series. Thus TQM“ =Tn +AnT; +0 (A102. Now from Eq. (8.9)
T"=—fT' (for outer solution) and from our expansion, we obtain TQM“ >Tn when f <0
and 7,3,,“ <T,’1 when f>0. Since f is negative at and near the fuel surface and posi-
tive in the far free stream, T must increase in the fuel region and decrease in the oxy-
gen region. Continued reasoning in this way indicates T is a maximum at f =0, which
occurs at the inﬂection point for the streamwise velocity proﬁle. Thus, the ﬂame lies
at the inﬂection point nf =11. . Henceforth we refer to the inﬂection point with the
subscript f .

Between the wall and the ﬂame there is no oxygen, and we obtain from Eq.

(8.14), using T=1 and u=uf at nf,
r=rw+(1-T,,)i o $11311; , (8.16)
“r

with

Y .. l-T 1-u Tw
g—°—= ”‘94 f ) . (8.17)
V 11f

 

Between the ﬂame and the free stream there is no fuel, and we obtain from Eq. (8.15),

using T=T.. and u=1 at n—)oo,

 

 

l-u “‘“f
T=T--u—f-+Tf [ l—uf J 11f511<°° , (8.18)

with

1-T..uf —(1-uf )Tw
l-llf .

 

QYFW= (8.19)

Here subscripts w , f and co represent conditions at the wall, ﬂame and free stream,
respectively. From Eqs. (8.17) and (8.19) one obtains the stoichiometric index 4) =
va/Yo, = uf/(l—uf ). Solutions for T (Eqs. 8.16 and 8.18) and T' are also plotted
in Fig. 8.2. We see that T, is discontinuous at n=r|f since it is the "outer" tempera-
ture proﬁle, to which the inner (reaction zone) proﬁle must be matched. Solutions for
inner-zone temperature proﬁles are plotted in Fig. 8.3 for various B. Parameters
chosen and calculated are tabulated in Table 8.1. Values of B chosen to obtain the

various temperature proﬁles of Fig. 8.3 are presented in Table 8.2.

This completes the solution for velocity and temperature proﬁles, which are
employed as mean or "basic-state" values in the disturbance equation described in the
next section. We use T; =295 K , T; =630 K ("vaporization" temperature for PMMA)
and Tf=2065 K (adiabatic ﬂame temperature for PMMA burning in air (chapter 2) in
Eqs. (8.16) and (8.18). This completes the solution for velocity and temperature
proﬁles, which are input as mean or "basic-state" values in the disturbance equation

described in the next section.

189

 

1.0

0.9-l

NON—DIMENSIONAL TEMPERATURE

 

 

 

 

Figure 8.3a Inner zone (reaction region) temperature proﬁles, B =0.275. Curves 1-6
represent proﬁles for different B’s

190

 

 

 

NON-DIMENSIONAL TEMPERATURE

 

 

Figure 8.3b Inner zone (reaction region) temperature proﬁles, B =1.125. Curves 1-6
represent proﬁles for different B’s

191

 

1.00
0.951
0.90%
0.86~ 6
an;

0375‘

0.70%

NON -DIMENSIONAL TEMPERATURE

 

 

0.65 v 1 I l ' l
1.00 1.40 1.80 2.20

 

Figure 8.3c Inner zone (reaction region) temperature proﬁles, B =1.6. Curves 1-6
represent proﬁles for different B’s

192

Table 8.1 Table of parameters used. Parameters a -b and QYO ,Jv are obtained from
Eqs. (8.A.2) and (8.17) respectively.

 

 

7;”; B , uf u,’ n, a-b o Ql'gg/v
3 0.275 0.298 0.414 0.733 2.54 0.424 1.33
7 1.125 0.448 0.354 1.405 1.82 0.811 1.71

15 1.600 0.479 0.340 1.630 1.65 0.920 1.87

 

T;=295 K, 13:63:; K, u=1, Pr=l, g=9.8m/s2, v..=16x10-6 m2/s, somb=7

Table 8.2 Values of the nondimensional activation energy B used for the different

 

 

B numbers.
Temp. Proﬁle # B =0.275 B =1.125 B =l.6
1 Infinite“ Infinite“ Infinite‘l
2 4.0 4.0 4.0
3 3.0 4.0” 3.0
4 2.0 2.5 2.0
5 1.5 1.3 1.0
6 1.0 0.8 0.75
7 - 0.6 -
8 - 0.6” -

 

“ Temperature Proﬁles calculated using Eqs. (8.16) and (8.18). b Temperature
Proﬁles calculated as described in Appendix 8.B. Remainder of the temperature
proﬁles calculated as described in Appendix 8.B.

