ABSTRACT

Title of dissertation: DEVELOPMENT OF A LAGRANGIAN
-EULERIAN MODELING FRAMEWORK
TO DESCRIBE THERMAL DEGRADATION
OF POROUS FUEL PARTICLES IN
SIMULATIONS OF WILDLAND FIRE
BEHAVIOR AT FLAME SCALE

Mohamed Mohsen Ahmed
Doctor of Philosophy, 2023

Dissertation Directed by: Professor Arnaud Trouvé
Department of Mechanical Engineering

The dynamics of wildland fires involve multi-physics phenomena occurring at

multiple scales ranging from sub-millimeter scale representative of small vegetation

particles to several kilometers representative of meteorological scales. The objective of

this research is to develop an advanced physics-based computational tool for detailed

modeling of the coupling between the solid-phase and the gas-phase processes that

control the dynamics of flame spread in wildland fire problems. This work focuses

on a modeling approach that resolves processes occurring at flame and vegetation

scales, i.e., the formation of flammable vapors from the porous biomass vegetation

due to pyrolysis, the subsequent combustion of these fuel vapors with ambient

air, the establishment of a turbulent flow because of heat release and buoyant

acceleration, and the thermal feedback to the solid biomass through radiative and



convective heat transfer. A modeling capability called PBRFoam is developed in this

dissertation based on the general-purpose Computational Fluid Dynamics (CFD)

library OpenFOAM and an in-house Lagrangian Particle Burning Rate (PBR) model

that treats drying, thermal pyrolysis, oxidative pyrolysis, and char oxidation using

a one-dimensional porous medium formulation. This modeling capability allows

the description of fire spread in vegetation fuel beds comprised of mono- or poly-

dispersed porous particles including thermal degradation processes occurring during

both flaming and smoldering combustion.

The modeling capability is calibrated for cardboard and pine wood using

available micro- and bench-scale experimental data obtained. Then it is applied to

simulate the fire spread across the idealized fuel beds made of laser-cut cardboard

sticks that have been studied experimentally at the Missoula Fire Sciences Laboratory.

The simulations are conducted with prescribed particle and environmental properties

(i.e., fuel bed height, fuel bed packing, particle size, moisture content, and wind

velocity) that match the experimental conditions. The model is first validated against

experimental measurements and observations such as the rate of spread of the fire

and the flame residence time. The modeling capability is then used to provide

insights into local as well as global behavior at the individual particle level and the

fuel bed level with variations of the fuel packing.

The modeling capability is also applied to simulations of fire spread across

idealized vegetation beds corresponding to mixed-size cylindrical-shaped sticks of pine

wood under prescribed wind conditions. Depending on the particle size distribution,

the simulations feature complete fuel consumption with a successful transition



from flaming to smoldering combustion or partial fuel consumption with no or

limited smoldering. These simulations show the existence of either a mixed mode

of heat transfer through convection and radiation for small particles or a radiation-

dominant heat transfer mode for larger particles. The results are interpreted using

a novel diagnostic called the Pseudo Incident Heat Flux (PIHF) and 2-D maps

that characterize single particle response as a function of the PIHF and the flame

residence time.



DEVELOPMENT OF A LAGRANGIAN-EULERIAN MODELING
FRAMEWORK TO DESCRIBE THERMAL DEGRADATION OF
POROUS FUEL PARTICLES IN SIMULATIONS OF WILDLAND

FIRE BEHAVIOR AT FLAME SCALE

by

Mohamed Mohsen Ahmed

Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment

of the requirements for the degree of
Doctor of Philosophy

2023

Advisory Committee:
Professor Arnaud Trouvé, Chair/Advisor
Professor James Baeder, Dean’s Representative
Dr. Mark Finney
Professor Johan Larsson
Professor Stanislav Stoliarov
Professor Peter Sunderland



© Copyright by
Mohamed Mohsen Ahmed

2023



Dedication

To my dear parents, my lovely sister, my darling Radwa, and our beautiful Nelli.

ii



Acknowledgments

The completion of this Ph.D. dissertation would not have been possible without

the generous support and encouragement of so many people, for which I am deeply

grateful.

First, I would like to express my deepest gratitude to my advisor, Dr. Arnaud

Trouvé. His guidance, mentorship, and expertise have been invaluable throughout

my research journey. His insightful comments and feedback on my work were

instrumental in refining my ideas and improving the quality of my research. I am also

grateful for the opportunities he provided me to present my research at conferences

and workshops, which helped me gain exposure and build my professional network.

His support in securing funding for my research projects was also critical in enabling

me to complete my research.

I am also grateful to the faculty members at the University of Maryland who

dedicated part of their precious time to serve as an advisory committee for my

dissertation. Thanks to Dr. James Baeder, Dr. Johan Larsson, Dr. Stanislav

Stoliarov, and Dr. Peter Sunderland for their insightful comments and constructive

discussion, which helped me to refine my research.

Special thanks to Dr. Mark Finney from the U.S. Forest Service for allocating

some of his valuable time to be on my advisory committee as well as for his ultimate

support. I am very thankful to him for sharing the data of his experiments and for

the invaluable comments and feedback that he provided at different phases during

this project.

iii



I would also like to extend special thanks to Mr. Jason Forthofer from the U.S.

Forest Service for the thoughtful and deep discussions we exchanged about model

implementation and code development. His guidance was very critical in improving

the quality of the computational tools developed in this project.

I would like to acknowledge all my former and present colleagues at the ME

and FPE departments especially: Salman Verma, Rui Xu, Yiren Qin, Yujeong Kim,

Bouaza Lafdal, Zishanul Hussain, Anagh Dave, Kelliann Lee, Jeffrey Bors and Sofia

Le Braddock. I am also grateful to the administrative staff of the FPE department:

Christine O’Brien and Nicole Hollywood.

I am thankful for the financial support from the USDA Forest Service, Missoula

Fire Sciences Laboratory, with Dr. Mark Finney as Program Manager. I am also

thankful for the partial financial support provided by the U.S. National Science

Foundation (NSF, CBET Program, Award #1604907) and by FM Global. I would

like also to acknowledge the supercomputing resources made available by UMD

(https://www.glue.umd.edu/hpcc) and by NSF (ACCESS Program, Grant #TG-

CTS140046, https://access-ci.org).

Finally, I would like to express my deep appreciation to my family. Their

ultimate love, encouragement, and support have been my source of strength and

inspiration throughout my academic career. Their sacrifices and unwavering belief

in my potential have made this accomplishment possible.

iv

https://www.glue.umd.edu/hpcc
https://access-ci.org


Table of Contents

Dedication ii

Acknowledgements iii

List of Tables x

List of Figures xi

Nomenclature xxiii

1 Introduction 1
1.1 Wildland fire behavior modeling . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Scales of wildland fire behavior dynamics . . . . . . . . . . . . 3
1.1.2 Semi-empirical approach to wildfire behavior modeling . . . . 5
1.1.3 Physics-based approach to wildfire behavior modeling . . . . . 6

1.2 Motivation and scope of this dissertation . . . . . . . . . . . . . . . . 9
1.3 Literature review of relevant experimental/modeling work in the

vegetation-flame scale . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Objectives and contributions . . . . . . . . . . . . . . . . . . . . . . . 13

1.4.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.2 Author’s contributions . . . . . . . . . . . . . . . . . . . . . . 15
1.4.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 The Computational Modeling Framework 22
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Particle Burning Rate (PBR) Model . . . . . . . . . . . . . . . . . . 24

2.2.1 Conservation equations for species/total mass inside the solid
phase of an individual particle . . . . . . . . . . . . . . . . . . 29

2.2.2 Rate of change of the particle volume (constant/shrinking/swelling) 32
2.2.3 Conservation equations for species/total mass inside the gas

phase of an individual particle . . . . . . . . . . . . . . . . . . 33
2.2.4 Conservation equation for energy inside an individual particle 35
2.2.5 Outputs of the PBR model . . . . . . . . . . . . . . . . . . . . 39

v



2.2.6 Particle burning state . . . . . . . . . . . . . . . . . . . . . . . 40
2.3 Fuel bed model and the coupled Lagrangian-Eulerian formulation . . 42

2.3.1 Conservation equations for species/total mass in the gas-phase 43
2.3.2 Conservation equations for momentum in the gas-phase . . . . 46
2.3.3 Conservation equation for energy in the gas-phase . . . . . . . 46
2.3.4 Radiation transfer in the multiphase fuel bed . . . . . . . . . . 47

2.4 Convective transfer model . . . . . . . . . . . . . . . . . . . . . . . . 49
2.5 Drag model and particle motion . . . . . . . . . . . . . . . . . . . . . 52
2.6 Gas-phase homogeneous combustion model . . . . . . . . . . . . . . . 54
2.7 Sub-grid scale turbulence model . . . . . . . . . . . . . . . . . . . . . 57
2.8 Gas-phase radiation absorption and emission model . . . . . . . . . . 59

2.8.1 The Prescribed Global Radiant Fraction (PGRF) approach . . 60
2.8.2 The Weighted-Sum-of-Gray-Gases (WSGG) approach . . . . . 61

2.9 Soot modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3 Verification and Evaluation of the Computational Framework 67
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2 Verification tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.2.1 In-depth heat conduction of solid particles . . . . . . . . . . . 68
3.2.2 Momentum exchange between the solid- and gas-phases . . . . 69
3.2.3 Pressure-velocity coupling inside the fuel bed . . . . . . . . . . 72
3.2.4 Radiation absorption and emission inside the fuel bed . . . . . 74
3.2.5 Smoldering propagation due to radiation penetration . . . . . 78

3.3 Calibration of the thermal degradation model . . . . . . . . . . . . . 80
3.3.1 Micro-scale thermal degradation of pine wood . . . . . . . . . 80
3.3.2 Micro-scale thermal degradation of cardboard . . . . . . . . . 84

3.4 Evaluation of the PBR model against bench-scale experiments . . . . 88
3.5 Representative results of complete consumption and the issue of vol-

ume change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4 Thermal Feedback in Canonical Pool Fire Configurations 101
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2 Medium-scale methanol pool fire . . . . . . . . . . . . . . . . . . . . . 102

4.2.1 Numerical configuration . . . . . . . . . . . . . . . . . . . . . 103
4.2.2 Flame structure . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.2.3 Thermal feedback . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.3 Turbulent ethylene burner with controlled coflow (FM-Burner) . . . . 115
4.3.1 Numerical configuration . . . . . . . . . . . . . . . . . . . . . 116
4.3.2 Flame structure . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.3.3 Radiant emissions . . . . . . . . . . . . . . . . . . . . . . . . . 123

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

vi



5 Simulations of Thermal Degradation of Biomass Particles under Prescribed
External Gas Conditions 130
5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.2 Pyrolysis of biomass particles under steady and oscillatory heating

conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.2.1 Particle response time . . . . . . . . . . . . . . . . . . . . . . 134
5.2.2 Response of thick charring particles to radiative heating . . . 134
5.2.3 Response of thick non-charring (shrinking) particles to radia-

tive heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.2.4 Response of thin non-charring (shrinking) particles to radiative

heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.2.5 Response of thin particles to convective heating . . . . . . . . 140
5.2.6 Response of thick particles to convective heating . . . . . . . . 142

