Understanding Soot Particle Growth Chemistry and Particle Sizing Using a Novel Soot Growth and Formation

Model

by

Armin Veshkini

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy

Department of Mechanical and Industrial Engineering University of Toronto

© Copyright by Armin Veshkini, 2015

ii

Understanding Soot Particle Growth Chemistry and Particle

Sizing Using a Novel Soot Growth and Formation Model

Armin Veshkini

Doctor of Philosophy

Department of Mechanical and Industrial Engineering University of Toronto

2015

Abstract

Research efforts are focused on advancing the understanding of soot modeling by computing

soot formation in laminar flames using a detailed sectional aerosol dynamic model. Toward an

end goal of developing a robust model of soot formation applicable to a wide range of

conditions, soot coalescence models are introduced, a correlation for the surface reactivity is

proposed, PAH contributions to soot formation in premixed and nonpremixed flames are

investigated, and a condensation efficiency model is developed and validated.

The effects of the soot coalescence process on soot particle diameter predictions are studied. Two

coalescence models based on different merging mechanisms are implemented into the soot

model. The models are applied to a laminar ethylene/air diffusion flame, and comparisons are

made with experimental data to validate the models. The implementation of coalescence models

significantly improves the agreement of prediction of particle diameters with the experimental

data.

A comprehensive study follows in which a function for surface reactivity of soot particles is

developed to eliminate tunable constants, and have a single model able to predict soot in many

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coflow ethylene/air flames. This study investigates how the surface reactivity of soot particles

varies with particle thermal age. The surface reactivity function is applied to coflow diffusion

flames with varying fuel/air ratios and fuel dilution, and to partially premixed coflow flames for

a range of equivalence ratios. Comparisons are made with experimental data to validate the

model. Very good agreement is seen between numerical predictions and experimental

measurements for soot volume fraction on the annular regions of the flames.

The final part of this thesis explores the role of PAH-soot modeling on burner stabilized

stagnation premixed flames and a coflow diffusion flame. Two chemical mechanisms are

employed to model both flames. It is found that one of the mechanisms gives more accurate

description of the PAH chemistry in premixed flames while the other improves the agreement of

soot predictions in diffusion flames and the results and conclusion are drastically effected by the

choice of PAH mechanism. An equilibrium based condensation efficiency model is developed

and combined with a reversible nucleation model to predict soot formation in both premixed and

nonpremixed flames. Compared to the measured data, soot PSDs are reasonably well predicted.

Effects of different soot formation processes on PSD predictions are characterized. In the

diffusion flame, soot predictions with the developed soot model are comparable with the

previous soot model predictions. However, employing a reversible nucleation model leads to a

delay in onset of soot formation in the diffusion flame.

iv

Acknowledgments

I would like to express my deepest gratitude to my supervisors, Professor Murray J. Thomson

and Professor Seth B. Dworkin for their constant guidance and support through my studies at the

University of Toronto. Thank you both for being a wise guide, a passionate leader, and a good

friend.

Much appreciation to Professor Ömer L. Gülder and Professor Markus Bussmann for being

members on my PhD supervisory committee. I also thank Professor James S. Wallace for serving

on my examination committee. It was an honour to have Professor Andrea D’Anna from the

University of Naples Federico II serving as my external examiner. His insights into my work

were invaluable.

I would also like to thank Dr. Nadezhda Slavinskaya and Professor Uwe Riedel of the German

Aerospace Center (DLR) for providing the chemical reaction mechanism, thermodynamic data,

and transport data for ethylene combustion and PAH formation.

Gratefulness to all my colleagues in the Combustion Research Group, specifically,

Mohammadreza Kholghy, Babak Borshanpour, Amir Alikhanzadeh, Kaveh Khalilian, Milad

Zarghami, Sina Moloodi, Dr. Tommy Tzanetakis, and Dr. Victor Chernov. I also owe a debt of

gratitude to Dr. Meghdad Saffaripour for his friendship through many years of collaboration. My

contemporary, Nick Eaves, deserves a special recognition for sharing his knowledge and fruitful

discussions.

Lastly, I owe my gratitude to my parents who are unwavering sources of encouragement and

support. I also want to express my gratitude to Leila whose love and support have made this

thesis possible.

Computations were performed on the Ryerson University Sandy Bridge computing cluster and

the GPC supercomputer at the SciNet HPC Consortium. SciNet is funded by: the Canada

Foundation for Innovation under the auspices of Compute Canada; the Government of Ontario;

Ontario Research Fund – Research Excellence; and the University of Toronto.

v

Table of Contents

Acknowledgments ........................................................................................................................................ iv

Table of Contents .......................................................................................................................................... v

List of Tables ................................................................................................................................................. ix

List of Figures ................................................................................................................................................. x

List of Appendices ....................................................................................................................................... xix

Chapter 1 Introduction .................................................................................................................................. 1

1.1 Motivation .......................................................................................................................................... 1

1.2 Literature Review ............................................................................................................................... 3

1.2.1 Soot Characteristics ..................................................................................................................... 3

1.2.2 Soot Formation Pathways ........................................................................................................... 7

1.2.3 Soot Modeling ........................................................................................................................... 11

1.3 Objectives and Outline of Subsequent Chapters ............................................................................. 13

Chapter 2 Mathematical Model .................................................................................................................. 16

2.1 Overview .......................................................................................................................................... 16

2.2 Gas-Phase Governing Equations ...................................................................................................... 16

2.2.1 Conservations of Mass and Momentum ................................................................................... 17

2.2.1.1 The Two-Dimensional Cylindrical Coordinates.................................................................. 17

2.2.1.2 The One-Dimensional Similarity Solution .......................................................................... 18

2.2.2 Conservation of Energy ............................................................................................................. 20

2.2.2.1 Radiation Heat Transfer .................................................................................................... 21

Optically thin approximation (OTA) ........................................................................................... 21

Discrete-ordinate method (DOM) .............................................................................................. 23

2.2.3 Conservation of Species Mass ................................................................................................... 24

2.2.3.1 Chemical mechanism ......................................................................................................... 24

DLR mechanism .......................................................................................................................... 25

KAUST mechanism ..................................................................................................................... 26

2.3 Soot Aerosol Dynamics Model ......................................................................................................... 28

2.3.1 The sectional aerosol dynamics model ..................................................................................... 29

2.3.1.1 Nucleation model .............................................................................................................. 31

vi

2.3.1.2 Condensation model ......................................................................................................... 32

2.3.1.3 Chemical surface growth and oxidation models ............................................................... 33

2.3.1.4 Coagulation model ............................................................................................................ 35

2.3.1.5 Fragmentation model ........................................................................................................ 36

2.4 Transport Properties ........................................................................................................................ 37

2.4.1 Diffusion coefficients ................................................................................................................. 38

2.5 Numerical Methods ......................................................................................................................... 40

2.5.1 2D coflow diffusion flame .......................................................................................................... 40

2.5.1.1 Boundary conditions ......................................................................................................... 43

2.5.2 Premixed stagnation flame ........................................................................................................ 43

2.5.2.1 Boundary conditions ......................................................................................................... 45

Chapter 3 Soot Particle Coalescence ........................................................................................................... 47

3.1 Overview .......................................................................................................................................... 47

3.2 Introduction ..................................................................................................................................... 47

3.2.1 The Collision-Coalescence Mechanism ..................................................................................... 48

3.3 Rate of Coalescence ......................................................................................................................... 50

3.3.1 Viscous Flow Transport .............................................................................................................. 51

3.3.2 Transport by Diffusion ............................................................................................................... 51

3.4 Coalescence Model .......................................................................................................................... 52

3.4.1 Cut-off Model (Model I) ............................................................................................................. 53

3.4.2 Sintering Model (Model II) ........................................................................................................ 54

3.5 Methodology .................................................................................................................................... 55

3.5.1 Numerical Model ....................................................................................................................... 56

3.6 Results and Discussion ..................................................................................................................... 57

3.6.1 Annular Pathline Comparison .................................................................................................... 58

3.6.2 Centerline Comparison .............................................................................................................. 64

3.6.3 Sensitivity analysis ..................................................................................................................... 67

3.6.3.1 Cut-off Diameter ............................................................................................................... 68

3.6.3.2 Coalescence Characteristic Time ....................................................................................... 69

3.6.3.3 Coalescence and Oxidation ............................................................................................... 71

3.7 Conclusions ...................................................................................................................................... 73

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Chapter 4 Soot Particle Surface Reactivity .................................................................................................. 75

4.1 Overview .......................................................................................................................................... 75

4.2 Introduction ..................................................................................................................................... 75

4.3 Numerical Model ............................................................................................................................. 80

4.4 Methodology .................................................................................................................................... 81

4.4.1 Soot Surface Reactivity .............................................................................................................. 81

4.4.2 Thermal Age ............................................................................................................................... 84

4.5 Results and Discussion ..................................................................................................................... 87

4.5.1 Surface Reactivity Analysis ........................................................................................................ 89

4.5.2 Parameter Study ........................................................................................................................ 92

4.5.2.1 Gas phase chemistry parameters ...................................................................................... 93

4.5.2.2 Soot model parameters ..................................................................................................... 96

4.6 Conclusions ...................................................................................................................................... 97

Chapter 5 Reversibility of Nucleation and Condensation ........................................................................... 98

5.1 Introduction ..................................................................................................................................... 98

5.2 Methodology ..................................................................................................................................102

5.2.1 Burner and Flame Description .................................................................................................102

5.2.2 Model Description ...................................................................................................................104

5.2.2.1 Sectional aerosol dynamic model ...................................................................................104

5.2.2.2 Reversible nucleation ......................................................................................................104

5.2.2.3 Condensation Efficiency ..................................................................................................108

5.2.2.4 Soot models .....................................................................................................................111

5.3 Results and Discussion ...................................................................................................................112

5.3.1 PAH Chemistry .........................................................................................................................113

5.3.2 Reversible Nucleation Model ..................................................................................................120

5.3.3 Condensation Efficiency ..........................................................................................................129

5.3.3.1 Sensitivity analysis ...........................................................................................................131

5.3.4 Diffusion Flames ......................................................................................................................133

5.4 Conclusions ....................................................................................................................................141

Chapter 6 Conclusions and Future Work ...................................................................................................143

6.1 Summary and Conclusions .............................................................................................................144

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6.2 Original contributions ....................................................................................................................147

6.3 Recommendations for future work ...............................................................................................149

Appendices ................................................................................................................................................153

Bibliography ...............................................................................................................................................163

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List of Tables

Table 2.1 HACA–based soot surface growth and oxidation reactions [86], 𝑘 = 𝐴𝑇𝑏𝑒 −𝐸𝑎𝑅𝑇 . .................................................................................................................... 34

Table 4.1 HACA–based soot surface growth and oxidation reactions [86], 𝑘 = 𝐴𝑇𝑏𝑒 −𝐸𝑎𝑅𝑇 . .................................................................................................................... 77

Table 4.2 Proposed functional forms of 𝛼 for models based on the HACA mechanism. ....... 80

Table 4.3 Proposed functional forms of 𝛼 for models based on the HACA mechanism. ....... 82

Table 4.4 Flames used to derive a function for surface reactivity and the optimized 𝛼 for each flame that reproduces the most accurate soot concentration on the wings. ........... 84

Table 5.1 Difference between nucleation and condensation models used to simulate flames .............................................................................................................................. 112

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List of Figures

Figure 1.1 TEM images of soot particle samples along the centerline of a coflow diffusion flame of a surrogate for Jet A-1 at different heights above the fuel tube exit (Source: Reprinted from ref. [35]). .......................................................................... 4

Figure 1.2 TEM image of soot sample formed from ethylene pyrolysis in a flow reactor at 1475 K in the presence of nitrogen oxides (specifically N2O); (Source: Reprinted from ref. [72]). .......................................................................................................... 5

Figure 1.3 Schematic diagram of soot formation (Source: Reprinted from ref. [89]). .............. 8

Figure 2.1 Schematic representation of a coflow flame, including coordinate orientation and computational domain (not drawn to scale). .......................................................... 18

Figure 2.2 Schematic representation of a burner stabilized stagnation flame, including coordinate orientation. ............................................................................................ 19

Figure 2.3 Schematic representation of the major reaction pathways for the formation of large PAHs considered by the DLR chemical kinetic mechanism. ................................. 26

Figure 2.4 Schematic representation of the major reaction pathways for the formation of large PAHs considered by the KAUST chemical kinetic mechanism. ........................... 27

Figure 2.5 Processes shaping the particle size distribution function in a small volume element of gas. Diffusion and sedimentation involve transport across the walls of the element. Coagulation, nucleation, and growth take place within the element. (Source: Reprinted from ref. [106]) ....................................................................... 28

Figure 2.6 Illustration of armchair sites on the surface of a soot particle. ............................... 33

Figure 2.7 Coflow code solver program structure. ................................................................... 42

Figure 2.8 Schematic of the coflow diffusion flame boundary conditions and the non-uniform structured mesh. ...................................................................................................... 43

Figure 3.1 Schematic of coalescence process of two colliding particles. ................................ 48

Figure 3.2 TEM images of soot particle samples along the centerline of a coflow diffusion flame of a surrogate for Jet A-1 at different heights above the fuel tube exit (Source: Reprinted from ref. [35]). ........................................................................ 49

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Figure 3.3 Schematic representation of aggregate formation with cut-off coalescence. ......... 54

Figure 3.4 Schematic representation of the sintering model for soot particle coalescence. ..... 55

Figure 3.5 Schematic representation of burner configuration of Santoro flame [58]. [Courtesy of Dr. Meghdad Saffaripour, University of Toronto.] ............................................ 56

Figure 3.6 Comparison of the predicted soot volume fraction along the annular pathline exhibiting the maximum soot volume fraction using a sintering coalescence model (solid line), the cut-off coalescence model (dashed line) and no coalescence (dot-dashed line) with the experimental measurements by [58]. ................................... 59

Figure 3.7 Comparison of the predicted soot volume fraction along the annular pathline exhibiting the maximum soot volume fraction using a sintering coalescence model (solid line), cut-off coalescence model (dashed line) and no-coalescence (dot-dashed line) with the experimental measurements by [192]. ................................. 59

Figure 3.8 Comparison of the predicted average primary particle diameter along the annular pathline exhibiting the maximum soot volume fraction, using a sintering coalescence model (solid line), cut-off coalescence model (dashed line) and no coalescence (dot-dashed line) with the experimental measurements by [56]. ....... 60

Figure 3.9 Comparison of the predicted primary particle number density along the annular pathline exhibiting the maximum soot volume fraction using a sintering coalescence model (solid line), cut-off coalescence model (dashed line) and no coalescence (dot-dashed line) with the experimental measurements by [39,57]. .. 61

Figure 3.10 Comparison of the predicted aggregate number density along the annular pathline exhibiting the maximum soot volume fraction using a sintering coalescence model (solid line), cut-off coalescence model (dashed line) and no coalescence (dot-dashed line) with the experimental measurements by [57,192]. ............................ 63

Figure 3.11 Comparison of the predicted average number of primary particles per aggregate along the annular pathline exhibiting the maximum soot volume fraction using a sintering coalescence model (solid line), cut-off coalescence model (dashed line) and no coalescence (dot-dashed line) with the experimental measurements by [57,192]. ................................................................................................................. 63

Figure 3.12 Comparison of the predicted average primary particle diameter along the centerline using a sintering coalescence model (solid line), cut-off coalescence model (dashed line) and no coalescence (dot-dashed line) with the experimental measurements by [37]. ........................................................................................................................ 65

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Figure 3.13 Comparison of the predicted soot volume fraction along the centerline using a sintering coalescence model (solid line), cut-off coalescence model (dashed line) and no coalescence (dot-dashed line) with the experimental measurements by [37,38,58] (a log scale is used so that comparisons can be made at heights less than 4 cm). ...................................................................................................................... 66

Figure 3.14 Variation of surface to volume ratio along the centerline using a sintering coalescence model (solid line), cut-off coalescence model (dashed line) and no coalescence (dot-dashed line). ................................................................................ 67

Figure 3.15 Comparison of the predicted average primary particle diameter using different cut-off diameter coalescence models a) along the annular pathline exhibiting the maximum soot volume fraction with the experimental measurements by [56] and b) along the centerline with the experimental measurements by [37]. ................... 68

Figure 3.16 Variation of the characteristic coalescence time of a 10 nm soot particle with temperature with four different activation energies. .............................................. 70

Figure 3.17 Comparison of the predicted average primary particle diameter along the annular pathline exhibiting the maximum soot volume fraction using a) different activation energy and b) different pre-exponential factor for the sintering coalescence model with the experimental measurements by [56]. ........................................................ 70

Figure 3.18 Effect of reduction of characteristic time on the predicted maximum primary particle diameter along the annular pathline exhibiting the maximum soot volume fraction. ................................................................................................................... 71

Figure 3.19 Computational isotherms (left panel) and isopleths of O2 mole fraction (right panel) in the Santoro coflow diffusion flame. ................................................................... 72

Figure 3.20 Comparison of the predicted average primary particle diameter along the annular pathline exhibiting the maximum soot volume fraction using different sintering coalescence models with an oxidation cut-off, and the experimental measurements by [56]. ................................................................................................................... 73

Figure 4.1 Illustration of armchair sites on the surface of a soot particle. ............................... 76

Figure 4.2 Total mass yield (𝑔𝑠𝑜𝑜𝑡/𝑔𝑚𝑖𝑥) by all soot growth processes, HACA surface growth, and inception plus PAH condensation for a soot particle travelling a) along the centerline and b) along the pathline of maximum soot on the wings, for the Santoro flame [58] (SA). ..................................................................................................... 82

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Figure 4.3 Comparison of computed peak soot volume fractions on the wings using 𝛼 = 0.45 for all SM and SA flames with experimental data from [192] and [41] for coflow diffusion ethylene-air flames. ................................................................................. 83

Figure 4.4 Comparison of computed peak soot volume fractions on the wings using an optimized average 𝛼 for each flame (The value of 𝛼 for each flame is shown below the computed result) with experimental data from [192], [41] and [212] for coflow diffusion ethylene-air flames. ................................................................................. 84

Figure 4.5 Average soot particle surface reactivity, 𝛼, as a function of a) peak flame temperature and b) instantaneous temperature at the peak soot concentration on the wings. ..................................................................................................................... 85

Figure 4.6 a) Average soot particle surface reactivity, 𝛼, as a function of thermal age at the location of peak soot concentration on the wings (the line is the correlation for 𝛼, Eq. 4.5). b) The integral of 𝛼, as a function of thermal age at the location of peak soot concentration on the wings (the line is the integral of the correlation for 𝛼, Eq. 4.6). .................................................................................................................. 87

Figure 4.7 Comparison of computed peak soot volume fractions on the wings using the 𝛼 function based on thermal age (Eq. 4.6), with experiments from [29,41,192,212].88

Figure 4.8 Isopleths of soot volume fraction (ppm) of the SM40 (left panel), SM80 (middle panel) and SA (right panel) flames. The left side of each panel is the model computed with the new 𝛼 function. The right side is the experimental data ([41] and [212]). .............................................................................................................. 89

Figure 4.9 Variation of surface reactivity and soot volume fraction as a function of soot particle residence time along the wings for SA and SM60 flames. ....................... 91

Figure 4.10 Variation of surface reactivity and soot volume fraction as a function of soot particle thermal age along the wings for SA, SM80 and SM40 flames. ................ 91

Figure 4.11 Comparison of computed (left panel) and experimental (right panel, from [192]) isotherms of the SA flame. ..................................................................................... 93

Figure 4.12 Comparison of numerical and experimental (from [215] and [37].) temperature profiles along the centerline of the flames, as a function of axial height. .............. 94

Figure 4.13 Comparison of the computed (lines) and experimental (symbols) a) concentrations of acetylene at the 𝑧 = 7 mm and 𝑧 = 20 mm axial heights as a function of radial distance from the centreline for the SA flame (measurements from [34]) b)

xiv

concentrations of acetylene on the centreline for the SM40 and SM80 flames (measurements from [215]) c) concentrations of benzene on the centreline for the SM40, and SM80 flames (measurements from [215]). .......................................... 95

Figure 5.1 Schematic representation of a burner stabilized stagnation flame, including coordinate orientation. .......................................................................................... 103

Figure 5.2 Condensation efficiency (Eq. 5.1) variation with temperature. ............................ 112

Figure 5.3 Comparison of experimental data (symbols) from [19] and calculated (lines) centerline temperature profiles at several separation distances between the burner and stagnation surface. Temperature measurement uncertainties and the positional uncertainty are shown with bars. .......................................................................... 115

Figure 5.4 Main species profiles computed with the KAUST mechanism (solid lines), and with the DLR mechanism (dashed lines) for a burner–stagnation surface separation of 𝐻𝑝 = 1.0 cm. ................................................................................................. 115

Figure 5.5 Main radicals and small aromatic molecules profiles computed with the KAUST mechanism (solid lines), and with the DLR mechanism (dashed lines) for a burner–stagnation surface separation of 𝐻𝑝 = 1.0 cm. .................................................... 116

Figure 5.6 Comparison of computed soot volume fraction (of which the particle diameter, D > 2.5 nm) of the KAUST and DLR mechanisms with Model 1 as a function of separation distance with experimental data [21]. ................................................. 117

Figure 5.7 Computed benzo(a)pyrene (A5) mass fraction profiles with the DLR mechanism as a function of height above the burner for six different burner stabilized stagnation flames. .................................................................................................................. 118

Figure 5.8 Computed anthanthrene (A6) mass fraction profiles with the KAUST mechanism as a function of height above the burner for six different burner stabilized stagnation flames. ................................................................................................. 118

Figure 5.9 Comparison of soot particle number density (of which the particle diameter, D > 2.5 nm) computed with constant efficiency nucleation (Model 1), reversible nucleation and constant efficiency condensation (Model 2), and reversible nucleation and temperature dependent condensation efficiency (Model 3) as a function of separation distance, with experimental data [21]. ............................. 121

Figure 5.10 Comparison of soot volume fraction (of which the particle diameter, D > 2.5 nm) computed with constant efficiency nucleation (Model 1), reversible nucleation and constant efficiency condensation (Model 2), and reversible nucleation and

xv

temperature dependent condensation efficiency (Model 3) as a function of separation distance, with experimental data [21]. ............................................... 122

Figure 5.11 Comparison of computed soot particle size distributions using reversible nucleation and constant efficiency condensation (Model 2), and reversible nucleation and temperature dependent condensation efficiency (Model 3) at several separation distances between the burner and stagnation surface, with experimental data [21]. .............................................................................................................................. 123

Figure 5.12 Equilibrium constant for dimerization of PAHs employed in the reversible nucleation model with different average vibration frequencies as a function of temperature. .......................................................................................................... 124

Figure 5.13 Comparison of computed soot particle size distribution using different intermolecular vibrational frequencies for the reversible nucleation model and a constant efficiency condensation (𝛾𝐶𝑜𝑛𝑑 = 5%) at several separation distances between the burner and stagnation surface with experimental data [21] (effect of vibrational frequencies on Model 2 predictions). ................................................. 126

Figure 5.14 Computed anthanthrene (A6) mass fraction profiles as a function of height above the burner for the 𝐻𝑝 = 1.2 cm burner stabilized stagnation flame using three models: without soot, with dimerization frequency of 26 cm-1, and with dimerization frequency of 14 cm-1. ...................................................................... 126

Figure 5.15 Comparison of computed soot particle size distribution with reversible nucleation model and different constant efficiencies for condensation (𝛾𝐶𝑜𝑛𝑑) at several separation distances between the burner and stagnation surface with experimental data [21] (effect of condensation on Model 2 predictions). ........... 128

Figure 5.16 Comparison of computed soot particle size distribution using different coagulation efficiencies for the reversible nucleation model and constant efficiency condensation (𝛾𝐶𝑜𝑛𝑑 = 5%) at several separation distances between the burner and stagnation surface, with experimental data [21] (effect of coagulation on Model 2 predictions). .......................................................................................................... 129

Figure 5.17 Comparison of (a) soot particle number density and (b) soot volume fraction (of which the particle diameter, D > 2.5 nm) computed with reversible nucleation and equilibrium based condensation efficiency (Model 4), and reversible nucleation and a constant efficiency condensation (Model 2), as function of separation distance, with experimental data [21]. .................................................................. 130

Figure 5.18 Comparison of computed soot particle size distribution using reversible nucleation and equilibrium based condensation efficiency (Model 4), and reversible

xvi

nucleation and a constant efficiency condensation (Model 2), at several separation distances between the burner and stagnation surface, with experimental data [21]. .............................................................................................................................. 131

Figure 5.19 Comparison of effects of (a) dimerization binding energy, (b) dimerization vibrational frequency, (c) surface reactivity, and (d) condensation vibrational frequency on computed soot particle size distribution using reversible nucleation and equilibrium based condensation efficiency (Model 4) for the 0.8 cm separation distances between the burner and stagnation surface flame with experimental data [21]. ...................................................................................................................... 132

Figure 5.20 Isopleths of soot volume fraction (ppm) of the Santoro ethylene/air coflow diffusion flame [58] computed using Models 1, 2, and 4 and experimental data from [212]. ............................................................................................................ 134

Figure 5.21 Computed contours of particle number density (cm-3) with Model 1, Model 2, and Model 4 of the Santoro ethylene/air coflow diffusion flame [58]. ....................... 135

Figure 5.22 Computed contours of anthanthrene, A6, mole fraction with Model 1, Model 2, and Model 4 of the Santoro ethylene/air coflow diffusion flame [58]. ....................... 136

Figure 5.23 Comparison of the predicted a) soot volume fraction, b) average primary particle diameter, c) primary particle number density, and d) aggregate number density along the annular pathline exhibiting the maximum soot volume fraction of the Santoro ethylene/air coflow diffusion flame [58] using Model 1 (dot‐dashed line), Model 2 (dashed line), and Model 4 (solid line), with the experimental measurements by [39,57,58,192]. ........................................................................ 138

Figure 5.24 Comparison of the predicted a) soot volume fraction, b) average primary particle diameter, c) primary particle number density, and d) aggregate number density along the centerline of the Santoro ethylene/air coflow diffusion flame [58] using Model 1 (dot‐dashed line), Model 2 (dashed line), and Model 4 (solid line), with the experimental measurements by [37–39,58,192]. ............................................ 140

Figure 5.25 Isopleths of soot volume fraction (ppm) of the Santoro ethylene/air coflow diffusion flame [58] computed using the KAUST and DLR mechanisms and soot Model 4, with experimental data from [212]. ...................................................... 141

Figure a. 1 Predicted particle size distribution functions at different axial heights above the burner along the annular pathline of the maximum soot volume fraction, and along the centerline. ....................................................................................................... 154

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Figure b. 1 Comparison of the predicted soot volume fraction along the wings using alpha function (solid line) and constant alpha (dashed line) with the experimental measurements by [58] .......................................................................................... 155

Figure b. 2 Comparison of the variation of predicted soot volume fraction with residence time along the wings using alpha function (solid line) and constant alpha (dashed line) with the experimental measurements by [192]. .................................................... 156

Figure b. 3 Comparison of the predicted average primary particle diameter along the wings using alpha function (solid line) and constant alpha (dashed line) with the experimental measurements by [56]. .................................................................... 156

Figure b. 4 Comparison of the predicted primary particle number density along the wings using alpha function (solid line) and constant alpha (dashed line) with the experimental measurements by [39,57]. .................................................................................... 157

Figure b. 5 Comparison of the predicted aggregates number density along the wings using alpha function (solid line) and constant alpha (dashed line) with the experimental measurements by [57,192]. .................................................................................. 157

Figure b. 6 Comparison of the predicted number of primary particles per aggregate along the wings using alpha function (solid line) and constant alpha (dashed line) with the experimental measurements by [33,57]. ............................................................... 158

Figure b. 7 Comparison of the predicted soot volume fraction along the centerline using alpha function (solid line) and constant alpha (dashed line) with the experimental measurements by [37,38,58]. ............................................................................... 158

Figure b. 8 Comparison of the predicted average primary particle diameter along the centerline using alpha function (solid line) and constant alpha (dashed line) with the experimental measurements by [37]. .................................................................... 159

Figure b. 9 Comparison of the predicted aggregates number density along the centerline using alpha function (solid line) and constant alpha (dashed line) with the experimental measurements by [192]. ....................................................................................... 159

Figure b. 10 Comparison of the predicted number of primary particles per aggregate along the wings using alpha function (solid line) and constant alpha (dashed line) with the experimental measurements by [37]. .................................................................... 160

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Figure c. 1 Comparison between experimental data from [233] and calculated mole fraction of major gaseous products. ....................................................................................... 161

Figure c. 2 Comparison between experimental data from [233] and calculated mole fraction of benzene and various PAHs. .................................................................................. 162

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List of Appendices

Appendix A .............................................................................................................................. 153

Appendix B .............................................................................................................................. 155

Appendix C .............................................................................................................................. 161

1

Chapter 1 Introduction

1.1 Motivation

More than 80% of the world’s energy supply comes from hydrocarbon sources including natural

gas, petroleum, and coal [1]. It is expected that the total demand for energy will increase steadily

throughout the world with particularly large increases in the demands from emerging economies.

Total world consumption of liquid fuels, as a sample of the world’s hydrocarbon consumption, is

estimated to increase by 33 MMbbl/d throughout the course of thirty years, starting from 2010,

which is equivalent to 30% of the current consumption [2]. Energy use has adverse

environmental and health consequences that have led to considerable restrictive regulations.

Particulate matter (PM) is a known pollutant and its health and environmental consequences are

linked directly to their size. Combustion-derived nano-particles, such as soot, are a significant

source of particles smaller than 2.5 µm (PM2.5) in urban areas. The role of the chemical

composition of the particles or the source of the particles on their adverse effects are yet to be

examined, however, health outcomes have a stronger correlation with exposure to combustion-

derived particulates than with particulates from other sources [3].

U.S., Canadian and Europe-based epidemiological studies have measured relationships between

exposure to PM2.5 and health outcomes including: cardiovascular morbidity, respiratory

symptoms, increases in hospitalization; mortality from cardiovascular and respiratory diseases

2

and from lung cancer, along with various other health complications [4]. The International

Agency for Research on Cancer recently listed the exhaust from diesel engines, and exposures to

some PAHs as carcinogenic [3]. Polycyclic aromatic hydrocarbons (PAH), as well as metals and

inorganic salts are among the constituent elements of soot particles. These components are

currently seen as responsible for the hazardous nature of combustion-driven particulate matter.

The environmental effects of particulate matter are mainly related to PM’s optical properties.

These effects of PM include impairment of visibility in rural and urban areas, effect on climate

by scattering incoming solar radiation and influencing cloud properties, and ecological effects

[4].

For these reasons, stricter regulations are now targeting particulate emissions in both automotive

(e.g., EURO 6) and aviation (e.g., ICAO) engines. Most of these regulations set limitations for

the cumulative particulate mass emissions over different periods of time. However, there are

growing concerns that potential effects of other particulate characteristics, such as particle

number, particle morphology, and detailed chemical speciation on the environment and health

should be considered [5–8]. In this way, a comprehensive understanding of the risks associate

with PMs may be achieved [9]. Thus, understanding the soot mass growth mechanisms as well as

formation of particle size distributions has received significant attention.

Controlling particulate emissions to abide with regulations while maintaining high efficiency has

been one of the challenges of combustion research and development. Novel combustion

strategies include low-temperature combustion (LTC) strategies as in homogeneous charge

compression ignition (HCCI), stratified-charge, compression-ignition (SCCI), and gasoline direct

injection (GDI) in internal combustion engines, and staged combustion in advanced gas turbines,

such as twin annular premixing swirler (TAPS) mixer technology [10]. These strategies offered a

significant fuel efficiency improvement and pollutant emissions reduction potential. To address

the challenges facing developing significantly more fuel efficient engines, it is crucial to advance

the science underpinning novel combustion strategies. Advancements needed relevant to soot

emission include: a fundamental understanding of soot formation in lean (diluted) fuel-air

mixtures at high pressure and temperature conditions representative of internal combustion

engine and gas turbine enviroments; robust soot models based on the fundamental chemical and

3

physical processes and their coupling in novel combustion regimes; a framework for developing

a multiscale model by combining the computational tools and methodologies [10].

Therefore, based on the perspective that has been projected for the environmental and industrial

research, the present work seeks to extend the development and use of soot formation models in

combustion simulation that are capable of predicting soot volume fraction, particle nanostructure

and size distribution and to advance computer modeling robustness toward capturing the changes

of flame temperature, mixing and residence time. Predictive models would allow engine

designers to tune various design and operating parameters without the need for costly

experimentation. However, there is an urgent need for a more fundamental understanding as

many soot formation and oxidation processes are poorly understood.

1.2 Literature Review

The remarkable advances on the kinetics of carbon nanoparticle formation and their final

properties, depending on the precursor, temperature, pressure, and concentration have been

comprehensively reviewed by Haynes and Wagner [11], Glassman [12], Kennedy [13], Richter

and Howard [14], Frenklach [15], D’Anna [16], Wang [17], and Eremin [18]. Build upon these

valuable studies, a brief review of soot particle characteristics, formation pathways, and

modeling will be presented in the following sections.

1.2.1 Soot Characteristics

Soot particles are generated in high temperature fuel rich regions of a combustion chamber when

burning a variety of fuels. Reported flame generated soot particles observed in a variety of

conditions including laminar premixed [19–28] and diffusion flames [29–45] as well as turbulent

flames [46–50] exhibited universal structures. Nonetheless, the nanostructure and aggregation

properties of soot particles present in a flame evolve in accordance to the type of the flame, and

the locations within a given flame. Figure 1.1 presents the structure evolution of a soot sample as

observed by transmission electronic spectroscopy (TEM) along the centerline of a coflow

diffusion flame of a surrogate for Jet A-1 at different heights above the fuel tube exit. The

diversity in nanostructure has been attributed to the evolutionary process which transforms

nascent soot particles into mature particles.

4

Figure 1.1 TEM images of soot particle samples along the centerline of a coflow diffusion flame of a surrogate for Jet A-1 at different heights above the fuel tube exit (Source: Reprinted from ref. [35]).

Dobbins and coworkers [30,39,51–58], D'Anna and coworkers [16,46,59–61], and Wang and

coworkers [17,62–64] have investigated the evolution and characteristics of nascent soot

particles in premixed and diffusion flames. The nascent soot particles, also referred to as

precursor nanoparticles (PNP) and nanoparticles of organic carbon (NOC), are nearly spherical

particles with sizes in the range 1-5 nm in diameter. Their spherical shape and lack of

aggregation are evidence of liquid-like behaviour and presumption of coalesce upon collision

[52]. The low contrast TEM images observed in [35,65] suggests that nascent soot particles are

semi-transparent to an electron beam and have low visible absorption. Chemical and

spectroscopic analysis through identification of the chemical bonds and C and H elements give

an indication of the chemical nature of the particles. Laser microprobe mass spectrometry

(LMMS) [53], gas chromatography/mass spectrometry (GC/MS) [66] and high-resolution

transmission electron microscopy (HRTEM) [22] measurements indicated that the nascent

particles can be thought of as polymer-like structures containing PAH molecules ranging in

molecular masses from 152 to 302 amu. Evidence of aliphatic and aromatic bonds and

occasionally oxygen have been detected by UV-visible absorption and fluorescence spectroscopy

and Fourier-Transformed Infrared (FTIR) spectroscopy [16,27,60]. Elemental analysis of nascent

soot particles shows that these particles have a relatively low atomic C/H ratio of ~ 1.6 – 4

[22,26,27,67] which can also be associated with their high chemical reactivity [22].

Nanoparticles have low coagulation rates at flame temperatures due to the weak Van der Waals-

interactions between particles relative to their thermal energy. The presence of functional groups

containing oxygen within the nanoparticles may also be related to the low coagulation efficiency

of the particles [16].

5

Simultaneous coagulation of the 1 – 5 nm particles, addition of compounds from the gas-phase,

and loss of H atoms direct particles towards gaining a graphitic structure, and eventually

transforms nascent soot particles to aggregate carbonaceous and hardened primary particles

[16,52]. The nascent particles may also be absorbed onto the surface of the aggregates upon

collision [52].

Mature soot particles, as illustrated in Figure 1.2, consist of small spherical units that are referred

to as primary particles. Primary particle diameters generally range from 20 to 60 nm, with

standard deviations of 15% – 25% [68]. The primary particles within an aggregate have nearly

identical diameters, and form chain-like aggregated structures that have broad distributions of the

number of primary particles per aggregate ranging from a few up to several thousand [39,68].

The elongated chain aggregate structure and the broad aggregation range of soot particles impose

the potential complexity in the characterization, and to a greater extent in simulating soot

particles. The complexity associated with aggregate structures is alleviated by the experimental

observations that soot aggregates exhibit a fractal-like structure. Aggregates produced in a wide

variety of flames exhibit a near universal fractal dimension of 𝐷𝑓 = 1.82 ± 0.06 [69] for

turbulent flames and around 1.8 for laminar flames [57,70], even when an aggregate consists of

only few primary particles [68,71]. The low fractal dimension of the soot particles indicates that

they have open structures as opposed to more compact near spherical structures. The fractal

dimension is also a measure of rate of change of aggregate size with the number of primary

particles per aggregate. In addition, the fact that soot aggregates have fractal-like structures,

allows the implementation of fractal aerosol theory in the modeling and laser diagnostics of soot

aggregates. Mature particles have a more opaque, black material optical properties [52].

Figure 1.2 TEM image of soot sample formed from ethylene pyrolysis in a flow reactor at 1475 K in the presence of nitrogen oxides (specifically N2O); (Source: Reprinted from ref. [72]).

6

Aggregated particles have high elemental carbon content. GC/MS measurements [66] verified

the existence of 2 to 4 ring PAHs, and liquid chromatography [73] measurements identified

existence of 5 to 10 ring polycyclic aromatic species as the constituents of mature soot particles.

The conversion of nascent soot particles to mature soot aggregates in flames is accompanied by

an increase of the carbon to hydrogen ratio (C/H), ranging from 6 to 20 [22,23,26,27,66]. The

mass density of mature soot material (𝜌𝑠= 1.77–2.0 g/cm3) [11] is also expected to be

substantially higher than that of nascent soot (𝜌𝑠= 1.2–1.5 g/cm3) [63,67].

Nascent soot particles are often observed at low heights in laminar diffusion flames. The chained

aggregates form in the higher flame region and a transition stage consisting of ill-defined,

composite particles separates the two particle regimes [35,53,62]. Hu and Köylü [69] reported

that if the flame is transformed to near turbulent or fully turbulent, all particle morphologies can

coexist in a diffusion flame. The coexistence of the singlet spheroids and the carbonaceous

aggregates also has been observed in particle size distribution (PSD) measurements in laminar

premixed flames [62,63]. In the later flames the bimodal PSD evolve from a unimodal PSD as a

function of time and height.

The bimodal particle size distribution is an indication of coexistence of nascent and mature soot

particles. Comparison of the measured PSD with the TEM results [62] and electrical mobility

measurements [74] indicates that the particles < 5 nm in diameter are associated with the nascent

soot particles (nucleation mode) which exhibit a distinctive behavior from the 10-50 nm particles

(the accumulation mode). Particles belonging to the accumulation mode, display the expected

soot properties that are characterized by light scattering and TEM: they gain mass and increase

their size due to surface growth and reduce in number due to coagulation as a function of

residence time. Meanwhile, the mean size and number density of the nucleation mode remains

nearly constant everywhere in the flame. Since the nascent particles grow and coagulate with

other particles, the consistent presence of the nucleation mode implies a continuous nucleation.

These observations link the shape of the particle size distribution to the morphology and mode of

particles.

7

1.2.2 Soot Formation Pathways

Emergence of the condensed-phase from the gas phase is known as nucleation. The newly

formed particles gain mass and grow in size through coalescence, surface reactions and

condensation of vapor species. The growth process continues by transforming the monomer

particles into fractal structures through aggregation. Finally, the soot particles lose mass and size

during oxidation and fragmentation processes. These processes mostly occur simultaneously in a

flame and over very short periods of time, as schematically illustrated in Figure 1.3. Many of the

underlying processes that control soot formation are not well understood. For each of these

processes, a model must be developed that captures the fundamental physics that is occurring

and interacts with other models too.