193

8.3. STABILITY ANALYSIS

As already mentioned, experiments on wind-aided ﬂame spread suggest that buoy-
ancy plays an important role in ﬂame spreading. The wind-aided spread of a ﬂame
across a ceiling is generally buoyantly unstable. We wish to determine the conditions
under which this instability can be suppressed. We perform a parallel inviscid stability
analysis with spatially growing disturbances. Michalke (1965) has suggested that spa-
tially growing disturbances accurately describe the instability properties of a disturbed
shear layer because some of these properties, such as ampliﬁcation of disturbances,
could not be described by linear inviscid temporal stability analysis. Previous
diffusion-ﬂame stability analyses (Buckmaster and Peters, 1986; Ellzey and Oran,
1990; Grant and Jones, 1975; Kimura, 1965; Puri and Hammins, 1990; Toong et al.,
1965) have all used parallel inviscid ﬂow. We believe that several important aspects
of our problem can be uncovered by an inviscid parallel-ﬂow stability analysis, with
the temperature and velocity (mean values) determined from the steady viscous-ﬂow
analysis of Sec. 8.2. Nevertheless, viscous and diffusive effects may be important
under many conditions. For instance, Weeratunga et al. (1990) suggest that hydro-
dynamic effects are responsible for the instability of plane ﬂames and diffusion of heat
and mass are responsible for the cellular instability. In the ﬁrst case diffusion effects
can be neglected but the density variation has to be retained (see Buckmaster and
Peters, 1986; Ellzey and Oran, 1990; Grant and Jones, 1975; Kimura, 1965; Puri and
Hammins, 1990; Toong et al., 1965). In the second limit the hydrodynamics (density

variation) can be neglected, but the diffusion terms must be retained.

We proceed with our inviscid parallel-ﬂow analysis. According to standard prac-
tice (Drazin and Howard, 1966) the governing equations are

 

89' J ag'u' . ap‘v‘

=0 . 8.20
at. 8x 3x ( )

t Bu' tut all. +psvt all“I = 31’.
3"“ Bx‘ By. ax’

 

, (8.21)

194

48V. 0 eaV‘ a 03V}. BP’ 4: 4- at
= _ .. , 8.22
at,.,+pu ax,+pvay, 8y, (P—P)g ( )

 

 

 

p

ar‘ W. 31" “.87"

. . . =0. (8.23)
at 3): 3y

 

 

 

 

Pe=paRaTt (824)
a t e a at: s a 1"2

297+“ a —;.—+V* 8%: 1 BF. 8p. BP“ 30* +[_p+:_]g t ’ (8.25)

at 81: By p 2 8y 3x 3x 3)’ P 2

where CEGv/ax —au lay) is the z-component of the vorticity. Equation (8.25) was
obtained by subu'acting the y-derivative of Eq. (8.21) from the x-derivative of Eq.
(8.22). The instability is analyzed in terms of a mean quantity (obtained from the
steady state boundary layer ﬂow of the previous section) and a "disturbance", generally
much smaller than the mean ﬂow quantity but still ﬁnite. Thus we deﬁne new vari-

ables

r*(x*.y*.z*)=u‘0’>+£’a‘ ; v*(x*,y*.:*)=e*v* ;
E‘tx*.y‘,r‘)=c‘0‘)+e*t' ; f‘<x*,y*.r*>=T*<y*)+e*f‘ ; (8.26)
l5*(x*.y‘.r‘)=p*<y‘>+e*p* ; ﬁ*(x*.y*.t*)=w*0‘>+e*é‘ ;

F'(x’.y',t')=P'(y*)+€‘ﬁ'.

where 6* =35" exp(i0t'x'-ito*t'). From Eqs. (8.20)—(8.26) one obtains the following

equation for the disturbance su'eam function 0‘

 

 

 

 

d 1 d6' .. a” 1 d 1 du“ 1 " d1“
s[ It a ]—9 42+ in it" a[ a at ]+ t at 2 8‘2 i: =0’ (8‘27)
dy T dy T u -uf dy T dy (u —uf) T dy