5.3 Maps of the pyrolysis and oxidation of porous biomass particles . . . 145
5.3.1 Effect of the particle size . . . . . . . . . . . . . . . . . . . . . 151

5.3.1.1 Maps of small particles (2 mm diameter) heated for
a long duration (τPIHF = 120 s) . . . . . . . . . . . . 151

5.3.1.2 Maps of large particles (20 mm diameter) heated for
a long duration (τPIHF = 120 s) . . . . . . . . . . . . 154

5.3.2 Effect of the residence time (τPIHF) . . . . . . . . . . . . . . . 158
5.3.3 Threshold value of the particle core temperature for complete

smoldering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6 Simulations of Fire Spread in Engineered Cardboard Fuel Beds 168
6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.2 Numerical configuration . . . . . . . . . . . . . . . . . . . . . . . . . 169

6.2.1 Computational domain and fuel bed setup . . . . . . . . . . . 169
6.2.2 Computational mesh resolution . . . . . . . . . . . . . . . . . 175
6.2.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 177
6.2.4 Sub-models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.2.5 Computational cost . . . . . . . . . . . . . . . . . . . . . . . . 178

6.3 Fire behavior in Burn 53 and Burn 67 . . . . . . . . . . . . . . . . . 179
6.3.1 Flow field at cold flow . . . . . . . . . . . . . . . . . . . . . . 179
6.3.2 Flow field during combustion . . . . . . . . . . . . . . . . . . 182
6.3.3 Flame structure . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.3.4 Fire intensity and rate of spread . . . . . . . . . . . . . . . . . 187
6.3.5 Local particle viewpoint . . . . . . . . . . . . . . . . . . . . . 190

6.4 The Pseudo Incident Heat Flux (PIHF) . . . . . . . . . . . . . . . . . 193
6.5 The flame residence time . . . . . . . . . . . . . . . . . . . . . . . . . 194
6.6 Fire behavior variation with the fuel bed packing . . . . . . . . . . . 197

6.6.1 Global fire behavior . . . . . . . . . . . . . . . . . . . . . . . . 197
6.6.2 Local particle viewpoint . . . . . . . . . . . . . . . . . . . . . 200

6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

vii



7 Simulations of Fire Spread in Idealized Uniform and Mixed Size Pine Wood
Fuel Beds 204
7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
7.2 Numerical configuration . . . . . . . . . . . . . . . . . . . . . . . . . 205

7.2.1 Computational domain and fuel bed setup . . . . . . . . . . . 207
7.2.2 Computational mesh resolution . . . . . . . . . . . . . . . . . 209
7.2.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 211
7.2.4 Sub-models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
7.2.5 Computational cost . . . . . . . . . . . . . . . . . . . . . . . . 212

7.3 Fire spread in the uniform fuel bed . . . . . . . . . . . . . . . . . . . 212
7.3.1 Global fire behavior . . . . . . . . . . . . . . . . . . . . . . . . 213
7.3.2 The structure of the fire . . . . . . . . . . . . . . . . . . . . . 216
7.3.3 Sensitivity to the grid resolution . . . . . . . . . . . . . . . . . 217

7.4 Fire spread in the mixed-size fuel patch . . . . . . . . . . . . . . . . . 224
7.4.1 Variations with the fuel bed height . . . . . . . . . . . . . . . 225
7.4.2 Local gas conditions at mid-elevation of the fuel bed . . . . . 228
7.4.3 Particle behavior in the wind-dominated fire . . . . . . . . . . 231
7.4.4 Particle behavior in the buoyancy-dominated fire . . . . . . . 237
7.4.5 The Pseudo Incident Heat Flux (PIHF) . . . . . . . . . . . . . 238
7.4.6 Interpretation of the particles’ behavior using the quasi-steady

2-D maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

8 Conclusion and Future Directions 249

A Numerical Algorithms of the Particle Burning Rate Model (PBR) 257
A.1 Discretization of the governing equations . . . . . . . . . . . . . . . . 260

A.1.1 Solid-phase mass conservation equations . . . . . . . . . . . . 260
A.1.2 The temperature equation . . . . . . . . . . . . . . . . . . . . 262
A.1.3 The gas-phase oxygen mass equation . . . . . . . . . . . . . . 267
A.1.4 The pressure equation . . . . . . . . . . . . . . . . . . . . . . 272

A.2 Iterative scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
A.3 Temporal discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 276
A.4 Deforming mesh and re-meshing capabilities . . . . . . . . . . . . . . 278

B Modified Eddy Dissipation Model for Diffusion Controlled Flames 280
B.1 Model derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
B.2 Simulations of two-dimensional laminar counter-flow methane-air flames284

C Extension of the LSP Soot Model to LES Framework 287
C.1 Conservation of soot mass . . . . . . . . . . . . . . . . . . . . . . . . 287
C.2 Soot formation and oxidation terms . . . . . . . . . . . . . . . . . . . 288
C.3 Closure model for turbulent soot formation and oxidation rates . . . . 289
C.4 Integration of the probability density function . . . . . . . . . . . . . 291

viii



D Verification of the WSGG Model Implementation in OpenFOAM 292
D.0.1 Radiation configuration 1 . . . . . . . . . . . . . . . . . . . . 293
D.0.2 Radiation configuration 2 . . . . . . . . . . . . . . . . . . . . 294
D.0.3 Radiation configuration 3 . . . . . . . . . . . . . . . . . . . . 295
D.0.4 Radiation configuration 4 . . . . . . . . . . . . . . . . . . . . 297

E Installation of PBRFoam on HPC Clusters and its Parallel Performance 299
E.1 Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
E.2 Parallel Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

Bibliography 303

ix



List of Tables

3.1 Analytical solutions of heat conduction inside solid fuel particles of
different geometries, where τ = αst/ζ

2, αs = ks/ρscs, Tg the external
gas temperature, and Ts,i the initial particle temperature. . . . . . . . 69

3.2 Calibrated model parameters for pine wood. . . . . . . . . . . . . . . 85
3.3 Calibrated chemical kinetic parameters for pine wood. . . . . . . . . . 85
3.4 Thermo-physical properties of cardboard. . . . . . . . . . . . . . . . . 87
3.5 Chemical kinetic parameters for thermal degradation of cardboard. . 87

5.1 Input parameters of the biomass particles used in the oscillatory
heating study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.2 Ignition and burnout times of particles exposed to quasi-steady (QS)
radiative heating and oscillatory radiative heating at 0.1 and 1 Hz.
The percentage difference in burnout time is calculated relative to the
QS case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.3 Ignition and burnout times of particles exposed to quasi-steady (QS)
convective heating and oscillatory convective heating at 0.1 and 1 Hz.
The percentage difference in burnout time is calculated relative to the
QS case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.1 Conditions of Burn 53 and Burn 67. . . . . . . . . . . . . . . . . . . . 175
6.2 Characteristics of Burn 53 and Burn 67. A comparison between

observed and simulated quantities. . . . . . . . . . . . . . . . . . . . 187

7.1 Design parameters of the buoyancy-dominated and wind-dominated
line fires in the uniform pine wood fuel bed. . . . . . . . . . . . . . . 206

7.2 Simulated parameters of the buoyancy-dominated and wind-dominated
line fires in the uniform pine wood fuel bed. . . . . . . . . . . . . . . 216

7.3 Summary of the simulation results corresponding to particles in the
mixed-size patch located at x = 40 m, y = 0.2 m, z = 0 position. . . 242

A.1 Definition of the mesh parameters. In these expressions, Arect is the
exposed surface area of rectangular particles, and Lcyl is the length of
the cylindrical-shaped particles. The subscript C refers to the center
of the cell; the subscript R refers to its right-boundary. . . . . . . . . 259

x



List of Figures

1.1 Scales of the wildland fire problem. Images sources: Smithsonian’s
National Zoo & Conservation Biology Institute [3]; Josh Edelson
via Getty Images; Greenpeace International [4]; and NASA Earth
Observatory [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 The different classes of CFD models used for simulating wildland
fire behavior: combustion solvers resolve dynamics at the vegetation
and flame scales; wildfire solvers resolve dynamics at the fireline and
geographical scales; atmospheric boundary layers (ABL) solvers resolve
dynamics at the meteorological scales. . . . . . . . . . . . . . . . . . 7

1.3 The larger framework of this project: chain of model development. . . 10

2.1 A schematic of fire spread in a fuel bed showing the characteristic
length-scales of the flame and the different zones identified based on
the particle burning state. . . . . . . . . . . . . . . . . . . . . . . . . 42

2.2 Schematic side view of a typical fuel bed showing fuel patches separated
by a given distance (left). The computational grid is shown in the
background. Each fuel patch is modeled by a collection of mono- or
poly-dispersed particles described at the sub-grid level (right). The
thermal degradation of each particle is tracked using the PBR solver
through the computational grid shown in the inset. . . . . . . . . . . 44

2.3 Variation of the convective heat transfer coefficient with: a) wind
velocity for particles with different sizes; and b) particle’s effective
diameter at a wind velocity of 1 m/s. . . . . . . . . . . . . . . . . . . 51

2.4 Variation of drag coefficient with Reynolds number for cylindrical-
and rectangular-shaped particles. . . . . . . . . . . . . . . . . . . . . 53

2.5 Temperature profile of laminar counter-flow diffusion flames under
adiabatic conditions obtained from well-resolved simulations using the
modified EDM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.6 Flowchart of the computational modeling framework: the chart of the
PBRFoam solver is highlighted in yellow; the chart highlight in gray
depicts an interface; the chart of the PBR solver is highlighted in blue. 65

3.1 Temperature evolution inside a semi-infinite rectangular-shaped solid
particle. The exposed surface is at ζ = 0. . . . . . . . . . . . . . . . . 70

xi



3.2 Temperature evolution inside a) cylindrical- and b) spherical-shaped
solid particles. The exposed surface is at ζ = 0.01 m. . . . . . . . . . 70

3.3 Simulations (solid lines) and analytical solution (symbols) of the mo-
mentum exchange between solid and gas-phases: a) transient velocity
of the solid and the gas-phases, b) momentum of the gas-phase and
total momentum of the system (gas+solid). . . . . . . . . . . . . . . . 72

3.4 Comparison between simulations with different grid resolutions (solid
lines) and the analytical solution (symbols) of the pressure variation
inside a fuel bed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.5 Simulations with different grid resolutions (solid lines) and analytical
solution (symbols) of the radiation absorption by the fuel bed: a)
radiation intensity variation with different packing ratios, b) spatial
variation of the radiation intensity along the fuel bed length. . . . . . 76