The initial step in soot formation from pure hydrocarbon flames is the pyrolysis and oxidation of

the fuel. In general, simple fuel pyrolysis and oxidation is relatively well known. Reasonably

accurate reaction mechanisms exist for the fuels of interest [75–78]. The next step involves the

formation of light aromatic hydrocarbon species in the gas phase from hydrocarbons generated

during fuel pyrolysis. Propargyl (C3H3) recombination and chemically activated isomerization is

the main route toward formation of the first aromatic ring [79,80]. Alternative routes for

formation of light aromatics are described by: cyclohexane dehydrogenation [81], formation of

naphthalene from cyclopentadienyl (cy-C5H5), allyl recombination, i-C4H3+C2H2, and i-

C4H5+C2H2 [82]. The identified growth pathways beyond the first aromatic ring to form larger

multi-ringed aromatic species (i.e., PAHs) are the hydrogen–abstraction–carbon–addition

(HACA) reaction sequence [83], free radical addition schemes, methyl substitution/acetylene

addition pathways [14], cyclopentadienyl moiety in aromatic ring formation [15,84], and

reactions between aromatic radicals and aromatic molecules [85]. Both fuel pyrolysis/oxidation

and PAH formation and growth pathways have been combined to generate reaction mechanisms

describing the formation of PAH species [84,86–88].

8

Figure 1.3 Schematic diagram of soot formation (Source: Reprinted from ref. [89]).

Onset of condensed phase materials follows the appearance of large PAH species in the gas

phase. Two well-received approaches to postulate a soot nucleation mechanism among others are

collision coagulation [90,91] and chemical coalescence [92,93]. The collision coagulation

hypothesis is that the Van der Waals interaction force becomes sufficiently large after PAH

growth to a certain size so that it can hold together a pair of PAHs during physical collision, thus

forming PAH dimers. The sequence of collisions among PAH dimers and PAH molecules leads

to the formation of PAH trimers, PAH tetramers and so on. Meanwhile, PAH species

constituting the PAH stacks keeps growing via molecular chemical reactions. Subsequently, the

PAH clusters evolve into solid particles. Most of the PAH-based soot models consider PAH

dimerization as the bridge from the gas phase to the solid phase [15]. The alternative hypothesis

is that aliphatic linking of 2-, 3-aromatic rings form 3-dimensional structures. Further growth of

these structures in this manner leads to emergence of nascent soot particles. Additional mass

growth as well as dehydrogenation of the nascent particles is marked as the emergence of the

solid state [92]. The latter mechanism is referred to as chemical coalescence.

9

Currently, experimental data characterizing the transition zone from the gas phase to the

condensed-phase are very limited due to the nature of the processes. Indirect experimental

evidence such as the observation of the bimodality in the size distribution functions of nascent

soot particles in premixed flames [19,63], supports both pathways as the initial nucleation step

[17]. Theoretical aspects of particle nucleation were discussed by Herdman and Miller [94] for

collision coagulation and by Violi and Venkatnathan [95] for chemical coalescence using large-

scale, statistical mechanics simulations and molecular dynamics. Both of these studies verified

the possibility of formation of condensed phase in a flame environment through the proposed

mechanisms. More recently studies which include a broader range of flames in terms of mixing

and temperature such as Chung and Violi [96] and D’Anna [16] showed that particle inception

can be considered as the result of both a chemical growth and a physical coagulation and

contribution of these two pathways to the particle inception rate varies according to the

combustion conditions. Wang [17], however, showed that neither of the current nucleation

theories are comprehensive enough to comply with the new findings with regards to the PAH

and nascent soot thermo/chemical characteristics, and proposed that more comprehensive

theories such as PAH coalescence through π-electron interactions, are required.

The growth of the soot particle can occur by the addition of small hydrocarbon species. This

process is currently described by the hydrogen–abstraction–carbon–addition (HACA)

mechanism [15,86,90]. The soot surface is assumed to consist of hydrogenated sites with a

predefined density. Mass growth on soot surface requires H-abstraction to form an aryl radical

site, followed by acetylene attack in a manner similar to the gas-phase PAH growth mechanism.

Observations have been made that cannot be explained in the context of the HACA mechanism:

the surface reactivity changes with time and temperature [97,98], the existence of aliphatic

compounds in nascent soot [45,62,99] and soot mass growth without the presence of gas-phase H

atoms [63,100]. These observations are indication of incompleteness of the HACA mechanism to

describe the entire process of soot surface growth.

Deposition of PAH species on the surface of the soot particles is also considered a viable growth

route for soot particles, which is referred to as PAH-soot surface condensation [90,101].

Molecular dynamics studies suggest that these adsorbed PAH species are not stable [102,103].

Yet the experiments suggest that PAH stacks are indeed the building block of soot [104,105]. A

better understanding is needed of the processes that stabilize these absorbed PAH species.

10

The final stage in the soot particle formation and growth mechanism is aggregation. The process

of formation of fractal-like aggregate structures as a result of particle collisions is termed

coagulation. Coagulation determinatively influences shaping of soot particle size distribution,

soot number density, and soot morphology. After collision, soot particles may experience

structural evolution. The restructuring processes is a function of particle state, surface property,

primary particle diameter, temperature, residence time, etc. [106]. The collision of liquid-like

nascent soot particles leads to complete merging of the colliding particles which is known as the

coalescence process [54]. The slow restructuring rate of the mature particles leads to the

formation of the fractal-like aggregate structure. Observation of neck formation at the contact

points of primary particles within an aggregate can be interpreted as partial coalescence or

surface growth obliteration [39]. The soot particles’ restructuring mechanisms are not well

understood. New models are needed to estimate the maturity of the particles as well as

comprehensive coagulation models that describe coalescence process, neck formation, and

aggregation.

The oxidation of the soot determines the amount of soot emissions. The soot is consumed

primarily by reactions with O, OH and O2. In near stoichiometric and fuel-rich conditions

oxidation by OH radical is the predominant mechanism for soot oxidation [107]. Under these

conditions some oxidation occurs via collisions with O. However, contribution from O is much

less in comparison to OH [108]. The rate of OH oxidation can be described by the fraction of

collisions of OH with soot particles that result in the removal of a carbon atom. The collision

efficiency of OH radicals with the soot particles reported to be 0.13 [107,108]. Although OH

oxidation is faster compared with O2, under fuel lean conditions oxygen plays a crucial role in

soot oxidation due to abundance of O2. Molecular oxygen oxidation has been represented by

power-law kinetics [108]; however, research has indicated that changes of both initial structure

of soot [22,109] and structure of soot during oxidation [108] complicates defining a universal

oxidation rate.

The structure of soot particles can also be affected by soot oxidation. An increase in particle

number has been reported by Neoh et al. [110] in lean premixed flames and by Xu et al. [111]

and Puri et al [57] in the oxidation region of diffusion flames. The increase in the numbers of

aggregates as well as the decrease in number of primary particles per aggregate was attributed to

fragmentation. Since the change in aggregate morphology is not seen for fuel-rich conditions, it

11

is linked to O2 oxidation. Although this phenomenon has been observed, the mechanism is

debated. One of the proposed mechanisms for fragmentation assumes that the aggregate chain

breaks at the bridges between particles which were weakened by oxidation. The other proposed

mechanism postulates that internal burning of soot particles by oxygen cause the break-up of

individual primary particles within an aggregate, dividing the aggregate into smaller aggregates

with fewer particles [108].

1.2.3 Soot Modeling

A broad range of length and time scales are involved in soot simulations. The relevant length

scales include:

- Angstroms for atomic and molecular level scales (10-10 m)

- Nanometers for dimers and soot particles (10-9 m)

- Millimeters for flow scales (10-3 m)

- Centimeters for burner geometry (10-2 m)

which make soot modeling a multiscale problem. The approach towards dealing with multiscale

problems is to model the processes at the smallest/shortest length/time scales based on

fundamental understanding and to resolve the larger/longer length/time scales. One of the

challenges of these systems is to keep a balance between simplicity of the model and loss of

accuracy and predictability.

The advancements made in early stages of developing soot models was reviewed by Kennedy

[13]. Based on the level of length/time scale to be resolved and the complexity of the models,

soot models were divided into three categories: empirical soot models, semi-empirical soot

models, and detailed soot models. Experimentally derived correlations are the essence of the

empirical soot models. The correlations include variation of different combustion parameters

such as pressure, equivalence ratio, and temperature, on soot formation/oxidation. These

correlations are embedded into empirical soot models to relate the amount of soot produced with

the operating conditions. The empirical models are mostly suitable for industrial applications.

Semi-empirical soot models attempts to add a level of sophistication to the soot modeling by

including rudimentary soot formation and oxidation mechanisms in the model. The two-equation

12

model by Fairweather et al. [112] is one of the most popular semi-empirical soot models. It

solves one transport equation for the soot mass fraction, and a second equation for the primary

particle number density. Inception, surface growth, oxidation and coagulation are the soot

processes that are considered in the Fairweather model which are empirically estimated. The

drawback of relying on empirical correlations is confinement of the model validity to the model

calibration cases.

The final category includes the detailed soot models. These are the most complex and

computationally expensive soot models. The detailed soot models are equipped with the most

advanced aerosol dynamics prediction tools which are capable of resolving a wide distribution of

polydispersed aggregate structures. State of the art chemical and physical mechanisms describing

PAH and soot formation/oxidation are incorporated into the detailed models to achieve a

rigorous description of processes involved with soot particles. These models can be employed to

provide detailed information regarding parameters influencing particles for a broad range of

conditions, which makes them a suitable tool for studying the fundamentals of particle

formation/oxidation.

In order to simulate combustion and soot particles in a flame, a detailed soot model needs to

model the flow field (solving the Navier-Stokes equations), predict temperature (solving the

energy equation), calculate gas phase composition (solving the gas-phase chemistry), and soot

(solving the aerosol dynamics equations) all of which are closely coupled.

Prerequisite of a detailed soot model is a detailed chemical kinetic mechanism that not only is

capable of describing the pyrolysis and oxidation of hydrocarbon fuels but also can model the

formation and growth of PAH species. Due to vast variation of species and pathways involved in

PAH formation and growth, the detailed chemical mechanisms designated for the simplest

hydrocarbon fuels include hundreds of species and thousands of reactions [86–88], which add a

substantial computational load to detailed soot simulations.

The aerosol dynamics models that are suitable for detailed soot models are moment methods

[90,113], stochastic methods [114], Galerkin methods [115,116] and sectional methods

[41,117,118]. These are efficient algorithms that with moderate computational costs can resolve

the majority of particle properties. However, modifications to these models to extract additional

chemical/physical resolutions exponentially increase their complexity and computational

13

expense. An example is the Monte Carlo (MC)/molecular dynamics (MD) calculations that have

been developed to bridge the time/length scales between molecular and particle levels in soot

formation [94,119,120]. The ability of these models to simultaneously resolve particle size

distribution, and morphology as well as chemical composition of the particles attracted a lot of

attentions in soot particle studies. The MC/MD models are viewed as a potential candidate for

development of the next generation of soot models. However, improving computational

capabilities and developing high efficiency algorithms for Monte Carlo methods are necessary

before application of these methods becomes feasible for flame simulations and soot particle

studies.

An advanced sectional aerosol dynamics model [121] is used in this thesis that can provide soot

morphology in addition to mean soot properties and the size distribution of particles. Two

equations, number densities of aggregates and primary particles, are solved per section which

allows resolving the formation and coagulation of the fractal-like soot aggregates as well as soot

polydispersity. Abilities of the sectional soot model to successfully simulate soot formation has

been demonstrated in plug flow reactors [121], shock tubes [122], and coflow diffusion flames

[123,124]. The sectional soot aerosol dynamic model is described in detail in Chapter 2.

1.3 Objectives and Outline of Subsequent Chapters

The objective of the subsequent chapters of this work will be to advance the field of

computational soot modeling by focusing on detailed laminar flame simulation using a sectional

soot method. The goal will be to move toward developing a robust model of soot formation that

can predict the mass, size distribution and aggregate structure of soot in laminar flames for a

wide range of conditions. This effort will include developing numerical models to simulate

processes which were not considered in the previous soot models, improving known weaknesses

in a commonly used soot model, increasing the soot modeling knowledge base by studying the

sensitivity of soot predictions to the involved processes and parameters, and extending the

applicability of the soot model to laminar diffusion, partially premixed as well as premixed

flames.

Chapter 2 will describe the governing equations that are necessary in combustion modeling and

the numerical methods to solve those equations. These equations will include the equations

governing the fluid dynamics, which are conservation of mass and conservation of momentum.

14

Conservation equations of species mass and energy will then be introduced to complete the set of

equations necessary to fully resolve the gas phase. The two configurations that will be

considered in the present work are the coflow diffusion flame and the burner stabilized

stagnation (BSS) premixed flame. The appropriate form of the governing equations that complies

with each of the flame configuration will be presented. The chapter will then proceed by

introducing the soot model to be used, including the equation of conservation of soot sectional

aggregate number density as well as primary particle number density. Next, the thermodynamic,

chemical kinetic and transport models that will be used in the present work will be stated, and the

chapter will conclude by describing the numerical solution procedures along with the boundary

conditions which will be incorporated in simulations of laminar ethylene/air flames. The soot

model development chapters are arranged in chronological order.

The objectives of Chapter 3 will be to introduce two particle coalescence models applicable to

soot particle simulations. The introduced coalescence models will be applied to a laminar coflow

ethylene/air diffusion flame, and comparisons will be made with experimental data to validate

the models. The effects of these coalescence models on predictions of soot particle morphology

will be quantified.

Chapter 4 will proceed as a comparative study of soot chemical growth for a variety of

ethylene/air flames, and it will specifically investigate how surface chemical reactivity can be

affected by temperature and residence time. Based upon this comparison, a function for surface

reactivity of soot particles based on temperature-time histories of particles will be proposed. The

sectional soot model with the new soot surface reactivity function is used to simulate multiple

coflow ethylene air flame. The coflow flames include several coflow diffusion ethylene/air

flames with varying fuel flow rate, and fuel dilution, and multiple partially premixed coflow

ethylene/air flames with varying equivalence ratios. Predictions of soot concentration will be

compared to experimental data for validation.

In Chapter 5 a more ambitious study is undertaken to present a model that is capable of

predicting soot in both premixed and diffusion flames. The role of PAH chemistry will be

investigated in PAH growth dominant flames which are the burner stabilized stagnation (BSS)

premixed ethylene flames. Prediction of soot particle size distribution with a reversible

nucleation model will be compared to efficiency based nucleation models in the BSS premixed

15

flames. The effects of dimerization equilibrium parameters as well as other soot formation

processes on predictions of PSDs are quantified. The chapter will then proceed by introducing a

novel model for PAH condensation that considers the possibility of PAH evaporation through

equilibrium conditions. Equilibrium parameter effects of the soot particle size distribution

predictions will be characterized. The chapter will conclude by evaluating the described model’s

performance in modeling soot formation in a diffusion coflow ethylene/air flame.

Finally, Chapter 6 presents a summary of the conclusions of the present work, as well as

recommendations for future investigations.

16

Chapter 2 Mathematical Model

2.1 Overview

This chapter will present the governing equations and state variable relationships that are

necessary for the chemically-reacting flow simulations in the present work. This will include

details on a sectional representation for modeling particulate (soot) formation. Two forms of

governing equations are employed in modeling different flames modeled in this work. The first

set of governing equations describes the two-dimensional reacting flow in cylindrical coordinates

which are utilized for modeling axisymmetric coflow diffusion flames. The second set of

governing equations is a similarity solution of the generalized governing equations, which casts

the governing equations as a one-dimensional boundary value problem valid along the centerline

of a stagnation flow. The burner stabilized premixed flames has been simulated using the latter

set of governing equations. The gas phase governing equations are presented in the next section.

In the subsequent section, the soot aerosol dynamic model is described. Finally, the numerical

methods used to solve the governing equations are described in Section 2.5.

2.2 Gas-Phase Governing Equations

The gas-phase governing equations include conservation of mass and momentum (Navier-

Stokes), conservation of energy and conservation of species. The solution to these equations

17

provides the flow field velocity, pressure, temperature and gas mixture composition. In addition,

species chemical kinetics, transport properties and thermodynamic properties have to be

evaluated. In the subsequent subsections all the conservation equations are presented followed by

the evaluation method of thermo-chemical properties.

2.2.1 Conservations of Mass and Momentum

The continuity equation in tensor form is presented in Eq. 2.

𝜕𝜌𝜕𝑡

+ 𝜕𝜕𝑥𝑘

(𝜌𝑢𝑘) = 0 ( 2.1)

Here, 𝜌 is the density of the mixture, is time, 𝑢𝑘 is the velocity component in the 𝑥𝑘 direction.

The general representations of the Navier-Stokes equations in tensor form are depicted in

Eq. 2.2.

𝜌𝜕𝑢𝑗

𝜕𝑡+ 𝜌𝑢𝑘

𝜕𝑢𝑗

𝜕𝑥𝑘= − 𝜕𝑝

𝜕𝑥𝑗+ 𝜕

𝜕𝑥𝑗 (𝜆𝜕𝑢𝑘𝜕𝑥𝑘) + 𝜕

𝜕𝑥𝑖 [𝜇 (𝜕𝑢𝑖𝜕𝑥𝑗

+𝜕𝑢𝑗

𝜕𝑥𝑖)] + 𝜌𝑓𝑗 ( 2.2)

where 𝜆 is the second viscosity coefficient, 𝜇 is the dynamic viscosity and 𝑓𝑗 is the net body

force.

2.2.1.1 The Two-Dimensional Cylindrical Coordinates

One of the flow configurations to be studied is that of a coflow laminar diffusion flame.

Figure 2.1 shows a schematic representation of the burner and flame geometry, with the

computational domain superimposed on the image. Since the flow is axisymmetric, the

governing equations become two-dimensional when they are expressed in cylindrical

coordinates. For axisymmetric flow (𝜕𝜕𝜃 = 0), the governing equations 2.1 and 2.2 are written in

cylindrical coordinates as:

1𝑟

𝜕𝜕𝑟

(𝑟𝜌𝑣) + 𝜕𝜌𝑢𝜕𝑧

= 0 ( 2.3)

𝜌𝑣 𝜕𝑢

𝜕𝑟+ 𝜌𝑢 𝜕𝑢

𝜕𝑧= − 𝜕𝑝

𝜕𝑧 + 1

𝑟𝜕𝜕𝑟 (𝑟𝜇 𝜕𝑢

𝜕𝑟) + 2 𝜕𝜕𝑧 (𝜇 𝜕𝑢

𝜕𝑧) − 23

𝜕𝜕𝑧 [

𝜇𝑟

𝜕𝜕𝑟

(𝑟𝑣)]

− 23

𝜕𝜕𝑧 (𝜇 𝜕𝑢

𝜕𝑧) + 1𝑟

𝜕𝜕𝑟 (𝑟𝜇 𝜕𝑣

𝜕𝑧) + 𝜌𝑔𝑧 ( 2.4)

18

𝜌𝑣 𝜕𝑣

𝜕𝑟+ 𝜌𝑢 𝜕𝑣

𝜕𝑧= − 𝜕𝑝

𝜕𝑟+ 𝜕

𝜕𝑧 (𝜇 𝜕𝑣𝜕𝑧) + 2

𝑟𝜕𝜕𝑟 (𝑟𝜇 𝜕𝑣

𝜕𝑟) − 23

1𝑟

𝜕𝜕𝑟 [

𝜇𝑟

𝜕𝜕𝑟

(𝑟𝑣)]

− 23

1𝑟

𝜕𝜕𝑟 (𝑟𝜇 𝜕𝑢

𝜕𝑧) + 𝜕𝜕𝑧 (𝜇 𝜕𝑢

𝜕𝑟) − 2𝜇𝑣𝑟2 + 2

3𝜇𝑟2

𝜕𝜕𝑟

(𝑟𝑣) + 23

𝜇𝑟

𝜕𝑢𝜕𝑧

( 2.5)

Here, 𝑟 and 𝑧 are the radial and axial coordinates; 𝑣 and 𝑢 are the radial and axial velocities; 𝑝 is

the pressure; 𝑔𝑧 is the axial gravitational acceleration.

Figure 2.1 Schematic representation of a coflow flame, including coordinate orientation and computational domain (not drawn to scale).

2.2.1.2 The One-Dimensional Similarity Solution

The second set of flame configurations are for the burner stabilized stagnation (BSS) premixed

flame which is shown schematically in Figure 2.2. The burner consists of a circular nozzle

carrying the premixed fuel and oxidizer toward a plate. This configuration produces an

axisymmetric flow field with a stagnation plane. The three dimensional partial differential

equations can be reduced to a one-dimensional boundary value problem by introducing a stream

Sym

met

ry

z

r

19

function in the form 𝜓(𝑧, 𝑟) = 𝑟2𝑈(𝑧) into the governing equations in the cylindrical coordinate

(Eq. 2.3 to 2.5). 𝑣/𝑟 and other variables become independent of 𝑟 when such a stream function is

assumed [125]. Following Kee et al.[126] the following variables have been defined:

𝐺(𝑧) = − 𝜌𝑣𝑟

( 2.6)

𝐹 (𝑧) = 𝜌𝑢2

( 2.7)

for which continuity, Eq. 2.3, reduces to

𝐺(𝑧) = 𝑑𝐹 (𝑧)𝑑𝑧

( 2.8)

Similarly, the radial momentum equation will be satisfied if

𝐻 = 1𝑟

𝜕𝑝𝜕𝑟

= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 ( 2.9)

and the axial momentum equation become

𝐻 − 2 𝑑𝑑𝑧 (

𝐹𝐺𝜌 ) + 3𝐺2

𝜌+ 𝑑

𝑑𝑧 [𝜇 𝑑𝑑𝑧 (

𝐺𝜌 )] = 0 ( 2.10)

Figure 2.2 Schematic representation of a burner stabilized stagnation flame, including coordinate orientation.

20

2.2.2 Conservation of Energy

The conservation of energy equation is presented in terms of temperature [127] in Eq. 2.11.

𝜌𝑐𝑝𝜕𝑇𝜕𝑡

+ 𝜌𝑐𝑝𝑣.̅ 𝛻𝑇 = 𝛻. (𝜆𝛻𝑇 ) − 𝜌 ∑ 𝑐𝑝,𝑘𝑌𝑘𝑣�̅�. 𝛻𝑇 − ∑ ℎ𝑘0�̇�𝑘𝑊𝑘 + �̇�𝑟

′′′ ( 2.11)

Here, the first term on the left hand side represents the temporal rate of change of temperature;

the second term represents convection heat transfer, and 𝑐𝑝 is the specific heat of the mixture

under constant pressure. On the right hand side of Eq. 2.11, the first term is the contribution from

conduction and 𝜆 is the thermal conductivity of the mixture; the second term is the heat flux rate

due to species diffusion, and 𝑣�̅� is the diffusion velocity of the kth species; the third term is the

rate of enthalpy production by chemical reaction; ℎ𝑘0 is the kth species specific enthalpy; and �̇�𝑟

′′′

is the change in energy density due to radiation from soot and gaseous species. All the terms on

the right hand side can be expressed as the sum of the effects of gas-phase species and soot

particles. Thus, the energy equation for a steady state axisymmetric condition in the cylindrical

coordinate is:

𝜌𝑐𝑝 (𝑣 𝜕𝑇𝜕𝑟

+ 𝑢 𝜕𝑇𝜕𝑧)

= 1𝑟

𝜕𝜕𝑟 (𝑟𝜆 𝜕𝑇

𝜕𝑟 ) + 𝜕𝜕𝑧 (𝜆 𝜕𝑇

𝜕𝑧) − 𝜌 ∑ 𝑐𝑝,𝑘𝑌𝑘 (𝑣�̅�,𝑟𝜕𝑇𝜕𝑟

+ 𝑣�̅�,𝑧𝜕𝑇𝜕𝑧)

𝐾𝐾

𝑘=1

− ∑ ℎ𝑘0�̇�𝑘𝑊𝑘

𝐾𝐾

𝑘=1− 𝜌𝑐𝑝,𝑠𝑌𝑠 (𝑣�̅�,𝑟

𝜕𝑇𝜕𝑟

+ 𝑣�̅�,𝑧𝜕𝑇𝜕𝑧) − ℎ𝑠

0�̇�𝑠𝑊𝑠 + �̇�𝑟′′′

( 2.12)

where subscript 𝑘 denotes those parameters related to gas species 𝑘 and subscript 𝑠 is used for

soot particles. 𝐾𝐾 represents the total number of species in the gas phase. Eq. 2.12 is solved for

predicting temperature for the coflow diffusion flames. For the BSS configuration, the

axisymmetric cylindrical energy equation, Eq. 2.12, is transformed similar to the momentum

equation into Eq. 2.13.

𝜌𝑐𝑝𝑢 𝜕𝑇𝜕𝑧

− 𝜕𝜕𝑧 (𝜆 𝜕𝑇

𝜕𝑧) + 𝜌 ∑ 𝑐𝑝,𝑘𝑌𝑘𝑣�̅�,𝑧𝜕𝑇𝜕𝑧

𝐾𝐾

𝑘=1+ ∑ ℎ𝑘

0�̇�𝑘𝑊𝑘

𝐾𝐾

𝑘=1+ 𝜌𝑐𝑝,𝑠𝑌𝑠𝑣�̅�,𝑧

𝜕𝑇𝜕𝑧

+ ℎ𝑠0�̇�𝑠𝑊𝑠

− �̇�𝑟′′′ = 0

( 2.13)

21

2.2.2.1 Radiation Heat Transfer

Radiation heat transfer has been recognized as an important flame heat-loss mechanism in

modeling laminar flames. Radiation heat transfer is not only necessary for prediction of

temperature but it is also coupled with soot and flame structure. Soot is the dominant source of

radiation in sooting flames which can noticeably influence flame temperature. However, most of

the processes involved in soot formation are endothermic. Radiation heat transfer lowers the rate

of soot formation therefore reduces heat-loss by radiation. The feedback loops from soot on

radiation and temperature and vice versa couple the soot formation with radiation heat transfer.

The radiative transfer equation (RTE) for an axisymmetric cylindrical system, considering the

medium to be in local thermodynamic equilibrium (LTE), is given in Eq. 2.14 [128].

𝜇𝜕𝐼𝜐𝜕𝑟

− 𝜂𝑟

𝜕𝐼𝜐𝜕𝜑

+ 𝜉𝜕𝐼𝜐𝜕𝑧

= −𝜅𝜐𝐼𝜐 + 𝜅𝜐𝐼𝑏𝜐 ( 2.14)

Here, 𝜇, 𝜂 and 𝜉 are directional cosines. Parameters 𝐼𝜐, 𝐼𝑏𝜐 and 𝜅𝜐 denote spectral intensity,

spectral blackbody intensity and the spectral absorption coefficient, respectively. The left hand

side refers to the rate of change of spectral intensity ((∇. 𝑠)𝐼𝜐). The first term on the right hand

side represents the reduction of radiant energy leaving an element of volume of matter due to

absorption. The last term on the right hand side of the RTE equation is the rate of emission by

the matter. The radiation heat transfer rate is calculated by integrating the RTE over all solid

angles and over the entire spectrum.

Since the radiation heat transfer equation is an integrodifferential equation, its solution is quite

difficult. Therefore, it is necessary to introduce some simplifying assumptions to solve the RTE.

In this work two methods has been adopted for estimation of the radiation heat transfer rate. The

optically thin approximation (OTA) is used to calculate the �̇�𝑟′′′ term in Eq. 2.13 for the BSS

premixed flames. The more sophisticated discrete ordinate method (DOM) is incorporated to

evaluate the radiation heat transfer rate for the coflow diffusion flame configuration.

Optically thin approximation (OTA)

Optical thickness is a dimension-less parameter which is a measure of the ability of a path length

of matter to attenuate radiation of a given wavelength. For a medium with uniform composition,

temperature and pressure optical thickness, 𝜏0𝜐, is defined as

22

𝜏0𝜐 = 𝜅𝜐𝐿 ( 2.15)

where 𝐿 is a characteristic dimension. In the condition of 𝜏0𝜐 ≪ 1, which refers to the optically

thin limit, the radiation emitted by a given fluid element will travel directly to the bounding

surfaces and any absorption by the fluid will be negligible. Therefore the radiation transfer

equation will become [129]:

�̇�𝑟′′′ = −4𝜎𝜅𝑃 (𝑇 4 − 𝑇∞

4 ) ( 2.16)

where 𝜎 is the Stefan-Boltzmann constant; 𝑇 and 𝑇∞ are the local and the ambient temperatures,

respectively; 𝜅𝑃 is the Plank mean absorption coefficient of the mixture. Liu et al. [130] by

comparing different radiation models concluded that the optically thin approximation could

predict the temperature field for a low sooting laminar flame reasonably well. Therefore

radiation heat transfer for the premixed flames has been estimated using the OTA method. In the

present work radiation from three species, CO, CO2, and H2O, and soot has been considered. For

the gaseous species the Plank mean absorption coefficient of the mixture, 𝜅𝑃 , is calculated from

Eq. 2.17.

𝜅𝑃 = 𝑃H2O𝜅H2O + 𝑃CO2𝜅CO2

+ 𝑃CO𝜅CO ( 2.17)

Here, 𝑃𝑖 and 𝜅𝑖 are the partial pressure and the Plank mean absorption coefficient of species 𝑖,

respectively. The Plank mean absorption coefficient is obtained as follows:

𝜅𝑖 = ∑ 𝐴𝑖𝑗𝑇 𝑗5

𝑗=0, 𝑖 = H2O, CO2 and CO ( 2.18)

where 𝐴𝑖𝑗 is the polynomial coefficient of a species expressed as a function of temperature [131].

Soot particles are assumed to be Rayleigh range absorber-emitters [132]. The Plank mean

absorption coefficient for soot was estimated according to [133] as

𝜅𝑃𝑠 = 3.83𝐶𝑓𝑣𝑇 ( 2.19)

Here, 𝑓𝑣 is the soot volume fraction and 𝐶 is constant taken to be [133]

𝐶 = 36𝜋 𝑛𝑘

(𝑛2 − 𝑘2 + 2)2 + 4(𝑛𝑘)2

( 2.20)

where, 1𝑛 + 𝑖𝑘 is the complex refractive index of soot, assumed to be 1.57 + 0.56𝑖 [134].

23

Discrete-ordinate method (DOM)

The radiation intensity is a function of the location, the direction of propagation of radiation and

of wavelength. One simplifying strategy to find radiation intensity is to divide the entire solid

angle (𝜙 = 4𝜋) into a finite number of ordinate directions and assume an average intensity within

given intervals of the solid angle. This assumption would discretize the radiation transfer

equation directionally into series of coupled linear differential equations. This procedure yields

the discrete-ordinates method [128]. DOM is developed without any presumption about the

opacity of the medium which makes it suitable for strong luminous and heavily sooting flames.

In addition, the DOM algorithm is highly compatible with the finite volume method and can be

readily incorporated into multi-dimensional finite volume codes. In terms of accuracy, DOM has

shown robustness comparable to the ones from more detailed and computationally intensive

Monte-Carlo methods [135].

Originally proposed by Liu et al. [130], the directional discretization has been obtained using a

T3 quadrature set [136] for the DOM radiation model. The RTE is written for each ordinate and

the integral terms are replaced by a Gaussian quadrature summed over each ordinate. The

directional cosines and weight functions of the T3 quadrature for the axisymmetric cylindrical

coordinate are taken from [130]. Radiation from CO, CO2, H2O and soot has been considered.

Mixture radiative properties have been evaluated using the statistical narrow-band-based

correlated-k (SNBCK) method [137]. The employed SNBCK divides the spectral band into nine

optimized nonuniform wide bands covering the spectral range from 150 to 9300 cm−1. The

radiative absorption characteristics for each band are approximated using an exponentially

decaying function [130,137,138]. The average radiation intensity at each narrow band is

determined by integrating the exponential function over the bandwidth which is numerically

estimated using the 4-point Gauss-Legendre quadrature scheme [139]. The spectral absorption

coefficient of soot is assumed to be 5.5𝑓𝑣𝜈 with 𝑓𝑣 being the soot volume fraction and 𝜈 the

wavenumber of each spectral band. The DOM equations are discretized using the finite volume

method. A total of 36 ordinate intensity equations are calculated to find the monochromatic

radiation passing through a volume element.

24

2.2.3 Conservation of Species Mass

In order to determine the chemical composition of a gas mixture in a reacting flow, where there

are numerous chemical species present, a conservation equation can be written for each of the

chemical species present. This mass transfer equation in axisymmetric cylindrical coordinates is

as follow

𝜌𝑣𝜕𝑌𝑘𝜕𝑟

+ 𝜌𝑢𝜕𝑌𝑘𝜕𝑧

= − 1𝑟

𝜕𝜕𝑟 (𝑟𝜌𝑌𝑘𝑉𝑘,𝑟) − 𝜕

𝜕𝑧 (𝜌𝑌𝑘𝑉𝑘,𝑧) + 𝑊𝑘�̇�𝑘 ( 2.21)

𝑘 = 1, 2, … , 𝐾𝐾

where 𝑌𝑘 is the 𝑘𝑡ℎ species mass fraction; 𝑉𝑘,𝑟 and 𝑉𝑘,𝑧 are the 𝑘𝑡ℎ species radial and axial

diffusion velocities, respectively; 𝑊𝑘 is the 𝑘𝑡ℎ species molecular weight; 𝐾𝐾 is the total

number of gaseous species; �̇�𝑘 is the 𝑘𝑡ℎ species molar production rate per unit volume and for

non-three-body reactions can be calculated by

�̇�𝑘 = ∑ 𝜈𝑘𝑖 (𝑘𝑓𝑖 ∏[𝑋𝑗]

𝜈′𝑗𝑖

𝐾𝐾

𝑗=1− 𝑘𝑟𝑖 ∏[𝑋𝑗]

𝜈"𝑗𝑖

𝐾𝐾

𝑗=1 )

𝑁𝑅

𝑖=1 ( 2.22)

where

𝜈𝑗𝑖 = 𝜈"𝑗𝑖 − 𝜈′𝑗𝑖 ( 2.23)

𝑁𝑅 is the total number of reactions; 𝑘𝑓𝑖 and 𝑘𝑟𝑖 are the forward and reverse rate of reaction 𝑖,

respectively. The interactions between soot formation/oxidation and gas-phase chemistry is

included in the chemical reaction source term, �̇�𝑘.

The species conservation equation 2.21 in the stagnation flow takes the following form

𝜌𝑢𝜕𝑌𝑘𝜕𝑧

+ 𝜕𝜕𝑧

(𝜌𝑌𝑘𝑉𝑘) − 𝑊𝑘�̇�𝑘 = 0 𝑘 = 1, 2, … , 𝐾𝐾 ( 2.24)

2.2.3.1 Chemical mechanism

The gas-phase chemical kinetics has been described using two chemical kinetics mechanisms

both utilizing recently advanced PAH formation pathways. The first chemical mechanism has

been developed by the German Aerospace Center (DLR) chemical kinetics department and it

will be referred to as DLR mechanism hereafter. The other chemical kinetic mechanism used in

this work has been developed by the Clean Combustion Research Center at King Abdullah

25

University of Science and Technology (KAUST) and this mechanism will be referred to as the

KAUST mechanism hereafter. A brief description of each of these mechanisms with an emphasis

on PAH formation pathways is given in the following sections.

DLR mechanism

Details of the DLR mechanism can be found in [87,140,141]. This chemical kinetic model,

developed for methane and ethane-fueled flames, contains 93 species and 719 reactions. The

mechanism provides growth and oxidation of PAH species up to five-ring aromatic species. The

C0–C2 chemistry in the DLR mechanism is based on the Leeds model [142] with updates from

[75]. The dominant routes for formation of the first aromatic ring in the DLR mechanism based

on [87] are the following reactions:

𝑖−C4H5 + C2H2 ⇐⇐⇐⇐⇐⇐⇒ A1 + H ( 2.25)

C3H3 + C3H3 ⇐⇐⇐⇐⇐⇐⇒ A1 ( 2.26)

H2CCCCH + C2H3 ⇐⇐⇐⇐⇐⇐⇒ A1 ( 2.27)

The growth mechanism considered for aromatic species beyond benzene as it is shown in

Figure 2.3 are: HACA, hydrogen atom migration yielding the five- and six-member rings,

interconversion of five- and six-member rings and zigzag aromatic edges; free radical addition

schemes, methyl substitution/acetylene addition pathways, cyclopentadienyl moiety in aromatic

ring formation and reactions between aromatic radicals and molecules. Several small radicals

and small molecules containing one to six carbon molecules were employed in the mentioned

PAH molecule growth and for the H atom abstraction from hydrocarbons. The hydrogen atom

migration was considered as part of the HACA reaction set.

Most of the PAH reactions are multi-step elementary reaction sequences including a lot of

intermediate species and are studied in most cases only qualitatively. These sequences have been

included in this model as lumped reactions (e.g., aromatic + aromatic/cyclic). Reaction rates of

heavy PAH molecules have been estimated based on the reaction rates of analogy reactions of

one- and two ring-molecules. The estimation of the reaction rates of aromatic + aromatic/cyclic

reactions has been done by increasing the frequency factors in the Arrhenius expression. For the

reactions of heavy PAHs with small radicals and molecules corresponding reaction rates for

small PAH molecules were adopted.

26

The reaction mechanism has been validated for flame speeds of methane and ethylene;

concentrations profiles of small molecules and radicals, medium size and high molecular mass

rings and of soot volume fractions in laminar premixed flames as well as at shock tube

conditions [140], counterflow non-premixed flames [87], and a laminar coflow diffusion flame

[141].

Figure 2.3 Schematic representation of the major reaction pathways for the formation of large PAHs considered by the DLR chemical kinetic mechanism.

KAUST mechanism

The KAUST mechanism is developed for modeling C1–C4 fuel oxidation [88]. The mechanism

contains 202 species and 1351 reactions. The PAH growth up to the formation of coronene (A7)

is included in this mechanism. The fuel pyrolysis/oxidation and molecular growth up to benzene

were based on USC Mech [143]. The A1 growth pathways considered in this mechanism involve

propargyl (C3H3) recombination, addition of C2Hx on C4Hy molecules, and addition of CH3 on

CH

CH2

CH2

CH

H, O

, OH

, C2 H

,

C2 H

2 , C4 H

, C4 H

2

+C2H (-H)

+C4 H

5 (-H, -H2 )

CH3 (-H)

O(-CO)

H, O, OH

H2, OH, H2O

CH3 (-H2)

H

+C2H2 (-H)

H, OHH2 , H

2 O

HACA

HACA

HACA

HA

CA

HACA

HACA

HACA

CH2

CH

C4H3, C4H2

C3 H

3

HACA

+C4 H

4

+H (-H2)

H, OH,C2H

H2, H2O,C2H2

+H

(-H

2)

+ ,

,

+C4H4,

C 4H2

CH

CH

C4H2

27

cyclopentadienyl (C5H5) radicals. The reactions for the growth of PAHs larger than benzene are

HACA, reactions involving species with odd-carbon number species such as indenyl (C9H7),

benzyl (C6H5CH2), C5H5 and C3H3 and the addition of C4H4 to large PAH radicals.

Figure 2.4 Schematic representation of the major reaction pathways for the formation of large PAHs considered by the KAUST chemical kinetic mechanism.

The reaction rates for PAH molecule reactions that were not present in the literature were

determined through quantum calculations using the density functional theory along with the

transition state theory. The rate constants for PAH reactions were obtained in the high pressure

limit, as PAH molecules are large in size and their reactions do not exhibit substantial pressure

dependence. The KAUST mechanism has been validated in several laminar premixed and

counterflow flames, where a reasonable agreement between the observed and simulated PAH

concentrations were obtained [88].

CH

CHCH

CH

H, C

H3 ,

C3 H

3

C4H4

2C2 H

2 , H

C2H4

C 3H3

H, C

H3 ,

C3 H

3 2C2H2, H

C 3H3

H, C

H 3, C 3H

3

CH

C2H2

C 2H2

C2 H

2

C2H2

C 2H 2

H, CH3, C3H3

2C2H

2, H

C4H

4

H, C2H2

CH3, C2H2

C3H3, C2H2 C3H3, C2H2

H, C2H2

CH3, C2H2

C4H4

C4 H

4

28

2.3 Soot Aerosol Dynamics Model

Soot particle size, concentration, and interaction with the gas phase are usually the soot

properties of most interest. For a soot particle confined in an infinitesimal volume of gas, the

physical and chemical processes shaping the size distribution are summarized in Figure 2.5.

These processes could be divided into two groups. The first group is the collection of those

processes occurring inside the element including gas-to-particle conversion and coagulation. The

second group are external processes that transport particles across the boundaries of the element

such as diffusion and thermophoresis. A general dynamic equation (GDE) for the particle

number density, 𝑛(𝑣, r, 𝑡), that includes all of these processes can be derived from the

Smoluchowski equation [144]. This equation is also referred to as a population balance equation.