195

where c'=0)*/a'. The above equation can also be obtained by from the full govern-
ing eqns (like Eqs. (8.1)-(8.7)), after discarding the diffusion terms and higher order (2
and more) nonlinear terms in 8'. We note that when T = constant =1 Eq. (8.27)
reduces to the classical "Rayleigh equation" for the stability of inviscid parallel ﬂow

(Michalke, 1965). Using non-dimensional parameters c=c'/U:,, (u=co‘L'lUL,

a=or'L‘ and 0:0'IUL, z’= p'dy’, n'=z*/‘J2x' and the parameters deﬁned previ-

V

O

ously (after Eq. 8.7) we obtain the nondimensional disturbance equation

 

9' ' 2 1 u" l 7"
—-60t+——+ —=O, 8.28
[ ] ° u-C[T2] Fro(u-c)2 T2 ( )

where 09:015. (00:0)‘127 , Fro—142x704, where Fr is the Froude number
(=U12/g‘L‘ ). The disturbances must vanish in the free stream and at the wall. Since
ti‘=dé"/dy‘ (which is directly related to 0') and the streamwise disturbance in the
velocity vanishes both at the surface and in the free stream, we have the following

boundary conditions for Eq. (8.28):
9(0)=e(~)=e'(0)=e'<oo)=o. (8.29)

8.4. RESULTS AND DISCUSSION

 

Presented in Fig. 8.4 is a plot of the Froude number, Fro , as a function of
streamwise distance x’ and free stream velocity U L. For laboratory-scale combustion
tunnel experiments ( L~l m and U ..=0.6-2.0 m/s) Fro varies over one order of mag-

nitude from 4 to 35.

196

 

 

 

 

 

 

 

1.50 . , . . . ‘, . ,
’t'n‘ Fro=20
\
E .
v
E; 10
5 1.00-
S
m 5
> .
H
a: 0.50 1
5..
(ll
[:1
tr: 0.1
at:
h ,
0.00l . , . I . . . ,
0 1 2 3 4

STREAMWISE DISTANCE (m)

Figure 8.4 Froude number F r0 as a function of free sueam velocity and streamwise
distance

 

197

In the limit Fro —>oo, where there are no buoyant inﬂuences, the disturbance Eq.
(8.28) is similar to the equation obtained by Buckmaster and Peters (1986); when
Fro _,.. and T’=0 (isothermal ﬂuid) it is identical to Michalke’s equation (1965). The
solution of Eq. (8.28) requires existing solutions for the basic-state quantities u, u', u",
T and T'. From Fig. 8.3 it is seen that for the inﬁnite-reaction-rate temperature proﬁle
(#1), T ' is undeﬁned at the ﬂame location (11f). For the neutral stability case, c’=u;
[Buckmaster and Peters, 1986; Michalke, 1965], 0' is not continuous at 11, although 0
is. In this case the eigenvalue 010 is obtained by integrating Eq. (8.28) from both
sides, n=0 and n—>oo, until 1],; enforcement of the condition 0;! =0,}; (continuity of 0)
gives (10 . The integration is performed using the Runge-Kutta-Gill method; the solu-
tion procedtue was tested by reproducing the results of Buckmaster and Peters (1986)
and Michalke (1965). This solution procedure is applied for temperature proﬁles #1-6
for B=0.275 and for B=l.6 and #1,2,4-7 for B=1.125. Since in all these cases the last
term of Eq. (8.28) becomes inﬁnite as ln-nfl -—)0, as discussed in Appendix 8.B, they
cannot be integrated continuously from one side to the other across the ﬂame. For
proﬁle numbers 3 and 8 (B =1.125) the singularity does not exist (see Appendix 8.B)
and hence the disturbance equation can be integrated across the ﬂame. As a check for
Eq. (8.28), temperature proﬁle #8 is integrated from T]—)°°, which we take to be 11
=10, to 11:0 and a0 is obtained by enforcing the conditions 0(0)=0 and 0:0. This
procedure gave the same O-proﬁle and 010 as by integrating from the wall and the free

stream to n, and matching values there.

We note that, strictly speaking, the proper means for ﬁnding a solution for 0 is to
integrate for complex a0 , then to progressively pass to the limit do has —)0. The
solution should converge to the 010 “mg -+0 limit. It is found, however, that solutions
obtained in this way are iilglgigﬂ to the ones obtained by patching the two separate
solutions at nf. Hence, this limiting process is not employed here, nor in fact do any

of the previous authors employ it (Buckmaster and Peters, 1986, Michalke, 1965).