3.6 Spatial variation of the radiation intensity inside a fuel bed that
emits and self-absorbs radiation. The solid lines refer to the numer-
ical simulations, while the symbols refer to the analytical solution.
The emission contribution obtained from a case where absorption is
neglected is also shown. . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.7 Smoldering propagation in idealized charred pine wood bed. The
gas-phase is colored by the mass fraction of CO2. The fuel bed is
colored by a) the percentage mass with red indicating 100% char
and blue indicating 100% ash, b) the positive net radiative heat flux
indicating heating zones. . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.8 Smoldering propagation in idealized charred pine wood bed: a) time
evolution of the back and front edges of the smoldering zone; b)
smoldering intensity. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.9 Variation of the mass loss rate of a micro-scale pine wood sample with
the particle temperature at a) heating rates of 5 and 10 K/min in
the inert environment; and b) heating rate of 10 K/min in oxidative
environments. Lines: PBR simulations using the kinetic parame-
ters of Lautenberger and Fernandez-Pello [44]. Symbols: the TGA
measurements of Ref. [80]. . . . . . . . . . . . . . . . . . . . . . . . . 83

3.10 Variation of the mass loss rate of a micro-scale pine wood sample with
the particle temperature at a) heating rates of 5 and 10 K/min in
the inert environment; and b) heating rate of 10 K/min in oxidative
environments. Lines: PBR simulations using the kinetic parameters
of Anca-Couce et al. [80]. Symbols: the TGA measurements of Ref. [80]. 84

3.11 Evolution of the cardboard particle normalized mass loss rate (top row)
and normalized mass (bottom row) with temperature. Comparison of
the simulated thermal degradation of cardboard using the calibrated
PBR model (solid lines) and the TGA measurements of Refs. [81, 82]
(symbols) at a heating rate of 10 K/min in a) 100% N2 environment,
b) 10% O2 − 90% N2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

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3.12 Comparison of the simulated heat flow (solid line) and the DSC
measurements of Ref. [81] collected at a heating rate of 10 K/min in
an inert environment. . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.13 Time evolution of the mass loss rate of the 3.8 cm pine wood particle
in inert and oxidative environments at a) 25 kW/m2 and b) 40 kW/m2

irradiation. Comparison between simulations of the PBR model (lines)
and experimental data of Kashiwagi et al. [85] (symbols). . . . . . . . 91

3.14 Spatial variation of exposed surface and in-depth temperatures of the
3.8 cm pine wood particle at 40 kW/m2 irradiation in: a) 100% N2,
b) 89.5% N2 − 10.5% O2, and c) air. Comparison between simulations
from the PBR model (lines), experimental data of Kashiwagi et al. [85]
(symbols). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.15 Spatial variation of exposed surface temperature and in-depth tem-
peratures of the 3.8 cm pine wood particle at 25 kW/m2 irradiation
in 89.5% N2 − 10.5% O2. Comparison between simulations using the
PBR model (lines), experimental data of Kashiwagi et al. [85] (symbols). 94

3.16 Predictions, using different grid resolutions, of a) MLR and b) the
temperature at 5 mm depth inside the particle under 40 kW/m2

irradiation in ambient air. . . . . . . . . . . . . . . . . . . . . . . . . 95
3.17 Variation of a 10 mm diameter pine wood particle mass loss rate (a)

and mass (b) with time obtained using PBR simulations. The particle
is exposed to 60 kW/m2 radiative flux for 60 s in ambient air flowing
at 1 m/s. The vertical dashed line indicates the end of the external
heating period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.18 Two snapshots of spatial profiles of temperature (a) and volume
fraction of char and ash (b) inside a 10 mm diameter pine wood
particle exposed to 60 kW/m2 radiative flux for 60 s in ambient air
flowing at 1 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.19 Two snapshots of spatial profiles of oxygen mass fraction (a) and
char oxidation reaction rate (b) inside a 10 mm diameter pine wood
particle exposed to 60 kW/m2 radiative flux for 60 s in ambient air
flowing at 1 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.20 Temporal evolution of the volume of 10 mm diameter pine wood
particle exposed to 60 kW/m2 radiative flux for 60 s in ambient air
flowing at 1 m/s. The dashed line represents simulations conducted
using the char and ash yields listed in Table 3.2. The solid line
represents simulations conducted with modified char and ash yields of
0.3 and 0.005, respectively. . . . . . . . . . . . . . . . . . . . . . . . . 99

4.1 A 3-D view of the computational domain of the methanol pool fire
(left), and 2-D sections showing an enlarged view of the medium mesh
resolution near the fuel pan (right). . . . . . . . . . . . . . . . . . . . 103

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4.2 A view of the virtual heat flux gauges used in the numerical config-
uration: a) side view of the computational domain near the burner
lip and the gauges; b) top view of the burner surface and the gauges.
The gauges are colored in black. The computational domain and the
burner surface are colored in red. . . . . . . . . . . . . . . . . . . . . 106

4.3 Sequence of instantaneous snapshots of the methanol pool flame taken
at selected times during a representative instability cycle. The flame
is visualized using volume rendering of the high-temperature region
(defined as the region where temperatures are larger than 800 K). The
time between successive snapshots is 0.06 s and the total duration
of the sequence is 0.42 s. The fuel pan and liquid pool surface are
colored grey. The simulation was performed with the fine mesh and
with PGRF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.4 Sequence of instantaneous snapshots of the flame-flow configuration
taken at selected times during a representative instability cycle and
plotted in a central vertical plane. The flame is visualized using
isocontours of temperature; the flow is visualized using velocity vectors.
The fuel pan and liquid pool are colored white. See the caption of
Fig. 4.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.5 The power spectrum of the temporal variations of the simulated heat
release rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.6 Centerline vertical variations of: mean temperature, T ; mean vertical
velocity, u; mean radial velocity, v; rms temperature, Trms; rms
vertical velocity, urms; rms radial velocity, vrms; cross-correlation
between vertical velocity and temperature, u′T ′. Comparisons between
experimental data (symbols) and simulation results obtained with a
coarse mesh (dashed blue line), a medium mesh (dash-dotted black
line), and a fine mesh (solid red line). Simulations performed with
PGRF. The vertical bars denote the uncertainties in the experimental
data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.7 Radial variations of the mean gauge heat flux at 13 mm above the
pool surface: (Left) radiative heat flux, q̇′′rad; (Right) total heat flux,
q̇′′tot. Comparisons between experimental data (square symbols) and
simulations performed with the medium mesh and with PGRF (dashed
red line with diamond symbols) or with WSGG (dashed blue line
with circle symbols). The vertical bars denote the uncertainties in the
experimental data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.8 Radial variations of the mean radiative heat flux at 13 mm above
the pool surface. Comparisons between experimental data (square
symbols) and simulations performed with the medium mesh and with
the WSGG-Bordbar model (dashed blue line with circle symbols),
WSGG-Cassol (solid red line with square symbols), and PMC-LBL
(dotted purple line with x symbols). The vertical bars denote the
uncertainties in the experimental data. . . . . . . . . . . . . . . . . . 115

xiv



4.9 Comparisons between experimental data (square symbols) and simula-
tions performed with the WSGG-Bordbard model with 36 solid angles
(dashed red line with diamond symbols), 64 solid angles (dashed blue
line with circle symbols), or 100 solid angles (dotted purple line with
square symbols). See the caption of Fig. 4.7. . . . . . . . . . . . . . . 116

4.10 Photographs of the FM-Burner taken during experiments conducted
at different oxygen concentrations [88]. . . . . . . . . . . . . . . . . . 117

4.11 A 3D representation of the computational domain of the FM-Burner
(left), and a slice showing the computational mesh near the burner
(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.12 Contours of the instantaneous and mean values of: the temperature
(T and T )), the mixture fraction (Z and Z)), and the soot volume
fraction (fv and fv)). The white isocontour represents fv = 10−6. The
results correspond to the FM-Burner at 20.9% O2 coflow. . . . . . . . 120

4.13 Contours of the instantaneous and mean values of: the volumetric
soot formation rate (ω̇′′′

sf and ω̇′′′
sf )), and the volumetric soot oxidation

rate (ω̇′′′
sf and ω̇′′′

sf)). The results correspond to the FM-Burner at
20.9% O2 coflow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

4.14 Radial profiles of mean and rms temperature at different elevations
above the fuel surface in the FM-Burner at 20.9% O2 coflow. Compar-
ison between experiments (symbols) and simulations with PGRF (red
dash-dot line), WSGG with gas and soot absorption/emission (blue
dashed line), and WSGG with gas radiation only (solid black line). . 124

4.15 Radial profiles of mean and rms soot volume fraction at different
elevations above the fuel surface of the FM-Burner at 20.9% O2 coflow.
Comparison between experiments (symbols) and simulations with the
WSGG model (blue dashed line). . . . . . . . . . . . . . . . . . . . . 125

4.16 Axial variations of the mean radiative power of the FM-Burner at
20.9% O2 and 15.2% O2 concentrations in the coflow. Comparisons
between experimental data (symbols) and simulations performed with
the PGRF model (dotted red line), the WSGG model with gas and soot
contributions (dashed blue line), and the WSGG with gas contribution
only (solid black line). . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.17 Axial variations of the radiation absorption and emission of the FM-
Burner at 20.9% O2 concentrations in the coflow. The results are
estimated using the WSGG model accounting for gas and soot radiation.127

5.1 Response of a 20 mm-thick charring particle to radiative heating:
a) particle surface and core temperatures; b) particle mass loss rate.
Comparison between particle response to quasi-steady forcing (QS)
(lines with symbols) and oscillatory forcing at f = 0.1 Hz or f = 1 Hz
(lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.2 Response of a 20 mm thick non-charring particle to radiative heating:
a) particle size; b) particle mass loss rate. See caption of Fig. 5.1. . . 138

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5.3 Response of the 2 mm non-charring particle to radiative heating: a)
particle surface and core temperatures; b) particle mass loss rate. See
caption of Fig. 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.4 Response of a 2 mm non-charring particle to convective heating: a)
particle surface and core temperatures; b) particle mass loss rate.
Comparison between the particle response to quasi-steady forcing
(QS) (lines with symbols) and oscillatory forcing by oscillations in Tg
at f = 0.1 Hz or f = 1 Hz (lines). . . . . . . . . . . . . . . . . . . . . 142

5.5 Response of a 2 mm non-charring particle to convective heating: a)
particle surface and core temperatures; b) particle mass loss rate.
Comparison between the particle response to quasi-steady forcing
(QS) (lines with symbols) and oscillatory forcing by oscillations in
both Tg and ug at f = 0.1 Hz or f = 1 Hz (lines). . . . . . . . . . . . 144

5.6 Mass loss rate response of the a 20 mm thick particle to radiative
heating: a) non-charring; b) charring. See caption of Fig. 5.5 . . . . . 145

5.7 Response of a 10 mm particle to an external irradiation of G =
60 kW/m2 for 60 s, and ambient air flow at ug = 1 m/s: a) the net
total heat flux; b) the surface and core temperatures; c) the convective
heat transfer coefficient; d) the surface emissivity. The duration of
the external irradiation is highlighted in red. . . . . . . . . . . . . . . 148

5.8 Response of a 10 mm particle to an external irradiation of G =
60 kW/m2 for 60 s, and ambient air flow at ug = 1 m/s: a) the
reactions rates of Rd-Rop; b) the normalized solid constituent mass.
The duration of the external irradiation is highlighted in red. . . . . . 149