For the number density of the particles in a volume range between 𝑣 and 𝑣 + 𝑑𝑣, 𝑛𝑣, the general

dynamic equation for the particles contained in a large chamber with a sufficiently small surface-

to-volume ratio to neglect deposition on the walls and sedimentation, is expressed by [106].

𝜕𝑛𝑣𝜕𝑡

+ 𝛻. 𝑛𝑣𝕧 = 𝛻. 𝐷𝛻𝑛𝑣 + [𝜕𝑛𝑣𝜕𝑡 ]𝑔𝑟𝑜𝑤𝑡ℎ

+ [𝜕𝑛𝑣𝜕𝑡 ]𝑐𝑜𝑎𝑔

+ [𝜕𝑛𝑣𝜕𝑡 ]𝑓𝑟𝑎𝑔

− 𝛻. 𝑐𝑛𝑣 ( 2.28)

In this expression, the diffusion coefficient 𝐷 is a function of particle size and 𝑐 is the particle

velocity resulting from the external force field; the second term on the right-hand side is the

summation of the growth terms; the third term on the right-hand side represents collisions

between particles; the fourth term on the right-hand side represents the change in number density

due to the fragmentation process.

Figure 2.5 Processes shaping the particle size distribution function in a small volume element of gas. Diffusion and sedimentation involve transport across the walls of the element. Coagulation, nucleation, and growth take place within the element. (Source: Reprinted from ref. [106])

29

In general, an infinite number of discrete particle sizes are present in an aerosol-containing

environment. In addition, the GDE is a nonlinear, partial integrodifferential equation. Thus,

numerical modeling is required. One of the numerical solution procedures for the dynamic

aerosol balance equation is finite sectional approximation [145]. This method is used to

approximate the virtually continuous size spectrum by a set of size classes, or sections, within

which all particles are assumed to be of the same size or the functional form of the size

distribution within the section is specified. By dividing the entire particle size domain into

sections and dealing only with one integral quantity in each section, the number of conservation

equations required is simply equal to the number of sections. There is also the possibility to track

multiple integral quantities per section. For example in addition to particle number density,

particle surface area, number of particles per aggregate, and composition of the particles can be

tracked within each section. For each independent quantity a GDE should be solved per section.

In the sections that follow, the sectional method used in this thesis is described and the

mathematical methods for characterizing aerosol size and chemical properties are discussed.

2.3.1 The sectional aerosol dynamics model The sectional soot aerosol dynamics model used in this thesis is based on the fixed pivot

approach in the classical sectional description of the particle population balance equation [146].

The mass range of the fractal-like solid soot aggregates is divided into a number of discrete

sections (i.e., particle mass bins). Each section represents a collection of aggregates with a fixed

prescribed mass. The representative mass of sections is a geometric progression with common

ratio 𝑓𝑠, also called the sectional spacing factor, and the scale factor equal to the mass of a dimer,

𝑈𝐷𝐼𝑀 . Eq. 2.29 shows the relationship between the mass of each aggregate in section 𝑖, 𝑈𝑖 (g #⁄ ),

the common ratio and scale factor.

𝑈𝑖 = 𝑈𝐷𝐼𝑀 × 𝑓𝑠𝑖−1 ( 2.29)

All soot aggregates in a section are assumed to be of similar enough character that they can be

modelled using mean characteristics. Soot aggregates fall into individual sections according to

their mass. A transport equation for the number density of soot aggregates is constructed and

solved in each section. The nucleation step is the process of formation of dimers from the gas-

phase incipient species. The soot dimers are assumed to be spherical and are added to the first

section. Processes which increase the mass of aggregates (i.e., coagulation and surface growth)

30

move particles from lower sections to the higher sections. On the other hand, higher section

particles move to lower sections by oxidation or fragmentation.

In addition to the aggregate number density equation, a transport equation for primary particle

number density is considered for each section. By conserving the primary particles within the

aggregates, the additional transport equation enables the model to predict the experimentally

observed fractal-like aggregate structures of soot particles [39,56,57]. Some simplifying

assumptions have been made to derive the primary particle number density equation. Primary

particles are considered to be spherical. Similar to the aggregates, it is assumed that the primary

particles within aggregates of the same section are similar enough that they can be modelled

using mean characteristics and they are connected together by point contact (i.e., particle necking

has been neglected). Another simplifying assumption is having a universal fractal dimension,

𝐷𝑓 , of 1.8 for aggregates larger than the primary spherule mass; whereas smaller particles are

assumed to be dense spheres (𝐷𝑓 = 3.0) [57,68,70]. A constant fractal dimension is a common

assumption in aerosol dynamics modeling under concurrent particle nucleation, coagulation, and

surface growth processes [113,124,147]. The structure of an aggregate could now be completely

determined by knowing the fractal dimension, the mass of a single aggregate in the section, the

primary particle number density, and the aggregate number density. The soot properties that can

be extracted are as follow: particle size distribution (PSD), soot volume fraction, primary particle

diameter, aggregate surface area, and number of primary particles per aggregate.

Based on the above descriptions, the conservation equations for aggregate number density and

primary particle number density in an axisymmetric cylindrical coordinate in each section are as

follows:

𝜌𝑣𝜕𝑁𝑖

𝑎

𝜕𝑟+ 𝜌𝑢

𝜕𝑁𝑖𝑎

𝜕𝑧= − 1

𝑟𝜕𝜕𝑟 (𝑟𝜌𝑁𝑖

𝑎𝑉𝑖,𝑟𝑎 ) − 𝜕

𝜕𝑧 (𝜌𝑁𝑖𝑎𝑉𝑖,𝑧

𝑎 ) + 𝜌𝑆�̇�𝑎 ( 2.30)

𝜌𝑣

𝜕𝑁𝑖𝑝

𝜕𝑟+ 𝜌𝑢

𝜕𝑁𝑖𝑝

𝜕𝑧= − 1

𝑟𝜕𝜕𝑟 (𝑟𝜌𝑁𝑖

𝑝𝑉𝑖,𝑟𝑝

) − 𝜕𝜕𝑧 (𝜌𝑁𝑖

𝑝𝑉𝑖,𝑧𝑝

) + 𝜌𝑆�̇�𝑝 ( 2.31)

(𝑖 = 1, 2, … , 𝑀𝑆)

Here, superscripts 𝑎 and 𝑝 refers to those parameters associated with aggregates and primary

particles, respectively; 𝑁𝑖 is the number of 𝑖𝑡ℎ sectional soot particles per unit mass of the

gaseous mixture; 𝑀𝑆 is the total number of soot sections; 𝑉𝑖 is the diffusive velocity of soot

particles in section 𝑖, and 𝑆�̇� contains the source and sink terms associated with the rate of change

of sectional mass and can be expressed in terms of soot process:

31

𝑆�̇� = (𝜕𝑁𝑖𝜕𝑡 )𝑛𝑢

+ (𝜕𝑁𝑖𝜕𝑡 )𝑐𝑜𝑛𝑑

+ (𝜕𝑁𝑖𝜕𝑡 )𝑠𝑔

+ (𝜕𝑁𝑖𝜕𝑡 )𝑜𝑥

+ (𝜕𝑁𝑖𝜕𝑡 )𝑐𝑜𝑎𝑔

+ (𝜕𝑁𝑖𝜕𝑡 )𝑓𝑟

( 2.32)

where, the processes considered are inception ( ), surface condensation ( ), chemical

surface growth ( ), oxidation ( ), coagulation ( ) and fragmentation ( ). Inception is

considered only for the first section. For the 1-D stagnation flow, Eqs. 2.30 and 2.31 transform

to:

𝜌𝑢𝜕𝑁𝑖

𝑎

𝜕𝑧+ 𝜕

𝜕𝑧 (𝜌𝑁𝑖𝑎𝑉𝑖

𝑎) − 𝜌𝑆�̇�𝑎 = 0 ( 2.33)

𝜌𝑢

𝜕𝑁𝑖𝑝

𝜕𝑧+ 𝜕

𝜕𝑧 (𝜌𝑁𝑖𝑝𝑉𝑖

𝑝) − 𝜌𝑆�̇�

𝑝 = 0 ( 2.34)

(𝑖 = 1, 2, … , 𝑀𝑆)

2.3.1.1 Nucleation model

In view of the PAH-based soot formation pathways, the formation and growth of aromatic

species bridges the main combustion zone chemistry and soot formation. This assumption is

founded based on evidence of dependency of the existence of small soot particles on PAH

species [11,148,149]. Therefore, nucleation is modeled based on dimerization of a pair of PAH

molecules. The rate of formation of dimers is considered to be proportional to the rate of

collision of PAH species [86,91]. A sticking efficiency has been used to calculate nucleation rate

from the collision rate. Nucleation is determined by the rate of collision of the nucleating PAH

molecules in the free-molecular regime as:

(𝜕𝑁1𝜕𝑡 )𝑛𝑢

=𝐴𝑣

2

𝜌 √8𝜋𝑘𝐵𝑇𝐶𝑚𝑎𝑠𝑠 ∑ ∑ 𝜂𝑘𝑗√

𝑁𝐶,𝑘 + 𝑁𝐶,𝑗

𝑁𝐶,𝑘𝑁𝐶,𝑗 (𝑑PAH,𝑘 + 𝑑PAH,𝑗

2 )

2[PAH]𝑘[PAH]

𝐾PAH

𝑗=𝑘

𝐾PAH

𝑘=1( 2.35)

(𝜕𝑁𝑖𝜕𝑡 )𝑛𝑢

= 0 (𝑖 = 2,3, … , 𝑀𝑆)

In this expression, 𝑘𝐵 is the Boltzmann constant; 𝐶𝑚𝑎𝑠𝑠 is the mass of a carbon atom; 𝑁𝑐 is the

number of carbon atoms in the incipient PAH species; 𝑑PAH is the diameter of the incipient PAH

species; 𝐴𝑣 is Avogadro's number; [PAH] denotes the mole concentration of the incipient PAH

species. 𝐾PAH is the total number of nucleating species; 𝜂𝑘𝑗 is the sticking efficiency of the two

colliding PAHs. Different PAH nucleating species has been used in this work ranging from

pyrene to coronene. Pyrene is the most widely used PAH nucleating species in soot modeling

[86,91,150,151]. Recent experimental studies determined the PAH molecules that participate in

32

soot production have a mean fringe length of 0.65 nm [104] and a mass range of 202 amu to 374

amu [53]. These findings suggest that dimerization of PAH molecules from 20 carbon atoms to

30 carbon atoms is plausible. More details about the nucleation model and the PAH species is

provided in the following chapters.

2.3.1.2 Condensation model

One of the heterogeneous gas-to-particle conversions is the growth of the particle due to

adsorption of gas phase species to the surface of particle which is referred to as condensation.

Similar to nucleation, condensation is modeled based on collision of condensing species and the

surface of particles [91]. The PAH molecules allowed to condense are assumed to be identical to

the PAH molecules that form dimers. The rate of change of mass in section is calculated by

𝐼𝑐𝑜𝑛𝑑,𝑖 = ∑ 𝛾𝑖𝑘𝛽𝑖𝑘𝑁𝐶,𝑘𝐶𝑚𝑎𝑠𝑠[PAH]𝑘𝑁𝑖𝑎

𝐾PAH

𝑘=1 ( 2.36)

where 𝐼𝑐𝑜𝑛𝑑,𝑖 is the rate of mass growth of the 𝑖𝑡ℎ section soot aggregates due to condensation in

the unit of gs/gmix/sec, and is always non-negative; 𝛽𝑖𝑘 is the collision kernel of the 𝑘𝑡ℎ

condensing species and the 𝑖𝑡ℎ section soot aggregate; 𝛾𝑖𝑘 is the sticking probability which takes

into account the probability of the molecules bouncing off the surface after collision.

The evaluated mass growths have to be interpreted in the terms of the sectional model. Since the

mass of aggregates are fixed, the growth of mass of an aggregate in section 𝑖 is reflected in the

sectional model by transferring the equivalent amount of added mass in terms of number of

aggregates from section 𝑖 to section 𝑖 + 1. In order to conserve the primary particle numbers the

growth term for the primary particle is multiplied by the number of primary particles per

aggregate in the transport equations for primary particles (e.g., Eq. 2.31). The above descriptions

have been shown in Eqs. 2.37 and 2.38.

(𝜕𝑁𝑖

𝑎

𝜕𝑡 )𝑐𝑜𝑛𝑑=

⎩⎪⎪⎪⎨⎪⎪⎪⎧−

𝐼𝑐𝑜𝑛𝑑,1

𝑚2 − 𝑚1𝐼𝑐𝑜𝑛𝑑,𝑖−1

𝑚𝑖 − 𝑚𝑖−1−

𝐼𝑐𝑜𝑛𝑑,𝑖

𝑚𝑖+1 − 𝑚𝑖𝐼𝑐𝑜𝑛𝑑,𝑀𝑆−1

𝑚𝑀𝑆 − 𝑚𝑀𝑆−1

if 𝑖 = 1

( 2.37)if 𝑖 = 2, … , 𝑀𝑆 − 1

if 𝑖 = 𝑀𝑆

33

(

𝜕𝑁𝑖𝑝

𝜕𝑡 )𝑐𝑜𝑛𝑑

=

⎩⎪⎪⎪⎨⎪⎪⎪⎧−

𝐼𝑐𝑜𝑛𝑑,1

𝑚2 − 𝑚1𝐼𝑐𝑜𝑛𝑑,𝑖−1

𝑚𝑖 − 𝑚𝑖−1𝑛𝑝,𝑖−1 −

𝐼𝑐𝑜𝑛𝑑,𝑖

𝑚𝑖+1 − 𝑚𝑖𝑛𝑝,𝑖

𝐼𝑐𝑜𝑛𝑑,𝑀𝑆−1

𝑚𝑀𝑆 − 𝑚𝑀𝑆−1𝑛𝑝,𝑖−1

if 𝑖 = 1

( 2.38)if 𝑖 = 2, … , 𝑀𝑆 − 1

if 𝑖 = 𝑀𝑆

Here, is the representative mass of the section aggregate; , is the number of primary

particles per aggregate of the section and it is equal to / . It has to be noted for the first

section, condensation would cause particles to leave this section therefore the growth rate is

always negative, and since this section only contains soot monomers the rate for primary

particles and aggregates are equal. In contrast, the growth term of the last section is always

positive. It should be emphasized that the sum of all the growth terms are equal to zero to ensure

that no new particles are numerically formed due to growth processes.

∑ (𝜕𝑁𝑖

𝑎

𝜕𝑡 )𝑐𝑜𝑛𝑑

𝑀𝑆

𝑖=1= ∑ (

𝜕𝑁𝑖𝑝

𝜕𝑡 )𝑐𝑜𝑛𝑑

𝑀𝑆

𝑖=1= 0 ( 2.39)

2.3.1.3 Chemical surface growth and oxidation models

The heterogeneous reactions of soot particle surfaces with the gas phase considered in this thesis

are detailed in Table 2.1. The soot mass growth and oxidation by oxygen, O2, is based on the

well-known hydrogen–abstraction–carbon–addition (HACA) scheme [86,91]. In the HACA

scheme, the kinetics of the surface reactions are described using the concept of surface sites (an

armchair site, which is a site with four carbon atoms as illustrated in Figure 2.6), which are

carbon atoms either saturated (Csoot–H) or dehydrogenated (Csoot ∘ ) on the surface of soot

particles. The concentration of saturated sites, [Csoot–H] (mole/cc), is calculated by Eq. 2.40

[Csoot– H] =𝐴𝑠𝐴𝑣

𝜒Csoot–H ( 2.40)

Figure 2.6 Illustration of armchair sites on the surface of a soot particle.

Armchair�site

34

Table 2.1 HACA–based soot surface growth and oxidation reactions [86], 𝑘 = 𝐴𝑇 𝑏𝑒−𝐸𝑎 𝑅𝑇⁄ .

No. Reaction A (𝑐𝑚3

𝑚𝑜𝑙.𝑠) b Ea (𝑘𝑐𝑎𝑙𝑚𝑜𝑙 )

S1 Csoot–H+ H

Csoot∘ + H2 4.2×1013 0.0 13.0

S2 Csoot–H+ OH

Csoot∘ + H2O 1.0×1010 0.73 1.43

S3 Csoot∘ + H ←←←←←←←←→ Csoot–H 2.0×1013 0.0 0.0

S4 Csoot∘ + C2H2 ←←←←←←←←→ Csoot–H+ H 8.0×107 1.56 3.8

S5 Csoot∘ + O2 ←←←←←←←←→ 2CO + product 2.2×1012 0.0 7.5

S6 Csoot–H+ OH ←←←←←←←←→ CO + product γOH=0.13

where – is the number of sites per unit soot surface area and estimated to be 0.23 site/Å2

[86]; 𝐴𝑠 (cm2/cc) is the surface density of soot particles and 𝐴𝑣 is Avogadro’s number. The

concentration of dehydrogenated sites [Csoot ∘ ] is similarly calculated with 𝜒Csoot° as the number

of dehydrogenated sites (Csoot ∘ ) per unit surface area. Finally by assuming a steady state for

Csoot ∘ , 𝜒Csoot∘ can be calculated from Eq. 2.41 and substituted to find the S4 and S5 rates

𝜒𝐶𝑠𝑜𝑜𝑡∘ =(𝑘1[𝐻] + 𝑘2[𝑂𝐻])𝜒𝐶𝑠𝑜𝑜𝑡–𝐻

𝑘−1[𝐻2] + 𝑘−2[𝐻2𝑂] + 𝑘3[𝐻] + 𝑘4[𝐶2𝐻2] + 𝑘5[𝑂2] ( 2.41)

Thus, the mass growth rate due to HACA and mass reduction rate due to O2 oxidation are

𝐼C2H2,𝑖 = 2𝛼𝐶𝑚𝑎𝑠𝑠𝐴𝑠,𝑖

𝐴𝑣

(𝑘1[H] + 𝑘2[OH])𝜒𝐶𝑠𝑜𝑜𝑡–𝐻 𝑘4[C2H2]𝑁𝑖𝑝

𝑘−1[H2] + 𝑘−2[H2O] + 𝑘3[H] + 𝑘4[C2H2] + 𝑘5[O2] ( 2.42)

𝐼O2,𝑖 = 2𝛼𝐶𝑚𝑎𝑠𝑠𝐴𝑠,𝑖

𝐴𝑣

(𝑘1[H] + 𝑘2[OH])𝜒Csoot–H𝑘5[O2]𝑁𝑖𝑝

𝑘−1[H2] + 𝑘−2[𝐻2𝑂] + 𝑘3[H] + 𝑘4[C2H2] + 𝑘5[O2] ( 2.43)

where 𝐴𝑠,𝑖 is the primary particle surface area in the 𝑖𝑡ℎ section; 𝛼 is the surface reactivity

parameter. As it is a focus on this thesis, a complete description of 𝛼 is provided in Chapter 4.

Soot oxidation by the OH radical is modeled based on kinetic theory with a probability, 𝛾OH, the

portion of collisions that result in reaction, of 0.13 [86,107,108].

𝐼OH,𝑖 = 𝛾OH𝛽OH,𝑖𝐶𝑚𝑎𝑠𝑠[OH]𝑁𝑖𝑎 ( 2.44)

(𝜕𝑁𝑖𝜕𝑡 )𝑠𝑔

is evaluated using Eqs. 2.37 and 2.38 by substituting 𝐼𝑐𝑜𝑛𝑑,𝑖 with 𝐼C2H2,𝑖. The source

terms due to surface oxidation are calculated as follows:

35

(𝜕𝑁𝑖

𝑎

𝜕𝑡 )𝑜𝑥=

⎩⎪⎪⎪⎨⎪⎪⎪⎧

𝐼𝑜𝑥,2

𝑚2 − 𝑚1−

𝐼𝑜𝑥,1

𝑚1

𝐼𝑜𝑥,𝑖+1

𝑚𝑖+1 − 𝑚𝑖−

𝐼𝑜𝑥,𝑖

𝑚𝑖 − 𝑚𝑖−1𝐼𝑜𝑥,𝑀𝑆

𝑚𝑀𝑆−1 − 𝑚𝑀𝑆

if 𝑖 = 1

( 2.45)

if 𝑖 = 2, … , 𝑀𝑆 − 1

if 𝑖 = 𝑀𝑆

(

𝜕𝑁𝑖𝑝

𝜕𝑡 )𝑜𝑥

=

⎩⎪⎪⎪⎨⎪⎪⎪⎧

𝐼𝑜𝑥,2

𝑚2 − 𝑚1𝑛𝑝,2 −

𝐼𝑜𝑥,1

𝑚1𝐼𝑜𝑥,𝑖+1

𝑚𝑖+1 − 𝑚𝑖𝑛𝑝,𝑖+1 −

𝐼𝑜𝑥,𝑖

𝑚𝑖 − 𝑚𝑖−1𝑛𝑝,𝑖

𝐼𝑜𝑥,𝑀𝑆

𝑚𝑀𝑆−1 − 𝑚𝑀𝑆𝑛𝑝,𝑀𝑆

if 𝑖 = 1

( 2.46)if 𝑖 = 2, … , 𝑀𝑆 − 1

if 𝑖 = 𝑀𝑆

The difference between the way the growth source terms are evaluated and the way oxidation

terms are evaluated lies in the fact that oxidation moves particles from high sections to low

sections while the growth terms do the opposite.

2.3.1.4 Coagulation model

Coagulation, which is the joining together of two soot particles when they collide, increases soot

aggregate size, effectively increasing soot aggregates number in a higher-mass section while

decreasing soot aggregate concentration in lower-mass sections. In total, coagulation decreases

the total number of aggregates while having no effect on the total number of primary particles.

The coagulation rates is estimated to be equal to the binary collision rate between soot

aggregates calculated in the entire Knudsen number regime [121,122] with a sticking probability

[118]. The coagulation terms for aggregates and primary particles in section are calculated as:

(𝜕𝑁𝑖

𝑎

𝜕𝑡 )𝑐𝑜𝑎𝑔= ∑ ∑ (1 −

𝛿𝑗𝑘

2 ) 𝜂𝑖𝑗𝑘𝛽𝑗𝑘𝜉𝑗𝑘𝑁𝑗𝑎𝑁𝑘

𝑎𝑘𝑗

− 𝑁𝑖𝑎

∑ 𝛽𝑖𝑚𝜉𝑖𝑚𝑁𝑚𝑎

𝑀𝑆

𝑚=1 ( 2.47)

(

𝜕𝑁𝑖𝑝

𝜕𝑡 )𝑐𝑜𝑎𝑔

= ∑ ∑ (1 −𝛿𝑗𝑘

2 ) 𝜂𝑝,𝑖𝑗𝑘𝜂𝑖𝑗𝑘𝛽𝑗𝑘𝜉𝑗𝑘𝑁𝑗𝑎𝑁𝑘

𝑎𝑘𝑗

− 𝑁𝑖𝑝

∑ 𝛽𝑖𝑚𝜉𝑖𝑚𝑁𝑚𝑎

𝑀𝑆

𝑚=1 ( 2.48)

{∀𝑘 ∈ [1, 𝑖] ∧ 𝑗 ∈ [𝑘, 𝑖]|𝑚𝑖−1 < 𝑚𝑗 + 𝑚𝑘 < 𝑚𝑖+1}

In this expression, 𝛿𝑗𝑘 is the Kronecker delta; 𝛽𝑗𝑘 is the collision kernel of two aggregates from

the 𝑗𝑡ℎ and the 𝑘𝑡ℎ sections [121,122,152] and 𝜉𝑗𝑘 is the coagulation efficiency of this collision

[118]. In order to conserve the mass and number of aggregates during the coagulation modeling

36

the newly formed aggregates are transferred into two consecutive sections. This division has

been accomplished using parameter 𝜂𝑖𝑗𝑘 which defined as follows [121,122]:

𝜂𝑖𝑗𝑘 =

⎩⎪⎪⎨⎪⎪⎧

𝑚𝑖+1 − (𝑚𝑗 + 𝑚𝑘)𝑚𝑖+1 − 𝑚𝑖

𝑚𝑖−1 − (𝑚𝑗 + 𝑚𝑘)𝑚𝑖−1 − 𝑚𝑖

0

if 𝑚𝑖 ≤ 𝑚𝑗 + 𝑚𝑘 < 𝑚𝑖+1

( 2.49)if 𝑚𝑖−1 < 𝑚𝑗 + 𝑚𝑘 < 𝑚𝑖

else 𝜂𝑝,𝑖𝑗𝑘 in Eq. 2.48 assigns primary particles to two adjacent sections based on the mass average of

the number of primary particles per aggregates. 𝜂𝑝,𝑖𝑗𝑘 and 𝜂𝑖𝑗𝑘 together in Eq. 2.48 ensure that the

primary particle size is conserved [121,122].

𝜂𝑝,𝑖𝑗𝑘 =𝑚𝑖

𝑚𝑗 + 𝑚𝑘(𝑛𝑝,𝑗 + 𝑛𝑝,𝑘) ( 2.50)

2.3.1.5 Fragmentation model

Fragmentation is the process of breakage of the aggregate chain connecting primary particles. In

this work, only oxidation-driven fragmentation has been considered. The model assumes that

aggregates break into two daughter aggregates with equal mass and no fragmentation occurs for

an aggregate containing fewer than two primary particles. Based on these assumptions the

fragmentation rate of the aggregates in the section is expressed as [123,153]

(𝜕𝑁𝑖

𝑎

𝜕𝑡 )𝑓𝑟=

⎩⎪⎨⎪⎧

𝛤+𝑆2𝑁2𝑎

(𝛤 − 1)𝑆𝑖𝑁𝑖𝑎 + 𝛤+𝑆𝑖+1𝑁𝑖+1

𝑎

(𝛤 − 1)𝑆𝑀𝑆𝑁𝑀𝑆𝑎

if 𝑖 = 1

( 2.51)if 𝑖 = 2, … , 𝑀𝑆 − 1

if 𝑖 = 𝑀𝑆

(

𝜕𝑁𝑖𝑝

𝜕𝑡 )𝑓𝑟

=

⎩⎪⎪⎨⎪⎪⎧𝛤+𝑆2𝑁2

𝑎𝑛𝑝,2

𝑓𝑠

(𝛤 − 1)𝑆𝑖𝑁𝑖𝑎𝑛𝑝,𝑖 +

𝛤+𝑆𝑖+1𝑁𝑖+1𝑎 𝑛𝑝,𝑖+1

𝑓𝑠(𝛤 − 1)𝑆𝑀𝑆𝑁𝑀𝑆

𝑎 𝑛𝑝,𝑀𝑆

if 𝑖 = 1

( 2.52)if 𝑖 = 2, … , 𝑀𝑆 − 1

if 𝑖 = 𝑀𝑆

Here, 𝛤 and 𝛤+ are breakage distribution functions that distribute the newly formed aggregates

into two adjacent sections such that the number and mass of aggregates are conserved. The

breakage distribution functions are calculated as [153]:

𝛤 =𝑓𝑠 − 2𝑓𝑠 − 1

( 2.53)

𝛤+ =𝑓𝑠

𝑓𝑠 − 1 ( 2.54)

37

In Eq. 2.51 𝑆𝑖 is the fragmentation rate per aggregate and is taken from [153]

𝑆𝑖 = 1.0 × 105𝑟𝑜𝑥,𝑖(𝑛𝑝,𝑖)1 𝐷𝑓⁄ ( 2.55)

where, 𝑟𝑜𝑥,𝑖 is the rate of oxidation on a mass basis of soot aggregates in section 𝑖 per unit surface

area; 𝐷𝑓 denotes the aggregate fractal dimension.

2.4 Transport Properties In order to solve the governing equations outlined in previous sections, transport properties of

the gas and soot particles need to be evaluated. The diffusion velocities of the 𝑘𝑡ℎ gaseous

species (𝑉𝑘 in Eq. 2.21) and soot particles (𝑉𝑖 in population balance equation, Eqs. 2.30– 2.34)

are calculated using a mixture-average formulation. This approximation for the diffusion

velocities implements only a Fickian description of diffusion for each component of the mixture.

In this matter the interactive diffusive effects caused by concentration gradients of different

mixture component on each other are neglected. In order to ensure the diffusion velocities do not

violate conservation of mass, a correction velocity, as detailed in [154], is added to the

expression for the diffusion velocity. Thus, the diffusion velocity is calculated as:

𝑉𝑘,𝑥𝑖= 𝑉𝐷,𝑥𝑖

+ 𝑉𝑇 ,𝑥𝑖+ 𝑉𝑐,𝑥𝑖 ( 2.56)

where 𝑉𝐷,𝑥𝑖 and 𝑉𝑇 ,𝑥𝑖

are the ordinary diffusion and thermal diffusion velocities, respectively

and 𝑉𝑐,𝑥𝑖 is the correction diffusion velocity. The ordinary diffusion velocity and thermal

diffusion velocity of the 𝑘𝑡ℎ species are obtained by:

𝑉𝐷,𝑥𝑖= −

𝐷𝑘𝜒𝑘

𝜕𝜒𝑘𝜕𝑥𝑖

( 2.57)

𝑉𝑇 ,𝑥𝑖= −

𝐷𝑘𝑇

𝜌𝑌𝑘

1𝑇

𝜕𝑇𝜕𝑥𝑖

( 2.58)

where 𝜒𝑘 is the 𝑘𝑡ℎ species mole fraction; 𝐷𝑘𝑇 is the 𝑘𝑡ℎ species thermal diffusion coefficient; 𝐷𝑘

is the mixture diffusion coefficient for the 𝑘𝑡ℎ species. The ordinary diffusion velocity and

thermal diffusion velocity (also known as thermophoretic velocity) of the 𝑖𝑡ℎ section soot

aggregates are obtained by

𝑉𝐷𝑠,𝑥𝑖𝑎 = −

𝐷𝑖𝑎

𝑁𝑖𝑎

𝜕𝑁𝑖𝑎

𝜕𝑥𝑖 ( 2.59)

𝑉𝑇𝑠,𝑥𝑖= −𝐷𝑇 ,𝑖

𝑎 1𝑇

𝜕𝑇𝜕𝑥𝑖

( 2.60)

38

where 𝐷𝑇 ,𝑖𝑎 is the 𝑖𝑡ℎ section thermal diffusion coefficient; 𝐷𝑖

𝑎 is the diffusion coefficient of the

𝑖𝑡ℎ section aggregate. The thermophoretic velocity for the primary particles and aggregates are

the same. The ordinary diffusion velocity is calculated as follows:

𝑉𝐷𝑠,𝑥𝑖

𝑝 = −𝐷𝑖

𝑎

𝑁𝑖𝑝

𝜕𝑁𝑖𝑝

𝜕𝑥𝑖 ( 2.61)

Note that the same 𝐷𝑖𝑎 appears in both primary particle and aggregate diffusion velocities. This is

because the aggregates are composed of the primary particles.

The diffusion velocity represents the velocity with which each species moves relative to the bulk

fluid velocity. The diffusion velocities must thus satisfy the conservation expression:

∑ 𝑌𝑗𝑉𝑗

𝐾𝐾+𝑀𝑆

𝑗=1= 0 ( 2.62)

By substituting diffusion velocities for species and soot into the above equation, the following

expression for the correction velocity is obtained:

𝑉𝑐,𝑥𝑖= ∑ 𝐷𝑘

𝜕𝑌𝑘𝜕𝑥𝑖

+𝐷𝑘

𝑇

𝜌1𝑇

𝜕𝑇𝜕𝑥𝑖

𝐾𝐾

𝑘=1+ ∑ 𝐷𝑖

𝑎𝑚𝑖𝜕𝑁𝑖

𝑎

𝜕𝑥𝑖+ 𝐷𝑇 ,𝑖

𝑎 𝑚𝑖𝑁𝑖𝑎 1

𝑇𝜕𝑇𝜕𝑥𝑖

𝑀𝑆

𝑖=1 ( 2.63)

In all simulations, the thermal diffusion is retained only for H2 and H and is neglected for the

other species.

2.4.1 Diffusion coefficients For the gaseous species the mixture diffusion coefficient, 𝐷𝑘, for the 𝑘𝑡ℎ species is calculated as

[155]:

𝐷𝑘 =

1 − 𝑌𝑘

∑𝜒𝑗

𝐷𝑗,𝑘𝐾𝐾𝑗=1,𝑗≠𝑘

( 2.64)

where 𝐷𝑗,𝑘 is the binary diffusion coefficient. The 𝑘𝑡ℎ species thermal diffusion coefficient 𝐷𝑘𝑇 is

evaluated from

𝐷𝑘𝑇 = 𝐷𝑘𝛩𝑘 ( 2.65)

where 𝛩𝑘 is the thermal diffusion ratio [156].

Two different approximations have been used to evaluate the diffusion coefficients for the soot

particles. For modeling the coflow diffusion flame the binary diffusion coefficient of soot

aggregates, 𝐷𝑖𝑎, is given as:

39

𝐷𝑖𝑎 =

𝑘𝐵𝑇 𝐶𝑐(𝐾𝑛)3𝜋𝜇𝑑𝑚

( 2.66)

where 𝑘𝐵 is the Boltzmann constant; 𝜇 is the gas viscosity; 𝑑𝑚 is the mobility diameter; 𝐶𝑐(𝐾𝑛)

is the Cunningham slip correction factor as a function of the Knudsen number 𝐾𝑛 and is

calculated as [157]

𝐶𝑐(𝐾𝑛) = 1 + 1.612𝐾𝑛 ( 2.67)

The Knudsen number 𝐾𝑛 is defined as:

𝐾𝑛 =2𝜆𝑚𝑓𝑝

𝑑𝑚 ( 2.68)

with 𝜆𝑚𝑓𝑝 being the mean free path of the gas. The mobility diameter and the absorbing cluster

radius have been studied by many researchers. In the current sectional aerosol dynamics model,

the calculation of mobility diameter is as follows:

𝑑𝑚 =⎩⎪⎨⎪⎧

2𝑟𝑝𝑛𝑝0.43

2𝑅𝑓 (𝐷𝑓 − 1

2 )

0.7 free-molecular regime

( 2.69)continuum regime

where 𝑟𝑝 is the primary particle radius; 𝑛𝑝 is the number of primary particles per aggregate; and

the outer radius of an aggregate 𝑅𝑓 is defined as:

𝑅𝑓 = 𝑟𝑝(𝑓𝑛𝑝)1 𝐷𝑓⁄ ( 2.70)

with 𝑓 being the volume filling factor which accounts for the fact that even in a closely packed

structure, the spherical monomers cannot occupy the whole available volume [158].

The thermal diffusion coefficient of soot aggregates 𝐷𝑇 ,𝑖𝑎 are calculated according to Talbot et al.

[159] as follows:

𝐷𝑇 ,𝑖𝑎 = 0.55𝜇

𝜌 ( 2.71)

In modeling the stagnation flame, as discussed by Abid et al. [19], the diffusion velocities are the

main drivers of particles and species as they approach the stagnation plate. Therefore, the particle

diffusion coefficients are determined through a more sophisticated expression proposed by Li

and Wang [160]. The binary diffusion coefficient has the form similar to Einstein’s diffusion

coefficient expression:

𝐷𝑖𝑎 = 3

2 √𝑘𝐵𝑇2𝜋𝑚𝑟

(1 + 𝛼′−1.143)0.875

𝑁𝑑𝑚2 𝛺𝑎𝑣𝑔

(1,1)∗ ( 2.72)

40

Here, 𝑚𝑟 is the reduced mass of the gas molecule and particle, 𝑚𝑟 = 𝑚𝑔𝑚𝑝 (𝑚𝑔 + 𝑚𝑝)⁄ and 𝑚𝑝 is the

mass of the particle; 𝑁 is the number density of the gas; 𝛺𝑎𝑣𝑔(1,1)∗

and 𝛼′ are the average reduced

collision integral and the correction factor taken from [160].

The thermal diffusion coefficient for soot aggregates are taken from [161].

𝐷𝑇 ,𝑖𝑎 =

⎝⎜⎜⎛1 − 6

5𝛺𝑎𝑣𝑔

(1,2)∗

𝛺𝑎𝑣𝑔(1,1)∗

⎠⎟⎟⎞ 𝜆

𝑁𝑘𝐵 ( 2.73)

where 𝜆 is the thermal conductivity of gas; 𝛺𝑎𝑣𝑔(1,2)∗

is the average reduce collision integral

determined based on expression given in [161].

2.5 Numerical Methods

Two numerical approaches are used to find the solutions to the governing equations described in

the previous sections for various reacting flows studied in this work. Discretization of the

governing equations for the coflow diffusion flames is done using a control volume scheme.

Parallel computing has been utilized to speed up the calculation for these flames. The premixed

boundary value problem is solved numerically based on the finite difference framework. The

details of the modeling methodology for the coflow diffusion flames and the premixed stagnation

flame are presented in the following sections.

2.5.1 2D coflow diffusion flame

The gas-phase governing equations and the sectional soot equations are discretized based on the

finite volume method on a staggered grid for the coflow diffusion flames. The Semi-Implicit

Method for Pressure Linked Equations (SIMPLE algorithm) is used to handle the pressure and

velocity coupling [162]. The coupling between pressure and velocity in the SIMPLE algorithm is

achieved by transforming the continuity equation into the pressure correction equation. The

diffusive terms are discretized using the second order central difference scheme while the

convective terms are discretized by the power law scheme [162]. Pseudo-time marching is used

to arrive at the converged steady state solution from the initial guess.

The equations of mass, momentum, species, energy, and sectional soot are highly coupled within

themselves and through detailed thermodynamic and transport relations and chemical kinetics.

However, to alleviate the strong interaction between the flow and combustion, and to avoid

41

saturating memory capacity by simultaneously solving this system of partial differential

equations, the governing equations are solved in a semi-coupled way. In this method, the

conservative quantities are divided into three categories: the fluid flow, the gas phase and the

aerosol dynamic. Quantities in each category are solved separately and will be updated in the

next iteration.

Since the flow field acts as the carrier of the gas phase and the solid phase, it can be anticipated

that a fast established flow field will provide a stable base for the reactions and therefore make

the species equations easy to converge. The gas phase and the aerosol dynamic that involve

multi-species, multi-step, chemical reactions are sensitive and stiff systems, and account for most

of the CPU time in the computations. The most effective approach to minimize the

computational costs is to reduce the iteration number by implementing efficient CFD methods

which are compatible with parallel computing. Therefore, the efficient Tri-Diagonal Matrix

Algorithm (TDMA) has been used to solve radial momentum, axial momentum, pressure

correction and energy equations. In order to overcome the stiffness of the soot and species

equations the source term is treated implicitly. In this method the source term, , is estimated

using the Taylor series expansions [163]:

𝑅𝛼𝑛+1 = 𝑅𝛼

𝑛 + ∑𝜕𝑅𝛼𝜕𝑌𝑚

𝜕𝑌𝑚𝑚

+ ∑ 𝒪(𝜕𝑌𝑚2)

𝑚 ( 2.74)

Neglecting the second and higher order terms, the source terms are linearized using Eq. 2.74.

The resulting Jacobian matrices are obtained by the perturbation method [164]. The Gaussian

elimination method is used to solve the resulting linear system at each control volume. The

species equations are solved control-volume-by-control-volume until the whole computational

domain is covered. Then the sectional soot aggregates and number densities are solved in the

same fashion as the species equation.

Offering a potential solution to the computationally intensive combustion simulations, the

Coflame code takes advantage of parallel computing by dividing most of the computational load

between several computational processing units. Since most of the computational load is from

species and sectional soot equations, these parts are parallelized. The parallelization has been

done using the domain decomposition method (DDM) [165]. The computational domain is

decomposed into NUMP sub-domains. Each sub-domain is consisted of a fixed set of

42

computational nodes with boundaries extending in the radial direction. Each sub-domain is then

assigned to a processor for calculation and the calculations in all sub-domains are carried out

simultaneously which makes NUMP the number of computing processors used. The parallel

programming has been performed using message passing interface (MPI) [166].

The structure of the code to solve the system of equations is depicted in Figure 2.7. The

numerical procedures solve for axial velocity 𝑢, radial velocity 𝑣, pressure correction 𝑝′, gaseous

species mass fractions 𝑌𝑘 (𝑘 = 1, 2, … , 𝐾𝐾), sectional soot aggregate number densities

𝑁𝑖𝑎(𝑖 = 1, 2, … , 𝑀𝑆), sectional soot primary particle number densities 𝑁𝑖

𝑝(𝑖 = 1, 2, … , 𝑀𝑆)

and finally temperature T. Convergence is deemed to be achieved when the maximum relative

error of flame temperature, species concentration, and soot volume fraction are all less than 10-4.

Sandia's CHEMKIN [167] and TRANSPORT [168] libraries are incorporated to calculate the

gaseous species thermal properties, transport properties and chemical reaction rates from the

database associated with the selected reaction mechanism.