198

Figure 8.5 presents the stability curves for different Froude numbers for various
-B -numbers. Plotted is the non-dimensional frequency 0) versus F r0 . Generally in pre-
vious stability analyses a single proﬁle is analyzed, such as the inﬁnite-reaction-rate
proﬁle in (Buckmaster and Peters, 1986), or a representative inner-outer proﬁle as in
(Puri and Hammins, 1991); the numerical results are then compared to the experi-
ments. Our approach, as already stated, is somewhat different. We wish to understand

how the differences in the T-proﬁle inﬂuence the ﬁnal stability properties.

Figure 8.5 suggests the existence of three regions, purely stable, purely unstable
and an overlap region, which could be stable or unstable depending on the speciﬁc
temperature proﬁle. Thus, experimental results lying in this region could be predicted
differently depending on the choice of the temperature proﬁle. In Fig. 8.5b the
inﬁnite-reaction-rate temperature proﬁle (1) is the most unstable. Proﬁle 8, which is
ﬂat across the reaCtion zone, is the most stable. This is also obvious if we exanrine
proﬁles 3 and 4. Proﬁle 3 has a higher temperature than proﬁle 4 but is ﬂatter and
hence more stable. Also, from Fig. 8.5 we observe that the inﬁnite-rate proﬁle has a
much higher frequency for the onset of the stable region than the most stable ﬂame
(proﬁle 8). Moreover, for experimental design purposes the stability criteria obtained
for the inﬁnite-rate temperature proﬁle could be used as a lower bound. Comparison
between different B -numbers suggests ﬂame ﬂickering increases with increase in B -

number.

There has been a question as to what the numerical value of the streamwise coor-
dinate it should be to quantify the ﬂickering frequency of the ﬂame (Buckmaster and
Peters, 19860. In Fig. 8.6 we present the dimensional frequency as a function of the
streamwise distance. The frequency is given by (0* =t00 Prof/U; . The streamwise
distance is obtained from the deﬁnition of the Froude number (x ' =0.5U,,",,s lv..g '2Fr02).
We choose U i=1 m/s and put g*=9.8 m/sz. The results are presented up to

x1 =7 m. After that the governing equations are not valid because the ﬂow becomes

199

 

N
L

stable

H
l

 

 

unstable
' r . l ' l ' J ' l . r r

0 p 5 10. 15 20 25 30 35
FROUDE NUMBER Fro

O

 

NON -DIMENSIONAL FREQUENCY coo

 

 

Figure 8.5a Stability curve; non-dimensional frequency versus Froude number,
B =0.275. Curves 1-6 are for different inner temperature proﬁles of Fig.
8.3a

 

 

 

 

 

 

 

 

 

 

 

3°7 ‘

B ‘l

26‘

£53 ~l 1

CV5"? 2

E -‘. 3
I

:4“ 4 stable
It: 5 .

2. g

o _l

'9' ‘4 3

E2~i‘\

5. 9O 8

T .

z

o ~~-_.

z I . t . t . l

 

20 25 30 35
FROUDE NUMBER Fro

Figure 8.5b Stability curve; non-dimensional frequency versus Froude number,
B =1.125. Curves 1-8 are for different inner temperature proﬁles of Fig.
8.3b

201

 

 

 

 

 

NON —DIMENSIONAL FREQUENCY coo

 

l ' I ' l

0'5'10‘15'210'25 30 35
FROUDE NUMBER Fro

Figure 8.5c Stability curve; non—dimensional frequency versus Froude number, B =1.6.
Curves 1-6 are for different inner temperature proﬁles of Fig. 8.3c

202

 

 

FREQUENCY (0 (Hz)

 

 

 

 

 

 

 

 

 

x (In)

Figure 8.6a Frequency curve; neutral frequency versus streamwise distance, B =0.275.
Curves 1-6 are for different inner temperature proﬁles of Fig. 8.3a

 

203.