5.9 Maps of a 2 mm pine wood particle for τPIHF = 120 s: a) degree
of completion of the drying reaction Rd; b) degree of completion of
the pyrolysis reaction Rp+Rop; c) degree of completion of the char
oxidation reaction Rco. The gray zone indicates PIHF < PIHF0. . . . 153

5.10 Maps of a 2 mm pine wood particle for τPIHF = 120 s: a) peak value
of the pyrolysis rate; b) pyrolysis duration. The red solid iso-contour
depicts the flaming threshold of 1 g/s/m2. The gray zone indicates
PIHF < PIHF0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

5.11 Maps of a 2 mm pine wood particle for τPIHF = 120 s: a) the peak
particle temperature Rd; b) the temperature at the core of the particle
at t = τPIHF. The gray zone indicates PIHF < PIHF0. . . . . . . . . . 155

5.12 Maps of a 20 mm pine wood particle for τPIHF = 120 s: a) degree
of completion of the drying reaction Rd; b) degree of completion of
the pyrolysis reaction Rp+Rop; c) degree of completion of the char
oxidation reaction Rco. . . . . . . . . . . . . . . . . . . . . . . . . . . 156

5.13 Maps of a 20 mm pine wood particle for τPIHF = 120 s: a) peak value
of the pyrolysis rate; b) pyrolysis duration. The red solid iso-contour
highlights the flaming threshold of 1 g/s/m2. . . . . . . . . . . . . . . 157

5.14 Maps of a 20 mm pine wood particle for τPIHF = 120 s: a) the peak
particle temperature Rd; b) the temperature at the core of the particle
at t = τPIHF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

xvi



5.15 Maps of the degree of completion of pyrolysis (left) and the degree of
completion of char oxidation (right) for a 2 mm pine wood particle
heated for different periods: a) τPIHF = 5 s, b) τPIHF = 15 s, and c)
τPIHF = 30 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

5.16 Maps of the particle peak temperature (left), and the particle core
temperature at the end of the heating period (right) for a 2 mm
pine wood particle heated for different periods: a) τPIHF = 5 s, b)
τPIHF = 15 s, and c) τPIHF = 30 s. . . . . . . . . . . . . . . . . . . . . 162

5.17 Maps of the degree of completion of pyrolysis (left) and the degree of
completion of char oxidation (right) for a 20 mm pine wood particle
heated for different periods: a) τPIHF = 15 s, b) τPIHF = 30 s, and c)
τPIHF = 60 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

5.18 Maps of the particle peak temperature (left), and the particle core
temperature at the end of the heating period (right) for a 20 mm
pine wood particle heated for different periods: a) τPIHF = 15 s, b)
τPIHF = 30 s, and c) τPIHF = 60 s. . . . . . . . . . . . . . . . . . . . . 164

5.19 Scatter plot of the char oxidation degree of completion against the
particle core temperature achieved at the time corresponding to the
end of the heating duration τPIHF. . . . . . . . . . . . . . . . . . . . . 165

6.1 Images from the experimental burns of fire spread in cardboard fuel
beds conducted at the Missoula Fire Sciences Laboratory (courtesy
of Dr. Mark Finney): a) top view of an experiment conducted on an
inclined surface, featuring flaming and glowing zones; b) back view
of a buoyancy-dominated fire experiment in the wind tunnel (Burn
53), featuring a uniform flame front with peak and trough structures;
c) isometric view of the wind-dominated fire in the wind tunnel at
high speed (Burn 67), featuring flaming and glowing zones as well as
a curved flame front. . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

6.2 Schematic of the wind tunnel facility at Missoula Fire Sciences Labo-
ratory (adapted from Ref. [105]). The dimensions on the image are
obtained from private communication with Dr. Mark Finney. The
blue dashed box contours the simulated region. . . . . . . . . . . . . 171

6.3 The computational domain used in the simulations of the cardboard
fire spread. The fuel bed is colored in yellow. . . . . . . . . . . . . . . 172

6.4 Variation of the fuel bed packing ratio with the axial spacing between
particle arrays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

6.5 Contours of the velocity magnitude at the vertical center-plane of the
wind tunnel (top slice) and at a horizontal plane at y = 0.025 m eleva-
tion (bottom slice). The contours correspond to cold flow simulations
of a) Burn 53 and b) Burn 67. . . . . . . . . . . . . . . . . . . . . . . 180

6.6 Profiles of the mean velocity at different locations in the streamwise
direction in a) the low-velocity case (Burn 53), and b) the high-velocity
case (Burn 67). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

6.7 Profiles of the root-mean-square velocity. See the caption of Fig. 6.6. 181

xvii



6.8 Streamwise variation of ∆y+w in: a) the low-velocity case; b) the high-
velocity case. The dashed line represents instantaneous values and
the solid line represents mean values. The cardboard fuel bed starts
at x = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

6.9 Tunnel flow field in the case of wind-dominated fire (Burn 67). Con-
tours of the axial velocity in the tunnel mid-plane. The velocity
vectors are scaled by velocity magnitude. . . . . . . . . . . . . . . . . 183

6.10 Tunnel flow field in the case of a buoyancy-dominated fire (Burn 53)
with a) a prescribed inlet velocity, b) a prescribed total pressure. See
the caption of Fig. 6.9. . . . . . . . . . . . . . . . . . . . . . . . . . . 184

6.11 A 3-D rendering of the simulated fires: a) back view of Burn 53
showing the iso-surface of hot gases at 1000 K and the cardboard
particles colored by the mass loss rate (fire spreads into the page); b)
isometric view of Burn 67 showing the iso-volume of hot gases > 550 K
and the cardboard particles colored by their surface temperature (fire
spreads from left to right). . . . . . . . . . . . . . . . . . . . . . . . . 186

6.12 Global fire behavior of Burn 53 and Burn 67 in terms of a) rate spread,
and b) fire line intensity. The dashed lines represent the slope from
the measured rate of spread. . . . . . . . . . . . . . . . . . . . . . . . 188

6.13 Time evolution of the pyrolysis front obtained from simulations with
different boundary conditions: a) effect of inlet boundary in Burn 53;
b) effect of the side walls on Burn 67. . . . . . . . . . . . . . . . . . . 189

6.14 Time evolution of the pyrolysis front obtained from simulations with
the baseline mesh and a refined mesh. . . . . . . . . . . . . . . . . . . 190

6.15 Local viewpoint of the gas conditions in Burn 67 at the top of the
particle located at x = 3 m, z = 0: a) local gas temperature, b) local
gas velocity, c) local oxygen mass fraction; and d) local irradiation. . 192

6.16 The behavior of the particle located at x = 3 m, z = 0 in Burn 67: a)
particle surface and core temperature, b) particle constituents masses,
c) particle mass loss rate; and d) net heat flux at the particle’s exposed
surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

6.17 The convective and radiation contributions to the pseudo incident heat
flux (PIHF) extracted at the top of the fuel bed at x = 3 m, z = 0 in
a) Burn 53, and b) Burn 67. . . . . . . . . . . . . . . . . . . . . . . . 195

6.18 Flame residence time based on a) gas temperature, b) particle flaming
duration, and c) duration of intense PIHF. The red solid line is a
spline fit of the raw data. The vertical dashed blue lines delimit the
start and the end of the residence time. The data correspond to the
particle located at x = 3 m, z = 0 in Burn 67. . . . . . . . . . . . . . 196

6.19 A 3-D rendering of the flaming zone variation with the packing ratio
βs. The flaming particles are colored by the gaseous fuel release rate
(GFRR) (i.e., the production rate of fuel from the pyrolysis reactions
Rp and Rop). The hot gases at 1000 K are represented by the white
iso-surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

xviii



6.20 Variations of the a) fire intensity, b) the flame zone depth, c) the
fire rate of spread, and d) the flame height with the fuel bed packing
ratio. The symbols represent the simulated cases and the solid line
represents a linear fit to the simulated data. . . . . . . . . . . . . . . 199

6.21 The behavior of 36 particles in Burn 67, starting from x = 3 m, z = 0
and separated by 0.2 m in the x−direction and 0.25 m in the z−
direction: a) particle surface temperature, b) normalized particle
masses, c) particle state, and d) particle mass loss rate. . . . . . . . . 201

6.22 Variation of the residence time with the packing ratio: a) the particle
flaming and smoldering times, b) the PIHF time, and c) the flame
residence time based on the gas temperature. The symbols represent
the average of 36 particles, the bars represent the standard deviation,
and the solid lines represent linear fit. The dashed blue line represents
the experimental fit of Finney et al. [2]. . . . . . . . . . . . . . . . . . 202

7.1 A 3-D view of the computational domain used in the simulations
of fire spread in pine wood fuel beds: a) the uniform fuel bed with
2 mm particles (βs = 0.005); b) the fuel bed with a patch of mixed
small/large particles (Dp = 2 and 10 or 20 mm) located at 38 ≤ x ≤
42 m (βs = 0.0068 or 0.012). . . . . . . . . . . . . . . . . . . . . . . . 208

7.2 Propagation of the wind-dominated fire in the uniform bed: a) time
evolution of the simulated flame, pyrolysis and smoldering zones (solid
and dashed lines) and the estimated spread from Rothermel model
(dotted line); b) time evolution of the simulated fire intensity from
homogeneous gas-gas reactions (black line) and from heterogeneous
solid-gas reactions (blue line). . . . . . . . . . . . . . . . . . . . . . . 214

7.3 Global behavior of the buoyancy-dominated fire in the uniform bed
at 1 m/s external wind speed. See the caption of Fig. 7.2 . . . . . . . 215

7.4 Visualization of the wind-dominated fire in the uniform bed: a) iso-
metric view (fire spreads from left to right) of; b) back view of the hot
gases only (fire spreads into the page). The hot gases in (a) and (b)
are rendered by an iso-volume of T > 550 K and colored by the gas
temperature. The glowing solid particles in (a) are colored by their
surface temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

7.5 Visualization of the buoyancy-dominated fire in the uniform bed: a)
isometric view (fire spreads from left to right) of; b) back view of the
hot gases only (fire spreads into the page). The hot gases in (a) and
(b) are rendered by an iso-volume of T > 550 K and colored by the
gas temperature. The glowing solid particles in (a) are colored by
their surface temperature. . . . . . . . . . . . . . . . . . . . . . . . . 219

7.6 Visualization of the solid particles colored by their respective burning
state. The unburned particles are colored in green, the burned particles
are colored in black, the flaming particles are colored red, and the
smoldering particles are colored in yellow. . . . . . . . . . . . . . . . 220

xix



7.7 Global fire behavior in the buoyancy-dominated fire predicted using
three grid resolutions of the gas-phase: a) pyrolysis front, b) fire
intensity from homogeneous gas phase combustion, and c) fire intensity
from heterogeneous particle reactions. The particle resolution is fixed
at 25 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

7.8 Global fire behavior in the wind-dominated fire predicted using three
grid resolutions of the gas-phase. See the caption of Fig. 7.7 . . . . . 222