Figure 2.7 Coflow code solver program structure.

Initial Guess

Solve Axial Momentum

Solve Radial Momentum

Solve Pressure Correction

Correct Velocities and Pressure

Solve Species Mass

Solve Sectional Soot

Solve Energy

Update Mixture Density

Check Convergence Criteria

Solution file

Chemical Mechanism

CHEMKIN Interpreter

CHEMKIN Link FileCHEMKIN Librery

TRANSPORT Fitting

TRANSPORT Link File

TRANSPORT Library

Transport Data

Thermodynamic Data

No

Yes

43

2.5.1.1 Boundary conditions

Inlet conditions are specified for the fuel and air streams at the 𝑧 = 0 boundary. Symmetry

conditions are enforced at the centerline, i.e., at 𝑟 = 0. Free-slip conditions are assumed at the

outer radial boundary (e.g., at = 4.709 cm). Zero-gradient conditions are enforced at the exit

boundary. The mesh and boundary conditions are illustrated schematically in Figure 2.8.

Figure 2.8 Schematic of the coflow diffusion flame boundary conditions and the non-uniform structured mesh.

2.5.2 Premixed stagnation flame

The described soot sectional aerosol dynamic model has been added to the OPPDIF code [169]

in order to simulate soot formation in the premixed stagnation flame. Discretization of the

differential equations in the OPPDIF code uses finite differencing techniques for nonuniform

mesh spacing. The discretization of the sectional aggregate number density and primary particle

number density has been carried out similar to the species conservation equation discretization.

Convective terms are discretized using the second order central difference approximation with

the option to switch to the first order windward differences for better convergence.

44

The diffusive term in the species conservation equation and the sectional soot number density

equations are approximated using an average-central difference approximation. The ordinary and

thermal diffusion velocities for soot and species are approximated at the 𝑗 ± 1/2 positions. The

correction velocity 𝑉𝑐 is computed using Eq. 2.63 at the midpoints by summation of the diffusion

velocities for all the species and soot particles. Upon calculation of the correction velocity the

full diffusion velocities at midpoint is determined by adding the correction velocity to the

diffusion velocity. Then the diffusion term is evaluated with the following difference

approximation:

𝑑𝑑𝑧

(𝜌𝑌𝑘𝑉𝑘)𝑗 ≈(𝜌𝑌𝑘𝑉𝑘)𝑗+1 2⁄ − (𝜌𝑌𝑘𝑉𝑘)𝑗−1 2⁄

𝑧𝑗+1 2⁄ − 𝑧𝑗−1 2⁄ ( 2.75)

All the non-differentiated terms, such as the chemical production rate terms, are evaluated at the

mesh points 𝑗. Coefficients not appearing within derivatives are also evaluated at the mesh

points.

For the implementation of the Newton’s method solution of the governing equations, once the

coupled, nonlinear system of equations has been discretized, the system of equations is cast in

residual form as follows:

𝐹 (𝑣) = 0 ( 2.76)

in which 𝑣 is the vector of all unknowns and 𝐹 (𝑣) is the vector of all equations. If 𝑣•, a collection

of approximate solution vectors, are chosen for the unknowns, the equations 𝐹 likely will not

vanish. Instead, the vector of residuals 𝐹 (𝑣•) is formed by evaluating the functions 𝐹 :

𝐹 (𝑣•) = 𝑅𝐸𝑆 ( 2.77)

The objective is to seek values, 𝑣⋆, with zero residuals, 𝐹 (𝑣⋆) = 0. OPPDIF uses the modular

solver routine TWOPNT to solve the boundary value problem. TWOPNT uses modified damped

Newton’s method to attempt solution of the steady-state equations, and resorts to time integration

when the Newton iteration is not converging [164]. After time integration evolves the solution

toward the steady state, TWOPNT returns to Newton’s method to rapidly converge on the steady

solution. From the initial estimate, 𝑣0, Newton’s method produces a sequence {𝑣(𝑛)} that

converges to the solution of the nonlinear equations:

45

𝑣(𝑛+1) = 𝑣(𝑛) + (𝜕𝐹𝜕𝑣 )𝑣(𝑛)

−1𝐹 (𝑣(𝑛)) ( 2.78)

This algorithm is computationally intensive and suffers from lake of robustness. Evaluation of

the Jacobian matrices 𝜕𝐹 /𝜕𝑣 is time consuming, and convergence to the solution usually requires

a very good initial guess 𝑣0. The modified Newton method necessitates the following

refinements to the original method. First, the Jacobian matrix is only updated after a finite

number of iterations as Jacobian evolution is the most costly component of the algorithm, and the

changes in the linear system is minimal from one iteration to the next. Second, so as to

conservatively adjust the solution in each iteration, and reduce the likelihood of divergence, a

damping parameter 𝜆(𝑛) has been introduced for the evaluation of 𝑣(𝑛+1) from 𝑣(𝑛). In this way the

iteration becomes:

𝑣(𝑛+1) = 𝑣(𝑛) + 𝜆(𝑛)(𝐽 (𝑛))−1𝐹 (𝑣(𝑛)) ( 2.79)

where the damping factor decreases geometrically.

𝜆(𝑛) = 2−0.5, 2−1.0 , …, 2−2.5 ( 2.80)

The elements of the Jacobian are formed by finite difference perturbations in the manner

suggested by [170]. For more details of OPPDIF code, numerical method and modified Newton

method please refer to [124,164,169,171].

2.5.2.1 Boundary conditions

The boundary conditions at the nozzles are:

𝐹 =𝜌𝐼 𝑢𝐼

2 ( 2.81)

𝐺 = 0 ( 2.82)

(𝑑𝐻𝑑𝑧 )𝐼

= 0 ( 2.83)

𝑇 = 𝑇𝐼 ( 2.84)

𝜌𝑢𝑌𝑘 + 𝜌𝑉𝑘𝑌𝑘 = (𝜌𝑢𝑌𝐾)𝐼 ( 2.85)

𝜌𝑢𝑁𝑖 + 𝜌𝑉𝑖𝑁𝑖 = (𝜌𝑢𝑁𝑖)𝐼 ( 2.86)

46

The inflow boundary condition specifies the total mass flux, including diffusion and convection,

rather than the species fraction (𝑌𝑘 = 𝑌𝑘,𝐼 ). If gradients exist at the boundary, these conditions

allow diffusion into the nozzle.

The boundary conditions at the stagnation wall are:

𝐹 = 0 ( 2.87)

𝐺 = 0 ( 2.88)

(𝑑𝐻𝑑𝑧 )𝑊

= 0 ( 2.89)

𝑇 = 𝑇𝑊 ( 2.90)

𝜌 (𝑑𝑉𝑘𝑌𝑘

𝑑𝑧 )𝑊= 𝑊𝑘�̇�𝑘 ( 2.91)

(𝑑𝑁𝑖𝑑𝑧 )𝑊

= 0 ( 2.92)

𝑢, 𝑣, and 𝑉𝑘 are all zero at the stagnation wall as a no–slip condition is assumed. The stagnation

wall has a temperature 𝑇𝑊 equal to the measured temperature. The axial convective velocity was

assumed to vanish, leading to an diffusive flux equal to that of the chemical source term for each

species at the stagnation surface – an assumption expected to be valid so long as the free radical

concentrations are negligible immediately below the stagnation surface, as suggest by [19].

47

Chapter 3 Soot Particle Coalescence

3.1 Overview Soot comprises fractal-like chains of order of 100 small spherical particles. Soot aerosol

morphology properties of interest include primary particle size (and/or size distribution) and

number of primary particles per aggregates. Agglomerates are not rigid structures. Evidence of

internal restructuring of aerosol agglomerates and the flexibility of nanoparticle chains is

discussed in this chapter. Methods have been developed for relating particle properties to process

conditions and the properties of the material composing the particles, namely the solid or liquid-

state diffusion coefficient, surface energy, and particle density. The collision-coalescence

mechanism of particle growth discussed in this chapter is thought to control primary particle size

in the flames. Two coalescence models are proposed for predicting soot particle morphology in

laminar coflow diffusion flames in this chapter. Finally, effect of different coalescence model

parameters on prediction of primary particle diameter is investigated.

3.2 Introduction

The final stage in the soot particle formation and growth mechanism is aggregation. The process

of formation of fractal-like aggregate structures as a result of particle collisions is termed

‘coagulation’. Coagulation has a determining effect on the shaping of soot particle size

distributions, soot number density, and soot morphology. After collision, soot particles may

48

experience structural evolution. The aggregate form may change due to (a) condensation and

evaporation from its surface, (b) heating, and (c) mechanical stresses. The ability of aggregates

to change their shape has important implications for aggregate transport and light scattering, as

well as specific surface area, which plays a critical role in particle growth mechanisms. Thermal

restructuring of soot aggregates is the focus of this chapter.

The restructuring processes is depends on particle state, surface property, primary particle

diameter, temperature, residence time, etc. [106]. The collision of liquid-like nascent soot

particles leads to complete merging of the colliding particles which is known as the coalescence

process [54]. The slow restructuring rate of the mature particles leads to the formation of the

fractal-like aggregate structure. Observation of neck formation at the contact points of primary

particles within an aggregate can be interpreted as partial coalescence or surface growth

obliteration [39]. Figure 3.1 schematically presents the three stages in coalescence of particles.

Although coalescence itself does not change the total mass of soot particles, it changes soot

morphology, soot number density, and the soot particle size distribution. Therefore, it plays an

important role in the structural evolution of soot particles.

Figure 3.1 Schematic of coalescence process of two colliding particles.

3.2.1 The Collision-Coalescence Mechanism

Aggregate formation is based on a series of steps assumed to proceed as follows:

- Formation of particle precursor and condensable species

- Nucleation

- Collision-coalescence of nascent particles (the particles may behave in a liquid-like or solid-like manner during the coalescence period)

- Termination or significant deceleration of coalescence due to increased particle size and/or reduction of temperature

- Agglomeration of fractal-like structures as coalescence ceases from subsequent collision

49

- Continuous coalescence and neck formation of particles within the agglomerate structures

Some of these processes may go on simultaneously. In addition, particles continuously gain mass

through different physical and chemical growth processes. Therefore, the primary particles

composing the agglomerates become considerably larger than the nascent particles. Particle

diameter is a function of the temperature, growth and oxidation history that influence particle’s

thermo/chemical as well as geometrical properties. In general, the rate of particle coalescence is

directly proportional to temperature, producing large singlet particles at high temperatures with a

low specific surface area [106].

Based on experimental observations, three structures for soot particles produced in flames, as

illustrated in Figure 3.2, can be identified: A cloud of individual spherical particles (Left panel of

Figure 3.2), Fractal-like agglomerates (Right panel of Figure 3.2), and a continuum of states

between these two limiting cases. From the mechanistic point of view, the difference between the

rate of collision and coalescence shape the final structure of particles. The presence of a

spectrum of particle structures at different stages in the flame is the evidence of variation of the

rate of coalescence versus collision. In order to parametrize the collision and coalescence

processes, two characteristic times are defined. The characteristic time of coagulation or collision

is the average time between binary particle collisions, 𝜏𝑐 , and the characteristic time of

coalescence is the time for two particles to coalesce into a single sphere after making contact, 𝜏𝑓 .

The formation of spherical particles is the outcome of having the coalescence time 𝜏𝑓 much

smaller than 𝜏𝑐 . When colliding particles cease to coalesce and 𝜏𝑓 ≫ 𝜏𝑐 , particles with

agglomerate structures will be produced. Allowing for a finite rate of coalescence once two

particles have collided will provide the basis for analyzing the structural evolution of particles.

Figure 3.2 TEM images of soot particle samples along the centerline of a coflow diffusion flame of a surrogate for Jet A-1 at different heights above the fuel tube exit (Source: Reprinted from ref. [35]).

50

Dworkin et al. [141] have shown that the sectional soot model described in Chapter 2 combined

with the developed mechanism (the DLR mechanism) is capable of accurately predicting soot

volume fraction in an ethylene/air coflow diffusion flame. However, the model performance to

predict the primary particle properties was unsatisfactory. Major underprediction of the primary

particle diameter followed by overprediction of number density of primary particles was

obtained using the sectional soot model. These results are an indication of a deficiency in

modeling soot primary particles. One of the processes involving primary particles that was not

considered in the Dworkin et al. [141] study is particle coalescence. The coalescence process

increases the diameter of the primary particle by merging the primary particles in contact, which

also reduces the total number of primary particles. Therefore, in this chapter coalescence models

that are suitable for sectional soot modeling have been developed.

A limited number of soot coalescence models can be found in the literature. Most soot models

that consider the coalescence process rely on the assumption of instantaneous particle merging

for small particles [41,172,173]. Ulrich and Subramanian [174] represents one of the first

modeling approaches that highlighted the importance of a finite coalescence rate on prediction of

soot particle structures. A coalescence model has been proposed in the work by Ulrich and

Subramanian [174] and was employed for prediction of flame generated silica particles. Sander

et al. [175] also proposed a coalescence model and characteristic time for SiO2 particles which

were further used by Sander et al. [176] and D’Anna et al. [177] to predict soot particle

formation and their size distribution in premixed flames.

In the sections that follow, the coalescence processes are incorporated in a model applicable to

the sectional primary particle number density equation. Expressions are derived for 𝜏𝑓 in terms

of material properties and process conditions from the collision-coalescence theory. The

resulting models have been used to predict particle morphology in a coflow diffusion

ethylene/air flame.

3.3 Rate of Coalescence

The coalescence mechanism for liquid particles and solid particles are different. For liquids, the

mechanism of coalescence usually considered is viscous flow. For solids, diffusion and

evaporation-condensation are the most common mechanisms for nanoparticle coalescence. These

51

mechanisms can be incorporated in the primary particle number density conservation equation

through suitable expressions for the loss of primary particles due to coalescence considering its

characteristic time. The coalescence rate can be derived from the linear rate law for decrease in

the surface area [178]. Considering an agglomerate particle composed of primary particles,

the coalescence rate can be expressed as follows [179]:

𝑑𝑛𝑃𝑑𝑡

= − 3𝜏𝑓 (𝑛𝑝 − 𝑛𝑝

23) ( 3.1)

3.3.1 Viscous Flow Transport

For liquid particles, coalescence takes place by viscous flow. After two droplets are in contact,

the surface tension forces the doublet shape to change and reach its equilibrium state. The

deformation continues to minimize surface free energy. The shear forces, however, resist against

fluid layer motions to approach a spherical shape. Thus, for these particles, the characteristic

coalescence time of two equal-sized spheres of diameter, 𝑑𝑝, is given by [180]:

𝜏𝑓 =𝜋𝜇𝑑𝑝

𝜎 ( 3.2)

where 𝜇 is the viscosity and 𝜎 is the surface energy.

3.3.2 Transport by Diffusion

Unlike liquid particle, the equilibrium form for solid particles in contact is not predetermined.

The exact shape corresponding to the minimum surface Gibbs free energy should be estimated

by a Wulff construction [181] involving complex calculations of crystal plane rearrangements.

One common assumption to avoid the cumbersome calculations is that the particles are spherical

and their properties are isotropic. Thereafter, the characteristic time, , can be obtained as [182]

𝜏𝑓 = 364𝜋

𝑘𝐵 𝑇 𝑉𝐷𝜎𝑣𝑚

( 3.3)

where is the Boltzmann constant; is the temperature; is the particle volume; is the

surface tension; is the molecular volume, and is the solid-state diffusion coefficient. The

value of D corresponds to the dominant transport route – for example, lattice, grain boundary, or

surface diffusion [106]. Nanosized particles like soot have high ratios of surface area to volume,

and it is expected that surface diffusion is the dominant diffusion route for these particles. The

52

driving force for surface diffusion is the gradient of the chemical potential along the surface.

Therefore, the diffusion coefficient is a function of surface free energy and the width of the

surface layer which makes it depend strongly on the temperature. An Arrhenius form with an

activation energy can be used to describe the temperature dependency of the diffusion coefficient

[106].

3.4 Coalescence Model

The coalescence mechanisms proposed for solid and liquid particles suggest that as the

temperature increases, the rate of coalescence increases exponentially [179,180,182,183]. Most

coalescence mechanisms are based on the assumption that at high temperatures the particles are

liquid and coalesce instantaneously. As the temperature decreases, the particles become solid and

the rate of coalescence dramatically reduces. There is also a transition state between the liquid

phase and solid phase [183]. For soot particles however, a different pattern has been observed

[30,53,184,185]. These studies on the evolution of soot particles suggest that nascent soot

particles have liquid-like behaviour. The soot particles at early stages will present as one

spherical droplet in the flames and show no sign of aggregation [35,184,185]. This behaviour can

be interpreted as being of high coalescence rate for young soot particles. As these particles

traverse the flame, and experience higher temperatures, they transform to solid particles and

form fractal-like aggregates.

The solidification of soot particles has been attributed to the carbonisation process [35,184–186].

Carbonisation is a collection of chemical activities of the inter-particle elements and

rearrangements of the internal structures near the surface of soot particles, which results in

solidification of the particles and alteration of the surface chemical reactivity. The phase change

part of the carbonisation process is the focus of this chapter. The effect of the carbonisation on

the surface reactivity will be discussed in the next chapter.

Kholghy et al. [35] observed an abrupt change of soot particles from liquid-like droplets to

fractal-like aggregates around 1500 K in a diffusion flame, suggesting a chemical reaction with

an activation energy that is overcome at that temperature. Reilly et al. [185] and Dobbins et al.

[53], by measuring soot particle carbon and hydrogen content observed an increase in carbon to

hydrogen ratio (C/H) as the particles went through the carbonization process. Therefore, these

53

studies imply that the carbonization reaction involves hydrogen release and carbon-carbon bond

formation. In spite of all efforts put into studying the carbonization process, the chemical

mechanism of carbonization is not well understood and further investigation needs to be

conducted to assess the reaction rates and other thermo/chemical properties of the process.

Nonetheless, two approaches to model coalescence of soot particles are proposed here. The first

approach is the simpler model to implement. This model only takes into account the dependency

of coalescence rate on the primary particle diameter and this model will be called the cut-off

model hereafter. The second model, which will be called the sintering model hereafter, is based

on the neck growth model, Eq. 3.1, with a characteristic time as a function of primary particle

diameter and temperature.

3.4.1 Cut-off Model (Model I)

The cut-off model is based on the idea of immediate merging of colliding particles having

particle diameters less than a finite value, as it is displayed schematically on the left side of

Figure 3.3. The assumption of instantaneous fusion has been applied by Fenimore and Jones

[187], Howard et al. [188], and Smooke et al. [41] to describe soot particle disappearance rates

in flames. Such an assumption is valid if particles rapidly coalesce between collisions. The

assumption is consistent with the observations of single, discrete, spherical particles in the

electron micrographs of small soot samples by Bonne et al. [24] and Homann [25]. The cut-off

diameter model in the sectional soot aerosol dynamic model has been implemented by modifying

parameter 𝜂𝑝,𝑖𝑗𝑘, Eq. 2.50, in the primary particle coagulation model:

𝜂𝑝,𝑖𝑗𝑘 ={

1 𝑚𝑖𝑚𝑗 + 𝑚𝑘

(𝑛𝑝,𝑗 + 𝑛𝑝,𝑘) if 𝑑𝑝,𝑖 < 𝑑𝐶𝑢𝑡

( 3.4)if 𝑑𝑝,𝑖 > 𝑑𝐶𝑢𝑡

where 𝑑𝐶𝑢𝑡 is the cut-off diameter. The cut-off diameter represents the diameter at which the

particles experience phase change and transfer from a liquid-like state into a solid state. Smooke

et al. [41] choose 25 nm as the cut-off diameter for modeling soot formation in laminar diffusion

flames. Woods and Haynes [97] suggest that all colliding particles must coalesce until their sizes

exceed 20 nm . The cut-off diameter represents the size of which the particles experience a phase

change from liquid-like to solid. Therefore in this study cut-off diameter has been chosen to be

𝑑𝐶𝑢𝑡 = 20 nm.

54

Figure 3.3 Schematic representation of aggregate formation with cut-off coalescence.

3.4.2 Sintering Model (Model II)

The sintering model allows merging of the colliding soot particles with a finite residence time.

The residence time is a function of local temperature and particle diameter. Figure 3.4 depicts the

coalescence mechanism considered in the sintering model. The neck growth model described by

Eq. 3.1 determines the rate of coalescence of primary particles within a single aggregate. In order

to find the total rate of coalescence for particles present in the section, the rate by the neck

growth model is

(

𝜕𝑁𝑖𝑝

𝜕𝑡 )𝑆𝑖𝑛𝑡

= − 3𝜏𝑓 (𝑛𝑝 − 𝑛𝑝

23) 𝑁𝑖

𝑎 ( 3.5)

In order to have an accurate expression for characteristic coalescence time, , it is necessary to

identify the different regimes of soot coalescence, verify the transition conditions from liquid-

like to solid-state, and know particle thermo/chemical properties such as its structure and

chemical composition. Unfortunately, such information is unavailable due to the lack of

fundamental understanding of part of the kinetics of soot particles. Therefore, assumptions have

to be made for the form of the characteristic time. The model does not distinguish solid and

liquid particles and a single coalescence mechanism has been used for all the particles. The

characteristic time has been assumed to be proportional to the forth power of primary particle

diameter, [179]. The dependency on the forth power diameter is typical for solid particles

55

[179], and it was enforced here to ensure formation of fractal-like aggregates when the primary

particle diameters are large enough (e.g., 20 nm). An Arrhenius function has been used to

account for the temperature dependency of the diffusion coefficient in Eq. 3.3 [106]. The

activation energy and pre-exponential terms are adjusted to allow small particles to merge.

𝜏𝑓 = 7.44 × 108𝑑𝑝4 𝑇 𝑒𝑥𝑝 (

3.31 × 104

𝑇 ) ( 3.6)

Figure 3.4 Schematic representation of the sintering model for soot particle coalescence.

3.5 Methodology

The coalesce models are implemented in the sectional soot model to predict soot particle

formation in the atmospheric pressure, non-smoking, coflow laminar ethylene/air diffusion

flame, first studied by Santoro et al. [58] (referred to as the Santoro flame hereafter). The

Santoro burner configuration is schematically depicted in Figure 3.5. The coflow burner consists

of an 11.1 mm inner diameter fuel tube at the center of the burner surrounded by the cylindrical

air passage with an inner diameter of 102.0 mm. Gaseous C2H4 fuel flows at a mean velocity of

3.98 cm/s (flow rate 3.85 cm3/s) and the air flows at a mean velocity of 8.9 cm/s (flow rate 713.3

cm3/s) at room temperature conditions. A ceramic honeycomb structure is installed into the air

annulus straightening the air flow. Although the fuel and air flows are at atmospheric

temperature and pressure, due to the anchoring of the flame around the fuel tube, some heating of

the fuel tube and fuel flow does occur. In order to reconcile the fuel tube preheating, the inlet

56

fuel flow temperature boundary has been increased to 650 K as recommended by [189–191]. The

flow configuration generates a stable, sooting, nonsmoking flame, with a visible flame height of

approximately 88 mm. This particular flame has been chosen because extensive experimental

measurements of soot particles have been obtained during over 30 years of studies of this flame.

These measurements include soot volume fraction, average primary particle diameter, aggregate

number densities, primary particle number densities, fractal dimension, and average number of

primary particles per aggregate [33,37,38,56–58,192]. Most relevant to coalesce are those

experimental data on soot particle morphology, i.e., average primary particle diameter, primary

particle number densities, and average number of primary particles per aggregate.

Figure 3.5 Schematic representation of burner configuration of Santoro flame [58]. [Courtesy of Dr. Meghdad Saffaripour, University of Toronto.]

3.5.1 Numerical Model

For the gas phase, the fully coupled elliptical conservation equations for mass, momentum,

energy, and species mass fraction are solved. The model utilizes the axi-symmetrical nature of

the flame, and equations are solved in the two-dimensional (z and r) cylindrical co-ordinate

system. A detailed description of the governing equations, boundary conditions, and solution

methodology can be found in Chapter 2. The DLR chemical mechanism (see chemical

Air Air

Fuel

57

mechanisms in Chapter 2) with 93 species and 719 reactions was applied to describe the

oxidation of the fuel and the formation of PAHs.

Soot is modeled using the detailed fully coupled sectional aerosol dynamics model discussed

in Chapter 2. In this approach the continual soot particle mass distribution is divided into a

discrete number of soot clusters, each with an assigned mass. For this study, the soot particle

mass range is divided into 35 discrete sections that describe the soot particle diameter ranging

between 0.86 nm and 3.3 μm. Conservation equations of soot aggregate number densities, and

primary particle number densities are solved for each soot section.

Nucleation is modeled based on the collision of PAH molecules with 5 aromatic rings, i.e.,

benzopyrene and benzo[ghi]flouranthene, in the free-molecular regime [40,193], which serves as

a connection between the gas phase reaction mechanism and the first soot section, with collision

efficiency of 100%. The HACA mechanism [86] is used to describe soot particle surface growth

with a constant surface reactivity, α, of 0.45 for the soot models with coalescence. While this

parameter is the subject of detailed investigation later in this thesis, here it is held constant to

isolate the effect of coalescence modelling. PAH condensation is modelled based on collision

theory between 5–member ring PAH molecules and aggregates, with a collision efficiency of

0.5 [147]. A constant coagulation efficiency, ξ, of 0.2 is chosen based on the recommendation of

Zhang et al. [118].

3.6 Results and Discussion

In order to test the coalescence models, the Santoro flame [58] has been simulated using two

models. Included in the first model is the cut-off coalescence model with the 20 nm diameter

chosen as the limiting factor for coalescence. The second set of computations employed the

sintering model with a characteristic time described by Eq. 3.6. The predicted soot properties

using these models are compared to measured soot properties. The soot properties of interest are

soot volume fraction, soot aggregate number density, primary particle number density, average

number of primary particles per aggregate, and average primary particle diameters. The

predictions of soot formation in the Santoro flame [58] using the model without any coalescence

(Model 0) are also included for comparison. For Model 0, the surface reactivity parameter, 𝛼, of

0.076 has been used based on the results of Dworkin et al. [141]. The predicted soot results will

58

be presented over two regions in the flame. The soot concentration peaks at the annular region in

the Santoro flame [58], therefore, the soot properties along the streamline passing through the

maximum soot concentration location, also known as ‘flame wing’ will be presented. The soot

predictions along the centerline of the flame are also included. The importance of these two

regions in the flame, as will be discussed in the next chapter, is in the difference between soot

growth mechanisms. Soot formation on the wings is dominated by chemical surface growth

whereas along the centerline, soot growth via PAH addition is the main soot growth route.

Finally, the effect of different parameters in the coalescence models has been investigated.

3.6.1 Annular Pathline Comparison

The predicted soot volume fraction along the annular pathline exhibiting the maximum soot

concentration as a function of height above the fuel tube, and residence time, are depicted in

Figure 3.6 and Figure 3.7, respectively. The soot predictions verify that all three soot models are

able to predict the soot concentration within the uncertainty range of experimental measurements

from the literature [58,192]. The agreement of predicted soot volume fractions with experimental

data was expected since the parameter 𝛼 for each model was deliberately chosen to correctly

reproduce the maximum soot volume fraction in the flame. The reason being, now that all the

models have the same amount of mass of soot in their system, a more sensible assessment of

their abilities to predict particle morphology could be made.

59

Figure 3.6 Comparison of the predicted soot volume fraction along the annular pathline exhibiting the maximum soot volume fraction using a sintering coalescence model (solid line), the cut-off coalescence model (dashed line) and no coalescence (dot-dashed line) with the experimental measurements by [58].

Figure 3.7 Comparison of the predicted soot volume fraction along the annular pathline exhibiting the maximum soot volume fraction using a sintering coalescence model (solid line), cut-off coalescence model (dashed line) and no-coalescence (dot-dashed line) with the experimental measurements by [192].

Improving predictability of the soot aerosol dynamics model for average particle diameter is the

pivotal milestone of adding the coalescence process. As presented in Figure 3.8, the model

without coalescence underpredicts the primary particle diameter everywhere along the pathline

exhibiting maximum soot concentration. The no-coalescence model predicts the maximum

[58]

[192]

60

average primary particle diameter to be 3.79 nm whereas the experimental data from the

literature [56] shows the maximum primary particle diameter to be in the range of 29–38 nm.

Addition of a coalescence model with either a cut-off diameter or sintering profoundly improved

the particle diameter predictions. The maximum particle diameter predicted by the cut-off model

and the sintering model are 22 and 30 nm, respectively.

Figure 3.8 Comparison of the predicted average primary particle diameter along the annular pathline exhibiting the maximum soot volume fraction, using a sintering coalescence model (solid line), cut-off coalescence model (dashed line) and no coalescence (dot-dashed line) with the experimental measurements by [56].

Although both of these coalescence models improved the predictions of average diameter, they

present distinctive behaviour which can be explored to gain a better understanding of the nature

of soot particle growth. The cut-off model shows a rapid growth in particle diameter in the

regions closer to the fuel tube before the diameter of primary particles hits the 20 nm cut-off

limit, which arrest the coalescence process. In this region, 𝑧 < 3 cm, predicted particle diameter

by the cut-off model shows better agreement with the experiments. The underprediction of

particle diameter by the sintering model and the reasonable predictions of the cut-off model in

the lower heights of the flame suggest that the apparent sintering rate is much faster than the rate

used in the model in this region. The results are reminiscent of the liquid-like behaviour observed

by Kholghy et al. [35] in this region in the diffusion flame. An increase in temperature (see

Eq. 3.6) and number of primary particles per aggregate (see Eq. 3.5, and will be discussed later

with regard to Figure 3.11) at heights above 𝑧 = 3 cm accelerates the sintering rate in this region

[56]

61

which results in primary particle diameters as large as 30 nm. Finally, above the 𝑧 = 4 cm height,

the soot particles enter the oxidation zone and particle diameter decreases due to lose of mass.

An important observation which is similar between the cut-off model and the model without

coalescence is the negligible increase of particle diameter in the regions where coalescence is not

present. This observation becomes more intriguing when the diameter profile, Figure 3.8, is

compared with the soot volume fraction profile, Figure 3.6. While the diameter is modestly

increased, significant mass has been added to the soot particles from the gas phase. The soot

volume fraction for the cut-off model increased from 4.4 ppm, at 𝑧 = 2 cm where the average

particle diameter is 20 nm, to 12 ppm at 𝑧 = 4 cm where the particle diameter reaches only 22

nm. In other words, the average diameter only increased by 10%, while soot mass has been

almost tripled. The primary particle number density profile, depicted in Figure 3.9, can be used

to further elucidate the situation. When entering the region where coalescence has ceased, the

primary particle number density vastly increases. Since the growth mechanisms are surface

dependent, most of the additional mass will be absorbed by the small particles, which have a

higher surface to volume ratio. Therefore, the mass addition, instead of growing the existent

particles, will be distributed among the newly incipient soot particles. Thus, the increase in the

soot mass will barely result in an increase in the average particle diameter.

Figure 3.9 Comparison of the predicted primary particle number density along the annular pathline exhibiting the maximum soot volume fraction using a sintering coalescence model (solid line), cut-off coalescence model (dashed line) and no coalescence (dot-dashed line) with the experimental measurements by [39,57].

[39] [57]

62

The number of primary particles is controlled by the nucleation process and there are

assumptions in the soot model which make nucleation more favourable. Nucleation and

condensation are the two processes that compete to absorb PAHs into the condensed phase.

Nucleation is modeled as being 100% efficient where condensation efficiency is considered to be

20% [147]. Also the shape of the condensed phase matter present in the first few sections is

considered to be a complete sphere, where in reality these are stacks of PAHs. Since the surface

of area of mass equivalent sphere is appreciably less than the PAH stacks, the spherical shape

assumption under-represents the area of the small particles. Condensation is a surface dependent

process, therefore the dimer shape assumption further supresses condensation. The final outcome

of these assumptions is that most of the PAH growth will be contribute to an increase in the

number of particles (Figure 3.9) as opposed to an increase of the existing particle volume

(Figure 3.8). These observations are consistent with the results of Saffaripour et al. [40] and

Eaves et al. [193]. For more discussion on nucleation and condensation please refer to Chapter 5.

Similar to the particle diameter predictions, the cut-off coalescence model in the lower flame

heights shows good agreement with the experimental data from [39,57] while the sintering model

overpredicts the number density of primary particles. Upstream in the flame, the sintering model

predicted particle number density to drop within the uncertainty range of experimental data,

whereas the cut-off model now overpredicts the particle number density. Both particle diameter

and number density results imply that a combination of these two models may be necessary in

order to predict soot morphology along the wings.

Predicted aggregate number density and number of primary particles per aggregate along the

wings are plotted in Figure 3.10 and Figure 3.11, respectively. Both of these properties are

directly dependent on the coagulation and fragmentation processes. Both coalescence models

were able to predict aggregate number density along the pathline of maximum soot within the

uncertainty of the experimental data. It should be noted that in the presented modeling results,

only aggregates larger than 5 nm in diameter, which is the threshold for the experimental

measurements have been considered in calculating total number of aggregates. While not

changing the coalescence models results substantially, neglecting the particles smaller than 5 nm

is the primary reason for the undeprediction of aggregate number density of the model without

coalescence.

63

Figure 3.10 Comparison of the predicted aggregate number density along the annular pathline exhibiting the maximum soot volume fraction using a sintering coalescence model (solid line), cut-off coalescence model (dashed line) and no coalescence (dot-dashed line) with the experimental measurements by [57,192].

Figure 3.11 Comparison of the predicted average number of primary particles per aggregate along the annular pathline exhibiting the maximum soot volume fraction using a sintering coalescence model (solid line), cut-off coalescence model (dashed line) and no coalescence (dot-dashed line) with the experimental measurements by [57,192].

The number of primary particles per aggregate ( ) results that are depicted in Figure 3.10

further emphasize the observations made earlier from the particle diameter and number density

results. The cut-off model underpredicts downstream of the flame while the sintering model

overpredicts compared to the experimental data in these areas. This result suggests that the

[192]

[33][57]

[57]

64

soot particle coalescence characteristic time should be somewhere between the instantaneous

cut-off model and the sintering model lower in the flame. However, overprediction of the

number of primary particles by all three models on the higher heights above the fuel tube

suggests that too many particles are forming. The coalescence process being dependent on the

rate of collision as well as thermophysical properties of particles is not solely accountable for

dissipation of primary particles. An alternative mechanism for controlling particle formation

would be to limit the nucleation process. In the models employed in this section it is assumed

that all the collisions between PAH molecules are 100% efficient in nucleating particles.

However, recent studies by Sabbah et al. [102] and Wang [17] on PAH dimerization suggest that

the dimerization process is not thermodynamically favored and is highly reversible. The studies

by Saffaripour et al. [40] and Eaves et al. [193] also confirm that if a very low nucleation

efficiency is employed, or the nucleation process is modeled as fully reversible, the relevant

average soot morphological parameters along the wings and centerline can be predicted

reasonably. Indeed, a combination of both of these pathways would be more representative of the

nature of soot particle formation.

3.6.2 Centerline Comparison

For further validation and comparison of the coalescence models, the soot particle predictions

along the centerline of the Santoro flame [58] are presented here. Only the primary particle

diameter and soot volume fraction results are presented here due to similarities of the prediction

trends observed on the wings. The average primary particle diameter predictions along the

centerline for the cut-off model, sintering model, and the no-coalescence model with the

experimental data from Koylue et al. [37] are shown in Figure 3.12. Both coalescence models

improved the diameter predictions substantially compared to the model without coalescence,

which similar to the wings results, underpredicts the primary particle diameter. The most

distinctive difference between the predictions of primary particle diameter along the centerline

with those along the wings is that the cut-off model predicts larger particles all along the

centerline. However, on the wings the maximum diameter predicted by the sintering model was

larger compared to the cut-off model, and it was closer to the measured diameter. The high

temperature dependency of the sintering model underlies its underperformance. The soot

formation on the centerline starts in the inner regions of the flame, where the temperature that the

soot particle experiences is lower than the flame temperature. The temperature does not exceed

65

1600 K on the centerline until after 𝑧 = 8.5 cm where soot formation has ceased and the particles

have entered the oxidation zone (see Figure 3.13). The temperature profile experienced by the

particles on the wings is completely contrary. The soot formation starts near the flame front

where the temperature is above 1700 K and the temperature never drops below 1500 K. The high

sensitivity of the sintering rate to the temperature and the difference between the temperature

profiles along the wings and centerline caused the sintering model to underperforme along the

centerline.

Figure 3.12 Comparison of the predicted average primary particle diameter along the centerline using a sintering coalescence model (solid line), cut-off coalescence model (dashed line) and no coalescence (dot-dashed line) with the experimental measurements by [37].

The soot volume fraction profiles along the centerline are depicted in Figure 3.13. The cut-off

model and the no-coalescence model predict the soot volume fraction reasonably well compared

to the experimental data by [37,38,58]. The predicted soot volume fraction by the sintering

model is lower than the predictions of the two other models and it is lower than the experimental

data. The soot formation along the centerline is dominated by PAH growth processes, i.e.,

nucleation and condensation as discussed by Thomson and coworkers [141,194,195]. The PAH

condensation rate as described in Chapter 2 is modeled based on the collision of PAH molecules

in the gas with the soot particles, and the collision rate is a function of soot surface area. For a

given mass of soot, if the set of particles consists of smaller particles, there will be a higher

chance for PAH molecules to collide and adsorb onto the soot particles. In other words, PAH

adsorption will be supressed when the surface to volume ratio is lower. The surface to volume

ratio profiles along the centerline are illustrated in Figure 3.14. The sintering model has the

[37]

66

lowest surface to volume ratio, therefore, predicts a lower soot volume fraction and it is not as

efficient as the two other models in adsorption of the gas phase PAHs. It should be noted that the

surface to volume ratio is calculated based on the weighted average of the particles’ surface to

volume ratio at each location. For the cut-off model this value is substantially higher than the

mean surface to volume ratio. The difference between the two values is an indication of the

presence of a great number of small particles in the particle size distribution which has low

contribution to the overall mass.

Figure 3.13 Comparison of the predicted soot volume fraction along the centerline using a sintering coalescence model (solid line), cut-off coalescence model (dashed line) and no coalescence (dot-dashed line) with the experimental measurements by [37,38,58] (a log scale is used so that comparisons can be made at heights less than 4 cm).

[58] [38] [37]

67

Figure 3.14 Variation of surface to volume ratio along the centerline using a sintering coalescence model (solid line), cut-off coalescence model (dashed line) and no coalescence (dot-dashed line).

Studying the particle size distribution (PSD) function can help identifying the roles of different

processes in forming aggregates. The predicted PSDs along the wings and centerline on the

Santoro flame are provided in Appendix A for various heights above the burner. Both on the

wings and centerline, unimodal distributions are found at lower heights, where the aggregate

number density is dominated by small aggregates, suggesting that nucleation is the dominant

process. The unimodal distribution becomes bimodal due to the growth and coagulation

processes as the height increases. Meanwhile, the curve widens, indicating large aggregates are

formed. As particles enter the oxidation region, the distribution becomes unimodal again and the

area under the curve reduces.

3.6.3 Sensitivity analysis

In this section the effect of different coalescence model parameters on the soot particle diameter

predictions will be examined. Three different coalescence parameters will be analysed. First to

be studied is the effect of the choice of cut-off diameter. The effect of activation energy in the

sintering characteristic time function, Eq. 3.6, on the predicted particle diameter will be

discussed next. Finally, a discussion on the effect of coalescence on particle diameter in the soot

oxidation zone will be provided.

68

3.6.3.1 Cut-off Diameter

The only assigned parameter in the cut-off coalescence model is the cut-off diameter for which

the boundary between instantaneous merging of the colliding particle and agglomeration is

defined. In order to investigate the effect of the choice of the cut-off diameter, soot formation in

the Santoro flame [58] is simulated using three different cut-off diameters, namely 15, 20, and 25

nm. The predicted primary particle diameter along the wings and centerline are presented in

Figure 3.15a, and Figure 3.15b, respectively. The particle diameter profiles in Figure 3.15 show a

strong dependence of predicted maximum diameter with the choice of cut-off diameter. In all

three cases the predicted maximum diameter both on the wings and the centerline are very close

to the cut-off diameter. The observed dependency of the predicted diameter on the cut-off

diameter, as was discussed in Section 3.6.1, is mostly due to the highly efficient nucleation

model which forms a vast number of small particles. Since the smaller particles have a high

surface to volume ratio, they have a higher tendency to absorb the available mass from the gas

phase in competition with the larger particles. Thus, the mass addition, instead of growing the

existent particles, will be distributed among the newly incipient soot particles and will barely

result in an increase in the particle average diameter.