 

 

 

 

 

 

FREQUENCY at) (Hz)
h OI G
O O O
l l 1 L I
I5 03

 

 

 

5
30‘

. 7 M
20" 8 ﬁ
10 l I I ' I I I

0 1 2 3 4 5 6

X(m)l

Figure 8.6b Frequency curve; neutral frequency versus streamwise distance, B =1.125.
Curves 1-8 are for different inner temperature proﬁles of Fig. 8.3b

 

204

125

 

H

O

O
l

W .
“l

75-

 

 

50+

 

 

 

25 ' I ' I I I

x (In)

Figure 8.6c Frequency curve; neutral frequency versus streamwise distance, B =1.6.
Curves 1-6 are for different inner temperature proﬁles of Fig. 8.30

 

205

turbulent. For higher B -numbers the disturbance frequency essentially remains con-
stant with the streamwise distance but changes substantially with the choice of tem-
perature proﬁle (density variation at the ﬂame). For lower B -numbers, where there is
n0t much density variation, the frequency decreases with the increase in streamwise
distance. Thus characteristic frequency strongly depends on the density variation in

the vicinity of the ﬂame.

To obtain the growth rate curve, the real and imaginary parts of do are obtained
for the frequencies to below the neutral frequency. Figure 8.7 shows a plot of Imtxo
versus 0) for B =1.125. It is observed that at lower Froude numbers the growth rate is
higher, corresponding to our previous observations. Results are plotted for proﬁle #5
for Fro = 2 and 4. For the more-buoyant ﬂame (Fro =2) the maximum growth rate
occurs at a larger frequency ((00 =03, —Im (10:0.238) than for the less-buoyant ﬂame

(Fro=4), where (00:0.14 and -Im eta =.18.

In Appendix 8.C the disturbance Eq. (8.28) is analyzed. An inﬂection point in
the velocity proﬁle is a necessary condition for instability. Hence, in the wind-aided
ﬂame spread problem, the ﬂame overhang (underﬂame preheat zone, see chapter 2)
will be stable if the pyrolysis zone ﬂame is stable, i.e, the ﬂame will be most unstable
near the leading edge, in the pyrolysis zone. The instability starts at the ﬂame locus;

the unstably suatiﬁed region between ﬂame and wall is the source of the instability.

8.5. CONCLUSIONS

 

In this study we have analyzed the hydrodynamic stability of the wind-aided
ﬂame spread problem in the ceiling conﬁguration. The basic state solution employs
x‘m blowing, the boundary layer assumption, and the restriction to self-similar proﬁles
of T; this last condition requires that the ﬂame temperature along the entire length of
the ﬂame be constant, which is valid for Re—roo. The inviscid parallel ﬂow analysis

introduced the Froude number as the sole global nondimensional parameter. For

206

 

 

 

0.3 . I ' I ' I ' I I I ' I
Temp profile# 5
31.125
0.2‘ ﬂ "'
is" . .
' /
0.1—J . ..
Fro=4 Fr°=2 ‘
0.0 t . l ' I ' I ' I ' I ' I

 

0.0 0.2 0.4 0.5 0.8 1.0 1.2 ' 1.4

Figure 8.7 Variation of frequency ((00) with —Im(oro ). Curves plotted are for tempera-
ture proﬁle #5 for Fro equal to 2 and 4; B=1.125. .

207

higher Froude numbers ( > 8) the ﬂames are essentially stable and the detailed tem-
perature proﬁle has little effect on the stability of the ﬂame. For lower Froude
numbers (< 8) the buoyancy effect is appreciable and the form of the temperature
proﬁle is crucial to the stability calculation. The inﬁnite-reaction rate ﬂame tempera-
ture proﬁle predicts very high frequency ﬂames which increases with sueamwise dis-
tance. Thus, for low Froude numbers a parameter representing the detailed ﬂame tem- I
perature distribution is necessary; the exuemely localized ﬂame vicinity dramatically
inﬂuences the stability of the whole problem.

208

APPENDIX 8.A: REACTION ZONE SOLUTION FOR TEMPERATURE

A brief discussion is given here. Details can be found in Linan (1974). By using
suetched variables §=B(n—nf )la and T=(9+c §)/Bb Eq. (8.9) reduces to

1m
O§§=8[92—§2]e b , (8.A.1)

as B—too, where de/d §=:tl as C—>ioo and 8(x )=2va2xD /(bQ B3) is the reduced Dam-
kohler number. Here

_-(ao+ap) _£( _ )
._ -——(ao_ap) , a—b (10 OF

 

d“ (0f) -1
dn '

and B is the nondimensional activation energy, B=E ' (Tf—T; )/(R I T ,3 2). When c=0
the heat losses from the ﬂame toward both sides are equal. Since the objective here is
to examine the effect of temperature proﬁles on stability, in a qualitative sense, we will
assume for simplicity that c=0. This gives (using deﬁnitions of a0, ap, b0 and 1),.-

given in Sec. 8.2)

l—Tw
"f " 2—Tw -—T,, '