7.9 Global fire behavior in the buoyancy-dominated fire predicted using
three mesh resolutions of the particles. The gas-phase resolution is
10 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

7.10 Behavior of the particle located 20 m downstream of the ignition
burner in the buoyancy-dominated fire in terms of a) the temperature
at the core of the particle; b) the particle mass. The results are
obtained using three mesh resolutions of the particles, and a gas-phase
resolution of 10 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

7.11 Snapshots of the wind-dominated fire while spreading across the mixed-
size patch of 2 and 10 mm particles. The flame is rendered by an
iso-volume of gases at Tg > 550 K. The solid particles are colored by
their surface temperature. . . . . . . . . . . . . . . . . . . . . . . . . 226

7.12 The variation of a) the core temperature and b) the normalized mass
of the lower, middle and upper parts of the 20 mm diameter particle
located at x = 40 m, z = 0 in the mixed-size patch of the wind-
dominated fire. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

7.13 The variation of a) the core temperature and b) the normalized mass
of the lower, middle and upper parts of the 10 mm diameter particle
located at x = 40 m, z = 0 in the mixed-size patch of the wind-
dominated fire. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

7.14 External conditions around the particle evaluated at x = 40 m, y =
0.2 m, z = 0 position in the wind-dominated fire: a) local gas
temperature; b) local irradiation; c) local oxygen mass fraction; d)
local gas velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

7.15 Time variation of the surface and core temperatures of the particles in
the mixed-size patch with 2 mm and 10 mm particles (a, b), and in
the mixed-size patch with the 2 mm and 20 mm particles (c, d). The
data correspond to particles located at x = 40 m, y = 0.2 m, z = 0
in the wind-dominated fire. . . . . . . . . . . . . . . . . . . . . . . . . 232

7.16 Time variation of the normalized mass loss rate of particles in mixed-
size patch of the wind-dominated fire. See caption of Fig. 7.15. . . . . 233

7.17 Time variation of the normalized solid constituents masses of particles
in mixed-size patch of the wind-dominated fire. See caption of Fig. 7.15.234

7.18 Spatial distribution of: a) temperature; b) volume fraction of char
and ash; c) oxygen mas fraction; d) char oxidation reaction rate inside
the 10 mm particle located at x = 40 m, y = 0.2 m, z = 0 in the
mixed-size patch of the wind-dominated fire. . . . . . . . . . . . . . . 236

xx



7.19 Spatial distributions inside the 20 mm particle. See the caption of
Fig. 7.18. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

7.20 Time variation of the normalized solid constituents masses of particles
in the mixed-size patch with 2 mm and 10 mm particles (a, b), and in
the mixed-size patch with the 2 mm and 20 mm particles (c, d). The
data correspond to particles located at x = 40 m, y = 0.2 m, z = 0
in the buoyancy-dominated fire. . . . . . . . . . . . . . . . . . . . . . 239

7.21 Time variation of the a) total heat flux, b) convective heat flux, and c)
radiative heat flux at the surface of the 2 and 10 mm particles located
at x = 40 m, z = 0 in the mixed-size patch of the wind-dominated
fire. The heat fluxes are evaluated at mid-elevation. . . . . . . . . . . 240

7.22 Time variation of the instantaneous values of the PIHF (black solid
line), the moving average (red solid line), and the time integral of
the intense PIHF of the 10 mm particle located at x = 40 m, y =
0.2 m, z = 0 position in the wind-dominated fire. . . . . . . . . . . . 242

7.23 Time variation of the convective and the radiative components of the
PIHF of particles in mixed-size patch of the wind-dominated fire. See
caption of Fig. 7.15. . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

7.24 Time variation of the convective and the radiative components of the
PIHF of particles in mixed-size patch of the buoyancy-dominated fire.
See caption of Fig. 7.20. . . . . . . . . . . . . . . . . . . . . . . . . . 245

7.25 Maps of Rco degree of completion for 2 mm particles (a,b), 10 mm
particles (c,d), and 20 mm particles (e,f). The cross symbol represents
conditions from the buoyancy-dominated fire listed in Table 7.3. The
circles represent similar conditions from the wind-dominated fire. . . . 246

A.1 One-dimensional computational mesh for particles with: (a) rectangu-
lar slab geometry; (b) cylindrical or spherical geometry . . . . . . . . 258

A.2 Thermal boundary condition at the exposed surface of the particle. . 267
A.3 3D rendering of a fuel bed with discrete fuel particles colored by their

corresponding time-step during fire spread. . . . . . . . . . . . . . . . 277

B.1 Predictions of the heat release rate per unit surface area of counter-flow
methane/air diffusion flames under different mixing conditions. . . . . 285

B.2 Profiles of mixture fraction (left) and temperature (right) at the
centerline distance between the fuel and oxidizer nozzles for an under-
resolved flame using EDM and a well-resolved flame using single-step
Arrhenius chemistry. . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

B.3 Scatter of the estimated values of the HRRPUA using a representative
value of χst = 1 s−1 in Eq. B.13 . . . . . . . . . . . . . . . . . . . . 286

D.1 Comparison between the numerical simulations (lines) and the LBL
solution (symbols) of radiation configuration 1. . . . . . . . . . . . . . 294

D.2 Comparison between the numerical simulations (lines) and the LBL
solution (symbols) of radiation configuration 2. . . . . . . . . . . . . . 295

xxi



D.3 Comparison between the numerical simulations (lines) and the LBL
solution (symbols) of radiation configuration 3. . . . . . . . . . . . . . 296

D.4 Comparison between the numerical simulations (lines) and the LBL
solution (symbols) of radiation configuration 4. . . . . . . . . . . . . . 298

E.1 Test of the scaling performance of PBRFoam on the HPC cluster
Expanse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

xxii



Nomenclature

Abbreviations

ABL Atmospheric Boundary Layer
CFD Computational Fluid Dynamics
DL Deep Learning
DSC Differential Scanning Calorimetry
EDM Eddy Dissipation Model
FMC Fuel Moisture Content
GFRR Gaseous Fuel Release Rate
GIS Geographic Information System
HPC High Performance Computing
HRR Heat Release Rate
HRRPUA Heat Release Rate Per Unit Area
LBL Line-by-line
LES Large Eddy Simulation
LSP Laminar Smoke Point
MaCFP Measurement and Computation of Fire Phenomena
MLR Mass Loss Rate
MPI Message Passing Interface
MRTE Multiphase Radiative Transfer Equation
PBR Particle Burning Rate
PDF Probability Density Function
PGRF Prescribed Global Radiant Fraction
PIHF Pseudo Incident Heat Flux
PMC Photon Monte Carlo
QS Quasi-steady
RHS Right Hand Side
ROS Rate of Spread
TGA Thermogravimetric Analysis
UMD University of Maryland
WALE Wall Adapting Local Eddy-viscosity
WSGG Weighted-Sum-of-Gray-Gases

xxiii



Symbols

α heat diffusivity

β packing ratio

χrad radiant fraction

∆H heat of reaction/combustion

∆y+w vertical wall unit

δ size (half-thickness) of rectangular-shaped particles

δik Kronecker symbol

∆LES LES filter size

ω̇′′′ volumetric mass reaction rate of gas-phase species/soot

ḣ′′′th volumetric rate of heat transfer

ṁ′′′ solid-phase volumetric mass reaction rate

ṁ′′ mass flux

q̇′′′comb heat release rate of the gas-phase combustion

q̇′′′rad net radiative power density of the gas-phase

q̇′′ heat flux

Q̇′′
rad radiative heat flux

Q̇′ fire line intensity

ϵ emissivity

ϵTurb subgrid-scale turbulent dissipation rate

η mass yield

γ effective heat conductivity due to pore radiation

κ radiation absorption coefficient

µ dynamic viscosity

ν kinematic viscosity

Ω radiation solid angle

ω inverse of periodic time

ϕ coordinate factor

ψ porosity

ρ density

σ Stefan-Boltzmann constant

σp particle surface area to volume ratio

τ characteristic time-scale/residence time

τw wall shear stress

xxiv



Θ normalized temperature

h̃th thermal enthalpy of the gas

ζ general coordinate

A pre-exponential factor

A⋆ projected surface area of the particle

Ap the area of the particle’s exposed surface

Arect surface area of rectangular particles

Bi Biot number

c heat capacity

CD aerodynamic drag coefficient

CL aerodynamic lift coefficient

cp specific heat at constant pressure

CA shape factor

CFL Courant–Friedrichs–Lewy number

D mass diffusivity

Deff effective diameter

E activation energy

Em radiation emission of the gas-phase mixture

Es radiation emission of the solid-phase

ERco degree of completion of the char oxidation reaction

ERd degree of completion of the drying reaction

ERp+Rop degree of completion of the pyrolysis reactions

f frequency

f−
i , f

+
i flux across cell’s upstream or downstream faces

F ′′′
i aerodynamic force vector component (per unit volume)

fv soot volume fraction

FD,j component of the aerodynamic drag force

FL,j component of the aerodynamic lift force

FO Fourier number

G irradiation

gi gravity vector component in index notation

h heat transfer coefficient

h0f enthalpy of formation

I radiation intensity

xxv



K permeability

k thermal conductivity

kTurb subgrid-scale turbulent kinetic energy

Lcyl length of cylindrical particles

lsp laminar smoke point height

Le Lewis number

m mass

m′′
f fuel loading

n reactant exponent

NC Byram convection number

Np number of particles

NuD Nusselt number based on effective diameter

P presumed probability density function

p pressure

Pr Prandtl number

R radius of cylindrical and spherical particles/ideal gas constant

r radial coordinate

ReD Reynolds number based on effective diameter

S−
i , S

+
i surface area of cell’s upstream or downstream faces

Sc Schmidt number

T temperature

t time

u velocity

uj velocity vector component in index notation

V volume

X volume fraction

x rectilinear coordinate

xj coordinate in index notation

Y mass fraction

y rectilinear coordinate

Z mixture fraction

z rectilinear coordinate

Zst stoichiometric mixture fraction

xxvi



Subscripts

∞ freestream or ambient

a ash

ad adiabatic

bed a control volume of solid vegetation and the gas-phase

C evaluated at the cell center

c char

conv convective

ds dry solid

g gas

g→s gas to solid

hrr heat release rate

i cell or face index

k species/reaction index

L evaluated at the cell’s upstream face

m mixture of gases

p particle

R evaluated at the cell’s downstream face

rad radiative

Rco char oxidation reaction

Rd drying reaction

Rop oxidative pyrolysis reaction

Rp pyrolysis reaction

s solid

sf soot formation

sg solid to gas

so soot oxidation

surf evaluated at the particle surface

tot total

Turb subgrid-scale turbulent quantity

w evaluated at the wall

ws wet solid

xxvii



xxviii



1 Introduction

Wildland fires are a natural phenomenon that can occur in any region of the

world where there is sufficient vegetation (e.g., forest fires, grassland fires, etc.) and

a favorable climate (e.g., hot dry weather and high wind conditions). These fires

can be caused by unintended actions such as lightning strikes and human activities.