(a) (b) Figure 3.15 Comparison of the predicted average primary particle diameter using different cut-off diameter coalescence models a) along the annular pathline exhibiting the maximum soot volume fraction with the experimental measurements by [56] and b) along the centerline with the experimental measurements by [37].

[56] [37]

69

3.6.3.2 Coalescence Characteristic Time

The next parameter that is the subject of study is the coalescence characteristic time, 𝜏𝑓 , present

in the sintering model. The Arrhenius function for the characteristic time, Eq. 3.6, has a pre-

exponential factor and an activation energy that needs to be estimated, ideally based on the

comprehensive study of the surface characteristics of soot particles under a flame environment.

However, such knowledge of the soot particle surface is not available. Instead, by analysing the

effect of pre-exponential and activation energy parameters on the prediction of soot particles, an

estimated range for these parameters can be identified. Decreasing any of these parameters will

decrease the characteristic time, meaning that the time needed for particles to merge reduces.

Therefore, the coalescence process becomes more efficient and the existence of larger particles is

expected to be predicted.

In order to quantitatively evaluate the influence of 𝜏𝑓 on the predicted particle size, the sintering

model with four different activation energies has been employed to model soot particles in the

Santoro flame [58]. The 𝜏𝑓 profiles for a 10 nm soot particle as a function of temperature with

the four activation energies are shown in Figure 3.16. The highest activation energy for the

characteristic time is 𝐸𝐴1 = 3.31 × 104 (1/K) and the following characteristic coalescence times,

each has an 8% lower activation energy than the preceding characteristic coalescence time. The

predicted particle diameter profiles along the wings using these four activation energies are

shown in Figure 3.17a. The characteristic time reduces by an average factor of 3.5 when the

activation energy reduces from 𝐸𝐴1 = 3.31 × 104 to 𝐸𝐴2 = 3.10 × 104. This reduction of the

characteristic time leads to an increase of the maximum primary particle diameter predicted from

30 nm to 43 nm (43% increase) on the wings. Similar trends are obtained when the pre-

exponential factor in the characteristic time has been altered. Shown in Figure 3.17b is the

predicted particle diameter using different pre-exponential factors. When the pre-exponential

factor is reduced from 𝐴1 = 7.44 × 108 (s/nm4) by a factor of two, the predicted maximum

particle diameter is increased by 20%.

70

Figure 3.16 Variation of the characteristic coalescence time of a 10 nm soot particle with temperature with four different activation energies.

(a) (b) Figure 3.17 Comparison of the predicted average primary particle diameter along the annular pathline exhibiting the maximum soot volume fraction using a) different activation energy and b) different pre-exponential factor for the sintering coalescence model with the experimental measurements by [56].

The effect of reducing the characteristic time on the predicted maximum particle diameter is

graphically depicted in Figure 3.18. The coalescence process is dependent on particle collision.

This graph is showing that if the coalescence residence time is further decreased, the particle

diameter will not increase linearly and the particle diameter reaches its maximum limit, which is

the diameter of the aggregate; meaning that all the primary particles within the aggregate are

[56] [56]

71

completely merged, and the particle is now a singlet sphere. This result further emphasizes the

dependency of the coalescence process on particle collision. If there were insufficient particle

coagulations, the rate of coalescence would become irrelevant.

Figure 3.18 Effect of reduction of characteristic time on the predicted maximum primary particle diameter along the annular pathline exhibiting the maximum soot volume fraction.

3.6.3.3 Coalescence and Oxidation

The treatment of coalescence processes in the oxidation region is the focus of the final section of

this chapter. A vast number of studies have been carried on in the literature focused on soot

particle evolution and examination of their internal structure. Based on extensive studies of soot

formation in different flames, a number of scenarios have been proposed for soot particles

evolution. Some of these scenarios related to soot formation in diffusion flames have been

reviewed by Kholghy et al. [35]. Although there is an ongoing debate about the early stages of

soot formation, the soot evolution studies unanimously conclude that soot particles eventually

reach a rigid state [30,35,185,196]. The rigidifying process of soot particles is attributed to

carbonization. Carbonization is described as a complex process involving formation of activated

complexes, molecular rearrangement, polymerization, and dehydrogenation [55,197]. However,

the current coalescence models are not sensitive to the carbonization of particles.

Moreover, both coalescence models overpredict the particle diameters in the oxidation regions

(higher heights of the flame) while the models predict the soot volume fraction in agreement

with the experiments (see Figure 3.6). This overprediction of diameter is caused by the fact that

72

the coalescence models continue to merge particles during their oxidation with the same rate as

the nascent soot particles. To address this deficiency, a coalescence termination criterion has

been added to the sintering model. Those studies of soot particle evolution in diffusion flames

suggest that the soot particles are carbonized by the time they enter the oxidation region

[35,51,53,58,185]. Therefore, the coalescence process has been set to cease when entering the

oxidation zone. Since molecular oxygen is the main oxidizer of the soot particles in the diffusion

flames, the concentration of oxygen in the mixture has been used to identify the oxidation region

and the sensitivity of the results to the oxidation concentration is investigated.

Three different cases have been modeled with a different O2 concentration considered to define

the oxidation region in each case. The O2 mole fraction isopelths are depicted on the right panel

of Figure 3.19, where the boundaries of the oxidation zone for each case are identified with the

black solid line. The corresponding O2 mole fractions for each of these boundaries are 0.02,

0.002, and 2×10-4, respectively. Computed isotherms of the Santoro flame [58] are also included

on the left panel of Figure 3.19 to clarify the locations of each of the boundaries with respect to

the flame front.

Figure 3.19 Computational isotherms (left panel) and isopleths of O2 mole fraction (right panel) in the Santoro coflow diffusion flame.

73

The predicted particle diameter for the three cases with a coalescence termination in the

oxidation region and the original sintering model along with the experimental data is presented in

Figure 3.20. The predicted diameter profiles confirm the effectiveness of the additional

termination criteria in the oxidation region. While the predicted soot diameter in the growth

region remains unchanged compared to the original sintering model, the predicted diameters are

reduced in the oxidation region and show more consistency with the measured particle diameters

in both magnitude and the slope of diameter reduction. The third case with the O2 mole fraction

of 2×10-4 as the oxidation boundary is underpredicting particle diameters, suggesting that the

reduction of the coalescence region has been too intense. However, the first two cases predict

particle diameters that are within the uncertainties of the experimental data in the oxidation

region. Therefore, an O2 mole fraction in the range of 0.02 and 0.002 can be considered as a

reasonable estimation for identifying the soot oxidation region in this flame and termination of

the coalescence process.

Figure 3.20 Comparison of the predicted average primary particle diameter along the annular pathline exhibiting the maximum soot volume fraction using different sintering coalescence models with an oxidation cut-off, and the experimental measurements by [56].

3.7 Conclusions

In this chapter, two different soot particle coalescence models have been implemented into a two

dimensional flame code to explore soot formation and oxidation in the non-smoking laminar

coflow ethylene/air diffusion flame of Santoro [58], in combination with a PAH-based soot

model and a detailed sectional aerosol dynamics model. The first coalescence model considered

[56]

74

instantaneous merging of particles upon collision if the diameters of the colliding particles are

less than a cut-off diameter. The second coalescence model was based on gradual sintering of

particles through surface diffusion. The rate of coalescence in the second model is a function of

temperature, particle size, as well as number of particles per aggregate. The predicted soot results

have been compared with the results of the model without coalescence and the experimental data

from literature.

The reported soot morphology properties included primary particle diameter, particle number

density, aggregate number density, and number of primary particles per aggregate along the

annular pathline exhibiting maximum soot concentration and centerline of the flame. Both

coalescence models exhibited significant improvement in predicting soot particle morphology.

The cut-off model in the lower heights of the flame predicted soot particle properties that are in

closer agreement with the experiment data, while the sintering model predicted profiles that are

more consistent with the measured properties in terms of overall shape and magnitude.

Sensitivity of the soot prediction to the coalescence parameters has been analysed. The

coalescence parameters studied were the cut-off diameter in the first model, and the

characteristic time of coalescence in the sintering model. Finally, an update to the coalescence

model based on experimental observations of soot particles on the flame oxidation regions has

been introduced to improve its predicting capabilities. The updated model terminates the

coalescence during the soot oxidation which leads to improvement of particle size predictions in

this region. Soot particle coalescence is shown to be a complicated phenomenon. However,

because it may significantly affect soot structure that in turn affects soot properties such as its

health effect and environmental effect, this phenomenon is worth detailed investigation. In the

future, more detailed theoretical and experimental studies should be conducted to gain a

fundamental understanding of soot particles surface properties and chemical characteristic

evolution.

75

Chapter 4 Soot Particle Surface Reactivity

4.1 Overview The effect of soot surface reactivity, in terms of the evolution of sites on the soot particles’

surface available for reaction with gas phase species, is investigated via modeling numerous

ethylene/air flames, using a detailed combustion and sectional soot particle dynamics model. A

new measure of soot particles’ age is introduced. A methodology has been developed to study

soot particle surface reactivity. Subsequently, it is investigated if the surface reactivity can be

correlated with the particle age. An exponential function giving a smooth transition of surface

activity with particle age is employed to model a variety of ethylene/air flames, which differ in

fuel stream dilution levels, fuel stream premixing, and burner configurations. Excellent

agreement with measured soot volume fractions of a variety of flames, burners, and datasets

could be obtained with this approach. The newly developed function based on particle age

eliminates the need to fit soot surface growth parameters to each experimental condition. Finally,

the applicability and limitation of the new surface reactivity function for use in detailed soot

formation models is discussed.

4.2 Introduction

Several stages have been identified during soot formation and oxidation. Soot formation starts

with inception, which is the appearance of the first nano-scale soot particles. The newly incipient

76

soot particles can grow through surface growth via surface chemical reaction and polycyclic

aromatic hydrocarbons (PAH) condensation, and through particle coagulation. Finally, soot

particles lose mass and size during oxidation and fragmentation processes.

Among all the different processes, surface growth is known to be responsible for most of the soot

mass yield in many systems [198]. As a soot particle traverses hot fuel rich regions, the surface

of the particle reacts with the gas phase. The chemical kinetics of the soot surface has been the

subject of several studies. These studies concluded that acetylene is the primary growth species

independent of the fuel type [11,198]. Based on this observation and the fact that the formation

of soot proceeds via PAHs, it has been proposed that the reaction sequence for the build-up of

PAHs and soot should be analogous. The most widely used theoretical model to describe the

formation and growth of aromatics is the hydrogen–abstraction–carbon–addition (HACA)

mechanism [11,199]. The HACA mechanism consists of a repetitive sequence of radical site

formation by hydrogen abstraction, followed by carbon addition, most often by acetylene

bonding, forming an additional aromatic ring. It is proposed that soot growth in flames also

occurs at active sites.

The reaction scheme used to account for surface growth and oxidation is detailed in Table 4.1.

The kinetics of the surface reactions are described using the concept of surface sites (an armchair

site is a site with four carbon atoms as illustrated in Figure 4.1), which are carbon atoms either

saturated (Csoot–H) or dehydrogenated (Csoot ∘ ) on the surface of soot particles. The

concentration of saturated sites, [Csoot–H] (mole/cc), is calculated by Eq. 4.1:

[Csoot– H] =𝐴𝑠𝐴𝑣

𝜒Csoot–H ( 4.1)

Figure 4.1 Illustration of armchair sites on the surface of a soot particle.

Armchair�site

77

where χCsoot–H is the number of sites per unit soot surface area; As (cm2/cc) is the surface density

of soot particles and 𝐴𝑣 is Avogadro’s number. The concentration of dehydrogenated sites

[Csoot∘ ] is similarly calculated with χCsoot° as the number of dehydrogenated sites (Csoot ∘ ) per

unit surface area. Finally by assuming a steady state for Csoot∘ , 𝜒Csoot∘ can be calculated from

Eq. 4.2 and be used to find the individual rate of each of the soot reactions listed in Table 4.1.

𝜒Csoot∘ =(𝑘1[H] + 𝑘2[OH])𝜒Csoot–H

𝑘−1[H2] + 𝑘−2[H2O] + 𝑘4[C2H2] + 𝑘5[O2] ( 4.2)

Table 4.1 HACA–based soot surface growth and oxidation reactions [86], 𝑘 = 𝐴𝑇 𝑏𝑒−𝐸𝑎 𝑅𝑇⁄ .

No. Reaction A (cm3

mol.s) b Ea (kcalmol)

S1 Csoot–H+ H ⇐⇐⇐⇐⇒ Csoot∘ + H2 4.2×1013 0.0 13.0

S2 Csoot–H+ OH ⇐⇐⇐⇐⇒ Csoot∘ + H2O 1.0×1010 0.73 1.43

S3 Csoot∘ + H ←←←←←←←←→ Csoot–H 2.0×1013 0.0 0.0

S4 Csoot∘ + C2H2 ←←←←←←←←→ Csoot–H+ H 8.0×107 1.56 3.8

S5 Csoot∘ + O2 ←←←←←←←←→ 2CO + product 2.2×1012 0.0 7.5

S6 Csoot–H+ OH ←←←←←←←←→ CO + product 𝛾𝑂𝐻 = 0.13

It was experimentally observed that the reactivity of surface sites changes with increasing

particle growth or age [11,198,200,201]. Hence, this process is often called surface ageing. It

was attributed to a decrease of active surface sites, i.e., sites that are accessible for reaction.

Other experimental studies [97,98,148,198] showed the dependency of soot ageing on

temperature. More recently, by analyzing surface growth pathways, Kronholm and Howard [202]

cast doubt on the monotonically decreasing behaviour of soot reactivity with residence time if

C2H2 is assumed to be the dominant soot surface growth reactant.

The notion of active sites on the soot particle surface was introduced into kinetic soot modeling

by Frenklach and Wang [199]. In conjunction with a decrease in concentration of C–H sites

[198,199], it was used as an explanation for the experimental observation of surface ageing. On a

mechanistic basis, Frenklach and co-workers [90,203,204] attributed surface ageing to the

formation of defects on the particles’ surface generated during surface growth. Surface ageing

was also attributed to the reversibility of the HACA surface growth scheme [205–207]. The

surface ageing effect was embedded into the HACA surface reaction scheme by introducing a

78

steric parameter, 𝛼, which is positive and less than unity. Therefore the reaction rate for an

individual reaction, for example S4 from the Table 4.1 above, becomes:

𝑅4 = 𝛼 𝑘4 [C2H2][Csoot ∘] ( 4.3)

A review of the assumptions made in the soot surface growth scheme clarifies the necessity of

the 𝛼 parameter. The number density of the Csoot–H sites, χCsoot–H, was estimated based on the

assumption that the surface is covered with stacks of benzene rings [86]. The distance between

the stacks is 3.51 Å and it was assumed that 2 C–H bonds are available per benzene ring length

(2.46 Å). Thus χCsoot–H was calculated to be 2/(3.51×2.46) = 0.23 site/Å2. Considering that all of

these sites are accounted for as armchair sites, this value is the theoretical maximum value of

soot surface site density. The nanostructure of soot particles has been experimentally studied in

[23,53,104]. All of these studies concluded that soot particles are composed of stacks of 4 to 8-

ring PAHs. If for estimation of χCsoot–H it was assumed that the surface of the soot particles is

covered with a 5-ring PAH such as benzopyrene (A5) in accordance with the recent findings, the

number of C–H bounds available per unit length on average would be 0.5 site/Å which results in

χCsoot–H = 0.5/3.51 = 0.14 site/Å2. Similarly, if it were assumed that the surface is covered with

layers of coronene (A7) as opposed to the classical benzene-surface assumption, the number of

C–H bounds available per unit length on average would be 0.4 site/Å and subsequently the

number density of the Csoot–H sites, χCsoot–H= 0.115 site/Å2. Thus, the estimated value of χCsoot–H

would be 25% to 50% less than the originally proposed value if the surface of the soot particles

is assumed to be covered by the layers of 4 to 8-ring PAHs. The rate coefficients of the

heterogeneous reactions, presented in Table 4.1, were estimated based on analogous gas phase

reactions of one-ring aromatics. The rate coefficient steric factor (A) of each of the Csoot–H sites

is assumed to be one sixth of the benzene molecule. Also the activation energy is chosen to be

constant for all the soot particles and 3 (kcal/mol) less than the corresponding gas phase

analogous reaction of one-ring aromatics. However, it has been shown by [11,23,109,208] that

the C/H ratio of soot particles which represents the carbonization or graphitization of soot,

increases with residence time of the soot particles, and results in less chemical reactivity. In

conclusion, the empirical ageing parameter, 𝛼, reconciles the inaccuracies of treating sites on the

soot surface as corresponding sites on gaseous PAH molecules.

79

While initially a constant fraction of active sites was used with the kinetic soot model [198,199],

it was later expressed as a function of flame temperature [209], and subsequently as a function of

flame temperature and mean particle size [86]. Several studies measured the ageing parameter, 𝛼,

based on the HACA surface growth scheme in laminar premixed and diffusion flames with

different fuels, pressures, and flame temperatures, and proposed a temperature dependent

function for the ageing parameter [28,31,32,36,44,150,208]. However, the predicted value of 𝛼

yielded by those forms is 1.0 for most of the sooting region of laminar diffusion flames, which is

quite close to the theoretical maximum value of available soot surface sites, and unrealistically

high.

Dworkin et al. [141] shows that if particle inception is enhanced by more accurate prediction of

PAH molecules in the gas phase, 𝛼 could be kept within a more realistic range to achieve

physically accurate values of soot volume fraction. By calculation of particle age distributions in

simulated premixed flames, Singh et al. [114] proposed two correlations for the fraction of active

sites. However, their attempt to relate particle ageing with flame temperature in order to find a

general expression for 𝛼 was unsuccessful. The various forms of 𝛼 proposed in the literature are

summarised in Table 4.2. The value for 𝛼 that is predicted for each of these functions at 1700 K,

which is close to the local temperature in most of the sooting region in the diffusion flames, is

included in this table. These values show a great discrepancy among different proposed functions

for 𝛼. In some of these studies, such as [150] and [141], despite the similarities in the flames

studied and soot surface growth models implemented, different values for 𝛼 have been employed

to achieve the same soot volume fraction predictions. This discrepancy is a consequence of using

different reaction mechanisms and obtaining different predictions for the soot precursors. A role

of the chemical mechanism in soot growth is to derive the concentration of the four species, H,

OH, C2H2 and the nucleating PAH, which are directly used by the soot growth model. Most of

the mechanisms are validated and perform well in prediction of small species (i.e., H, OH and

C2H2). However, as is comprehensively discussed in [141], due to the complexity and

uncertainties involved in growth pathways of heavy PAH molecules, the performance of

different chemical mechanisms in the prediction of heavy PAH molecule concentrations can be

vastly different. The role of the chemical mechanism and its interaction with the soot model has

been discussed in more depth in Chapter 5.

80

Table 4.2 Proposed functional forms of 𝛼 for models based on the HACA mechanism.

Proposed by Function 𝜶 at 1700K

Frenklach and Wang [199] 0.1 0.1

Appel et al. [86] 𝑡𝑎𝑛ℎ(𝑎 𝑙𝑜𝑔 𝜇1⁄ + 𝑏) § 0.93

El-Leathy et al. [36] 0.0017 𝑒𝑥𝑝(12100/𝑇 ) 1.0

Guo et al. [150] 0.0045 𝑒𝑥𝑝(900/𝑇 ) 0.9

Dworkin et al. [141] 0.078 0.078

Singh et al. [114] 1 𝑓𝑜𝑟 𝐴𝑝 ≤ 0.0120.2 𝑓𝑜𝑟 𝐴𝑝 > 0.012 † 1.0

Singh et al. [114] 0.02+0.8 exp (–CAp) ‡ 0.71 § where μ1 is the first size moment of the soot particle distribution, and a and b are fitted parameters and found to be 12.56 − 0.00563𝑇 , and −1.38 + 0.00068𝑇 , respectively.

† where 𝐴𝑝 is particle residence time.

‡ different values for 𝐶 have been used for each of the flames studied.

The aim of the present study is to propose a systematic method to define a function that relates

the reactivity of soot surface sites with flame properties. Employing a detailed sectional soot

model, several ethylene diffusion flames are studied. Thomson and coworkers [141,195,210]

have shown that for each diffusion flame, a constant value for 𝛼 could be implemented to predict

soot concentration with reasonable accuracy for different fuels, pressures, and burners. From the

knowledge gained through studying these flames, a novel approach to describe soot surface

reactivity is introduced. First, a definition of the ageing parameter is proposed. It is investigated

if the surface reactivity of the soot particles could be correlated with particle age. Using a

detailed sectional model and comparisons to experimental data in the literature, abilities and

limitations of these approaches are investigated.

4.3 Numerical Model A detailed description of the governing equations, boundary conditions, and solution

methodology can be found in Chapter 2. For the gaseous phase, the fully coupled elliptical

conservation equations for mass, momentum, energy, and species mass fraction are solved. The

model utilizes the axi-symmetrical nature of the flame, and equations are solved in the two-

dimensional (𝑧 and 𝑟) cylindrical co-ordinate system. The DLR chemical mechanism (see the

chemical mechanisms section in Chapter 2) with 93 species and 719 reactions was applied to

describe the oxidation of the fuel and the formation of PAHs.

81

Soot is modeled using the detailed fully coupled sectional aerosol dynamics model discussed

in Chapter 2. In this approach the continuum soot particle mass distribution is divided into a

discrete number of soot clusters, each with an assigned mass. For this study, the soot particle

mass range is divided into 35 discrete sections that cover the soot particle diameter ranging

between 0.86 nm and 3.3 μm. Conservation equations of soot aggregate number densities, and

primary particle number densities are solved for each soot section.

Nucleation is modeled based on the collision of two pyrene molecules in the free–molecular

regime [86,90], which serves as a connection between the gaseous phase reaction mechanism

and the first soot section with collision efficiency of 1. The HACA mechanism [86] is used to

describe soot particle surface growth. PAH condensation is modelled based on collision theory

between pyrene molecules and aggregates, with a collision efficiency of 0.5 [147]. A constant

coagulation efficiency, 𝜉, of 0.2 is chosen based on the recommendation of Zhang et al. [118].

Soot particle coalescence has been modeled using the cut-off model described in Chapter 3 with

a cut–off diameter of 20 nm.

4.4 Methodology Twelve different laminar diffusion and partially premixed ethylene flames differing in fuel inlet

dilution level, inlet velocities, and burner configuration were simulated. These flames were

chosen since they were the subjects of previous studies ([141] and [211]). The experimental

datasets of Santoro et al. [192] (SA), Smooke et al. [41] (SM) Shaddix and Smyth [212] (SY)

and Arana et al. [29] (PM) were used for comparison (see Table 4.3). Three flames, i.e., SM40,

SM80 and SA, were investigated in more detail as they differ markedly in their sooting

behaviour.

4.4.1 Soot Surface Reactivity Similar to observations by Dworkin et al. [141], soot forming on the centerline region of the

flame is less sensitive to 𝛼 than the soot forming on the annulus region (wings) near the edge of

the flame. Probing the contribution of different processes to the soot mass yield confirms that

inception and PAH condensation is the dominant mechanism for soot generation along the

centerline. It also shows that soot volume fraction on the wings is more representative of surface

growth and the role of 𝛼 in simulations of soot particles. As an example, contributions of

different soot growth processes on the centerline and wings for the Santoro flame [58] (SA) are

82

presented in Figure 4.2. This graph shows that 80% of the peak soot mass on the centerline

comes from PAH growth processes, which is based on physical collisions, and do not rely on

particle surface chemistry. On the contrary, the relative contribution of PAHs to peak soot mass

on the wings is less than 8%, and nearly 92% of the peak soot mass is from HACA surface

growth. Thus, the main focus of this study is on soot growth along the wings in order to

investigate and analyze soot particle surface reactivity.

Table 4.3 Proposed functional forms of 𝛼 for models based on the HACA mechanism.

Flame Designation

Fuel Volumetric Conc. (%)

Fuel Stream Equivalence

Ratio ( )

Inlet Velocity(𝐜𝐦/𝐬) Fuel tube diameter

(𝐜𝐦) Reference

Fuel Cold gas SA

100 ∞ 3.98 8.9

1.11 [192] SAM 5.05 13.3 SM80 80

∞

35 35

0.4 [41] SM80.2 80 17.5 17.5 SM60 60 35 35 SM40 40 35 35 SM32 32 35 35 SY41

100 ∞ 4.14 8.9

1.11 [212] SY46 4.58 10.6 SY48 4.76 10.6 PM10 41 10 9.66 8.9

1.11 [29] PM20 58 20 6.82 8.9

PM24 63 24 6.35 8.9

Figure 4.2 Total mass yield (𝑔𝑠𝑜𝑜𝑡/𝑔𝑚𝑖𝑥) by all soot growth processes, HACA surface growth, and inception plus PAH condensation for a soot particle travelling a) along the centerline and b) along the pathline of maximum soot on the wings, for the Santoro flame [58] (SA).

z (cm)

MassY

ield

0 4 8 120

0.005

0.01

0.015

Total Mass YieldMass From HACAMass From PAH

(a)

z (cm)

MassY

ield

0 4 8 120

0.03

0.06

0.09

Total Mass YieldMass From HACAMass From PAH

(b)

83

Dworkin et al. [141] showed that with a constant value for 𝛼 the model is able to predict soot

concentration on the wings of the SA flame. Therefore, the same value for 𝛼 that could predict

soot concentration on the wings of the SA flame has been employed here to simulate all the

flames as a base case for comparison. The maximum soot concentrations on the wings predicted

with this 𝛼 with experimental data from [192] and [41] with experimental uncertainty estimated

based on the experimental technique are shown in Figure 4.3. As the dilution level of inlet fuel

increases, the difference between computed and measured soot concentration become more

significant. This result emphasizes the necessity of a variable form for 𝛼.

Although for each flame, a constant 𝛼 can be found which leads to a precise prediction of

maximum soot concentration, this would merely be a curve fit and would not leverage the

knowledge base of surface ageing. Such a procedure however, is a precursor to our analysis. By

examining several 𝛼 values for each flame, a different value for 𝛼 for each flame that reproduces

the most accurate soot concentration on the wings is found. These values, representing average

surface reactivity of each flame, are then used to derive functions that are then tested in the

numerical algorithm in the context of the current knowledge base of ageing. The comparison of

the computed soot concentration with an optimum 𝛼 and the experimented data from

[41,192,212] are shown in Figure 4.4. The constant 𝛼 for each flame is tabulated in Table 4.4. In

the following sections, scenarios and procedures used to obtain 𝛼 functions will be described.

Figure 4.3 Comparison of computed peak soot volume fractions on the wings using 𝛼 = 0.45 for all SM and SA flames with experimental data from [192] and [41] for coflow diffusion ethylene-air flames.

SM32SM40

SM60 SM80.2SM80

SA

0.010.1110

Soot

Vol

ume

Frac

tion

(ppm

)

Flames

Computed with α = 0.45

Experimental by [40]

Experimental by [41]

[192]

[41]

84

Table 4.4 Flames used to derive a function for surface reactivity and the optimized 𝛼 for each flame that reproduces the most accurate soot concentration on the wings.

Flame Designation Optimized Average 𝜶

SA 0.45 SY41 0.47 SY46 0.48 SY48 0.49 SM80 0.32

SM80.2 0.33 SM60 0.25 SM40 0.24 SM32 0.16

Figure 4.4 Comparison of computed peak soot volume fractions on the wings using an optimized average 𝛼 for each flame (The value of 𝛼 for each flame is shown below the computed result) with experimental data from [192], [41] and [212] for coflow diffusion ethylene-air flames.

4.4.2 Thermal Age Experimental studies have indicated that soot surface reactivity is a function of temperature

[28,32,44,53,97,98,104,148,199,201,207–209]. Thus, as the first attempt to define a function for

𝛼, a comparison was made between the reference 𝛼 for each flame to the corresponding peak

flame temperature, and to the instantaneous temperature at the location of peak soot

concentration on the wings. This comparison is shown in Figure 4.5 for a variety of diffusion

ethylene-air flames. Consistent with Singh et al. [114], it is impossible to identify a unique

SA

SM32

SM40

SM60 SM80.2SM80

SY41 SY46 SY48

0.050.5550

Soot

Vol

ume

Frac

tion

(ppm

)

Flames

Computed with ref. αExperimental by [40]Experimental by [41]Experimental by [42]

[192][41] [212]

85

function of either peak flame temperature or local temperature at the location of peak soot

concentration that can cover reference 𝛼 values for all flames.

Figure 4.5 Average soot particle surface reactivity, 𝛼, as a function of a) peak flame temperature and b) instantaneous temperature at the peak soot concentration on the wings.

In addition to temperature, experimental studies signify age of a particle as the inducer of surface

reactivity variation [22,98]. Based on these observations, a new ageing parameter is introduced,

namely thermal age (𝑇𝑎). The thermal age is defined as the integral of temperature to which a

particle has been exposed with respect to time (a temperature-time history, Eq. 4.4) along the

particle pathway. The thermal age accommodates effects of both temperature and residence time.

This new definition of soot particle age inherently considers that the more time a particle spends

in a hotter region, the more its surface reactivity is subject to change.

𝑇𝑎 = ∫ 𝑇 𝑑𝑡𝑠

( 4.4)

To investigate correlations between 𝛼 and the thermal age of individual particles, the age

distribution of soot particles was obtained. The values of reference 𝛼 implemented for each flame

were plotted as functions of thermal age of a particle at the peak soot concentration of the

respective flames (shown in Figure 4.6a). Despite the differences in their measurement

techniques, and flame configurations, the nine flames align monotonically when surface activity

of these flames are compared with their corresponding thermal age. It was suggested that the

fraction of active surface sites varies exponentially with its age [3,8]. This idea was adopted and

86

an exponential function was used to correlate thermal age and fraction of active sites. The

correlated exponential function is presented in Eq. 4.5 and its variation with thermal age is

depicted in left side of Figure 4.6 with a solid line.

𝛼 = 0.6 𝑒𝑥𝑝 (– 25.4

𝑇𝑎 ) ( 4.5)

Although, Eq. 4.5 is fitted to the data points with minimal deviation (𝑅2 = 0.94), it is not

representative of the local available active sites on a soot surface. Since the average surface

reactivity has been used to develop Eq. 4.5, this function is only suitable to predict an average 𝛼

to model a flame. The surface growth rate (�̇�𝑠) is defined as the rate of increase of the soot mass

via heterogeneous reaction of the soot surface with the gas phase; as shown in Eq. 4.3, this mass

growth rate is proportional to the instantaneous surface reactivity of the soot particles (�̇�𝑠 ∝ 𝛼).

Soot formation on the wings, being dominated by surface chemistry [141], is a metric to examine

the surface growth model’s predictive capability. Since surface growth is the rate of mass

increase, the computed peak soot concentration on the wings could be interpreted as the

cumulative effect of all the chemical reactions having occurred on the surface of the soot particle

from nucleation to this point (𝑓𝑣 ∝ ∫ �̇�𝑠𝑑𝑡). Based on the proportionality of �̇�𝑠 and 𝛼 and the

assumption that 𝛼 is a function of thermal age, the proper strategy to assign a function for 𝛼 is

through comparison of the integral of 𝛼 with respect to thermal age. ∫ 𝛼𝑑𝑡 can be thought of as

the representative surface character of a soot particle spanning from its inception to any moment

in time. Such an integral takes into account the temporal variation in surface character of a

particle at its corresponding thermal age for each flame. The 𝛼 values presented in Table 4.4 are

integrated along the pathline of maximum soot on the wings in the growth region for each flame

with respect to thermal age and the results of this integral is demonstrated on the right side of

Figure 4.6. In order to find the instantaneous 𝛼, the exponential function fitted to the integrated 𝛼

has been differentiated. The function coefficients have been optimized for the most accurate

prediction of peak soot concentration on the wings. The final function is shown in Eq. 4.6 and

variation of the integral of this function with thermal age is depicted on the right side of

Figure 4.6 with a solid line.

𝛼 = 6974.6𝑇𝑎

2 𝑒𝑥𝑝 (– 88.06

𝑇𝑎 ) ( 4.6)

87

Figure 4.6 a) Average soot particle surface reactivity, 𝛼, as a function of thermal age at the location of peak soot concentration on the wings (the line is the correlation for 𝛼, Eq. 4.5). b) The integral of 𝛼, as a function of thermal age at the location of peak soot concentration on the wings (the line is the integral of the correlation for 𝛼, Eq. 4.6).

4.5 Results and Discussion

The newly developed function, Eq. 4.6, has been implemented in the sectional model. First,

conservation equations of momentum, energy and soot number density are solved. Then, at each

location in the flame, the trajectory of the soot particle is calculated based on the flow velocity,

and the corresponding path of a fluid parcel, and then corrected for soot transport. All necessary

properties are interpolated along the trajectory of the soot particle. Starting from the nucleation

point, the thermal age is calculated and integrated along the trajectory. A unique 𝛼 is calculated

at each streamwise location of a fluid parcel containing soot. At a given height above the fuel

tube, 𝛼 is likely to vary radially as each radius represents a different pathline on which velocities,

temperature, and thus 𝑇𝑎 may all vary. Simulations have been repeated for all the flames with

this function. In Figure 4.7, the peak experimental and computed wing soot volume fractions are

compared with experimental error bars estimated based on the techniques used. Results reported

in Figure 4.7 show excellent overall comparisons of peak soot concentrations for multiple flames

considering different experimental datasets, ranging three orders of magnitude of soot

concentration. It should be noted that the partially premixed flames (PY flames) and the SAM

flame were not used during the development of the function for 𝛼, and yet the surface reactivity

88

model is able to accurately predict the soot volume fraction for these flames without any further

model adjustment.

Figure 4.7 Comparison of computed peak soot volume fractions on the wings using the 𝛼 function based on thermal age (Eq. 4.6), with experiments from [29,41,192,212].

Computed and experimental soot volume fraction contours are presented side by side for the

SM40, SM80 and SA flames on Figure 4.8. The experiments depict a dramatic shift in the

location of maximum soot away from the centreline to the wings as ethylene concentration in the

fuel stream is increased. For instance, the peak soot concentrations occur on the wings near 𝑧 = 4

cm for the SM80 flame, contrarily, in the SM40 flame, peak soot concentration is on the

centerline near 𝑧 = 2.2 cm. Comparison of the model with experiments reveals that the model

prediction of the initial formation of the soot on the wings is in good agreement with the

experiments. The model captures the general shape and magnitude of the soot isopleths for all

the flames. Moreover, the model captures the extent of the soot along the wings and the peak

soot concentration both in magnitude and location on the wings. However, the model failed to

predict the transition of peak soot concentration from the wings toward centerline as fuel dilution

is increased. Similar behaviour of this model has been reported by Dworkin et al. [141] and

Eaves et al. [195]. Both of these studies suggest that the discrepancy is due to PAH chemistry.

As discussed earlier, centerline soot growth is driven highly by PAH based growth mechanisms

and any miss-representation of the PAH concentration in the gas phase directly affects the soot

concentration on the centerline. These results suggest that future studies to investigate new

[192] [41] [212] [29]

89

pathways to form PAH molecules in the gas phase chemistry are needed, which is beyond the

scope of this work.

Figure 4.8 Isopleths of soot volume fraction (ppm) of the SM40 (left panel), SM80 (middle panel) and SA (right panel) flames. The left side of each panel is the model computed with the new 𝛼 function. The right side is the experimental data ([41] and [212]).

4.5.1 Surface Reactivity Analysis

The variation of surface reactivity calculated using Eq. 4.6 with soot particle residence time

along with predicted soot volume fraction for the SA and SM60 flames are shown on Figure 4.9.

The form of the new function suggests that the soot particle surface reactivity increases in the

early stages of soot formation in the SA flame, then it reaches its maximum (which does not

coincide with the maximum rate of soot formation) and then gradually decreases as the particles

traverse the flame. There are processes that increase surface reactivity and processes that

SM40 SM80 SA

r (cm)r (cm)r (cm)

z(cm

)

-1 0 1-1 0 1-1 0 10

2

4

6

8

0

2

4

6

8

0

2

4

6

8

10

0

3.4

0.0

0.47

0.00

90

suppress surface reactivity. The shape of the 𝛼 curve represents the balance between these

competing processes.

The increase in the surface reactivity is partly due to the increase of the number density of the

Csoot–H sites. Recent studies of detailed Monte-Carlo simulations on graphene layer surface

reactions [213,214] concluded that during the HACA growth process, χCsoot–H increases. In

addition, deposition of PAH molecules on the surface adds new sites. It is shown in [113] that

PAH deposition is the main contributor to the increase of hydrogenated site density. A

comparison of the soot mass gained by PAH-based processes (Figure 4.2) on the wings with the

increase in 𝛼 in Figure 4.9 also confirms the observed relationship between PAH deposition and

an increase in the surface reactivity by [113].

Simultaneously, there are processes causing deceleration of the surface growth. One of these

processes is carbonization, which involves polymerization, dehydrogenation, and bond

formation/rearrangements between PAH layers forming the soot particles. The carbonization

process which has received much attention in both experimental and theoretical studies

[22,86,109,113,198,208], could be characterized by the carbon to hydrogen ratio (C/H) within a

soot particle. It is suggested [22,113,208] that C/H for nascent soot particles is close to 2.0,

which is the typical value for a five member ring PAH, and it is between 5 and 10 for the mature

soot particles. Analogous to PAH molecules, as the C/H ratio increases, it is expected that the

soot particles would tend to be more stable, and thus less chemically reactive. Another factor

which could affect solid particle reactivity is the size of the particles. When a small solid particle

with negligible vapour pressure is in chemical equilibrium with the gas phase, the equilibrium

constant is proportional to the particle internal pressure [106]. The internal pressure of the

particle is related to the particle size by the Laplace formula ∆𝑝 = 2𝜎/𝑟. Substituting pressure in

the equilibrium constant equation, the equation could be expressed as a function of particle size.

−𝑅𝑇 𝑙𝑛 𝐾𝑃 = ∆𝐺𝑇0 +

2𝜈𝑠𝑣�̅�𝜎𝑟

( 4.7)

where ν is the stoichiometric coefficient and v is the volume per mole of solid particles. From

Eq. 4.7, the effect of finer particles is to increase the equilibrium constant. The increase in

internal pressure caused by reducing particle size, which is evident in the Laplace formula, leads

to an increase in thermodynamic activity of the particle substance. Since the average primary

91

particle size increases in the growth region, the particle size will have a reducing effect on the

surface reactivity.

Figure 4.9 Variation of surface reactivity and soot volume fraction as a function of soot particle residence time along the wings for SA and SM60 flames.

Figure 4.10 Variation of surface reactivity and soot volume fraction as a function of soot particle thermal age along the wings for SA, SM80 and SM40 flames.

The initial value of 𝛼 still remains to be estimated. The incipient soot particle is a 0.86 nm

diameter sphere that consists of two pyrene molecules. Each pyrene molecule has 10 C–H sites.

Assuming that all of these sites are on the surface of soot particles, the calculated χCsoot–H is an

order of magnitude lower than the estimated value of 0.23 (#/Å2). Since the average value of 𝛼

is on the order of 0.1, based solely on the above calculation, the initial value for 𝛼 should be in

92

the order of 0.01. There is a need for further investigation to reach a better estimation of the

surface reactivity of incipient soot particles. However, the results are not sensitive to the initial

value of 𝛼 as long as it is in the order of 0.01.

The variation of 𝛼 for two different flames as a function of residence time is presented in

Figure 4.9. A comparison of these curves shows that at the same residence times in these two

flames different values of 𝛼 would be obtained with the proposed function. However, if the 𝛼

variation is plotted as a function of thermal age, as shown in Figure 4.10, the 𝛼 for all of the

flames converge to a single curve. The soot concentration variation predicted for different flames

along the wings shows that for some of the flames, the surface reactivity is dominated by the

processes that induces higher 𝛼 as the soot grows, such as in the SM40 flame, but this is not

always the case. For instance most of the soot mass in the SA flame is formed in the region of

decreasing 𝛼. Therefore one has to be cautious when it comes to studying surface reactivity of

soot particles. These results suggest that it is essential that every theory proposed for surface

reactivity should be tested in several different temperature and residence time conditions before

reaching any conclusions.

In section 4.4, it was stated that the focus has been on the regions where the soot growth is

dominated by acetylene. Therefore, this function represents the effect acetylene addition has on

surface growth. Based on the assumption that HACA growth via C2H2 is the only chemical

growth pathway for soot particles that is independent of the fuel, it is expected that the proposed

𝛼 function will perform equally well for predicting soot formation in other hydrocarbon-fuelled

flames as well. One barrier to test this hypothesis is the availability of a reliable chemical

mechanism.