 

For Tw=0.3046 (since T;=635 K for PMMA), T,.=l/7 (since 7.1/T; ~ 1n)we get
uf =0.448, which in turn requires B = 1.125. For the present case a=1.82/b with
Q=l4.12 (from Eq. (8.17) for Yo..=0.233). Now using the ﬁrst term of the expansion

8
5=50+__1_+... and putting 86’3b =7 and using the inner variables deﬁned above, we

5

209

obtain a single graph for 9 versus é. Now using B = 0.6 to 4 we obtain various
graphs in Fig. 8.3. Solutions for proﬁle numbers 3 and 8 are discussed in Appendix
8.B.

210

APPENDIX 8.B: ASYMPTOTIC APPROXIMATION FOR T' AS n—mf

Near n=nf the most important term in Eq. (8.28) is the last one since (u--uf)‘2
there becomes very small. The solution for 6 in this region thus depends strongly on
T'(u —uf )‘2. Using Taylor series and neglecting (u —u ”)4 and higher order terms we
obtain (u -uf)-2 = (ﬂ-nf)-2(f”)_2 since f”'(nf )=f (hf)=0 Use of the inner variables
deﬁned in appendix 8.A gives

 

T’ _ -92 are
WWLa]

We observe that 9 is symmetric in § since we have assumed c=0 (Appendix 8.A);
hence, in the expansion for 6 = 90 + 9,: + 92E2 +---, we have 61: 93= 95:0. Thus
the term in Eq. (88.1) becomes inﬁnite as §—>O. When 92:0, the above term
becomes ﬁrs—tang as §—>0. Thus when 6 = 90 + 94:4 +--- Eq. (8.B.1) is continuous
near nf; 9 (see Eq. (8.28)) will be continuous for 0Sn<oo.

For proﬁles 3 and 8 we used 9:2.25+0.0092§4 and B=0.6 and 4 respectively.

The inner solutions are matched to outer solutions with continuous slopes.

211

APPENDIX 8.C: ANALYSIS OF THE DISTURBANCE EQUATION

Rayleigh (1880) found that for a isothermal plane parallel ﬂow (T'=O and T=l in

Eq. 8.28), the necessary condition for instability is that the velocity proﬁle u (11) should

have a point of inﬂection. Fjortoft (1950) proved a stronger condition for instability,

namely tin/(u —c) < 0 somewhere in the ﬁeld of ﬂow. We presently derive correspond-

ing conditions for Eq. (8.28). The derivation follows the approach of Drazin and

Howard (1966).

We multiply Eq. (8.28) by its complex conjugate S and integrate from 11:0 to

n—r», to obtain the real and imaginary parts as

 

“ﬂ .a2 2 _” ' _
-Ir2 dn-a IIGIanIJ_I:-,;l2(uc)[ —]dn

1 |6|2
Fr 0; lu—c l4

 

T!
21.2
Kn WC) szdn=0.

 

 

lu-c I2

00 2 I ' 0‘ 2 I
c] IeI [u]dn+ 2 MI (u—c,)%dn=0,

Fro o 'u-cl4

(8.C.1)

(8.C.2)

respectively. Here x'=dx/dn and c=c,+ic,». From Eq. (8.C.2) cian requires that we

satisfy at least once in 05119» the condition

tetz _u_'_ - 2 <u—c.) T' A)
lu-cl2 T2 17% lu-cl2 T2

Since u=f' etc. this becomes

 

 

212

 

.. T’ _-2 04-9) ,
f [f+2-f]—Fro lu-clzT’ , (8.C.3)

which is identically satisﬁed at 11:11 f, where f (11f )=0 and T'mf )=0. Thus 11:11, may
be a locus of unstable points (disturbances at the ﬂame may grow). This is the
equivalent of the Rayleigh criterion. Now we know that since f (n, )=0 we must have
an inﬂection point in the velocity ﬁeld. At this. location f m=0, i.e, u"=0. Since the
vorticity is §=du ldy (easily related to du Idn) the vorticity has either a local maximum
or minimum. From Eq. (8.8) we have f ""=-f 'f ", i.e., um=-uu'. Since u>0, and
u'>o, we have u"'<o giving dZCIdy2<0, thus the vorticity has a local maximum at
11:11)“