Wildland fires also include prescribed burns that intend to clean dense forests and

manage the wildland. Uncontrolled wildland fires can spread rapidly, driven by

strong winds and/or inclined terrains, and can quickly consume large areas of forest,

grassland, or other types of vegetation, and sometimes lead to catastrophic disasters.

The impacts of wildland fire disasters can be severe, including damage to

property, loss of wildlife habitat, the displacement of people from their homes, as well

as economic losses. According to the National Interagency Fire Center (NIFC) [1],

wildfires have increased in occurrence by 223% since 1983, with 68,988 wildfires

that burned 7,577,183 acres in the United States (U.S.). In 2022 alone, 4.5 million

U.S. homes were identified to be at high or extreme risk of wildfire. Moreover,

approximately 4.4 billion U.S. dollars were spent on fire suppression costs in 2021 [1].

In addition to these immediate impacts, wildland fires can also have long-term effects

on the environment, including changes to soil fertility and water quality.

1



The increase in wildland fire risk in recent years is due mainly to: 1) climate

change: the rising temperatures and prolonged drought conditions make it easier

for fires to ignite and spread quickly; 2) improper land management practices: the

accumulation of fuel in many forests makes fires more intense when they occur; and 3)

urbanization: the expansion of urban communities to the wildland increases the risk

of fires caused by human activities as well as exposes communities at the wildland-

urban interface (WUI) to fire risk. As a result of the increasing wildland fire risk,

advanced wildland fire behavior modeling tools are critical to help land managers,

firefighters, and other responders plan and implement strategies to effectively predict,

control, and mitigate wildland fire risks.

This dissertation work is part of a larger research program sponsored by the

U.S. Forest Service that aims at developing an advanced dynamical fire spread model

in replacement of classical models and an operational tool for the real-time prediction

of wildland fires. We describe in the following section the categories of wildland fire

behavior models and how the current work compares to existing models and further

model developments.

1.1 Wildland fire behavior modeling

Wildland fire behavior refers to the physical and chemical processes that govern

the ignition, growth, and spread of wildfires [2]. These processes include the heat

transfer from the flame to the vegetation fuel or from a vegetation particle to another

within the vegetation fuel bed; the production of flammable vapors due to thermal

2



degradation of individual fuel particles (known as pyrolysis); the reaction of these

flammable vapors with ambient air to release heat, produce luminous turbulent

buoyant flames and smoke (known as combustion); the reaction of the charred

particles with ambient oxygen (known as glowing or smoldering); and the interaction

of the turbulent buoyant flames with the surrounding environment, including the

vegetation fuel, topography, and weather. Wildland fire behavior is affected by

various factors, including the characteristics of the fuels, such as fuel material, fuel

moisture content, and fuel arrangement, as well as weather conditions, such as

temperature, humidity, wind speed and direction, and atmospheric turbulence. The

topography of the landscape (e.g., flat or sloped terrain) also affects fire behavior, as

it can significantly influence the rate of spread of the fire and its direction.

Wildland fire behavior research is critical to understand the complex behavior

of wildfires and their interactions with the surrounding environment. The modeling

of wildland fire behavior combines various disciplines, including atmospheric science,

ecology, physics, and computer science, to develop comprehensive models that can

simulate the behavior of wildfires in different environments. These models are

essential for developing effective wildfire management strategies, reducing the risk of

wildfire damage, and protecting the safety of firefighters and local communities.

1.1.1 Scales of wildland fire behavior dynamics

Before reviewing the approaches to model wildland fire behavior, it is important

first to recognize that the dynamics of wildland fires involve multi-physics phenomena

3



occurring at multiple scales and different length scales are believed to play a role

in fire behavior. As shown in Fig. 1.1 the wildland fire behavior can be studied at

different scales: the vegetation scale characterized by the geometry of the biomass

fuel particles; the flame scales represented by a characteristic flame height and width,

as well as the length of the fire line that characterize the combustion and heat transfer

processes; the geographical scales that characterize the terrain topography and land

cover; and the meteorological scales represented by the depth of the atmospheric

boundary layer that characterizes atmospheric conditions.

Vegetation scales 
O(1 mm – 1 cm)

Flame scales O(1 m)

Geographical scales 
O(10s-100s m)

Meteorological scales 
O(1 km)

Figure 1.1: Scales of the wildland fire problem. Images sources: Smithsonian’s

National Zoo & Conservation Biology Institute [3]; Josh Edelson via Getty Images;

Greenpeace International [4]; and NASA Earth Observatory [5].

In wildland fire problems, the vegetation scale is on the order of a few millimeters

or centimeters; the flames scales are the order of a few meters; the geographical scales

4



are typically on the order of a few tens or hundreds of meters; and the atmospheric

boundary layer is on the order of kilometers. We focus in this dissertation on the

wildland fire behavior at the vegetation-to-flame scales.

1.1.2 Semi-empirical approach to wildfire behavior modeling

The first approach to wildland fire behavior modeling uses a semi-empirical

technique that combines empirical observations with simplified mathematical equa-

tions to describe fire behavior based on input data such as weather conditions, terrain,

and fuel load. Details of this approach are outside the scope of this dissertation,

but interested readers are invited to review the description of relevant tools such as

BEHAVE [6], FARSITE [7], ELMFire [8], and FlamMap [9]). In short, these models

rely on a classical empirical model that was proposed by Rothermel in 1972 [10]

to estimate the rate of spread of the fire. These models can also use Geographic

Information System (GIS) to create maps at regional scales that show areas at risk

of wildfire.

There are some limitations and drawbacks to the semi-empirical approach

which makes its application in a variety of scenarios questionable [11]. For example,

The Rothermel model is based on laboratory-scale observations of certain classes

of fuels and may not accurately predict fire behavior under extreme conditions,

such as very high winds, extreme slopes, or unusual fuel structures. Moreover, the

Rothermel model treats the fuel bed as a uniform entity that burns at a constant rate

to complete consumption. However, fire behavior, in reality, can be highly variable

5



depending on the characteristics of the fuel particles and the spatial heterogeneity of

the fuel bed. Therefore, there is a strong interest to replace the Rothermel model

with an advanced dynamical model based on physical principles [12].

1.1.3 Physics-based approach to wildfire behavior modeling

The second approach to wildland fire behavior modeling relies on the phys-

ical description of the governing processes of fluid dynamics, heat transfer, and

combustion. This physics-based approach has potential advantages over the semi-

empirical modeling approach, including reducing model-related errors in predictions

(by contrast to errors introduced from data or users) and explaining rather than

just correlating fire behaviors. Physics-based models are also expected to better

describe phenomena outside the range of current semi-empirical models, such as

spread/no spread thresholds, acceleration/deceleration, fluctuating wind conditions,

discontinuous and heterogeneous fuel beds, and conditions outside current laboratory

and field data sets (very strong winds, large flame dimensions, etc.).

The physics-based modeling has the potential to provide detailed information

on the interactions between physical phenomena occurring at any of the wildland

fire scales by means of Computational Fluid Dynamics (CFD). However, because

of computational cost, the domain of application of 3D CFD-based tools is limited

to a particular range of scales. Thus, current 3D CFD-based wildland fire models

are scale-specific and belong to one of the following three classes (see Fig. 1.2):

combustion solvers aimed at describing the coupling between pyrolysis, combustion,

6



radiation, and flow occurring at the vegetation and flame scales; wildfire solvers

aimed at describing the coupling between combustion and flow occurring at the

fire line scales and/or geographical scales; and atmospheric boundary layer (ABL)

solvers aimed at describing the coupling between combustion and flow occurring at

the meteorological scales.

W
flame

L
flame

L
fireline

L
topography

L
ABL

O(1 km)O(1 m) O(10s-100s m)

ABL solvers:
WRF-SFIRE, WRF-Fire
MESO-NH/ForeFire

Combustion solvers:
FDS, FIRESTAR3D,
OpenFOAM

WildfIre solvers:
FIRETEC, WFDS

L
vegetation

O(1 mm)
O(1 cm)

L
land_cover

CFD

A multi-scale problem

Figure 1.2: The different classes of CFD models used for simulating wildland fire

behavior: combustion solvers resolve dynamics at the vegetation and flame scales;

wildfire solvers resolve dynamics at the fireline and geographical scales; atmospheric

boundary layers (ABL) solvers resolve dynamics at the meteorological scales.

Examples of combustion solvers that have been developed for wildland fire

dynamics applications include a group of models known as multiphase models

[13–16]. These solvers provide a relatively fine-grained treatment of the fuel bed, the

combustion, and the heat transfer processes that are responsible for flame spread.

Simulations with these solvers are typically performed in small domains equivalent to

a field scale. There are, however, simplifications in existing solvers which we would

7



like to improve in our modeling approach. Particularly, we would like to include a

more sophisticated description of the in-depth thermal degradation processes through

a porous medium treatment of the solid fuel. Other examples of combustion solvers

include FDS [17] and FireFoam [18]; these solvers are well-established fire modeling

tools that are primarily used for building fire applications.

Examples of wildfire solvers include FIRETEC (see Refs. [19–21]) and WFDS

(see Ref. [22] and the recent development in Refs. [23,24]). These solvers provide a

coarse-grained treatment of unresolved vegetation-scale and flame-scale processes

through a simplified (but physics-based) combustion model. Simulations with these

solvers are typically performed in relatively larger domains (e.g., 1 km in size).

Examples of atmospheric boundary layer (ABL) solvers that have been de-

veloped for wildland fire dynamics applications include WRF-SFIRE and WRF-

Fire [25–28], and MESO-NH/ForeFire [29]. These ABL solvers typically provide a

macroscopic level treatment of unresolved vegetation-scale, flame-scale, fireline-scale,

and geographical-scale processes through a parameterized rate of spread model based

on the Rothermel model or simplified physical models of surface fire spread (e.g. [30]),

which may have uncertainty in their application to a variety of scenarios. Simulations

with ABL solvers are typically performed in arbitrary-size field-scale domains (from a

few kilometers to several tens of kilometers and beyond). A strength of ABL solvers

is that they are integrated with research-level or operational-level numerical weather

prediction capabilities (i.e. WRF and MESO-NH) and therefore incorporate detailed

descriptions of the fuel maps, topographic maps, and weather conditions.

8



1.2 Motivation and scope of this dissertation

As described in the previous section, the existing operational tools rely on sim-

plified semi-empirical models that are questionable, while there is a major challenge

in the ability to span a wide range of scales with physics-based models due to the

computational burden. Motivated by these challenges, this dissertation work is part

of a larger project sponsored by the Missoula Fire Science Laboratory that aims

to produce a dynamic physics-based dynamical fire spread model as a replacement

for the classical Rothermel model and to deploy this model into an operational

tool for real-time wildland fire simulations using Deep Learning (DL) techniques.