4.5.2 Parameter Study

It was shown in Figure 4.7 and Figure 4.8 how well the soot model is able to predict soot

concentration on the wings; however there are several prerequisites for a flame model to exploit

the proposed function. These requirements could be divided into two main groups; the

parameters that are derived from the gas phase chemistry and the soot parameters that are

predicted by the soot model.

93

4.5.2.1 Gas phase chemistry parameters

The most important parameters from the gas phase that affect surface reactivity are temperature,

acetylene concentration, hydrogen radical concentration, and to some extend PAH concentration.

Temperature plays two important roles in modeling surface growth. First, the reactions in the

HACA surface scheme (Table 4.1) are temperature dependent; especially the hydrogen

abstraction reaction, S1, which has the highest activation energy. The second role of the

temperature is in the 𝛼 function. The proposed 𝛼 is a function of thermal age which is derived

from temperature. In order to examine the performance of the model in predicting temperature,

computed and measured temperature contours for the SA flame are depicted in Figure 4.11 and

for a more quantitative comparison, temperature profiles on the centerline for the SM40, SM80

and SA flames are demonstrated in Figure 4.12. The model reproduces temperature very well,

particularly on the wings. The slight overprediction in the vicinity of the centerline which is also

evident in Figure 4.12, are in the regions where soot is underpredicted. The associated

underestimation of soot radiation caused the temperature to be overpredicted. As it was stated in

the previous sections, it would have been more relevant if all the analysis and discussion were

made on the flame wings; the reason comparisons were made on the centerline in Figure 4.12

and Figure 4.13 is because of unavailability of experimental data on the wings.

Figure 4.11 Comparison of computed (left panel) and experimental (right panel, from [192]) isotherms of the SA flame.

94

Figure 4.12 Comparison of numerical and experimental (from [215] and [37].) temperature profiles along the centerline of the flames, as a function of axial height.

Acetylene is a principal soot growth species. Hence, accurate prediction of acetylene is critical to

predictions of soot. Figure 4.13a compares the radial variation of the acetylene profile generated

by the model to experimental data from [34] at two different heights above the burner for the SA

flame. The overall shape, trend, and magnitude of the experimental data are well reproduced by

the model. As shown in Figure 4.13b, centreline acetylene profiles are also simulated well. The

experiments depict a slight shift in the location of maximum acetylene downstream in the flame

as ethylene is increased. These changes with decreasing dilution are also reproduced. It is worth

noting although most of the soot mass is derived from acetylene addition, the hydrogen

abstraction reactions (especially with H radicals in the diffusion flame) are rate limiting in the

HACA mechanism. Therefore, predictions of the concentration of H radicals and to some extent

OH radicals are more important for surface growth as compared to C2H2.

Hydrogen addition to the fuel stream has two effects on soot formation: a dilution effect and a

chemical effect [150]. The dilution effect has been investigated in this work. The chemical effect

has been investigated by Guo et al. [150]. It was shown that addition of hydrogen will not

change the H radical concentration in the flame. Since H radicals are the bottle neck of HACA

growth and one of the potential contributors to soot surface reactivity variations, it is expected

that hydrogen dilution not to have any direct effect on soot surface reactivity. It should be noted

that the increase in H2 concentration will shift the S1 reaction in Table 4.1 towards decreasing

hydrogen abstraction. However, Guo et al. [150] showed that its effect on the growth rate is

z (cm)

T(K

)

0 2 4 6 8400

1300

2200

SM40 Exp. bySM40 modelSM80 Exp. bySM80 modelSA Exp. bySA model

[215]

[37]

[215]

95

negligible. Thus, the proposed function should be able to predict soot formation in the mixture of

hydrocarbon/hydrogen fuels. However, further investigations would need to be done to verify

these hypotheses.

The surface growth model is linked to the PAH concentrations in two ways. Since the nucleation

of the soot particles is modeled based on collisions of heavy PAH molecules, the inception of

nascent soot particles is related to PAH formation in the gas phase. Therefore, inaccuracy in the

gas phase chemistry in predicting the PAH concentration results in calculation of an erroneous

particle age field, which has direct impact on the performance of the proposed 𝛼 function.

Another pathway that links PAH concentration to surface growth is because of the role of PAHs

in predicting the soot particle diameters [40]. The importance of particle diameters on surface

growth is discussed in the following paragraph (section 4.5.2.2). Benzene, being the first

aromatic molecule, is a gauge to demonstrate the performance of the gas phase mechanism in

prediction of PAH concentrations. The formation of heavier PAH molecules are more

complicated than that of benzene. Many mechanisms can predict benzene but fail to accurately

predict heavier PAH molecules [87,141]. However, if benzene is not well predicted, then higher

PAH molecules are unlikely to be correctly predicted. As depicted in Figure 4.13c, the model for

the diluted ethylene flames reproduces benzene concentrations along the centreline reasonably

well; significant increases in benzene observed experimentally with decreasing dilution are

simulated well.

Figure 4.13 Comparison of the computed (lines) and experimental (symbols) a) concentrations of acetylene at the 𝑧 = 7 mm and 𝑧 = 20 mm axial heights as a function of radial distance from the centreline for the SA flame (measurements from [34]) b) concentrations of acetylene on the centreline for the SM40 and SM80 flames (measurements from [215]) c) concentrations of benzene on the centreline for the SM40, and SM80 flames (measurements from [215]).

(a) (b) (c)

96

4.5.2.2 Soot model parameters

In addition to the gas phase chemistry, there are several parameters from the soot model that

influence the applicability of the 𝛼 function, among which the most crucial parameter is the

predicted surface area of the soot particles. By substitution of the concentration of hydrogenated

sites from Eq. 4.1 into the surface growth rate, Eq. 4.3, will become:

𝑅4 = 𝛼 𝐴𝑠𝑘4 [C2H2] 𝜒Csoot–H

𝐴𝑣 ( 4.8)

𝐴𝑠 = 𝜋4

𝑑𝑝2𝑁𝑝 ( 4.9)

where 𝐴𝑠 is the total area of the soot particles per unit volume in space, dp is the primary particle

diameter and 𝑁𝑝 is the total number of primary particles per unit volume in space. These

equations show that if the soot model failed to predict the soot particle diameter or number

density accurately, then a different value for 𝛼 would be needed to compensate for the effect that

area has on the surface growth rate. For instance, if the soot model predicts very low soot particle

diameters, for the same mass of soot, then too many soot particles would exist in the predictions

with a high surface to volume ratio. This high surface to volume ratio will increase the soot

HACA reaction rates substantially. Therefore a very low value for 𝛼 would be needed to obtain

the correct amount of soot mass. Such compensation is essentially what happened in the study by

Dworkin et al. [141], wherein the model is the same as that in the present study, except as it is

lacking any coalescence model, diameters were underpredicted; thus Dworkin et al. [141] had to

use a very low 𝛼 (0.078 comparing to 0.45 in this study) to predict the correct soot volume

fraction. There are two processes in soot modeling that are directly involved in determining the

particles’ diameter; one is the competition between nucleation and PAH condensation and the

other is coalescence. For a detailed analysis of the significance of these processes on particle

diameter prediction the reader is referred to [40].

The SA flame has the geometry and flame configuration that is most widely used to study soot

formation in diffusion flames. Numerous experimental techniques have been adapted by different

studies to measure soot particle properties in this flame [29,37,192,212] (and all other works

citied in Appendix B). The measured soot properties include soot volume fraction, primary

particle diameter, primary particle number density, aggregate number density, and number of

primary particles per aggregate. In addition to soot properties, experimental measurements for

97

temperature, velocity and some species concentration are also available for this flame. Based on

the availability and reliability of the measured soot and flame properties, this flame has been

considered as one of the target flames for the International Sooting Flame (ISF) Workshop [191].

To demonstrate the abilities of the soot model to predict different soot particle properties, the

simulation results of soot volume fraction, primary particle diameter, primary particle number

density, aggregate number density, and number of primary particles per aggregate for the SA

flame is compared with the experimental measurements and the comparison plots are available in

Appendix B. The predicted soot volume fraction as well as spatial and temporal expansion of

soot concentration along the wings are in perfect agreement with the experiments. On the

centerline soot volume fraction is well predicted with an underprediction in the vicinity of the

peak soot concentration. Finally, the model demonstrates the ability to simulate the right order

and trend of all measured soot morphology parameters both on the wings and on the centerline.

4.6 Conclusions

A new definition for particle age based on temperature-time history has been proposed. With the

calculated soot thermal age, it was investigated if particle age could be correlated with the

reactivity of surface sites. Surface reactivity was expressed as an exponential function of particle

age. Measured soot volume fractions could be well reproduced with this function for a variety of

ethylene flames of different dilution levels, premixing and burner configurations, without any

adjustment or tuning of the function. It was discussed how different parameters from the gas

phase and the soot model could affect the applicability of the proposed function. It should be

emphasized that the methodological study of soot particle surface growth proposed in this study

was the driving force behind this work, not just the coefficients provided for the function. The

proposed function has been derived based on the growth region of the atmospheric diffusion and

partially premixed ethylene/air flames; future efforts will seek to extend this study to include

elevated pressures, different fuels and the reactivity of the soot particles in oxidation regions.

98

Chapter 5 Reversibility of Nucleation and

Condensation

5.1 Introduction

The existence of condensable species in the gas phase will move the system into a

nonequilibrium state. Equilibrium can be achieved by formation of new particles (nucleation) or

deposition of condensable species on existing particles (condensation) [106]. Aromatic species

are theoretically the only species in flame environments that have the potential to form

condensable species. Therefore, in view of soot formation pathways, formation and growth of

aromatics link the gas phase chemistry to condensed phase formation.

One of the most acceptable mechanisms for formation of condensed-phase materials is based on

the hypothesis that the Van der Waals interaction forces of heavy PAH molecules are sufficiently

large that they can hold on together a pair of PAHs during physical collision, thus forming PAH

dimers. The sequence of collisions among PAH dimers and PAH molecules leads to the

formation of PAH trimers, PAH tetramers and so on. Meanwhile, individual PAH species within

the newly formed clusters keeps increasing in size via molecular chemical growth reactions.

Subsequently, the PAH clusters evolve into solid particles. In addition, soot particles can grow

via the collision and the resultant deposition of PAH species on the soot surface. This process is

often referred to as PAH–soot surface condensation.

99

Following the pioneering work by Frenklach and Wang [91], theoretical aspects of modest sized

PAH agglomeration have been investigated by several studies under flame conditions

[120,209,216,217]. This mechanism for nucleation is referred to as ‘collisional coagulation’.

Molecular dynamic models were employed by Schuetz and Frenklach [120] and more recently

by Herdman and Miller [94] to examine the possibility of the collisional coagulation mechanism

under flame conditions by determining the lifetimes of dimers. These studies concluded that the

onset of PAH dimerization in flame conditions is possible for a pair of PAH molecules as small

as pyrene.

Despite all theoretical studies, there is mostly indirect experimental evidence supporting

formation of condensed-phase materials through PAH dimerization. Dobbins et al. [53]

experimentally studied the chemical evolution and the PAH components of soot particles

extracted from the centerline of a laminar ethylene diffusion flame. They found that the

thermodynamically stable PAH species (stabilomers) with a mass range of 202 amu (C16H10) to

374 amu (C30H14) are the constituents of the soot precursor particles. However, C20H12, C22H12,

and C24H12 with atomic mass units of 252, 276, and 300, respectively, had the highest

concentrations. A similar observation has been made by Teini et al. [104]. The fringe length of

the PAH molecules accommodated within a soot particle, which is a strong function of the

number of carbon atoms in the molecule, has been measured by Teini et al. [104]. The soot

particles were created by the pyrolysis of acetylene in a rapid compression machine at 10 atm

with temperatures ranging from 1600 K to 2000 K. The mean fringe length of the soot forming

PAH molecules was found to be 0.65 nm which corresponds to a PAH molecule with 20 carbon

atoms.

As a practical measure, the formation of dimers is assumed by most of the PAH-based soot

models to be the inception of the solid particle phase [17,90]. For a collisional coagulation

nucleation mechanism, the nucleation rate can be estimated based on the collision rate of the two

condensable PAHs in the free-molecular regime. Similarly, for heterogeneous condensation, the

rate of soot mass addition is considered to be proportional to the rate of collisions between PAH

molecules and aggregates. An all–effective collision among condensable molecules is currently a

common assumption in many numerical simulations of soot formation in flames

[86,118,123,141,147,153,195,210,218]. This assumption is made despite the fact that an energy

barrier must be overcome to stabilize small PAH clusters for the inception of stable nuclei.

100

Efficiencies are used to account for the probability of sticking in each collision. For example,

Zhang et al. [147] assumed a 50% sticking probability in each collision for condensation, 𝛾 , in a

coflow ethylene/air diffusion flame. These values were chosen for the model to match the

predicted soot with the measured properties. In a recent study by Saffaripour et al. [40], the

effect of the relative contribution of nucleation versus condensation on soot particle size

predictions in laminar coflow diffusion flames was studied. It was shown that the result of

favoring nucleation by increasing its effectiveness is the prediction of a large number of small

primary particles which drastically decreased the average primary particle diameter, while

favoring condensation results in larger primary particle predictions. These observations were the

motivation to study sticking probability in the PAH nucleation and growth processes and develop

models that better represent the physics of these processes.

A significant number of recent investigations focused on improving modeling of nucleation and

condensation processes based on the nature of physical PAH nucleation and condensation.

Instead of a constant sticking probability, D’Alessio et al. [219] by studying nascent and mature

soot particles formed in ethylene–air premixed flames stabilized on a water-cooled capillary

burner proposed a function for collision efficiency of particles smaller than 10 nm. The Van der

Waals interactions of the nanoparticles were estimated based on gas kinetic theory and

neglecting collisional friction from interactions with the surrounding gas. The estimated

interactions of nanoparticles were employed to define the collision efficiency function which is

given by Eq. 5.1.

𝛾 = 1 − (1 +𝛷0𝑘𝑏𝑇 ) 𝑒𝑥𝑝 (−

𝛷0𝑘𝑏𝑇 ) ( 5.1)

Here, is the Boltzmann constant; is the temperature, and is the potential well depth

which is calculated using the Hamaker constant of the colliding elements [220]. The potential

well depth is linearly proportional to the reduced mass of collision participants. Therefore, the

proposed collision efficiency is a function of temperature and size of the particles. The proposed

function has been employed by D’Anna et al. [117,177,221] to model particle size distribution

(PSD) in rich premixed ethylene flames and soot formation in nonpremixed flames of methane

and ethylene. The collision efficiency function has been improved by Lindstedt and Waldheim

[222] to consider the collisional stabilization effect by the surrounding gas to model PSDs in a

burner-stabilized stagnation premixed ethylene/air flames.

101

Recent studies on PAH dimerization demonstrated that under flame conditions dissociation of

dimers is thermodynamically favored. In work by Sabbah et al. [102], dimerization of two

pyrene molecules was experimentally and theoretically explored. An equilibrium constant for the

dimerization of pyrene was proposed. Based on the estimated equilibrium constant, it was

concluded that dimerization must be highly reversible at high temperatures. It was asserted in the

work by Sabbah et al. [102] that any proposed collision efficiency should be correlated with

equilibrium concentration ratios. In addition to the study by Sabbah et al. [102], there have been

several attempts to describe equilibrium properties for the dimerization of PAHs which include:

defining an equilibrium constant for the dimerization of pyrene, ovalene, and circumcoronene by

Wang [17] utilizing binding energies from [94]; determining vibration modes and binding

energies for pyrene, coronene, ovalene, hexabenzocoronene, and circumcoronene dimers in the

work of Totton et al. [223]; an aggregation efficiency for PAH dimerization and PAH–PAH

cluster collisions (condensation) based on the equilibrium constant in the same work by Totton et

al. [223]; and estimating of the frequencies of the vibration modes for coronene stacks

containing up to eight coronenes and vibration modes for various PAH dimers by Rapacioli et al.

[224,225]. All of these works listed above reached the conclusion that PAHs much larger than

pyrene (ovalene and larger) would need to be present for physically realistic PAH attraction to

play a role in nucleation and condensation. Despite the participation of different PAH molecules

in the nucleation process, dimerization of PAH molecules lighter than circumcoronene (C54H18, a

PAH with 19 aromatic rings, A19) in the flame conditions is highly reversible. In addition, while

PAH stack sizes increase, as would occur during the condensation process, the deposition

process becomes more favorable.

The reversibility of nucleation and condensation processes has been developed by Eaves et al.

[193] into a novel nucleation and condensation model to study soot formation in the Santoro

coflow diffusion flame [58]. It has been concluded in the work by Eaves et al. [193] that a fully

reversible nucleation model and the condensation model with an efficiency based on equilibrium

constants can reasonably reproduce all relevant average soot morphological properties. Based on

the comparisons of different efficiency based models, neither single constant efficiencies nor the

functional form as in [223] for condensation will give satisfactory results for all soot

morphological parameters.

102

Measurements of the change in the size distribution function with time can be used to examine

the particle growth modes. The PSD data provides the details necessary for soot morphology

model validation as compared to most soot measurements in different flames, which only consist

of average quantities. Abid et al. [19] recently published a series of soot particle size distribution

measurements in the burner-stabilized stagnation (BSS) premixed ethylene/air flames. This set of

experiments includes temperature measurements and particle size distributions for six different

spacings between the burner nozzle and the stagnation surface. The burner-stabilized stagnation

flame provides well-defined boundary conditions which are necessary from the modeling point

of view. More importantly, in this flame the soot particles form in the post flame regions where

the concentration of H radicals are negligible [17]. The absence of H radicals significantly limits

the contribution of the HACA mechanism to soot particle formation. Therefore, soot growth in

this flame is controlled by nucleation and condensation processes. This unique situation allows

examining the nucleation and condensation models with more confidence by eliminating

uncertainties associated with HACA–based surface growth. Thus, in this chapter, different

aspects of PAH molecule contribution to soot formation and size distribution will be investigated

by modeling soot particle size distributions in BSS premixed ethylene flames. A discussion on

the role of PAH formation/oxidation chemistry on soot formation is presented. The necessity of

modeling nucleation as a reversible process is investigated. A computationally efficient

equilibrium–based PAH condensation model is proposed. Sensitivity of PSDs to differing growth

mechanisms and to dimer and PAH stacks equilibrium parameters are examined. Finally, the

performance of soot models in predicting soot formation in diffusion flames is tested to further

validate the proposed model and to study the differences in soot formation in premixed and non-

premixed environments.

5.2 Methodology

5.2.1 Burner and Flame Description

The burner-stabilized, stagnation flame is shown schematically in Figure 5.1. The burner consists

of a water-cooled sintered porous plug with an outer diameter of 7.6 cm. The fuel and oxidizer

mixture consisting of 16.3% ethylene–23.7% oxygen–argon (equivalence ratio 𝜙 = 2.07) flows

at a cold gas velocity of 8 cm/s (STP) into the burner nozzle. A shroud of nitrogen flowing at

43.6 cm/s (STP) through a concentric porous ring seperates the flame from the ambient air. This

103

configuration will maintain a flat flame at atmospheric pressure. A circular aluminum plate is

positioned in parallel to the burner surface at the separation distance, 𝐻𝑝, to form an

axisymmetric stagnation flow. The top of the plate is water cooled and the separation distance

can be resolved to within an accuracy of ±0.015 cm.

Temperature and soot particle size distributions have been measured for five different separation

distances, 0.55, 0.6, 0.7, 0.8, 1.0, and 1.2 cm, namely. The burner nozzle temperature has been

maintained at 353 ± 10 K. The temperature was measured using a type-S thermocouple. The

plate temperature is measured by a type-K thermocouple embedded at the bottom of the plate.

The soot particles are sampled through an orifice 127 μm in diameter placed on the central axis

of the burner. The soot sample is drawn into a probe and diluted with a cold nitrogen flow at 30

L/min (STP) to minimize the particle losses in the sampling line. The sampled soot particle PSD

has been measured using scanning mobility particle sizing (SMPS). PSD measurements also

have been used to determine the soot volume fraction and soot particle total number density at

the stagnation plate. This flame has been numerically studied only by Lindstedt and Waldheim

[222]. Similar to the Santoro flame [58], this flame also has been considered as one of the target

flames for the International Sooting Flame (ISF) Workshop [191]. More details about the burner

and measurements technics and uncertainties of measurements can be found in [17,19,226,227].

Figure 5.1 Schematic representation of a burner stabilized stagnation flame, including coordinate orientation.

104

5.2.2 Model Description

The BSS flames studied here have a separation-to-diameter ratio ≪ 1, for which it is appropriate

to use the pseudo one-dimensional formulation for stagnation/counter flow reacting jets of Kee et

al. [126]. These equations were later incorporated into the OPPDIF code [169] to be solved

numerically. Therefore, the OPPDIF code has been used as the base of the numerical solution for

this study. There have been modifications to the OPPDIF code to account for existence of soot

particles in the flow. The main modification to the species conservation equations has been

adding the chemical interaction of soot particles with the gas phase species as a source term to

make the model coupled. The effect of particle diffusion on the gas phase has been taken into

account by adding an additional term to the correction velocity calculation. The presence of

particles in the mixture is recognized when calculating mixture average properties, e.g., mixture

density, and mixture molecular weight. Finally, a radiation model based on the optically thin

assumption has been added to the energy equation to allow for particles and gas species

radiation. The governing equations of the stagnation reacting flow and the modified Newton

method used to solve these equations are described in Chapter 2.

5.2.2.1 Sectional aerosol dynamic model

In addition to the modifications to the gas phase governing equations, a detailed fully coupled

sectional aerosol dynamic model is employed to predict soot particle size distribution. The soot

particle mass range is divided into 55 to 65 discrete sections that cover the soot particle diameter

rang between 1 and 100 nm The spacing between the sections has been chosen to be consistent

with the sampling bins considered in measuring particle size distribution. Conservation equations

of soot aggregate number densities, and primary particle number densities are solved for each

soot section. The soot sectional model considers nucleation, surface growth, PAH surface

condensation, surface oxidation, coagulation, fragmentation, particle diffusion, thermophoresis,

and particle radiation. The details of the soot model can be found in the mathematical model

chapter (Chapter 2, Section 2.3).

5.2.2.2 Reversible nucleation

The reason for using efficiencies in the nucleation models is to account for the fact that the pair

of PAH molecules present in a dimer due to thermodynamic conditions can separate, which

105

according to Sabbah et al. [102] is very common at flame temperatures. Thus, to avoid dealing

with arbitrary or tuned efficiencies and to improve the nucleation model based on a fundamental

understanding of the dimerization process, the nucleation process has been allowed to be

reversible.

PAH + PAH ⇐⇐⇐⇐⇐⇐⇒ Dimer ( 5.2)

The forward rate of dimerization is determined by the rate of physical collision of the nucleating

PAH molecules in the free-molecular regime, similar to the non-reversible nucleation model. The

forward rate of dimerization and the forward rate coefficient (𝑘𝐹𝑊𝐷) for a dimer composed of

𝑃𝐴𝐻𝑗 and 𝑃𝐴𝐻𝑘 are calculated according to Eqs 5.3 and 5.4 respectively:

(𝜕𝑁𝐷𝐼𝑀

𝜕𝑡 )𝐹𝑊𝐷= 𝑘𝐹𝑊𝐷[PAHj][PAHk] ( 5.3)

𝑘𝐹𝑊𝐷 = 2.2𝜌 ⎷

√√√√8𝜋(𝑁𝐶,PAHj

+ 𝑁𝐶,PAHk)𝑘𝐵𝑇

𝐶𝑚𝑎𝑠𝑠𝑁𝐶,PAHj𝑁𝐶,PAHk

(𝑑PAHj+ 𝑑PAHk)

2 𝐴𝑣

2 ( 5.4)

where 𝑘𝐵 is the Boltzmann constant; 𝐶𝑚𝑎𝑠𝑠 is the mass of a carbon atom; 𝑁𝐶,PAH is the number

of carbon atoms in the incipient PAH species; 𝑑PAH is the diameter of the incipient PAH species;

𝐴𝑣 is Avogadro's number; and [PAH] denotes the molar concentration of the incipient PAH

species.

Following the work by Eaves et al. [193], the reverse rate coefficient ( ) is calculated from

the relation between dimerization equilibrium constant and rate coefficients, Eq. 5.5.

𝑘𝐹𝑊𝐷𝑘𝑅𝐸𝑉

= 𝐾𝑝,𝐷(𝑅𝑇 )∆𝑛 ( 5.5)

Assuming the dimer is a gaseous species leads to ∆𝑛 equal to 1. In order to determine the

equilibrium constant of dimerization, Eq. 5.6, the Gibbs free energy of dimerization has to be

evaluated, which is related to enthalpy and entropy through the following relation: ∆GD° =

∆𝐻𝐷° − 𝑇 ∆𝑆𝐷

° . The following equations can be derived utilizing statistical mechanical

principles [228] following the assumptions described in [17,193,223] to estimate the change in

entropy and enthalpy of the nucleation processes for any arbitrary PAH–PAH collision event:

106

Kp,D = exp(

−∆GD

°

RT ) ( 5.6)

∆𝐻𝐷 ≅ −𝐸0 − 4𝑘𝐵𝑇 + ∑ (12

+ 1𝑒ℎ𝑐𝑣𝑖/𝑘𝐵𝑇 − 1) ℎ𝑐𝑣𝑖

6

𝑖=1 ( 5.7)

∆𝑆𝐷𝑅𝑢

≅ 𝑙𝑛[(

𝑚3ℎ𝑐𝐵1̅𝐵2̅

2𝑚1𝑚2𝐵3̅ )

3/2ℎ3𝑃

𝜋2(𝑒1 𝑘𝐵𝑇 )4

𝜎1𝜎2𝜎3 ]

+ ∑ {ℎ𝑣𝑖/𝑘𝐵𝑇

𝑒ℎ𝑣𝑖/𝑘𝐵𝑇 − 1− 𝑙𝑛(1 − 𝑒−ℎ𝑣𝑖/𝑘𝐵𝑇 )}

6

𝑖=1

( 5.8)

Here, 𝛥𝐻𝐷 is the enthalpy change due to dimerisation, 𝛥𝑆𝐷 is the entropy change due to

dimerisation, 𝑅𝑢 is the universal gas constant, 𝑘𝐵 is the boltzman constant, 𝑇 is the gas

temperature, ℎ is Plank’s constant, 𝑐 is the speed of light, 𝑚1 and 𝑚2 are the masses of the two

colliding entities, 𝑚3 is the combined mass of the two entities, 𝜎𝑖 are the symmetry numbers,

with dimers assumed to have no symmetry (𝜎𝑖 = 1), and 𝐵𝑖 are the rotational constants, utilizing

the correlation presented in [229] and assuming the rotational constants of a dimer are half of a

monomer PAH [17].

The remaining parameters, influencing both the enthalpy and entropy change, are 𝐸0, the binding

energy, and 𝑣𝑖, the 𝑖𝑡ℎ (of 6 in total) vibration mode frequencies created when a nucleation

process occurs. It is assumed that the frequency of vibration modes of colliding entities are not

altered by the nucleation processes, and vibration mode frequencies here represent the 6 newly

created modes. This assumption indicates that no chemical bond is forming as two bodies collide

during the nucleation process [17,223].

According to Herdman and Miller [94], binding energy is linearly proportional to the reduced

mass of the colliding entities (whether they be individual PAHs, or a PAH colliding with an

existing PAH stack). Therefore, the binding energy for any two colliding PAHs can be

determined by comparing the reduced mass of the colliding entities to the reduced mass of PAH

dimers for which binding energy is known. Herdman and Miller [94], Sabbah et al. [102], and

Totton et al. [223] determined the binding energy for several pairs of PAH dimers ranging from

benzene to circumcoronene (C150H30) in size. Given the more accurate binding energies proposed

in [223], the binding energy of a coronene dimer, 69.2 kJ/mol, is used to calibrate the magnitude

107

of different PAH dimer binding energies. The study in [223] also provides all needed vibration

frequencies for the collision of five homo-molecular dimers, pyrene, coronene, ovalene,

hexabenzocoronene, and circumcoronene. These studies indicate that the frequencies do not vary

significantly between differing PAH pairs, with the intermolecular vibrational modes for

coronene ranging from 70 to 4 cm-1. In another related study by Sabbah et al. [102], based on the

equilibrium constant proposed for pyrene dimerization, an effective vibration frequency of 18

cm-1 can be inferred. Since the exact vibration frequencies for all pairs of PAH dimers

considered for nucleation are not available, given the range present in the literature, a value of 16

cm-1 is used for the nucleation process.

Once the entropy and enthalpy change have been determined, the equilibrium constant can be

expressed as a function of monomer vibrational frequencies, binding energy, and temperature.

Thus, the reverse rate of dimerization will be:

(𝜕𝑁Dim

𝜕𝑡 )𝑅𝐸𝑉=

𝑘𝐹𝑊𝐷𝑅𝑇 𝐾𝑝

[Dim] ( 5.9)

where [Dim] is the concentration of the dimers. To track dimer concentrations of different pairs,

additional transport equations are needed. The total number of dimer transport equations depends

on the number of PAH species that are considered to nucleate. If 𝑛𝑃𝐴𝐻 is the total number of

PAH species that nucleate, the number of possible pair of dimers is equal to a 2-combination

from 𝑛𝑃𝐴𝐻 and can be evaluated as follows:

𝑁DIMER = (𝑛PAH

2 ) =𝑛PAH(𝑛PAH − 1)

2 ( 5.10)

The transport equation for the dimers is similar to that of the first section soot particles. For the

one dimensional stagnation flame, the dimer transport equations are provided in Eq. 5.11.

𝜌𝑢

𝜕𝑁𝑖𝑑

𝜕𝑧+ 𝜕

𝜕𝑧 (𝜌𝑁𝑖𝑑𝑉𝑖

𝑑) − 𝜌𝑆�̇�𝑑 = 0 ( 5.11)

(𝑖 = 1, 2, … , 𝑁DIMER)

Here, 𝑁𝑖𝑑 is the number density of the 𝑖𝑡ℎ pair of PAH dimers; 𝑆�̇�

𝑑 contains the source and sink

terms associated with the rate of change of dimer mass and can be expressed in terms of soot

process:

108

𝑆�̇� =(

𝜕𝑁𝑖𝑑

𝜕𝑡 )𝑛𝑢–𝐹𝑊𝐷

+(

𝜕𝑁𝑖𝑑

𝜕𝑡 )𝑛𝑢–𝑅𝐸𝑉

+ [(𝜕𝑁1𝜕𝑡 )𝑐𝑜𝑛𝑑

+ (𝜕𝑁1𝜕𝑡 )𝑠𝑔

+ (𝜕𝑁1𝜕𝑡 )𝑜𝑥

+ (𝜕𝑁1𝜕𝑡 )𝑐𝑜𝑎𝑔

+ (𝜕𝑁1𝜕𝑡 )𝑓𝑟]

𝑁𝑖𝑑

𝑁1

( 5.12)

where, the first two terms represent the forward and reverse rate of dimerization of the 𝑖𝑡ℎ pair of

PAH dimers. The rest of the terms are calculated by multiplying the corresponding source term

for the soot particle’s first section by the ratio of the number density of the 𝑖𝑡ℎ pair of PAH

dimers to the number density of the first soot section (𝑁𝑖𝑑 𝑁1⁄ ).

5.2.2.3 Condensation Efficiency

Despite the necessity to improve the condensation model indicated in the previous section,

limited theoretical studies have been conducted to raise the current understanding of the PAH

deposition process. Following the pioneering condensation model of Frenklach and Wang [199]

which proposed deposition of PAH molecules on the surface of the soot particle upon collision, a

handful of studies have been directed to validate and improve the condensation model. A study

by Miller [230] is among the first theoretical studies that confirmed the possibility of formation

of PAH stacks as PAH molecules grow in the flame environments by determining the van der

Waals attractive potentials. D’Alessio et al. [219] added a collision efficiency function based on

the pairwise interaction between particles according to a Lennard–Jones attractive and repulsive

potential to explain the low coagulation efficiency measured in ethylene premixed flames. The

role of temperature in successful PAH collisions further was emphasized when studying PAH

mass spectra in different flame environments in [105]. The above findings suggest that the

collision efficiency may depend on the flame temperature, PAH diameter and/or the PAH mass.

Raj et al. [231] combined the above findings in a correlation for the collision efficiency, that was

then used to generate the soot mass spectra for a number of laminar premixed C2H4–O2 flames at

different pressures. In Totton et al. [223], molecular dynamics simulations were utilized to

propose a coagulation efficiency for PAH–PAH (nucleation) and PAH–PAH cluster

(condensation) collisions that had dependencies on temperature and collisional reduced mass. All

of these proposed efficiency models overlooked the equilibrium state and subsequently the role

of PAH concentration on progress of condensation process. The consequence of such an

109

assumption is complete depletion of all gas phase PAHs through the soot growth in a

thermodynamic favorable environment and endless growth of particles.

In the model proposed by Eaves et al. [193], the surface of the soot particle is assumed to be

covered with loose PAH molecules. The surface PAHs can leave the surface in order to reach

equilibrium with the gas phase, if the partial pressure of the corresponding PAH in the gas phase

drops significantly. Therefore the balance between PAH molecules depositing on the soot

surface and those leaving the surface determine the net growth of particles. The rate of

evaporation of surface PAHs was calculated based on the equilibrium constant in the same

fashion as the reverse rate of nucleation with the exception that negative condensation rates are

disallowed. This more fundamentally advanced model was shown to accurately predict

experimental data in a coflow diffusion flame.

The main drawback of the Eaves et al. [193] model is its cumbersome implementation and

additional computational cost. Since the concentration of surface PAHs are required to find the

reverse condensation rate, additional transport equations have to be solved. The number of

additional transport equations depends on the number of condensing species which makes the

total number of transport equation for surface PAHs more than the number of equations solved

for soot primary particles and aggregates in total. In addition, the stiffness the new transport

equations impose on the convergence history can increase the computational time of an already

computationally intensive soot model by a factor of five.

To make the condensation process sensitive to the equilibrium conditions and concurrently avoid

the computational burden of the reversible model, an alternative approach is proposed herein

which is an equilibrium based sectional condensation efficiency. The new condensation

efficiency model confines the growth rate via PAH addition as the gas phase PAH partial

pressure drops beyond the equilibrium concentration. The equilibrium concentration is defined

based on the growth of a particle between two subsequent sections defined by the reaction

described in reaction 5.13.

Soot𝑖 + 𝑛 PAH Soot𝑖+1 ( 5.13)

Here, 𝑛 is the total number of PAHs needed to move a particle from section 𝑖 to section 𝑖 + 1 and

is calculated by:

110

𝑛 =𝑊 𝑇𝑃𝐴𝐻

𝑈𝑖(𝑓𝑠 − 1)𝐴𝑣 ( 5.14)

where, 𝑊 𝑇PAH is the molecular weight of the PAH; 𝑈𝑖 is the representative mass of section 𝑖; 𝑓𝑠

is the sectional spacing factor, and 𝐴𝑣 is Avogadro’s number. The equilibrium concentration of

PAH species can be calculated based on Eq. 5.15.

𝐾𝑝,𝐶 = exp(

−∆GC

°

𝑅𝑇 )= exp

(−

∆𝐻𝐶° − 𝑇 ∆𝑆𝐶

°

𝑅𝑇 )=

𝑁𝑖+1𝑁𝑖 𝜒𝑒𝑞,PAH

𝑛 ( 5.15)

In order to determine the equilibrium constant and Gibbs free energy, the following equations

can be derived utilizing statistical mechanical principles [228] following the assumptions

described in [17,193,223] to estimate the change in entropy and enthalpy of the condensation

process for addition of 𝑛 PAH molecules to a soot particle:

∆𝐻𝐶° ≅ −𝑛𝐸0 − 4𝑛𝑘𝐵𝑇 + ∑ (

12

+ 1𝑒ℎ𝑐𝑣𝑖/𝑘𝐵𝑇 − 1) ℎ𝑐𝑣𝑖

6𝑛

𝑖=1 ( 5.16)

∆𝑆𝐶

°

𝑅𝑢≅ 𝑙𝑛

⎣⎢⎢⎢⎡

⎝⎜⎜⎛𝑚3(ℎ𝑐𝐵1̅)

𝑛𝐵2̅

2𝑛𝑚1𝑛𝑚2𝐵3̅ ⎠

⎟⎟⎞

32

(ℎ3𝑃

𝜋2(𝑒1 𝑘𝐵𝑇 )4)

𝑛 𝜎1𝑛𝜎2𝜎3

⎦⎥⎥⎥⎤

+ ∑ {ℎ𝑣𝑖/𝑘𝐵𝑇

𝑒ℎ𝑣𝑖/𝑘𝐵𝑇 − 1− 𝑙𝑛(1 − 𝑒−ℎ𝑣𝑖/𝑘𝐵𝑇 )}

6𝑛

𝑖=1

( 5.17)

The binding energy, 𝐸0, is a linear function of reduced mass [94] and the reduced mass of

particles increases as the PAHs are added to the surface. Therefore the average reduced mass is

used to calculate the binding energy. All the parameters are kept consistent with the reversible

nucleation model and those used by Eaves et al. [193]. Rapacioli et al. [224,225] looked at the

frequencies of the vibration modes for coronene stacks containing up to 8 coronenes and

vibration modes for various PAH dimers. Their conclusion was that in general, as PAH stack

sizes increase, as would occur during the condensation process, vibration frequencies reduced. In

the case of an octomer, the proposed vibrational frequencies range from 50 to 1.8 cm-1. Eaves et

al. [193] used an average of 0.5 cm-1 for the reversible condensation model with a note that the

vibrational frequencies should be relaxed for small cluster sizes. In the premixed stagnation

flame, as suggested by the measured particle size distribution, the particles are ranging from 1 to

60 nm in size, which are much finer than the typical particles in the diffusion flames ( the

111

average size of the particles in the Santoro flame [58], which was modeled by Eaves et al. [193],

is around 1 μm). Therefore, an average value of 7.5 cm-1 is used for the condensation vibrational

frequency in this work, which is higher than the vibration frequency of nucleation in accordance

with suggestions of [193,224,225].

The estimated equilibrium constant is substituted into Eq. 5.15 to define the PAH equilibrium

mole fraction. The equilibrium concentration of condensing PAH species is incorporated in a

Heaviside function to form a condensation efficiency function which is shown in Eq. 5.18.

𝛾𝐶𝑜𝑛𝑑. = 1

1 + 𝑒𝑥𝑝⎝⎜⎜⎛−2

⎝⎜⎜⎛4𝜒𝑃𝐴𝐻𝑗 (

𝐾𝑝,𝐶𝑗𝑁𝑖

𝑁𝑖+1 )

1𝑛

− 2⎠⎟⎟⎞ ⎠⎟⎟⎞

= 12

+ 12

𝑡𝑎𝑛ℎ 4⎝⎜⎜⎛𝜒𝑃𝐴𝐻𝑗 (

𝐾𝑝,𝐶𝑗𝑁𝑖

𝑁𝑖+1 )

1𝑛

− 0.5⎠⎟⎟⎞

( 5.18)

Here, 𝑛 is the PAH coefficient in reaction 5.13 and determined by Eq. 5.14; 𝜒𝑃𝐴𝐻𝑗 is the mole

fraction of the 𝑗𝑡ℎ condensing PAH species and 𝐾𝑝,𝐶𝑗 is the corresponding species equilibrium

constant for condensation; 𝑁𝑖 is the number density of soot particles in the 𝑖𝑡ℎ section.

5.2.2.4 Soot models

In order to assess the capabilities of soot nucleation and PAH condensation submodels to predict

soot formation in the flames, four different models are employed. In the first model (Model 1),

the nonreversible nucleation model and the PAH condensation model introduced in Chapter 2 are

used to simulate soot formation. Model 1 assumes a constant nucleation efficiency of 𝜂𝑁𝑢𝑐 =

10−5 and a constant condensation efficiency of 𝛾𝐶𝑜𝑛𝑑 = 0.1. The second model (Model 2) applies

the reversible nucleation model introduced in this chapter alongside a constant PAH

condensation efficiency. In the third model (Model 3), in addition to reversible nucleation, the

PAH-soot probability of sticking function proposed by D’Alessio et al. [219] is added to the

condensation model to take into account the thermal rebound effect. The collision efficiency

function given by Eq. 5.1 is dependent on temperature (see Figure 5.2) and potential well depth

of the colliding entities. The collision efficiency function has been adapted by D’Anna et al.