Now we derive equivalent of Fjortoft’s criterion. We multiply Eq. (8.C.2) by
(c, -uf)/c,-, and add it to the RHS of Eq. (8.C.1) to obtain

u, ' 1 T'
(u -uf ) [-13] + Fro lu -c l2 T2 [(u —c, )(u —uf )-(u —c, )(uf —u, )—c,-2] < O , (8.C.4)

 

somewhere in 0Sn<oo. At n=nf we have u=uf and T'=0; hence, inequality (8.C.4) is
not satisﬁed at 11f. Near 11f we write Eq. (8.C.4) as

u" zu’T' g 1 :r’ 2
(u—uf)[T2 T3 ]Tprolu_c|2F[(u—c,)(u-uf)—(u-c,)(uf-c,)—c,-] <0. (8.C.5)

 

Near the ﬂame the temperature gradients change much more rapidly than the velocity
gradients. Thus for slight displacements near 11f we are justiﬁed in putting u"=0.
' After dividing inequality (8.C.5) by IT’ITZI and neglecting terms multiplied by (u —uf),

213

we obtain

_ ')
_S&'1(_T_[(u -c,)2+ct2] < o , (8.C.6)

Fro lu-c I2

which is satisﬁed here since Fro >0, lu—c, |>O, (u -c, )2+c,-2>O, and sgn (T ')>0 when
n<11f (in the fuel region). Thus, in our case we expect the fuel region, in which den-
sity increases as we move upwards toward the wall (away from the ﬂame), to be the
source of the instability. When g —> - g (F r0 <0), the oxidizer region will be the

source of the instability.

CHAPTER 9

CONCLUSIONS

SYNOPSIS

 

In this chapter some conclusions are drawn and possible future work is suggested.

214

215

9.1. CONCLUSIONS

 

In this study a detailed theoretical and experimental investigation of wind-aided
ﬂame spread over vaporizing solids is performed. Effects of free stream velocity and
oxygen mass fraction, unsteadiness of the solid, surface radiative heat loss and surface
heat ﬂux near the leading edge on pyrolysis and the ﬂame front speed (location) are
investigated. A detailed investigation of the thermal degradation of a vaporizing solid
is performed. Also performed is a stability analysis of wind-aided ﬂame spread across
a ceiling.

In Chapter 2 a simpliﬁed heat-transfer model of wind-aided ﬂame spread was
developed. An explicit expression for the pyrolysis front speed was obtained:

2

 

 

l e e t t t 2
U; - U; p,‘).;c; T: - TI. erfc(-M W)

It was found that the pyrolysis front speed varies linearly with both free stream velo-
city and oxygen mass fraction. It compares favorably with the experiments of Chapter

4. Also an expression was obtained for ﬂame tip location (ﬂame reattachment point):

, , Y’ale‘
Uf = Upcosec2 £- 0 ,I PS. . (9.2)
2 1 + Y0a/VYFS

 

In Chapter 3 the model of Chapter 2 was further examined. Numerical results
were obtained for excess pyrolyzate and surface heat ﬂux in the preheat zone. The
numerical results were correlated to provide expressions for the surface heat ﬂux in the

underﬂame preheat zone:

216

 

 

0.6

grain/E =1+5[§gp_] , (9.3)

gTP(§p)\]E; gf-{P

and in the preﬂame preheat zone:
g (a): P. 3
T f

= — , 9.4
81‘,(§f)§f [ a] ‘ ’

The excess pyrolyzate was nearly 50% for all oxygen concentrations.

In Chapter 4 the pyrolysis and ﬂame front locations and production rates of major
chemical species were measured as functions of time for coﬂow ﬂame spreading over
PMMA samples for various ﬂow speeds and oxygen mass fractions. It was found that
both the pyrolysis front and the ﬂame front speeds vary linearly with the free stream
velocity, as predicted by the model of Chapter 2. With respect to the free stream oxy-
gen mass fraction, the pyrolysis front speed was also affected by internal reﬂections at
the higher oxygen concentrations. Under completely adiabatic conditions (no
reﬂections) the dependence approaches as predicted by Eq. (9.1). It was also observed

that the ﬂame front location was very close to pyrolysis front location.