Figure 1.3 shows the chain of model development in this project. As described in

Fig. 1.3, there are three pieces in this project that are connected together. The first

piece is a high-fidelity physics-based 3-D solver called PBRFoam that resolves the

dynamics of wildland fire behavior at vegetation-flame scale using the Large Eddy

Simulations approach (LES). The development of PBRFoam is the main objective of

this dissertation and will be discussed in detail in the following chapters. In short, the

tool PBRFoam supports the development of the second piece by providing simulation

data and detailed diagnostics of the parameters needed to drive the dynamical 1-D

fire spread model. For example, we would like to understand how the local gas

condition around an individual particle inside a fuel bed varies with the packing of

the fuel bed so that a reasonable boundary condition can be set in the dynamic 1-D

model. The second piece is a dynamic 1-D fire spread model called LIHTFire [31].

This model aims at providing burning rates, heat transfer, and ignition with explicit

9



accounting of heterogeneous fuels and time-varying weather. This tool uses the

Particle Burning Rate (PBR) model that we developed throughout this dissertation

work to describe the thermal degradation of individual fuel particles. It accounts for

the coupling between the solid particles and the gas-phase through a set of boundary

conditions imposed at the fuel particle surface given estimates of the conditions of

the gas-phase around the fuel bed. LIHTFire solves a single 1-D fire line spanning

domains of about 50− 100 m in less than a minute, but this is still considered too

slow to be implemented directly in real-time simulations of large fires. This takes us

to the third piece which aims at using advanced Artificial Intelligence techniques to

embed the fire model LIHTFire inside a Deep Learning (DL) model that accounts for

combinations of fuel, weather, and topographical conditions (see Finney et. al. [32]).

Boundary 

conditions
Data

High-resolution 

3-D physics-

based tool 

(PBRFoam)

Dynamical 1-D 

fire spread 

model 

(LIHTFire)

Deep learning 

operational tool

Figure 1.3: The larger framework of this project: chain of model development.

1.3 Literature review of relevant experimental/modeling work in the

vegetation-flame scale

As we are interested in developing a computational tool for detailed modeling

of wildland fire behavior at the vegetation-flame scale, we present here a literature

review of the experimental and modeling efforts that have been carried out to study

wildland fire behavior in this regime.

10



Vogel and Williams [33] found that the rate of flame propagation over horizontal

arrays of vertically oriented matchsticks in quiescent air depends on the spacing,

the length of the matchsticks, and whether the matchsticks are ignited from the

bottom or the top. Wolff et al. [34] investigated fire spread across fuel beds made

of arrays of toothpicks of uniform or mixed sizes in a movable ceiling wind tunnel.

They identified a relationship between the rate of spread of the fire and the ratio

between the wind speed to the fuel loading (i.e., the mass of fuel particles per unit

bed area). They also highlighted some important effects in wind tunnel testing of

fire spread such as the effect of flow circulation in closed ceiling tunnels, the edge

effects, and the inability to obtain an asymptotic one-dimensional fire front. More

importantly, they found the burnout time of individual fuel particles to vary linearly

with the fuel loading and to be independent of the wind speeds considered in their

tests. More recently, Di Cristina et al. [35] identified three regimes of flame spread

across discrete wooden dowels under different particle spacing and wind speeds in

the range of 2.2 to 3.4 m/s: 1) a continuous flame spread regime at a low spacing

in which a flame covers multiple fuel particles; 2) a discrete flame spread regime

at a high spacing in which a flame is only attached to an individual particle; 3) a

quenching regime at large spacing at which a flame could not be sustained. A more

practical technique to experimentally study flame spread in fuel beds comprised of

discrete particles at a larger scale was introduced by Finney et al [36]. In these

experiments, the fire spread over fuel beds made of laser-cut cardboard of different

arrangements was tested in the Missoula wind tunnel at wind speeds up to 1.34 m/s.

Similar to the observations made by Wolff et al. [34], these fire spread experiments

11



showed a linear relationship between the fuel packing ratio and the flame residence

time. Using the same experimental technique, He et al. [37,38] conducted fire spread

experiments in a smaller wind tunnel and at wind speeds between 1 and 3.8 m/s.

They found that the flame residence time increases linearly with the fuel packing

ratio. However, their data suggest that there is also a dependence of the residence

time on wind speed.

On the computational modeling side, the studies concerning the fire behavior

in fuel beds comprised of discrete particles with different fuel bed characteristics

are very limited. For example, Frangieh et al. [39] studied the flame structure of

the cardboard fires that have been studied experimentally at the Missoula Fire

Sciences Laboratory [36]. They were mainly interested in simulating the coherent

flame structure observed in line fires (i.e., the peaks and troughs) using numerical

simulations in a wind tunnel and in an open field. Awad et al. [40] conducted 2-D

simulations to investigate the fuel moisture content (FMC) threshold for spread and

no-spread conditions in grassland fires under different fuel bed loads (i.e., packing)

and wind speeds. They found a correlation between the threshold FMC and the

Byram number.

In summary, there are a number of studies that aimed at characterizing the

burning behavior of discrete fuel particles through experiments of fire spread in

idealized fuel beds of thin discrete particles. However, the behavior of fire spread in

mixed-size class fuel beds has received little to no examination to this point in the

wildland fire literature. Moreover, existing experimental diagnostics and modeling

efforts are very limited and there is a need for more studies concerning the burning

12



behavior in terms of the flame residence time, individual particle response, the degree

of fuel consumption, the dependence on fuel bed characteristics such as fuel particle

size and packing, the respective weights of radiative and convective heating, and the

modes of flaming and smoldering combustion.

1.4 Objectives and contributions

1.4.1 Objectives

The general objective of this project is to develop an advanced computational

modeling capability that simulates wildland fire behavior at vegetation and flame

scales with detailed descriptions of thermal degradation processes occurring during

both flaming and smoldering combustion. Our main focus here is on the development,

validation as well as testing of the computational modeling capability in canonical

configurations. As described in the previous section, this modeling capability will

serve in two ways: 1) providing data for the development of operational tools, and

2) developing our fundamental understanding of the wildland fire behavior through

setting the basis for parametric studies and sensitivity analyses.

Our specific objectives in this dissertation are to:

1. develop a particle burning rate (PBR) model that describes the thermal

degradation of individual solid biomass/vegetation fuel particles and implement

it as an object-oriented C++ tool

2. couple the model into a multiphase solver using an open-source library called

13



OpenFOAM [41] which allows the utilization of advanced models previously

developed in our group at the University of Maryland, as well as an easy

implementation of new models

3. validate the model against available experimental measurements and observa-

tions

4. simulate flame spread across surrogate vegetation beds comprised of mono-

/poly-dispersed particles with prescribed particle and environmental properties

(i.e., bed height, surface-to-volume ratio, packing ratio, moisture content, and

wind velocity)

5. study the global fire behavior in terms of flame structure, fire intensity, global

fuel consumption, and the individual particle response as a function of the

fuel packing and the duration of the thermal loading process, and the flame

residence time.

On the route to achieve these objectives, there are important intermediate steps

discussed in the following chapters. These steps focus primarily on evaluating the

ability of the models adopted and developed in this work to accurately describe two

main drivers of fire spread: 1) pyrolysis and oxidation of isolated biomass particles

at both micro- and bench-scales; 2) heat feedback in canonical fire flames in terms

of thermal radiation and flame structure.

14



1.4.2 Author’s contributions

The present work is part of a project sponsored by the U.S. Forest Service that

aims at two objectives: 1) developing a dynamical 1-D fire spread model, and 2)

studying the wildland fire behavior dynamics through high-fidelity 3-D simulations

of spreading line fires in vegetation fuel beds. Past work conducted at the University

of Maryland by Dr. Salman Verma [42] on high-fidelity 3-D simulations was focused

on studying the structure of non-spreading laboratory-scale line fires produced

from stationary gaseous burners in the presence of external wind or on inclined

surfaces. On the other hand, the previous version of the dynamical 1-D fire spread

model developed by Mr. Jason Forthofer at the Missoula Fire Sciences Laboratory

included a particle burning rate (PBR) model that describes only the drying and

the pyrolysis of thermally-thick non-porous particles, and it did not account for

smoldering combustion.

The present study extends the previous work by developing a model of thermal

degradation of porous biomass fuel particles, which includes a description of both

flaming and smoldering combustion, that can be implemented in the 1-D dynamical

fire spread model as well as in the high-fidelity 3-D simulations of spreading fires.

Several implementations and code development were conducted during this

Ph.D. work. The candidate started first by developing an early MATLAB version

of the PBR model that was originally created by Dr. Arnaud Trouvé to describe

the drying and pyrolysis of thermally-thin as well as thermally-thick rectangular-

shaped particles. The candidate extended this MATLAB code to account for the

15



thermal degradation of 1-D axisymmetric cylindrical- and spherical-shaped particles.

The candidate then used this MATLAB code as a stand-alone solver to simulate

the unsteady response of charring and non-charring biomass particles (in terms of

ignition, mass loss rate, and burnout) when exposed to conditions of the surrounding

gas that are fluctuating in time (i.e., surrounding gas temperature, velocity, and

irradiance) [43].

Following the work of Lautenberger and Fernandez-Pello [44], the MATLAB

version of the PBR model was then further developed by Dr. Arnaud Trouvé and

the candidate to account for smoldering combustion by treating the particles as a

porous-medium that includes in-depth oxygen diffusion. The candidate used this

new solver to generate 2-D maps that describe the burning behavior of individual

particles under prescribed thermal loads (Chapter 5).

The candidate created an object-oriented C++ code as stand-alone software

for the PBR model of porous biomass particles. The candidate also assisted with

the coupling of the C++ PBR code with the dynamical 1-D fire spread model

developed by Mr. Jason Forthofer at the Missoula Fire Science Laboratory. The

candidate is also the developer of the 3-D multi-phase fluid dynamics solver PBRFoam

that is based on the PBR code and the multi-purpose open-source C++ library

OpenFOAM [41] (Chapter 2 and Appendix A). The candidate constructed a two-way

coupling interface between the Lagrangian particle tracking library of OpenFOAM

and the newly developed PBR solver. This interface exchanges information that

allows the solid particles to evolve under the given gas-phase conditions and in

turn modify the gas-phase by exchanging information about the contribution of the

16



solid-phase to the mass, momentum, energy, and radiation of the gas-phase. To

construct the gas-phase solution algorithm in the PBRFoam solver, the candidate

adopted a similar approach to that has been adopted in the well-established fire

modeling tool FireFoam developed at FM Global [18], particularly, the description

of the pressure-velocity coupling using the PIMPLE algorithm [45].

Additionally, the candidate conducted intermediate model developments and

code implementations using the OpenFOAM library toward more accurate fire

modeling as part of the Measurement and Computation of Fire Phenomena (MaCFP)

working group. These efforts aimed at evaluating the accuracy of different fire

modeling approaches in predicting thermal feedback in non-sooting and sooting

buoyant turbulent diffusion flames. The candidate developed a modified version of

the Eddy Dissipation Model (originally proposed by Magnussen and Hjertag [46])

using laminar diffusion theory to dynamically estimate the rate of fuel consumption

based on laminar diffusion in fire regimes where laminar diffusion is dominating

over turbulent mixing such as near the flame base (see Appendix B). The candidate

also developed and implemented an extension of the Laminar Smoke Point (LSP)

soot model (originally proposed by Yao et al. [47]) to account for subgrid-scale

turbulent fluctuations in LES description of soot formation and oxidation rates using

a probability density function called β−PDF (Appendix C). Furthermore, based

on an original code implementation of a Weighted-Sum-of-Gray-Gases (WSGG)

framework implemented in the OpenFOAM library by Sikic et al. [48], the candidate

implemented a new version that uses a WSGG model proposed by Cassol et al. [49]

that accounts for soot radiation (Appendix D).