[117,177,221] and Lindstedt and Waldheim [222] to model particle size distribution in premixed

112

ethylene flames and soot formation in nonpremixed flames of methane and ethylene. Finally,

Model 4 combines the developed equilibrium based PAH condensation model with the reversible

nucleation model to predict soot formation in the flames. All other parameters are kept constant

between different models. These parameters include a constant coagulation efficiency, 𝜉, of 0.2

and a constant surface reactivity, 𝛼, of 1. All the models are summarized in Table 5.1.

Figure 5.2 Condensation efficiency (Eq. 5.1) variation with temperature.

Table 5.1 Difference between nucleation and condensation models used to simulate flames

Model designation Nucleation model Condensation model Surface

reactivity

Model 1 Constant efficiency nonreversible (𝜂𝑁𝑢𝑐 = 10−5)

Constant efficiency (𝛾𝐶𝑜𝑛𝑑 = 0.1) 𝛼 = 1

Con

stan

t eff

icie

ncy

coag

ulat

ion

𝜉=

0.2

Model 2 Reversible nucleation Constant efficiency (γCond = 0.05) 𝛼 = 2.4

Model 3 Reversible nucleation Temperature dependent efficiency (Eq. 5.1) 𝛼 = 2.4

Model 4 Reversible nucleation Equilibrium based efficiency (Eq. 5.18) 𝛼 = 1.9

5.3 Results and Discussion

Results and discussions on the contribution of PAH species on modeling PSD in BSS premixed

flames will be presented in the following order. First, the effect of PAH chemistry model on soot

formation is discussed. Next, the effect of nucleation model and the role of reversibility of

nucleation process on predicting soot particles are emphasized. Subsequently, the importance of

condensation, the other process involving PAH molecules, on predicting particle size distribution

is studied. Subsequently, an analysis on the sensitivity of the particle size distributions to

different nucleation and condensation parameters is presented. Finally, predictions of soot in the

0.0

0.2

0.4

0.6

0.8

1.0

0 500 1000 1500 2000 2500

.

T (K)

113

Santoro diffusion flame [58] employing the described models are compared to experimental data

to complete the soot formation analysis.

5.3.1 PAH Chemistry

The two chosen chemical mechanisms to study the effect of PAH chemistry on predicted soot in

the premixed stagnation flame are the DLR mechanism [87] and the KAUST mechanism [88].

The DLR mechanism has been employed to describe chemical kinetics and PAH formation to

predict soot in diffusion flames in several studies [40,193–195,210,232]. It has been shown by

Dworkin et al. [141] and Chernov et al. [210] that with the DLR mechanism the prediction of

soot in the centerline region of diffusion flames where the soot formation is dominant by PAH

growth will vastly improve compared to the predictions of an identical soot model that uses other

PAH growth mechanisms. The novelty of the DLR mechanism is in the new pathways proposed

for formation of the initial aromatic rings, which is the bottle neck for formation of larger PAH

molecules as well as new pathways for growth of larger PAH molecules. The new PAH growth

reactions and their reaction rates are mainly estimated based on analogy to smaller aromatic

molecules, e.g., benzene and phenyl. The advantage of the KAUST mechanism is that although it

shares most of the embedded PAH growth pathways with the DLR mechanism, the PAH reaction

rates not present in the literature were determined through quantum calculations using density

functional theory along with transition state theory.

PAH prediction for both of these mechanisms has been validated for premixed and diffusion

flames. As an example, predicted species and PAH concentrations using both the DLR and

KAUST mechanisms for the ethylene premixed flame of Castaldi et al. [233] along with the

measured species concentrations from [233] are shown in Appendix C. Computations are

performed for a premixed flame using the PREMIX code of CHEMKIN. It is evident from these

species profiles that both of the studied mechanisms are comparably successful in predicting the

measured small species as well as larger PAH species concentrations. Therefore, these two

mechanisms along with the sectional soot model (Model 1) have been employed to predict soot

particles in the BSS premixed flame.

All the model parameters are kept constant between the two simulations. The only difference

between the two soot models is in the nucleating/condensing PAH species. The three heaviest

PAH species in each mechanism has been chosen as the PAH species that interact with the soot

114

particles. For the DLR mechanism these PAH molecules are benzo(a)pyrene (BAPYR),

secondary benzo(a)pyrenyl (BAPYR*S), and benzo(ghi)fluoranthene (BGHIF). The three PAH

species at the end of the growth pathways in the KAUST mechanism are anthanthrene (A6),

benzo(ghi)perylene (BGHIPER), and A4R5. The differences in the nucleating/condensing

species is reflected on determining the inception or condensation rates by the collision kernel

which is a function of projected area of colliding entity.

The calculated temperature profiles along with measured data from [19] as a function of height

above the burner for six burner spacings are shown in Figure 5.3. The black line represents the

KAUST mechanism model and the grey line represents the DLR mechanism model. The

temperature profiles predicted by both models display reasonable agreement with the

experimental data for all six flames. For most spacings, the two temperature profiles have

overlapped and neither kinetic mechanism shows a distinguishable advantage over the other. As

for the major species, both mechanisms predict similar concentration profiles along the

centerline of the flame as a function of height above the burner. The major species concentration

profiles computed by the KAUST and DLR mechanisms for the 𝐻𝑝 = 1.0 cm flame are

illustrated in Figure 5.4 and H, OH, benzene, and naphthalene concentrations are depicted in

Figure 5.5. Although the magnitude of the computed concentrations for major species and

radicals differ between the two chemical kinetic models, predictions of the trends and the shape

of the profiles are similar. When it comes to the initial aromatic ring formation, deviation of the

two kinetic models starts to become more noticeable. Although the computed concentration of

benzene and naphthalene at the stagnation plate are close between the two models, the growth

pathways are completely different. For the model with the KAUST mechanism, the maximum

growth rate for benzene and naphthalene occurs near the height 0.1 cm above the burner for the

𝐻𝑝 = 1.0 cm flame. This height coincides with the H and OH radical peak concentration.

Conversely, the peak of the benzene and naphthalene growth rate for the model that employed

the DLR mechanism is at a height 0.7 cm where the H and OH concentration have dropped by

two orders of magnitude. H and OH radicals, being the two most abundant radicals near the

flame, are the drivers for the HACA growth scheme. Also in premixed combustion, formation of

most of the radicals either directly or indirectly depends on these two radicals. Therefore, H and

OH radicals can be used as a metric of the overall radical levels in the mixture. When the peak

115

formation rate of benzene and naphthalene are in the low H and OH areas, it can be inferred that

the dominant growth pathway is independent of radicals.

Figure 5.3 Comparison of experimental data (symbols) from [19] and calculated (lines) centerline temperature profiles at several separation distances between the burner and stagnation surface. Temperature measurement uncertainties and the positional uncertainty are shown with bars.

Figure 5.4 Main species profiles computed with the KAUST mechanism (solid lines), and with the DLR mechanism (dashed lines) for a burner–stagnation surface separation of 𝐻𝑝 = 1.0 cm.

0

0.05

0.1

0.15

0.2

0.01 0.1 1

Mol

e Fr

actio

n

Height Above Burner, H (cm)

H2×15

C2H2

CO2

H2O

CO

O2

C2H4

Hp = 1.0 cm

[19]

116

Figure 5.5 Main radicals and small aromatic molecules profiles computed with the KAUST mechanism (solid lines), and with the DLR mechanism (dashed lines) for a burner–stagnation surface separation of 𝐻𝑝 = 1.0 cm.

Looking at the predicted soot volume fraction at the stagnation plate magnifies the contrast

between the two mechanisms. In Figure 5.6, computed soot volume fraction at the stagnation

plate for all six flames using the KAUST mechanism and the DLR mechanism are compared to

the measured data from Camacho et al. [21]. For consistency with the experimental data, only

those particles larger than 2 nm have been considered when calculating soot volume fractions.

The soot model with both mechanisms overpredicts soot volume fraction which indicates that the

soot model needs to be improved. Besides the overprediction of soot volume fraction, the model

with the DLR mechanism fails to capture the trend of increasing soot volume fraction with the

burner spacing observed in the measured data by Camacho et al. [21].

117

Figure 5.6 Comparison of computed soot volume fraction (of which the particle diameter, D > 2.5 nm) of the KAUST and DLR mechanisms with Model 1 as a function of separation distance with experimental data [21].

Since PAH addition is responsible for most of the soot mass yield in this flame, further analysis

of the PAH formation will help to have a more complete picture of the role of mechanisms in

prediction of soot volume fraction. Hence, the computed mass fraction profiles of

benzo(a)pyrene for the DLR model and anthanthrene for the KAUST mechanism are presented

in Figure 5.7 and Figure 5.8, respectively. These two PAH molecules have the highest

contribution to soot formation in each of the soot models that use the DLR and KAUST

mechanisms. The soot model has been turned off when computing the PAH concentrations

presented in Figure 5.7 and Figure 5.8 to prevent depletion of these PAHs by the soot formation

processes and conversion into solid state. The predicted PAH concentrations are expected to

follow the same trend as with the soot volume fraction results. For the model with DLR those

flames with the lowest spacing between burner and the stagnation wall (i.e., 0.55 and 0.6 cm)

have the highest concentration of PAHs and soot at the top wall and the concentrations decline as

the distance between the burner and the stagnation plate increases while the model with KAUST

and the measured data suggest otherwise.

1.E-09

1.E-08

1.E-07

0.5 0.7 0.9 1.1 1.3

Soot

Vol

ume

Frac

tion

Separation Distance, Hp (cm)

Exp. byComp. KAUST IIComp. DLR

[21]

118

Figure 5.7 Computed benzo(a)pyrene (A5) mass fraction profiles with the DLR mechanism as a function of height above the burner for six different burner stabilized stagnation flames.

Figure 5.8 Computed anthanthrene (A6) mass fraction profiles with the KAUST mechanism as a function of height above the burner for six different burner stabilized stagnation flames.

Two growth stages are distinguishable for benzo(a)pyrene, based on the concentration profiles in

Figure 5.7: the initial stage for heights below 0.1 cm where the growth rate among all flames are

consistent, and the secondary stage for heights above 0.1 cm where the concentration profiles

from different flames deviate but follow the same trend. The initial stage according to the

119

temperature profiles (Figure 5.3) and OH profile (Figure 5.5) is in the vicinity of the flame front

where the mixture contains the highest radical population and the temperature is close to the

adiabatic flame temperature. Both high temperatures and rich radical concentrations set an

environment to trigger PAH growth through HACA, which according to Slavinskaya and Frank

[140] is the dominant high temperature PAH growth route. The growth mechanism in the second

stage is different from that of the first stage. The temperature as well as radical concentrations

decrease continuously as the flow reaches the stagnation plate. However, such decline in the

PAH growth rate is not predicted by the model. In contrast, the growth rate is boosted in the

second stage, reaching a maximum at 1500 K.

A sensitivity analysis has been done to identify the growth pathways of the DLR mechanism at

1500 K for the same equivalence ratio as the flame, using the constant volume homogenous

reactor model of Reaction Design’s Chemkin-Pro package. Consistent with Slavinskaya and

Frank [140], the result of the analysis held the PAH growth through C4H2 addition to be

responsible for most of the PAH mass yield in the conditions described above. The C4H2 addition

route, for which reactions 5.19 and 5.20 are examples, for pyrene and benzo(a)pyrene growth

respectively, is independent of the radical population, since C4H2 is directly added to the stable

molecules. In addition, these reactions have been made irreversible for stability purposes in the

DLR mechanism [87] which could also influence the growth rate through this route, and make it

more favorable.

The reactions for the PAH growth through C4H2 addition have been developed based on the

analogies to indene thermal decomposition mechanism proposed by Laskin and Lifshitz [234].

The intermediate reactions in the indene decomposition had been lumped together by

Slavinskaya and Frank [140] to form the global reaction 5.21 and the reaction rate value equal to

the limited step of the sequence was used. By analogy to the reverse of reaction 5.21, PAH

growth reactions by C4H2 have been developed. By lumping the intermediate reactions, several

of which contains unstable radicals or activated sites, those global reactions lost their sensitivity

to the radical population in the mixture, and in situations similar to the BSS flame can result in

inaccurate predictions. In addition, according to M. ElRachidi (personal communication,

December 12, 2013) the reverse of the indenyl decomposition mechanism reported by Laskin

and Lifshitz [234] may not be an plausible representative of the C5H5 + C4H2 reaction and for the

C4H2 addition to PAH molecules, H abstraction is necessary to have an active site.

120

In conclusion, although the DLR mechanism established promising results in predicting PAHs

and soot in premixed and diffusion flames by introducing novel PAH growth pathways, it has

some limitations. Based on the analysis provided above, the DLR mechanism is not an

appropriate model for modeling PAH formation in low temperature post flame environments.

C4H2 + A2R5 ⇐⇐⇐⇐⇐⇐⇐⇒ A4 ( 5.19)

C4H2 + A4 ⇐⇐⇐⇐⇐⇐⇐⇒ BAPYR ( 5.20)

INDENYL ⇐⇐⇐⇐⇐⇐⇒ C5H5 + C4H2 ( 5.21)

PAH growth via unabstracted sites similar to the C4H2 reactions described above are not present

in the KAUST mechanism, and as a result most of the PAH species form near the flame front

where the temperature is high and gas mixture is abundant in radicals. As the PAH species

approach the stagnation wall, the rate of formation decline and the profiles form a plateau near

the stagnation plate which comply with the endothermic nature of PAH formation [17]. The mass

of PAH formed in the gas phase by the KAUST mechanism are comparable to the amount of

PAH predicted to form by the DLR mechanism. In addition, the trend of the soot levels are

consistent with measured data. Therefore, the rest of the modeling and analysis will be carried

out using the KAUST mechanism.

5.3.2 Reversible Nucleation Model

Although the sectional soot model combined with the KAUST mechanism predicts the trend of

increasing soot volume fraction with the burner spacing, it fails to capture the correct magnitude

of soot volume fraction. In addition, the total number of particles, depicted in Figure 5.9 as

Model 1, is overpredicted for all premixed flames. None of the processes considered for

modeling soot particles influence the number of particles as much as the nucleation model does

[40]. Despite the fact that a very low nucleation efficiency (𝜂𝑁𝑢𝑐 = 10−5) has been considered

which tends to lower the total number of particles [40], employing a nucleation model with a

constant efficiency highly overpredicts the total number of particles. This overprediction of the

number density of particles indicates that the constant efficiency nucleation model is incapable

of predicting soot in these flames without empirically tuning, and the nucleation model is

suffering from a fundamental flaw.

To address the overpredictions of soot volume fraction, as well as number density of particles

predicted by Model 1, in addition to reversible nucleation, two condensation models are

121

employed. A constant condensation efficiency the same as Model 1 is used for calculating soot

particle size distributions in one of the models (Model 2). For the other model (Model 3) the

PAH-soot probability of sticking function proposed by D’Alessio et al. [219] is added to the

condensation model.

The predicted number density of particles and soot volume fraction using Models 1–3 and the

experimental data from Camacho et al. [21] for the six different burner spacings are presented in

Figure 5.9 and Figure 5.10, respectively. Models 2 and 3, which incorporate the reversible

nucleation for modeling the particle dimerization process displayed significant improvement in

prediction of soot overall properties over the previous non-reversible nucleation model (Model

1). Model 2 is able to capture the rise and fall of the total number of particles suggested by the

measured data as the distance between the burner and the stagnation plate increases. Number

density predictions of Model 2 and Model 3 are very similar for most of the flames except the

𝐻𝑝 = 1.0 cm and 𝐻𝑝 = 1.2 cm, where Model 3 overpredicts total number of particles while

Model 2 predictions better match the experimental data. Despite the improvement over Model 1,

Model 3 is unable to predict soot volume fraction for most of the flames in compare to the

experimental data.

Figure 5.9 Comparison of soot particle number density (of which the particle diameter, D > 2.5 nm) computed with constant efficiency nucleation (Model 1), reversible nucleation and constant efficiency condensation (Model 2), and reversible nucleation and temperature dependent condensation efficiency (Model 3) as a function of separation distance, with experimental data [21].

1.E+09

1.E+10

1.E+11

1.E+12

0.5 0.7 0.9 1.1 1.3

Tota

l Num

ber o

f Par

ticle

s (cm

-3)

Separation Distance, Hp (cm)

Exp. byModel 1Model 2Model 3

[21]

122

Figure 5.10 Comparison of soot volume fraction (of which the particle diameter, D > 2.5 nm) computed with constant efficiency nucleation (Model 1), reversible nucleation and constant efficiency condensation (Model 2), and reversible nucleation and temperature dependent condensation efficiency (Model 3) as a function of separation distance, with experimental data [21].

In order to assess the performance of the new models in more detail, predicted particle size

distribution profiles along with measured data [21] are illustrated in Figure 5.11. Reasonable to

good agreement is obtained with the reversible nucleation model. Both Model 2 and 3 are able to

predict the bimodal distribution of the PSDs for the 𝐻𝑝 = 1.0 cm and 𝐻𝑝 = 1.2 cm flames. The

models also show a transition from unimodal distribution to bimodal distribution as the

separation distance increases. The condensation efficiency has been tuned to minimize the

difference between PSD predictions and the experimental data. However, these models are

incapable to capture the curvatures in the measured PSD profiles. The reversible nucleation

prevents the nucleation rate from excessive increase in the high temperature environment by

increasing the rate of dissociation. But, if condensation be fast enough, the dimers may grow and

form trimers and larger particles before they dissociate which could result in fast transition from

unimodal to bimodal PSD. One of the functionalities of the temperature dependent condensation

efficiency, as it is depicted in Figure 5.2, is to reduce condensation at high temperatures.

However, the attempt to combine these strategies as was done in Model 3 to improve the

predictions, was unsatisfactory and the disagreement of the prediction PSDs of the temperature

dependent condensation model with the experiment data worsened. It should be noted that if the

overall condensation efficiency for Model 3 is increased, predictions similar to Model 2 could be

achieved. The purpose of using Model 3 with the settings used here was to demonstrate the effect

1.E-10

1.E-09

1.E-08

1.E-07

0.5 0.7 0.9 1.1 1.3

Soot

Vol

ume

Frac

tion

Separation Distance, Hp (cm)

Exp. byModel 1Model 2Model 3

[21]

123

of reduction of condensation efficiency with temperature on predictions of PSDs. Also, to further

highlight the fact that a constant condensation efficiency or temperature dependent efficiency are

insufficient to predict soot PSDs for these flames.

Figure 5.11 Comparison of computed soot particle size distributions using reversible nucleation and constant efficiency condensation (Model 2), and reversible nucleation and temperature dependent condensation efficiency (Model 3) at several separation distances between the burner and stagnation surface, with experimental data [21].

For in-depth insight on the effect of different processes involved in the formation of soot

particles, the sensitivity of the soot PSD results to different growth process rates has been

examined.

The first process to be explored is nucleation and the effect of reversibility on the soot

predictions. The reverse rate of dimerization is calculated based on the equilibrium constant of

dimerization. The enthalpy and entropy of dimerization required for 𝐾𝑝,𝐷 are functions of

vibrational frequencies and binding energies of the PAH molecules that are participating in the

dimerization. For a given binding energy, higher vibration frequencies results in higher dimer

entropy change; thus, increasing vibrational frequency will lower the equilibrium constant, 𝐾𝑝,𝐷

and favor dimerization, as shown in Figure 5.12. In order to investigate the effect of nucleation,

different vibration frequencies have been implemented to calculate soot particle size distribution.

1 10 100

Hp = 0.70 cm

1 10 100

Hp = 0.60 cm

1 10 100

Hp = 0.55 cm

1 10 100

Hp = 1.20 cm

1 10 100

Hp = 1.00 cm

1 10 100

Hp = 0.80 cm

1013

Part

icle

Size

Dist

ribut

ion,

dN/

dlog

Dp

(cm

-3)

Particle Diameter, Dp (nm)

1011

109

107

105

109

1011

1013

107

105

Exp. by Model 2 Model 3[21]

124

Figure 5.13 displays comparison of the predicted PSDs for all the flames with the experimental

data taken from [21].

Figure 5.12 Equilibrium constant for dimerization of PAHs employed in the reversible nucleation model with different average vibration frequencies as a function of temperature.

Based on PSD profile shapes, the flames can be divided into three groups. The first two flames

with the lowest distance between the burner and the stagnation plate (i.e., 𝐻𝑝 = 0.55 cm and

𝐻𝑝 = 0.6 cm) have a unimodal PSD profile. The effect of limiting the reversibility (lowering 𝜐)

for these flames is an upward shift of the peak of the PSD profile without any significant change

in the overall shape of the profile. Looking at the total number of particles and soot volume

fraction reveals that both of these quantities increased by lowering the reversibility for these two

configurations. The second category of the PSDs which includes, 𝐻𝑝 = 0.7 cm and 𝐻𝑝 = 0.8

cm flames represent a transition of the PSD profile from unimodal to bimodal. These flames

have the highest number of particles. Although the distribution of particles slightly conveyed

towards a bimodal distribution as the frequency decreases, increasing nucleation does not have a

substantial effect on the total number of particles or soot volume fraction when it is combined

with a constant efficiency condensation model. The final flame configuration has a distinctive

bimodal PSD profile. This category includes the 𝐻𝑝 = 1.0 cm and 𝐻𝑝 = 1.2 cm flames.

The result of promoting reversibility of dimerization (higher vibration frequencies) in these

flames is the shift of the PSD toward larger particles and stronger bimodality. The magnitude of

the peak slightly decreases with decreasing nucleation, which lowers the total number of

1.E-05

1.E-03

1.E-01

1.E+01

1.E+03

0 500 1000 1500 2000

K p

T(K)

ν = 14 cm-1ν = 18 cm-1ν = 24 cm-1

125

particles. The reason for the shift is interrelatedness of PAH condensation and nucleation. For a

fixed concentration of gas phase PAHs, higher vibrational frequencies will result in higher

reverse rates in high temperature regions. For the BSS configuration, the effect of vibrational

frequency considering the temperature profiles, Figure 5.3, is to postpone onset of soot

downstream closer to the stagnation plate, therefore reducing the growth residence time.

The shift of the onset of soot downstream of the flow can be visualized by the PAH profiles since

the gas phase PAH is the main source for soot formation in these flames. Figure 5.14 illustrates

anthanthrene (A6) mass fraction profiles computed with soot models that have dimerization

vibration frequency of 14 cm-1 and 26 cm-1. The A6 concentration for the model without soot is

also included in this figure as a base for comparison. Deviation of the A6 profiles from the base

line (no soot model) is caused by transformation of the PAH molecules to the solid phase starting

by the nucleation process. The shift caused by changing the vibrational frequency from 14 cm-1

to 26 cm-1 can be easily distinguished in this graph. On the other hand, when the nucleation

process is suppressed by increasing reversibility more PAH molecules will be available to be

absorbed through surface condensation. As a result, although fewer particles are formed, the soot

volume fraction is not drastically affected by the changes in nucleation which is the reason the

A6 profiles are converging as reaching the stagnation plate at 𝐻𝑝 = 1.2 cm.

Overall, the effect of increasing dimerization reversibility when the soot model is using a

constant efficiency condensation model is reduction of total number of particles. The increase in

reversibility has an insignificant effect on soot volume fraction for flames 𝐻𝑝 = 0.7 − 1.2 cm

and in the remaining flames the reduction of soot volume fraction (max reduction is a factor of 3)

is not enough to correct for the massive overpredictions.

126

Figure 5.13 Comparison of computed soot particle size distribution using different intermolecular vibrational frequencies for the reversible nucleation model and a constant efficiency condensation (𝛾𝐶𝑜𝑛𝑑 = 5%) at several separation distances between the burner and stagnation surface with experimental data [21] (effect of vibrational frequencies on Model 2 predictions).

Figure 5.14 Computed anthanthrene (A6) mass fraction profiles as a function of height above the burner for the 𝐻𝑝 = 1.2 cm burner stabilized stagnation flame using three models: without soot, with dimerization frequency of 26 cm-1, and with dimerization frequency of 14 cm-1.

The analysis would be incomplete without examining the effect of condensation. To do so, PSD

profiles have been calculated using three different constant condensation efficiencies, 1%, 5%,

and 10%, namely. The predicted PSD profiles for all six burner configurations are illustrated in

Figure 5.15. The condensation efficiency exhibits a considerable effect on the calculated PSD

profiles. From the comparison of the results in Figure 5.13 and Figure 5.15 it can be concluded

1 10 100

Hp = 0.70 cm

1 10 100

Hp = 0.60 cm

1 10 100

Hp = 0.55 cm

1 10 100

Hp = 1.20 cm

1 10 100

Hp = 1.00 cm

1 10 100

Hp = 0.80 cm

1013

Part

icle

Size

Dist

ribut

ion,

dN/

dlog

Dp

(cm

-3)

Particle Diameter, Dp (nm)

1011

109

107

105

109

1011

1013

107

105

Exp. by Vib. Frq. = 14 cm-1

Vib. Frq. = 26 cm-1 Vib. Frq. = 20 cm-1

0.E+00

1.E-04

2.E-04

0 0.4 0.8 1.2

A6 M

ass F

ract

ion

Height Above Burner, Hp (cm)

No Soot

Vib. Frq. = 26 cm-1

Vib. Frq. = 14 cm-1

[21]

127

that the concentration of smallest soot particles (those at the far left of the PSD) is strongly

correlated with the nucleation model, and condensation has a weaker influence on prediction of

these particles. The declining slope of particle number density as the particles grow and the

formation of bimodal PSD, on the other hand, is determined by the condensation process. The

1% efficiency however, is insufficient to increase the soot mass to a level that matches the

experimental data for all of the flames. As the distance between the burner and the stagnation

plate increases, higher condensation efficiencies are required in order for the predicted soot

volume fractions to agree with the measured soot volume fractions.

The dependency of the shape of the PSD profiles and formation of a bimodal distribution on

nucleation and condensation is more complicated. One noticeable trend is that for these flames,

as the condensation increases, the unimodal profile transfers to a bimodal distribution. If the

condensation is further increased, the branch on the left side of the PSD profile (the smaller

particles) will completely disappear. These results further emphasize that a constant efficiency

model is incapable of reproducing PSD profiles that match the measured data for all the flames.

The development of an advanced condensation model which integrates state of the art theories of

soot growth phenomena is needed in order to achieve higher predictability.

128

Figure 5.15 Comparison of computed soot particle size distribution with reversible nucleation model and different constant efficiencies for condensation (𝛾𝐶𝑜𝑛𝑑 ) at several separation distances between the burner and stagnation surface with experimental data [21] (effect of condensation on Model 2 predictions).

The final analysis seeks the effect of particle coagulation on the particle size distribution. Two

sets of PSD profiles are calculated using a model with 100% efficient particle coagulation and a

model with no particle coagulation and are depicted in Figure 5.16. These two models are

extreme cases that can maximize the effects of the coagulation process on the particle size

distributions. Even so, the coagulation does not have a distinctive effect on the PSD profiles for

flames 𝐻𝑝 = 0.55 − 0.8 cm. The reasons for the ineffectiveness of particle coagulation in these

flames are the low number density of particles and limited residence time for particles to

coagulate. It is not until 𝐻𝑝 = 0.8 cm that the number density of particles becomes in the order

of 1011 cm-3, which is the particle density at which coagulation starts to play a stronger role. It is

evident from the particle size distribution profiles of these flames that the majority of the soot

particles forming the size distribution belong to the smaller sized particles, which not only have

smaller projected area but also are formed downstream in the vicinity of the stagnation plate. The

small projected area lowers the chance of a particle to collide with other particles. Being formed

near the stagnation plate lowers the particle residence time to collide, which further reduces the

1 10 100

Hp = 0.70 cm

1 10 100

Hp = 0.60 cm

1 10 100

Hp = 0.55 cm

1 10 100

Hp = 1.20 cm

1 10 100

Hp = 1.00 cm

1 10 100

Hp = 0.80 cm

1013

Part

icle

Size

Dist

ribut

ion,

dN

/dlo

g D p

(cm

-3)

Particle Diameter, Dp (nm)

1011

109

107

105

109

1011

1013

107

105

Exp. by Cond. Eff. = 1%Cond. Eff. = 5% Cond. Eff. = 10%

[21]

129

chance of agglomeration. It is the 𝐻𝑝 = 1.2 cm flame that shows the highest dependency of the

PSD on coagulation where in this case the coagulation made the PSD become a stronger bimodal

distribution. The coagulation process also shifts the particle distribution further towards the

larger particles and also reduces the total number of particles, which appears in the PSD profile

as a downward shift.

Figure 5.16 Comparison of computed soot particle size distribution using different coagulation efficiencies for the reversible nucleation model and constant efficiency condensation (𝛾𝐶𝑜𝑛𝑑 = 5%) at several separation distances between the burner and stagnation surface, with experimental data [21] (effect of coagulation on Model 2 predictions).

5.3.3 Condensation Efficiency

Predicted total number of particles and soot volume fractions using the developed equilibrium

based condensation efficiency and reversible nucleation (Model 4) for all burner configurations

are provided in Figure 5.17a, and b, respectively. The calculated quantities using Model 2 and

the measured data from [21] are also included for comparison. The model with the equilibrium

based condensation efficiency demonstrates an improvement in prediction of number density for

most of the flames. Considering that the error bars on the experiment data are representing the

variation of the measurements, and that the uncertainties of the measurements are much higher

especially for the lower values, it is fair to claim that the predictions are within the uncertainties

or at least very close. Calculated PSD profiles are presented in Figure 5.18. Among all the

1 10 100

Hp = 0.70 cm

1 10 100

Hp = 0.60 cm

1 10 100

Hp = 0.55 cm

1 10 100

Hp = 1.20 cm

1 10 100

Hp = 1.00 cm

1 10 100

Hp = 0.80 cm

1013

Part

icle

Size

Dist

ribut

ion,

dN/

dlog

Dp

(cm

-3)

Particle Diameter, Dp (nm)

1011

109

107

105

109

1011

1013

107

105

Exp. by Coag. Eff. = 100% Coag. Eff. = 0%[21]

130

models that have been tested, Model 4 is the only model that can predict both shape and

magnitude of particle size distribution with good agreement with the experimental data. In

addition, Model 4 is the only model that captures the transition of unimodal distribution to

bimodal distribution region from 𝐻𝑝 = 0.55 cm to 𝐻𝑝 = 0.8 cm flames.

(a) (b) Figure 5.17 Comparison of (a) soot particle number density and (b) soot volume fraction (of which the particle diameter, D > 2.5 nm) computed with reversible nucleation and equilibrium based condensation efficiency (Model 4), and reversible nucleation and a constant efficiency condensation (Model 2), as function of separation distance, with experimental data [21].

1.E+09

1.E+10

1.E+11

0.5 0.7 0.9 1.1 1.3

Tota

l Num

ber o

f Par

ticle

s (cm

-3)

Separation Distance, Hp (cm)

Exp. byModel 2Model 4

1.E-10

1.E-09

1.E-08

1.E-07

0.5 0.7 0.9 1.1 1.3

Soot

Vol

ume

Frac

tion

Separation Distance, Hp (cm)

Exp. byModel 2Model 4

[21]

[21]

131

Figure 5.18 Comparison of computed soot particle size distribution using reversible nucleation and equilibrium based condensation efficiency (Model 4), and reversible nucleation and a constant efficiency condensation (Model 2), at several separation distances between the burner and stagnation surface, with experimental data [21].

5.3.3.1 Sensitivity analysis

Similar to Model 2, effects of different parameters on the predicted particle size distributions

have been investigated. The parameters of interest include: dimerization binding energy and

vibrational frequency for the nucleation process, surface reactivity, and vibrational frequency of

condensation. The effect of each of these parameters on the calculated PSD profile for the

𝐻𝑝 = 0.8 cm flame has been summarized in Figure 5.19. Dimerization binding energy, 𝐸0, and

vibrational frequency both influence nucleation by changing the equilibrium constant of

nucleation. Higher binding energy means that the net enthalpy and Gibbs free energy will reduce

more during dimerization. Therefore, nucleation becomes more favorable. This effect is unlike

vibrational frequency which lowers Gibbs free energy by increasing the net changes of entropy

of the system during dimerization. Ultimately, both of these parameters induce nucleation by

changing the equilibrium constant and subsequently the reverse rate of nucleation, which is why

their effect on particle size distributions are similar as it is depicted in Figure 5.19a and b.

1 10 100

Hp = 0.70 cm

1 10 100

Hp = 0.60 cm

1 10 100

Hp = 0.55 cm

1 10 100

Hp = 1.20 cm

1 10 100

Hp = 1.00 cm

1 10 100

Hp = 0.80 cm

1013

Part

icle

Size

Dist

ribut

ion,

dN/

dlog

Dp

(cm

-3)

Particle Diameter, Dp (nm)

1011

109

107

105

109

1011

1013

107

105

Exp. by Model 2 Model 4[21]

132

Overall, when the reverse rate is reduced, total number of particles increases and the PSD profile

shifts upward and slightly toward smaller particles. Surface growth plays a minor role in shaping

the particle size distributions as shown in Figure 5.19c. The vibrational frequency of

condensation, similar to its counterpart in the nucleation process, controls condensation

efficiency by changing the equilibrium constant. As the condensation increased, amount of

smaller particles reduces, and these particles transfer to the larger bins, therefore the PSD shift

towards the larger particles. Comparison of the results provided here in Figure 5.19 with those

obtained by Model 2 in Figure 5.13– 5.16 further underline the robustness of Model 4 in

preserving the distribution of the particles and shape of the PSD profile for different model

parameters.

Figure 5.19 Comparison of effects of (a) dimerization binding energy, (b) dimerization vibrational frequency, (c) surface reactivity, and (d) condensation vibrational frequency on computed soot particle size distribution using reversible nucleation and equilibrium based condensation efficiency (Model 4) for the 0.8 cm separation distances between the burner and stagnation surface flame with experimental data [21].

1 10 100

(d)

1013

Part

icle

Size

Dist

ribut

ion,

dN/

dlog

Dp

(cm

-3)

Particle Diameter, Dp (nm)

1011

109

107

105

109

1011

1013

107

105

(a) (b)

1 10 100

(c)

Exp. by= 14 cm-1

= 20 cm-1

= 26 cm-1

Exp. byE0 = 85 kJ/gE0 = 70 kJ/gE0 = 55 kJ/g

Exp. byExp. by

= 5 cm-1

= 11 cm-1

= 18 cm-1

[21] [21]

[21][21]

133

5.3.4 Diffusion Flames

So far in this chapter, the importance of reversibility and equilibrium of the nucleation and

condensation process for modeling particle size distribution in the premixed burner stabilized

stagnation flames has been highlighted. This flame configuration and accessibility of

measurements of the size distribution of soot particles provide a unique opportunity to study

nucleation and condensation models in such detail, which is otherwise unavailable. One of the

major drawbacks of these flames in terms of testing the soot models is that most of the soot is

formed in the regions where the temperature is below 1500 K, therefore all conclusions made are

reflective of the low temperature growth mechanism of soot particles.

To have a more comprehensive understanding of nucleation and condensation processes, the

analysis has been extended to include the effect of high temperature. This extension includes

using the models introduced in this chapter to predict soot in a coflow diffusion flame. In

addition to providing an environment to test high temperature soot growth, choosing a diffusion

flame as the target for the investigation adds an extra dimension to the analysis, which is the

effect of premixing on soot formation. As a reminder, the difference between the models are

summarized in Table 5.1. The described models (Models 1, 2, and 4) have been employed to

predict soot in the Santoro coflow ethylene/air diffusion flame [58]. The details of the flames

have been described previously in Chapter 3 and Chapter 4. It should be emphasized that this is

the first attempt the same model has been utilized to predict soot formation in both the BSS

premixed flames and the diffusion flame.

The predicted soot volume fraction contours and the experimental data from [212] are

demonstrated in Figure 5.20. All models predicted the maximum soot volume fraction to be 8.6

ppm, which agree well with the experimental data and the soot concentration predictions are

within the uncertainty range of experimental data. The 𝛼 parameter for each model was adjusted

accordingly, thus, all models have produced comparable soot mass. The contribution of chemical

surface growth to soot formation and the dependency of the predictions on surface reactivity in

the annular regions of the flame have been extensively discussed in Chapter 4. More important

than calculated soot concentration is the soot spatial spread predictions. The soot prediction by

all of the models is consistently extended beyond the height suggested by experimental

134

measurements. The lift of the onset of soot, predicted by Model 4, is also noticeable in

Figure 5.20.

Figure 5.20 Isopleths of soot volume fraction (ppm) of the Santoro ethylene/air coflow diffusion flame [58] computed using Models 1, 2, and 4 and experimental data from [212].

Identifying the differences in the soot particle growth pathway by comparing the models is the

first step to explain the discrepancy in the performance of the models in premixed and diffusion

flames. The analysis starts with particle number density evolution which is a good indicator of

particle growth history. Contours of particle number density calculated by Model 1, Model 2,

and Model 4 are presented in Figure 5.21. Observation of Model 1 results, which incorporate a

constant–efficiency nonreversible nucleation and condensation model, shows that inception of

soot particles starts low in the flame (near the fuel tube on the wings, and much lower than other

models on the centerline). The number density of particles also peaks in these lower regions

which indicates a high nucleation rate. In the annular regions these high number densities

coincide with high temperatures (see Figure 4.11). This configuration provides appreciable

[212]

135

surface area and residence time which both play key roles in soot particle growth via surface

reaction and condensation. Finally, the predicted peak particle concentration by Model 1 is

appreciably higher than those predicted by Model 2 and Model 4.

Model 2 and Model 4 both take advantage of the reversible nucleation model which caused the

onset of soot formation to shift away from the high temperature areas near the fuel tube towards

inner parts of the flame. The soot contours are no longer anchored to the fuel tube and are

predicted to be more lifted. A distinctive change in the particle pattern can also be observed by

comparing the three model predictions. The regions of concentration of particles on the wing and

on the centerline as two separate islands present in Model 1 predictions were vanished in the

Model 4 prediction and replaced by a continues high particle concentration zone spanning from

the centerline towards the wings. The displacements of particle concentration away from the high

temperature areas in the flame is induced by the high sensitivity of the nucleation reverse rate to

the temperature.

Figure 5.21 Computed contours of particle number density (cm-3) with Model 1, Model 2, and Model 4 of the Santoro ethylene/air coflow diffusion flame [58].

136

The predicted anthanthrene (abbreviated as A6), concentrations using Model 1, Model 2, and

Model 4, are depicted in Figure 5.22. Anthanthrene, predicted by the KAUST mechanism to be

the most abundant large PAH in the gas phase, can be viewed as a representative of the gas phase

PAHs that contributes to soot formation. The constant efficiency nucleation and condensation

model incorporated in Model 1, determine the rate of conversion of gas phase PAHs to the

condensed phase solely based on the rate of collision. The collision rate of two fixed colliding

quantities is a function of concentration of the participating matters and temperature. An increase

in any of these dependent variables will result in an increase in the rate of solidification. The

consequence of such an assumption is the complete obliteration of gas phase PAHs which is

reflected in anthanthrene mole fraction contours shown in Figure 5.22. The effect of the addition

of nucleation reversibility and equilibrium based condensation modeling on the A6 concentration

is substantial. The remaining A6 in the gas phase from Model 4 is two orders of magnitude

higher than the A6 concentrations from Model 1.

Figure 5.22 Computed contours of anthanthrene, A6, mole fraction with Model 1, Model 2, and Model 4 of the Santoro ethylene/air coflow diffusion flame [58].

137

For a more quantitative comparison of the model predictions, the predicted soot properties along

the pathline exhibiting maximum soot and along the centerline of the Santoro flame [58] are

displayed in Figure 5.23 and Figure 5.24, respectively. The soot properties are soot volume

fraction, average primary particle diameter, primary particle number density, and aggregate

number density. The corresponding experimental measurements are also included in each graph

for comparison.

On the wings, as was mentioned before, all models predict the location of the peak of soot

volume fraction downstream of the experimentally observed peak. The soot volume fraction

predictions by all models agree well with the experimental data on the wings and the soot

concentration predictions are within the uncertainty range of experimental data. The onset of soot

for Model 2 and Model 4 is further delayed. Model 1 and Model 4 predict the average primary

particle diameters fairly well in comparison to the experimental data which indicates that there is

a reasonable balance between nucleation and condensation within these models (as discussed in

[40]). However, Model 2 overpredicts the average particle diameters. Primary particle number

density is where the difference between model predictions is greatest. Model 1 and Model 4

predict the number density of particles reasonably well compared to the experimental data.