In Chapter 5, mass loss rate and in-depth temperature measurements were
obtained during the pyrolysis of PMMA samples in a nitrogen environment for various

heat ﬂuxes. An explicit expression was obtained for the mass loss rate:

 

 

. » 4;;(1) A0
z: 1— _ , 9.5
mp (X) L [ 14(1)] ( )

217

In Chapter 6 a model for wind-aided ﬂame spread over an unsteadily vaporizing
solid was developed. An explicit expression for the surface convective heat ﬂux under

the ﬂame was obtained:

, .. k U3 Re,“2 Yo...
qcom, (x ) = 0332-qu 'x— AH T-C pg (Tf -T..) (9.6)

Also obtained was an expression for the maximum ﬂame length as a) function of the

pyrolysis front location:

x_r_ __1:B_ "
xp -[1+Y0,Jv] ’ (9'7)

where n is equal to 2 for laminar ﬂow and 1.25 for turbulent ﬂow. It was found that
inclusion of surface radiative heat loss is necessary to result in a phenomena observed
during the experiments of Chapter 4, ﬂame spread stoppage before reaching the end of
the sample. I

In Chapter 7, burned PMMA samples were used to determine the surface heat

ﬂux distribution in the burning zone using a regression model. An explicit expression

for the regressed surface location was also obtained:

 

_ "" ki" _ '-
.(.) = 4m". 4M] . (9.3)

It was determined that surface convective heat ﬂux near the upstream leading edge

decreases instead of rising in proportion to 16"“2 as predicted by boundary layer

218

approximation. This, coupled with the surface radiative heat loss and the unsteadiness
of the solid are the main reasons for the shorter ﬂame overhang (not much excess

pyrolyzate) observed during experiments of Chapter 4.

In Chapter 8 a detailed inviscid stability analysis was performed which suggests
that in the ceiling conﬁguration the ﬂame is stable because of high rates of forced con-
vection. For smaller Froude numbers it is essential to include gas-phase ﬁnite reaction

rates in the analysis.

Thus, for development of comprehensive models it is essential to relax the boun-
dary layer approximation in the gas-phase, at least near the leading edge. It is also
important to include detailed surface boundary conditions and transient effects in the

solid.

Till now in the Ostudy we have not mentioned anything about the choice of ther-
mophysical properties. As can be seen from Eq. (9.1) the correct choice of gas-phase
properties plays an important role in comparisons with the experiments. Generally
they are kept constant for all free stream conditions. In Fig. 9.1 is plotted the product
of density, thermal conductivity and speciﬁc heat as a function of temperature for vari-
ous free stream oxygen mass fractions. Clearly in this temperature range an error of
20% can be encountered because of the choice of conditions under which the proper-
ties are selected. When comparing experiments to theory, an error of this order can

thus be treated as normal and can be attributed to inaccurate properties values.

kpcp/(kpcp)¢ . m

219

 

IL05

P
o
‘t’

l I I l

.°
a:
‘1”

I ' I I I ' I ' I

‘\ (kpcp). “I. 30.76 kP/m‘K'see

 

lJllJlllLlllljllj

IJLAILI

 

 

0.30 '

T r

I ' r ' I I I I
400 600 800 1000 1200

TEMPERATURE (K)

Figure 9.1 Variation of kpCp as a function of temperature

 

1400

220

9.2. SUGGESTIONS FOR FUTURE WORK

In the present study we have not included the effects of soot radiation, gas-phase

kinetics, turbulence and chairing of the solid.

Because of the inclusion of so many non-linear effects the next stage of modeling
should be oriented towards numerical modeling. Elliptic equations should be used in
the gas-phase which could be simpliﬁed to parabolic equations later downstream.
Because of this gas-phase kinetics should also be included. Modeling should not be
limited to determination of spread rates but should be able to predict species produc-
tion and depletion rates, which is very important from the pollution point of view.

Experimental efforts should also head in the same direction.

LIST OF REFERENCES

221

LIST OF REFERENCES

Abramowitz, M. and Stegun, I. A. (1972), Handbook of Mathematical Functions,
Dover, New York.

Abu-Zaid, M. (1988), Effects of water on ignition of cellulosic materials, PhD Thesis,
Michigan State University, East Lansing, MI.

Agrawal, S. (1992a),"0n thermal degradation of a vaporizing solid", Spring Technical
Meeting of Central States of the Combustion Institute, Columbus, OH, April.

Agrawal, S. (1992b), "Determination of surface heat ﬂux in the burning zone during
wind-aided ﬂame spread using burned PMMA samples", submitted.

Agrawal, S. and Atreya, A. (1992), "Wind-aided ﬂame spread over an unsteadily
vaporizing solid", Twenty-Fourth Symposium (International) on Combustion. The
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