17



In terms of model verification and calibration, the candidate conducted a series

of benchmark tests to evaluate the accuracy of the implemented models and to check

the establishment of a reliable computational framework (Chapter 3).

As part of the MaCFP workshop, the candidate performed Large Eddy Simula-

tions (LES) using FireFoam solver in two configurations that have been identified as

target experiments by the MaCFP working group for validation of fire models. The

emphasis in these simulations was on evaluating the ability of the EDM combustion

model, the Prescribed Global Radiant Fraction model (PGRF), and the WSGG

radiation modeling approach to predict thermal feedback in, first, a non-sooting

methanol pool fire, then in a sooting ethylene burner (Chapter 4).

Using the newly developed modeling capability (PBRFoam), the candidate

performed LES simulations of fire spread in fuel beds made of vertically-oriented

engineered cardboard sticks that have been studied experimentally at the Missoula

Fire Sciences Laboratory [36] (Chapter 6). The candidate created a C++ script to

generate the population of the cardboard particles inside the fuel beds using the

given information about the separation distances between arrays of particles. The

candidate constructed 3-D configurations of two experimental burns called Burn 53

and Burn 67, as well as 7 configurations corresponding to fuel beds with different

particle separation and fuel bed packing. The emphasis in these simulations was

on validating the model by comparing the simulated fire behavior in terms of ROS,

flame structure, and residence time with available experimental measurements and

observations.

The candidate also performed LES simulations of fire spread in surrogate fuel

18



beds comprised of cylindrical-shaped pine wood particles using PBRFoam. The

candidate constructed configurations that have either a uniform fuel bed of the same-

size particles or a fuel bed that features a patch of mixed-size particles (Chapter 7).

The focus of this study was on studying the degree of consumption of the particles,

the respective weights of convective and radiative heating, and the transition from

flaming to smoldering combustion.

1.4.3 Organization

In chapter 2, the computational framework is described in detail. The governing

equations for the thermal degradation of porous biomass particles are first introduced,

followed by a description of the multi-phase formulation that we adopted in our

implementation to describe the coupling between the solid biomass particles and

the gas-phase. The chapter also provides details of the physical sub-models adopted

to describe the convective heat transfer at the surface of the solid particles, the

multi-phase radiation transport, the gas-phase combustion, as well as the sub-grid

scale turbulence. This chapter is concluded with a summary of the implementation

of the computational framework.

In chapter 3, a series of benchmark test cases are presented to verify, calibrate

and evaluate the computational modeling framework. This chapter first discusses

verification test cases through simplified configurations where analytical solutions

can be derived at both the particle level as well as the fuel bed level. Then,

the calibration of the thermal degradation model with micro-scale experiments is

19



discussed. A discussion about the ability of the stand-alone PBR solver to predict

thermal degradation of bench-scale experiments is presented, followed by a discussion

for representative cases of particles that feature complete consumption (i.e. through

smoldering) and particles that feature volume-change (i.e., shrinking or swelling).

Chapter 4 presents numerical studies of flame structure and thermal feedback

in two canonical pool fire configurations. This chapter first presents LES results

from a medium-scale pool fire that is fueled by a non-sooting ethylene fuel, followed

by LES results from a turbulent buoyant flame produced from a circular burner

(called the FM-burner) that is fueled by sooting ethylene fuel under controlled coflow

conditions.

In chapter 5, a numerical study of the response of the biomass vegetation

particles to external heating conditions imposed at their exposed surface is presented.

The chapter discusses first the response of a selection of particles, including thin,

thick, charring, and/or non-charring particles, in terms of ignition and mass loss

rate due to pyrolysis to either steady or oscillatory external gas conditions. Then

the chapter presents 2-D maps that characterize the thermal degradation of porous

flaming and smoldering particles in terms of the degree of completion of the drying,

the pyrolysis, and the char oxidation processes as a function of the external thermal

load and the ambient wind.

Chapter 6 focuses on LES simulation results of fire spread in engineered fuel

beds made of cardboard sticks that have been studied experimentally at the Missoula

Fire Sciences Laboratory. The chapter first discusses the model predictions of the

fire behavior against experimental observations of two burns corresponding to a

20



buoyancy-dominated or a wind-dominated fire regime. The rest of the chapter focuses

on fire diagnostics related to the flame residence time and the rate of spread and

their variation with the packing of the fuel bed.

Chapter 7 presents LES simulation results of fire spread in idealized fuel beds

comprised of vertically-oriented cylindrical-shaped pinewood sticks under two wind

speeds corresponding to either a buoyancy-dominated fire or a wind-dominated fire.

The chapter discusses first the fire propagation in uniform fuel beds comprised of

small particles that are on the order of 1 mm-scale. Then, results from fire spread

simulations in fuel beds that feature a patch with larger particles (on the order of

1 cm-scale) that are added to the smaller ones. The last part of the chapter focuses

on an interpretation of the LES results regarding the degree of completion of thermal

degradation of the fuel bed using the 2-D maps constructed in chapter 5.

Chapter 8 presents a summary of the work conducted in this dissertation and

provides some concluding remarks. The chapter also discusses recommendations and

suggestions for a continuation of this work in the future.

21



2 The Computational Modeling Framework

2.1 Overview

We model the fuel bed as a population of porous particles with different

arrangements and with either unique or mixed sizes and geometries. Each porous

particle is described as a system with a solid-phase and a gas-phase, which allows

for a detailed treatment of the particle-to-external-gas outflow of volatile mass and

the external-gas-to-particle diffusion of oxygen mass; this detailed treatment is

required to account for in-depth oxidative pyrolysis and char oxidation. We consider

here thermally-thick and composition-thick particles featuring in-depth variations of

temperature and composition. The particle’s geometry can be modeled as rectangular

(leaves or sticks), cylindrical (needles or stems), or spherical (embers or firebrands).

This chapter discusses the development and implementation of a Lagrangian

Particle Burning Rate (PBR) solver to describe the thermal degradation of individual

solid fuel particles, and a coupled multiphase solver to describe the interaction between

a solid vegetation fuel bed and the surrounding gases in simulations of wildland fire

spread at flame scale.

In terms of exchange of information, the PBR solver uses estimates of the

22



gas-to-solid heat flux onto a population of particles as input quantities and produces

in return estimates of the particle mass, size, and energy as well as the production

rate of combustible gaseous volatiles and other gaseous products inside the fuel bed

as output quantities to the coupled multiphase solver. More precisely, the input

quantities to the PBR model are:

q̇′′g→s = h(Tg,∞ − Tp,surf ) + ϵp,surf (G− σT 4
p,surf ) (2.1)

and

ṁ′′
g→s,O2

= hmass(Yg,O2,∞ − Yg,O2,surf ) (2.2)

where h is the convective heat transfer coefficient, Tg,∞ the external gas temperature

in the vicinity of the particle under consideration, Tp,surf the temperature of the

exposed surface of that particle, ϵp,surf the particle surface emissivity (treated as an

opaque solid), G the averaged radiation heat flux incident on the particle external

surface (G designates the irradiation due to distant hot sources, e.g., the flame,

the plume, other particles, etc., as well as the irradiation due to ambient gas), σ

the Stefan-Boltzmann constant, hmass is the convective mass transfer coefficient

hmass = (h/c̄p,g) with c̄p,g the heat capacity of the gas at constant pressure, where

Yg,O2,∞ is the mass fraction of oxygen in the external gas in the vicinity of the particle,

and where Yg,O2,surf is the mass fraction of oxygen gas at the exposed surface of the

particles.

In the expression for q̇′′g→s, the input quantities are the external gas temperature

Tg,∞, the convective heat transfer coefficient h (treated through Nusselt-number-

correlations as a function of the external flow velocity and temperature, ug,∞ and

23



Tg,∞ and the irradiation G. In the expression for ṁ′′
g→s,O2

, the input quantity is

the external oxygen mass fraction in the gas in the vicinity of the particle Yg,O2,∞.

These quantities are functions of spatial location inside the fuel bed, i.e., presumably

functions of both the distance along the direction of fire spread and the vertical

elevation. We refer to these quantities as "external gas conditions" and we describe

their variations using the coupled multiphase formulation described in section 2.3.

2.2 Particle Burning Rate (PBR) Model

Thermal degradation of individual solid fuel particles exposed to the external gas

conditions is tracked individually in space and time following a Lagrangian viewpoint.

In a one-way coupled formulation, the external gas conditions are imposed, and the

model calculates the time-dependent response of the fuel particle to the heat loading

from the gaseous environment. In a two-way coupled formulation, the external

gas conditions (i.e., composition, velocity, temperature, and radiation) are updated

with terms representing the effects of the fuel particles. The description of the fuel

particle model presented in this section applies to both one-way and two-way coupled

formulations.

We follow the work of Lautenberger and Fernandez-Pello [44, 50] by describing

the fuel particle as a matrix of pores featuring a solid-phase and a gas-phase. We

consider each porous particle to experience four heterogeneous (gas/solid) reactions:

a drying reaction (Rd); a thermal pyrolysis reaction (Rp); an oxidative pyrolysis

reaction (Rop); and a char oxidation reaction (Rco):

24



[wet solid] −→ ηH2O,Rd [H2O] + ηds,Rd [dry solid] (Rd)

[dry solid] −→ ηv,Rp [fuel1] + ηc,Rp [char] (Rp)

[dry solid] + ηO2,Rop O2 −→ ηv,Rop [fuel2] + ηc,Rop [char] (Rop)

[char] + ηO2,Rco O2 −→ ηCO2,Rco CO2 + ηa,Rco [ash] (Rco)

where ηH2O,Rd and ηds,Rd are the mass yields of water vapor and dry solid in

reaction Rd, respectively, ηH2O,Rd + ηds,Rd = 1, where ηv,Rp and ηc,Rp are the mass

yields of volatile and char in reaction Rp, respectively, ηv,Rp+ηc,Rp = 1, where ηO2,Rop

is the oxygen-to-dry-solid mass ratio, and ηv,Rop and ηc,Rop are the mass yields of

volatile and char in reaction Rop, respectively, ηv,Rop + ηc,Rop = 1 + ηO2,Rop, and

where ηO2,Rco is the oxygen-to-char mass ratio, and ηCO2,Rco and ηa,Rco are the mass

yields of CO2 and ash in reaction Rco, respectively, ηCO2,Rco + ηa,Rco = 1 + ηO2,Rco.

Note that these reactions are written per unit kg of the reactant.

Consistent with the model proposed in Ref. [44], we do not differentiate between

the char products of the thermal and oxidative reactions Rp and Rop. While the

gaseous fuel produced from Rp and Rop may have different chemi