Contrarily, Model 2 underpredicts primary particle number density. Since nucleation is the only

source for particle formation, this underprediction indicates that the collision-coagulation

mechanism for nucleation may not be sufficient to describe the nucleation process, especially in

the high temperature areas. Probably, a chemical coalescence nucleation mechanism [92,93],

described in Chapter 1, would be a reasonable alternative for these regions, consistent with

suggestions by D’Anna [16]. The incapability of Model 2, which utilizes a reversible nucleation

model and a constant efficiency condensation, further accentuate Eaves et al. [193] conclusion

that both nucleation and condensation models must explicitly account for reversibility in order to

accurately predict experimental data.

Overall, Model 1 and Model 4 both display a reasonable performance in matching the

experimental data on the wings. However, it should be considered that Model 1 components

were included in simulations during years of calibrating the model to experimental

measurements in diffusion flames; [40,118,123,141,147,195,210] are examples of the efforts

made to improve the soot model that is introduced here as Model 1. Model 2 and Model 4 are

138

equipped with features that are based on fundamental physics of the soot formation process. In

addition, Model 1 has the worst agreement with the experimental data in the BSS flame.

Figure 5.23 Comparison of the predicted a) soot volume fraction, b) average primary particle diameter, c) primary particle number density, and d) aggregate number density along the annular pathline exhibiting the maximum soot volume fraction of the Santoro ethylene/air coflow diffusion flame [58] using Model 1 (dot-dashed line), Model 2 (dashed line), and Model 4 (solid line), with the experimental measurements by [39,57,58,192].

Figure 5.24 displays the predicted soot properties on the centerline. The contributions from

PAHs are the dominant soot growth process on the centerline of the diffusion flame as was

shown in Chapter 4. Thus, the centerline prediction is a better criterion for validation of

nucleation and condensation models. The predictions of the models are relatively similar on the

centerline. The soot volume fraction is underpredicted by all models. Similar to the wings, the

Height Above Burner (cm)

Soot

Volu

me

Frac

tion

(ppm

)

0 2 4 6 8 1010-1

100

101

Exp. by .Model 4Model 2Model 1

(a)

Height Above Burner (cm)

Prim

ary

Part

icle

Diam

eter

(nm

)

0 2 4 6 8 100

20

40

60

80Exp. by .Model 4Model 2Model 1

(b)

Height Above Burner (cm)Prim

ary

Part

icle

Num

berD

ensit

y(c

m-3

)

0 2 4 6 8 101010

1011

1012

Exp. by .Exp. byModel 4Model 2Model 1

(c)

Height Above Burner (cm)

Aggr

egat

eN

umbe

rDen

sity

(cm

-3)

0 2 4 6 8 10109

1010

1011

Exp. by .Exp. byModel 4Model 2Model 1

(d)

[58] [39]

[39][57]

[57][192]

139

onset of soot is delayed compared to the experimental data. The fact that all models are

underpredicting the soot volume fraction makes it extremely hard to draw any solid conclusions

about validity of the models. For instance, all models are predicting primary particle and

aggregate number density to be within the range of the uncertainty of the measured data; the

underprediction of the soot volume fraction, as suggested by Dworkin et al. [141], is probably

caused by underprediction of PAH formation. If the PAH level could be enhanced by developing

new PAH formation pathways then the particle number density predictions would drastically

change. An alternative option for overcoming the soot volume fraction underprediction is

introducing new soot surface growth reaction pathways that are not dependent on hydrogen

radicals. The chemical growth will not affect the number density of the particles. Thus, the

validity of the predictions will not be compromised.

140

Figure 5.24 Comparison of the predicted a) soot volume fraction, b) average primary particle diameter, c) primary particle number density, and d) aggregate number density along the centerline of the Santoro ethylene/air coflow diffusion flame [58] using Model 1 (dot‐dashed line), Model 2 (dashed line), and Model 4 (solid line), with the experimental measurements by [37–39,58,192].

Finally, for the sake of completeness of the discussion on the role of soot PAH growth pathways,

the effect of chemical mechanism is analyzed. Similar to the premixed flames, the KAUST and

DLR mechanisms are combined with Model 4 to predict soot in the Santoro diffusion flame [58].

The predicted soot volume fraction contours with the experimental data from [212] are illustrated

in Figure 5.25. The employment of the DLR mechanism boosted the predicted soot everywhere

in the flame by a factor of two compared to the predictions with the KAUST mechanism. The

predictions are in good agreement with the experiments as well. The location of the peak soot is

Height Above Burner (cm)

Soot

Volu

me

Frac

tion

(ppm

)

0 2 4 6 8 1010-3

10-2

10-1

100

101

Exp. byExp. by .Exp. byModel 1Model 2Model 4

(a)

Height Above Burner (cm)

Prim

ary

Part

icle

Diam

eter

(nm

)

0 2 4 6 8 100

10

20

30

40Exp. by .Model 1Model 2Model 4

(b)

Height Above Burner (cm)Prim

ary

Part

icle

Num

berD

ensit

y(c

m-3

)

0 2 4 6 8 10109

1010

1011

1012

1013

Exp. byExp. by .Model 1Model 2Model 4

(c)

Height Above Burner (cm)

Aggr

egat

eN

umbe

rDen

sity

(cm

-3)

0 2 4 6 8 10109

1010

1011

1012

Exp. by .Model 1Model 2Model 4

(d)

[58]

[37]

[39]

[192]

[38][37]

[37]

141

not accurately predicted by either model. The misrepresentation of the peak location may be a

boundary effect. Since the flame is anchored to the fuel tube there will be heat conducted to the

fuel tube and it is expected that the pyrolysis of the fuel starts inside the tube. Possibly, adding a

conjugate heat transfer model as shown by Eaves et al. [235] can improve the prediction of the

location of the peak.

Figure 5.25 Isopleths of soot volume fraction (ppm) of the Santoro ethylene/air coflow diffusion flame [58] computed using the KAUST and DLR mechanisms and soot Model 4, with experimental data from [212].

5.4 Conclusions

Soot particle size distribution in the BSS laminar premixed ethylene flames and the Santoro

diffusion flame [58] has been calculated using one set of soot formation and growth models. An

equilibrium based PAH condensation efficiency model has been developed. It has been shown

that the developed condensation model combined with a reversible nucleation model is capable

of predicting PSD profiles that are in good agreement with the experiment data. The nucleation

[212]

142

equilibrium was shown to have a direct effect on the total number of particles. It was shown that

the bimodality of the PSD would disappear if the nucleation and condensation rate were not in

balance. Also particle coagulation plays a minor role in forming the particle size distribution in

this flame.

The models developed and validated for the premixed flame, have been implemented into a

multi-dimensional flame code to explore soot formation and oxidation in the non-smoking

laminar coflow C2H4/air diffusion flame of Santoro [58]. The comparison of the soot predictions

suggests that as more fundamentally advanced models are developed, the necessity to explore

new soot and PAH growth pathways becomes more imminent.

Finally, the effect of PAH chemistry on soot predictions using different models has been

investigated in both premixed and nonpremixed flames. It has been shown that there are

limitations associated with chemical mechanism which should be considered when used for

modeling.

143

Chapter 6 Conclusions and Future Work

The present work has advanced the field of computational combustion, focusing on developing a

robust model of soot formation capable of predicting the mass, and structure of soot in laminar

flames for a wide range of conditions. Research has been conducted in the area of advancing soot

modeling by understanding the processes involves in soot formation in laminar flames. Attention

was first focused on soot particle diameter predictions by developing and implementing two

coalescence models. The new models were applied to a laminar coflow ethylene/air diffusion

flame. Soot formation was also modeled in several coflow flame configurations, and the

variation of soot surface reactivity was studied. Based upon the results of this study, a function

for surface reactivity of soot particles was proposed. Predictions of soot concentration in multiple

diffusion and partially premixed coflow ethylene/air flames were used to validate the surface

reactivity model. A further advancement in soot modeling came in Chapter 5, where simulations

of a set of burner stabilized stagnation premixed flames and a coflow ethylene/air diffusion flame

were conducted to investigate the significance of soot-PAH interaction modeling. This

investigation led to the development of a novel equilibrium based condensation model. The

condensation model was combined with a reversible nucleation model implemented to predict

soot particle size distribution and volume fraction in premixed and nonpremixed flames.

Conclusions from these studies, the major contributions, as well as recommendations for future

investigations, are summarized in the remainder of this chapter.

144

6.1 Summary and Conclusions

In Chapter 3, particle coalescence models applicable to sectional soot particle simulations were

introduced. The addition of the coalescence process was necessary to overcome the major

underprediction of the primary particle diameter accompanied by overprediction of number

density of primary particles obtained with the original soot model in diffusion flames. The first

of the two coalescence models, considers a step change from instantaneous coalescence to

aggregation at a certain particle size. The second model assumes a smooth transition from

complete merging to aggregation as a function of temperature, particle size, as well as number of

particles per aggregate.

The coalescence models were then applied to a well-studied laminar coflow C2H4/air diffusion

flame using detailed PAH-based combustion chemistry, a PAH-based soot formation/oxidation

model, and a detailed radiation model, along with the sectional aerosol dynamics model. The

computational model was validated by comparing predicted soot volume fraction and

morphology properties, to experimental data. The soot morphology properties used for the

validation were primary particle diameter, particle number density, aggregate number density,

and number of primary particles per aggregate. The comparisons to experimental data were

including the soot formation along the annular pathline exhibiting maximum soot concentration

and centerline of the flame. Both coalescence models significantly improved the predictions of

soot particle morphology. The cut-off model in the lower heights of the flame predicted soot

particle properties that are in closer agreement with the experiment data, while the sintering

model predicted profiles that are more consistent with the measured properties in terms of overall

shape and magnitude. These results supported the variation of particle structure and

consequential variation of the coalescence mechanism at different regions of the flame.

Comparisons were made for measured and computed primary particle diameters using different

cut-off and sintering coalescence models to investigate the sensitivity of the model prediction to

the coalescence parameters. Finally, an update to the coalescence model has been proposed to

improve the particle size predictions in the oxidation region. The updated model terminates

coalescence during soot oxidation based on experimental observations of soot characteristics in

these regions. The update led to improvement of particle size predictions in this region.

145

Chapter 4 of the present work investigated the variation of soot surface reactivity in laminar

coflow diffusion and partially premixed ethylene/air flames. The diffusion flames were

computed at four different nitrogen dilution levels in the Smooke/Long burner [41] (32%, 40%,

60% and 80%) and five different air and fuel mass flow rates in the Santoro burner [58]. The

partially premixed flames were computed at four different equivalence ratios in the Santoro

burner [58]: 10, 20, 24 and ∞. It was found computationally that the average soot surface

reactivity on the annular regions of the flames correlated well with particle thermal age, defined

as the integral of temperature to which a particle has been exposed with respect to time. A

methodology to study particle surface growth of soot was proposed. Surface reactivity was

expressed as an exponential function of particle thermal age. Measured soot volume fractions on

the wings could be well reproduced with this function for a variety of ethylene flames of

different dilution levels, premixing and burner configurations. The new function suggests that the

soot particle surface reactivity increases in the early stages of soot formation in diffusion flames

until it reaches a maximum and then gradually decreases as the particles traverse the flame.

Several physical processes that might cause the variation of surface reactivity were discussed.

The computational model was validated by comparing predicted gas temperature and major

species concentration profiles to experimental data.

In the study presented in Chapter 5, soot formation in the six burner stabilized stagnation (BSS)

premixed ethylene flames and the Santoro coflow diffusion flame [58] were computed to

comprehensively study the role of PAH chemistry and PAH-soot interaction modeling in soot

formation. The KAUST and DLR mechanisms were employed to describe the fuel

pyrolysis/oxidation and PAH formation. The gas temperature profiles in all six BSS premixed

flames computed by both mechanisms were in excellent agreement with the experimental

measurements from the literature. Both mechanisms predicted similar major species, H, and OH

radical concentration profiles. On the contrary, the PAH concentration profiles computed by the

DLR and KAUST mechanisms showed completely opposite trends with the KAUST predictions

more in line with the experimental soot volume fractions. It was found that the PAH growth via

unabstracted sites was the main reason for the unexpected PAH growth in the low temperature

regions in the DLR mechanism. In addition, the soot volume fraction and number density of

particles were overpredicted by the aerosol dynamics model, despite using very low nucleation

efficiency (Model 1). Since the constant efficiency nucleation model was incapable of predicting

146

soot in these flames, it was concluded that the nucleation model was suffering from a

fundamental flaw.

The reason for using efficiencies in the nucleation models is to account for the fact that the pair

of PAH molecules present in a dimer, due to thermodynamic conditions, can separate. Thus, to

improve the nucleation model based on a fundamental understanding of the dimerization process,

the nucleation process was allowed to be reversible. A reversible nucleation model based on the

work of Eaves et al. [193] was combined with two condensation efficiency models: a constant

efficiency condensation and a temperature dependent condensation efficiency model. The

predicted number density of particles, soot volume fraction and particle size distribution at the

stagnation plate were compared to the experimental data for validation. Despite the improvement

over Model 1, both models are overpredicting the number of particles and soot volume fraction

for most of the flames. However, both models were able to predict the bimodal distribution of the

PSDs for the 𝐻𝑝 = 1.0 cm and 𝐻𝑝 = 1.2 cm flames. The models also captured the transition

from unimodal distribution to bimodal distribution as the separation distance increases.

Consistent in all PSD predictions was the overprediction of the concentration of larger particles.

A sensitivity study of the effects of different growth process rates on the soot PSD results was

conducted. The effect of increasing nucleation rate was an increase in the total number of

particles for all flames and an increase in soot volume fraction for flames 𝐻𝑝 = 0.55 and 0.6 cm.

The coagulation did not have a distinctive effect on the PSD profiles for flames 𝐻𝑝 = 0.55, 0.6,

0.7, and 0.8 cm. For the remaining flames, the coagulation made the PSD have a stronger

bimodal distribution. It was found that condensation has a major role on the predicted soot

volume fraction and shaping of the PSD. Also, it was shown that reproducing PSD profiles that

match the measured data for all the flames with a constant efficiency model is not possible with

the current model.

An equilibrium based sectional condensation efficiency model was developed to improve the

particle size distribution predictions. The new model (Model 4) limited the growth rate via PAH

addition as the gas phase PAH concentration drops beyond the equilibrium concentration. The

estimation of the equilibrium constant was done using statistical mechanics based on the

assumption that the surfaces of the soot particles were covered by loose PAHs that could detach

from the surface. It was demonstrated that the model with the equilibrium based condensation

147

efficiency improved the prediction of both soot volume fraction and number density for most of

the flames. In addition, the new model was the only model that could predict both shape and

magnitude of particle size distribution with good agreement with the experimental data and the

only model that captures the transition of unimodal distribution to bimodal distribution as the

burner spacing increases from 𝐻𝑝 = 0.55 cm to 𝐻𝑝 = 0.8 cm. Physical processes that might

influence particle size distribution predictions of Model 4 were discussed and their effects on

PSD were investigated numerically.

Finally, the models developed and validated for the premixed flame, have been implemented into

a multi-dimensional flame code to explore soot formation and oxidation in a non-smoking

laminar coflow C2H4/air diffusion flame. Comprehensive comparisons were made among soot

volume fraction and morphology predictions, and experimental data. Using reversible nucleation

delayed the onset of soot and caused an underprediction of soot volume fraction on the wings.

On the centerline, all models performed similarly, underpredicting soot volume fraction. In

addition, the computed soot formation with the reversible nucleation and equilibrium

condensation using the KAUST and DLR mechanisms are compared with the measurement data

in the Santoro flame [58]. Employing the DLR mechanism improved the soot predictions and the

agreement with the experiments mostly on the centerline. The location of the peak soot was not

accurately predicted by both models which might be improved by adding conjugate heat transfer.

The implementation of reversible nucleation and equilibrium condensation efficiency diminish

the need for with arbitrary or tuned efficiencies, and replace them with parameters that have

physical meaning, which can be evaluated. The comparison of the soot predictions suggests that

as more fundamentally advanced models are developed, the necessity to explore new soot and

PAH growth pathways becomes more imminent.

6.2 Original contributions

The scientific contributions of this thesis work can be summarized as follows:

- Introduction of a soot particle coalescence model that describes particle merging rate as a

function of temperature, particle size, as well as number of particles per aggregate. The

addition of the coalescence significantly improved the predictions of soot particle

morphology in the laminar diffusion flame.

148

- Investigation of the variation of soot surface reactivity in laminar coflow diffusion and

partially premixed ethylene/air flames. It was found computationally that the average soot

surface reactivity on the annular regions of the flames correlated well with particle

thermal age.

- Development of a model for soot particle surface reactivity as an exponential function of

particle thermal age. Measured soot volume fractions on the wings could be well

reproduced with this function for a variety of ethylene flames of different dilution levels,

premixing and burner configurations.

- Recommendation of a methodology to study particle surface growth of soot in coflow

flames.

- Demonstration of performance of detailed sectional soot models in prediction of soot

formation in the BSS premixed flames and the Santoro diffusion flame. This was the first

time that a model was able to predict PSDs in the BSS premixed flames and the diffusion

flame reasonably well.

- Assessment of the nucleation reversibility on the PSD predictions in the six burner

stabilized stagnation premixed ethylene flames. The first PSD predictions of the BSS

flames using the reversible nucleation model were performed.

- Development of an equilibrium-based PAH condensation efficiency model to improve

the particle size distribution predictions in the BSS flames. The model with the

equilibrium based condensation efficiency combined with the reversible nucleation could

predict both shape and magnitude of particle size distribution with good agreement with

the experimental data

- Comparison of the effect of PAH chemistry on soot predictions using the KAUST and

DLR mechanisms in both premixed and nonpremixed flames. It was shown that there are

limitations associated with chemical mechanism which should be considered when used

for modeling.

149

In addition to the preceding research contributions, this thesis led to several computational

framework developments in order to achieve the research objectives. These developments

include:

- Development of a computational frame work to track any soot or gas properties along the

streamlines at every location within the computational domain. The computational

procedure was embedded into the Coflame code to evaluate surface reactivity of soot

particles.

- Implementation of a detailed sectional soot model compatible with the CHEMKIN

package. The callable units were combined by CHEMKIN’s OPPDIF code to predict soot

formation in the BSS premixed flames.

- Development of a computational framework for predicting concentrations of different

dimers. The dimer concentrations were necessary for modeling the nucleation process as

reversible.

6.3 Recommendations for future work

The complex multi-physics processes involved in soot formation as well as the entropy-driven

nature of soot precursor formation make it a challenging field for research. Despite the

tremendous progresses in understanding and modeling soot formation, significant improvements

have to be made to reach a robust model of soot formation that can predict the mass, size

distribution, and aggregate structure of soot in flames. Currently, there are many questions about

the processes involved in soot formation, which necessitate a more fundamental understanding of

soot particles, before they can be answered. In the paragraphs that follow, based on the detailed

modeling studies of soot formation/oxidation performed in this thesis, recommendations for

future studies are presented.

An interesting prospect for future investigations related to the coalescence model development

would be to study the mechanism of transformation of liquid-like particles into solid state. In the

present studies, the differences between solid versus liquid particle coalescence have been

discussed and only solid state coalescence has been imposed. If the transformation mechanism

became available, a model could be developed that discriminates the coalescence rate based on

the state of the colliding particles. It is expected that implementation of such a model would

150

improve the particle size and number distributions in the lower heights in diffusion flames. Using

an advanced nucleation/condensation model, such as the ones introduced in the present study, is

highly recommended.

Future studies could proceed by investigating the mechanisms behind the dependency of the soot

surface reactivity on thermal age. Possible functional parameters that have been discussed in this

work are as follows:

- Variation of the number density of the hydrogenated sites;

- Carbonization, which involves polymerization, dehydrogenation, and bond

formation/rearrangements between PAH layers;

- Carbon to hydrogen ratio (C/H);

- Size of the particles.

A future numerical experiment could test the significance of each of these parameters on the

reactivity of soot particles. Future efforts could also seek to extend the study of soot surface

reactivity to include elevated pressures, different fuels, and the reactivity of the soot particles in

oxidation regions and premixed flames.

Future investigations could also include studying the effect of addition of species other than

acetylene and new soot growth pathways. The chemical growth of the soot particle is currently

described by the Hydrogen Abstraction Carbon Addition (HACA) mechanism with the carbon

species being acetylene. A multitude of reaction pathways have been introduced for PAH

formation that remarkably elevated the predictions of PAH concentration. Considering that PAH

molecules are the constituents of soot particles, it is expected that growth pathways similar to PAH

formation be applicable to soot particles too. These pathways include methyl substitution/acetylene

addition pathways, carbon addition via C4H2, C4H3, and C3H3, and aromatic/cyclic addition. These

growth pathways could have major effects on soot predictions in the regions were H radicals are

scarce. Examples of such regions are the centerline of coflow diffusion flames and the burner

stabilized stagnation flame. A study of this kind could proceed by comparing computational results

with measurements of soot volume fraction and particle size distribution in these regions.

151

The primary objective of future work relating to the nucleation modeling is the addition of the

chemical coalescence route to the soot model. It has been shown in Chapter 5, that dimerization

of PAH molecules through collision coagulation in the high temperature zones is

thermodynamically unfavorable which leads to a delay in the onset of soot on the wings of the

coflow flame. A kinetically controlled nucleation route akin to chemical coalescence or PAH

cluster stabilization could be added to address this gap in the model and it would be a pathway

toward completeness of the soot formation puzzle. The effect of adding the kinetic route could

also be tested in predictions of the PSD in the BSS flame. The high temperature nucleation route

would only affect the high spacing flames and might be able to improve PSD predictions in these

flames.

Future theoretical studies are required to estimate thermo/chemical behaviors of large PAH

stacks and mature soot particles, especially vibrational frequencies and binding energy. These

studies will advance the perception of the equilibrium state of particles with PAH molecules.

Such studies would provide a better understanding of the condensation process and pave the way

for development of fundamental soot condensation models.

An intriguing observation by comparing the results presented in Chapter 3 and Chapter 5 is that

the two different approaches are predicting similar soot morphologies in a diffusion flame and

the predictions are in good agreement with the experimental data. Future studies could

investigate how the coalescence process should be integrated into the soot model with the

reversible nucleation/condensation. These investigations could include procedures for

identification of these processes and validation of the models.

There is an urgent need for a robust, comprehensive, and extensively validated PAH chemistry

sub-model. In the present studies, two of the most advanced PAH chemistry models have been

used to model soot formation in premixed and nonpremixed flames. The drastic difference

between the results obtained by comparisons of these models reiterates the dependency of the

soot predictions and subsequent flame analysis on the PAH chemistry. Credibility of all analyses

is directly determined by the validity of the PAH chemistry. Thus, developing PAH formation

models should be one of the priorities of soot studies.

Extensive experimental measurements should be conducted to provide more data for model

validation. The measurements in laminar coflow C2H4/air diffusion flames should include soot

152

volume fraction as well as soot aggregate structural data (primary particle size, number density,

number of primary particles per aggregate, aggregate morphology) and size distribution data

similar to the Santoro flame [58]. New experiment setups should be designed that target

individual soot formation processes such as the BSS flames which are suitable for studying

nucleation and condensation, or the two-stage burner which is suitable for studying O2 oxidation.

Modifications to the BSS setup could also provide invaluable information for improving soot

models. Currently, experimental data is available only at the stagnation plate. Species and soot

measurements before the stagnation plate could be used to measure the timescales of soot

formation. Information on the structure of individual particles (number of primary particles per

aggregate) at the stagnation plate could be used to examine the coagulation models. Finally, a

very interesting test would be to investigate the effect of stagnation plate temperature on the

particle size distribution. As of now, the stagnation wall temperature is around 600 K and most

of the soot is forming in the vicinity of the stagnation wall which is not representative of soot

formation in most parts of practical combustion devises.

153

Appendices

Appendix A

Predicted particle size distribution functions at different axial heights above the burner along the

annular pathline of the maximum soot volume fraction, and along the centerline in the Santoro

laminar coflow diffusion ethylene/air flame [58] predicted using three different soot coalescence

mechanism: sintering, cut-off, and no coalescence ( Chapter 3).

Maximum Soot Pathline Centerline

Sin

teri

ng m

odel

Part

icle

Size

Dist

ribut

ion,

dN

/dlo

g D p

(cm

-3)

Particle Mobility Diameter, Dp (nm)

1.E+06

1.E+08

1.E+10

1.E+12

1 10 100 1000 10000

HAB = 0.3 cmHAB = 0.5 cmHAB = 1.0 cmHAB = 4.0 cmHAB = 6.0 cm

1.E+06

1.E+08

1.E+10

1.E+12

1 10 100 1000 10000

HAB = 1.0 cmHAB = 2.0 cmHAB = 5.0 cmHAB = 9.0 cm

154

Cut

-off

mod

el

Part

icle

Size

Dist

ribut

ion,

dN

/dlo

g D p

(cm

-3)

No

coal

esce

nce

Particle Mobility Diameter, Dp (nm)

Figure a. 1 Predicted particle size distribution functions at different axial heights above the burner along the annular pathline of the maximum soot volume fraction, and along the centerline.

1.E+06

1.E+08

1.E+10

1.E+12

1 10 100 1000 10000

HAB = 0.3 cmHAB = 0.5 cmHAB = 1.0 cmHAB = 4.0 cmHAB = 6.0 cm

1.E+06

1.E+08

1.E+10

1.E+12

1 10 100 1000 10000

HAB = 1.0 cmHAB = 2.0 cmHAB = 5.0 cmHAB = 9.0 cm

1.E+06

1.E+08

1.E+10

1.E+12

1 10 100 1000 10000

HAB = 0.3 cmHAB = 0.5 cmHAB = 1.0 cmHAB = 4.0 cmHAB = 6.0 cm

1.E+06

1.E+08

1.E+10

1.E+12

1 10 100 1000 10000

HAB = 1.0 cmHAB = 2.0 cmHAB = 5.0 cmHAB = 9.0 cm

155

Appendix B

The simulation results of soot formation and growth in the Santoro laminar coflow diffusion

ethylene/air flame [58] using the developed surface reactivity function ( Chapter 4) and a constant

surface reactivity (𝛼 = 0.45). The predicted soot volume fraction, primary particle diameter,

primary particle number density, aggregate number density, and number of primary particles per

aggregate is compared with the experimental measurements

Figure b. 1 Comparison of the predicted soot volume fraction along the wings using alpha function (solid line) and constant alpha (dashed line) with the experimental measurements by [58]

156

Figure b. 2 Comparison of the variation of predicted soot volume fraction with residence time along the wings using alpha function (solid line) and constant alpha (dashed line) with the experimental measurements by [192].

Figure b. 3 Comparison of the predicted average primary particle diameter along the wings using alpha function (solid line) and constant alpha (dashed line) with the experimental measurements by [56].

157

Figure b. 4 Comparison of the predicted primary particle number density along the wings using alpha function (solid line) and constant alpha (dashed line) with the experimental measurements by [39,57].

Figure b. 5 Comparison of the predicted aggregates number density along the wings using alpha function (solid line) and constant alpha (dashed line) with the experimental measurements by [57,192].

158

Figure b. 6 Comparison of the predicted number of primary particles per aggregate along the wings using alpha function (solid line) and constant alpha (dashed line) with the experimental measurements by [33,57].

Figure b. 7 Comparison of the predicted soot volume fraction along the centerline using alpha function (solid line) and constant alpha (dashed line) with the experimental measurements by [37,38,58].

159

Figure b. 8 Comparison of the predicted average primary particle diameter along the centerline using alpha function (solid line) and constant alpha (dashed line) with the experimental measurements by [37].

Figure b. 9 Comparison of the predicted aggregates number density along the centerline using alpha function (solid line) and constant alpha (dashed line) with the experimental measurements by [192].

160

Figure b. 10 Comparison of the predicted number of primary particles per aggregate along the wings using alpha function (solid line) and constant alpha (dashed line) with the experimental measurements by [37].

161

Appendix C

Comparison of the DLR and KAUST mechanisms in laminar premixed flames:

Flame Properties:

- 21.3% C2H4 / 20.9%O2 / 57.8% Ar

- 𝜙 = 3.06, P = 1 𝑏𝑎𝑟

- Marinov et al. [233], Proc. Combust. Inst. 26 (1996) 693–702.

Results:

Figure c. 1 Comparison between experimental data from [233] and calculated mole fraction of major gaseous products.

0

0.05

0.1

0.15

0.2

0 0.5 1

C2H4

Exp.KAUSTDLR

-0.02

0.03

0.08

0.13

0.18

0 0.5 1

O2

Exp.KAUSTDLR

0

0.05

0.1

0.15

0.2

0.25

0 0.5 1

CO

Exp.KAUSTDLR

0.03

0.06

0.09

0.12

0.0 0.2 0.4 0.6 0.8 1.0

H2

Exp.KAUSTDLR

0

0.05

0.1

0.15

0 0.5 1

H2O

Exp.KAUSTDLR

0

0.01

0.02

0.03

0.04

0.05

0 0.3 0.6 0.9 1.2

C2H2

Exp.KAUSTDLR

162

Figure c. 2 Comparison between experimental data from [233] and calculated mole fraction of benzene and various PAHs.

1.E-06

1.E-05

1.E-04

1.E-03

0 0.5 1

A1

Exp.KAUSTDLR

1.E-08

1.E-06

1.E-04

0 0.5 1

A2

Exp.KAUSTDLR

1.E-10

1.E-08

1.E-06

1.E-04

0 0.5 1

A3

Exp.KAUSTDLR

1.E-10

1.E-08

1.E-06

1.E-04

0 0.5 1

A4

Exp.KAUSTDLR

1.E-10

1.E-08

1.E-06

1.E-04

0 0.5 1

A4R5-BGHIF

KAUST-A4R5DLR-BGHIFExp. A4R5Exp. BGHIF

1.E-07

1.E-06

1.E-05

1.E-04

0 0.5 1

A5

Exp.KAUSTDLR

163

Bibliography

[1] Key World Energy Statistics 2014. Paris: 2014.

[2] Conti J, Holtberg P, Napolitano S, Schaal AM. The International Energy Outlook 2014. Washington, DC: 2014.

[3] Health effects of particulate matter. Copenhagen, Denmark: 2013.

[4] Integrated Science Assessment for Particulate Matter. Research Triangle Park, NC: n.d. doi:EPA/600/R-08/139F.

[5] Glovsky MM, Miguel AG, Cass GR. Particulate Air Pollution: Possible Relevance in Asthma. Allergy Asthma Proc 1997;18:163–6. doi:10.2500/108854197778984392.

[6] Mauderly JL. Toxicological approaches to complex mixtures. Environ Health Perspect 1993;101 Suppl :155–65.

[7] Albritton DL, Greenbaum DS. Atmospheric observations helping build the scientific basis for decisions related to airborne particulate matter. Cambridge, MA: 1998.

[8] Lighty JS, Veranth JM, Sarofim AF. Combustion Aerosols: Factors Governing Their Size and Composition and Implications to Human Health. J Air Waste Manage Assoc 2000;50:1565–618. doi:10.1080/10473289.2000.10464197.

[9] Phalen R. Proceedings of the Third Colloquium on Particulate Air Pollution and Human Health : final report. Sacramento CA: California Environmental Protection Agency Air Resources Board Research Division; 1999.

[10] McIlroy A, McRae G, Sick V, Siebers DL, Westbrook CK, Smith PJ, et al. Basic Research Needs for Clean and Efficient Combustion of 21st Century Transportation Fuels. 2006. doi:10.2172/935428.

[11] Haynes BS, Wagner HG. Soot formation. Prog Energy Combust Sci 1981;7:229–73. doi:10.1016/0360-1285(81)90001-0.

[12] Glassman I. Soot formation in combustion processes. Symp Combust 1989;22:295–311. doi:10.1016/S0082-0784(89)80036-0.

[13] Kennedy IM. Models of soot formation and oxidation. Prog Energy Combust Sci 1997;23:95–132.

[14] Richter H, Howard JB. Formation of polycyclic aromatic hydrocarbons and their growth to soot—a review of chemical reaction pathways. Prog Energy Combust Sci 2000;26:565–608. doi:10.1016/S0360-1285(00)00009-5.

164

[15] Frenklach M. Reaction mechanism of soot formation in flames. Phys Chem Chem Phys 2002;4:2028–37. doi:10.1039/b110045a.

[16] D’Anna A. Combustion-formed nanoparticles. Proc Combust Inst 2009;32:593–613. doi:10.1016/j.proci.2008.09.005.

[17] Wang H. Formation of nascent soot and other condensed-phase materials in flames. Proc Combust Inst 2011;33:41–67. doi:10.1016/j.proci.2010.09.009.

[18] Eremin A V. Formation of carbon nanoparticles from the gas phase in shock wave pyrolysis processes. Prog Energy Combust Sci 2012;38:1–40. doi:10.1016/j.pecs.2011.09.002.

[19] Abid AD, Camacho J, Sheen DA, Wang H. Quantitative measurement of soot particle size distribution in premixed flames – The burner-stabilized stagnation flame approach. Combust Flame 2009;156:1862–70. doi:10.1016/j.combustflame.2009.05.010.

[20] Abid AD, Heinz N, Tolmachoff ED, Phares DJ, Campbell CS, Wang H. On evolution of particle size distribution functions of incipient soot in premixed ethylene–oxygen–argon flames. Combust Flame 2008;154:775–88. doi:10.1016/j.combustflame.2008.06.009.

[21] Camacho J, Liu C, Gu C, Lin H, Huang Z, Tang Q, et al. Mobility Size and Mass of Nascent Soot Particles in a Benchmark Premixed Ethylene Flame. Combust Flame 2015. doi:10.13140/RG.2.1.4340.3689.

[22] Alfè M, Apicella B, Barbella R, Rouzaud J-N, Tregrossi A, Ciajolo A. Structure–property relationship in nanostructures of young and mature soot in premixed flames. Proc Combust Inst 2009;32:697–704. doi:10.1016/j.proci.2008.06.193.

[23] Alfè M, Apicella B, Rouzaud J-N, Tregrossi A, Ciajolo A. The effect of temperature on soot properties in premixed methane flames. Combust Flame 2010;157:1959–65. doi:10.1016/j.combustflame.2010.02.007.

[24] Bonne U, Homann KH, Wagner HG. Carbon formation in premixed flames. Symp Combust 1965;10:503–12. doi:10.1016/S0082-0784(65)80197-7.

[25] Homann KH. Carbon formation in premixed flames. Combust Flame 1967;11:265–87.

[26] Russo C, Alfè M, Rouzaud J-N, Stanzione F, Tregrossi A, Ciajolo A. Probing structures of soot formed in premixed flames of methane, ethylene and benzene. Proc Combust Inst 2013;34:1885–92. doi:10.1016/j.proci.2012.06.127.

[27] Russo C, Stanzione F, Tregrossi A, Alfè M, Ciajolo A. The effect of temperature on the condensed phases formed in fuel-rich premixed benzene flames. Combust Flame 2012;159:2233–42. doi:10.1016/j.combustflame.2012.02.014.

165

[28] Xu F, Lin K-C, Faeth GM. Soot Formation in Laminar Premixed Methane/Oxygen Flames at Atmospheric Pressure. Combust Flame 1998;115:195–209. doi:10.1016/S0010-2180(98)00017-0.

[29] Arana CP, Pontoni M, Sen S, Puri IK. Field measurements of soot volume fractions in laminar partially premixed coflow ethylene/air flames. Combust Flame 2004;138:362–72. doi:10.1016/j.combustflame.2004.04.013.

[30] Dobbins RA, Megaridis CM. Morphology of flame-generated soot as determined by thermophoretic sampling. Langmuir 1987;3:254–9. doi:10.1021/la00074a019.

[31] El-Leathy AM, Kim CH, Faeth GM, Xu F. Soot surface reactions in high-temperature laminar diffusion flames. AIAA J 2004;42:988–96.

[32] El-Leathy AM, Xu F, Kim CH, Faeth GM. Soot surface growth in laminar hydrocarbon/air diffusion flames. AIAA J 2003;41:856–65.

[33] Iyer SS, Litzinger TA, Lee S-Y, Santoro RJ. Determination of soot scattering coefficient from extinction and three-angle scattering in a laminar diffusion flame. Combust Flame 2007;149:206–16. doi:10.1016/j.combustflame.2006.11.009.

[34] Kennedy IM, Yam C, Rapp DC, Santoro RJ. Modeling and measurements of soot and species in a laminar diffusion flame. Combust Flame 1996;107:368–82. doi:10.1016/S0010-2180(96)00092-2.

[35] Kholghy M, Saffaripour M, Yip C, Thomson MJ. The evolution of soot morphology in a laminar coflow diffusion flame of a surrogate for Jet A-1. Combust Flame 2013;160:2119–30. doi:10.1016/j.combustflame.2013.04.008.

[36] Kim CH, El-Leathy AM, Xu F, Faeth GM. Soot surface growth and oxidation in laminar diffusion flames at pressures of 0.1–1.0 atm. Combust Flame 2004;136:191–207. doi:10.1016/j.combustflame.2003.09.017.

[37] Köylü ÜÖ, McEnally CS, Rosner DE, Pfefferle LD. Simultaneous measurements of soot volume fraction and particle size / microstructure in flames using a thermophoretic sampling technique. Combust Flame 1997;110:494–507. doi:10.1016/S0010-2180(97)00089-8.

[38] McEnally CS, Köylü ÜÖ, Pfefferle LD, Rosner DE. Soot volume fraction and temperature measurements in laminar nonpremixed flames using thermocouples. Combust Flame 1997;109:701–20. doi:10.1016/S0010-2180(97)00054-0.

[39] Megaridis CM, Dobbins RA. Soot aerosol dynamics in a laminar ethylene diffusion flame. Symp Combust 1989;22:353–62. doi:10.1016/S0082-0784(89)80041-4.

[40] Saffaripour M, Veshkini A, Kholghy M, Thomson MJ. Experimental investigation and detailed modeling of soot aggregate formation and size distribution in laminar coflow

166

diffusion flames of Jet A-1, a synthetic kerosene, and n-decane. Combust Flame 2014;161:848–63. doi:10.1016/j.combustflame.2013.10.016.

[41] Smooke MD, Long MB, Connelly BC, Colket MB, Hall RJ. Soot formation in laminar diffusion flames. Combust Flame 2005;143:613–28. doi:10.1016/j.combustflame.2005.08.028.

[42] Smooke MD, McEnally CS, Pfefferle LD, Hall RJ, Colket MB. Computational and experimental study of soot formation in a coflow, laminar diffusion flame. Combust Flame 1999;117:117–39. doi:10.1016/S0010-2180(98)00096-0.

[43] Smooke MD, Puri IK, Seshadri K. A comparison between numerical calculations and experimental measurements of the structure of a counterflow diffusion flame burning diluted methane in diluted air. Symp Combust 1988;21:1783–92. doi:10.1016/S0082-0784(88)80412-0.

[44] Xu F, Faeth GM. Soot formation in laminar acetylene/air diffusion flames at atmospheric pressure. Combust Flame 2001;125:804–19. doi:10.1016/S0010-2180(01)00221-8.

[45] Cain JP, Laskin A, Kholghy M, Thomson MJ, Wang H. Molecular characterization of organic content of soot along the centerline of a coflow diffusion flame. Phys Chem Chem Phys 2014;16:25862–75. doi:10.1039/c4cp03330b.

[46] D’Anna A, Commodo M, Violi S, Allouis C, Kent J. Nano organic carbon and soot in turbulent non-premixed ethylene flames. Proc Combust Inst 2007;31:621–9. doi:10.1016/j.proci.2006.07.062.

[47] Faeth GM, Köylü ÜÖ. Soot Morphology and Optical Properties in Nonpremixed Turbulent Flame Environments. Combust Sci Technol 1995;108:207–29. doi:10.1080/00102209508960399.

[48] Köylü ÜÖ, Faeth GM. Structure of overfire soot in buoyant turbulent diffusion flames at long residence times. Combust Flame 1992;89:140–56. doi:10.1016/0010-2180(92)90024-J.

[49] Qamar NH, Nathan GJ, Alwahabi ZT, King KD. The effect of global mixing on soot volume fraction: measurements in simple jet, precessing jet, and bluff body flames. Proc Combust Inst 2005;30:1493–500. doi:10.1016/j.proci.2004.08.102.

[50] Hu B, Yang B, Köylü ÜÖ. Soot measurements at the axis of an ethylene/air non-premixed turbulent jet flame. Combust Flame 2003;134:93–106. doi:10.1016/S0010-2180(03)00085-3.

[51] Dobbins RA. Soot inception temperature and the carbonization rate of precursor particles. Combust Flame 2002;130:204–14. doi:10.1016/S0010-2180(02)00374-7.

167

[52] Dobbins RA. Hydrocarbon Nanoparticles Formed in Flames and Diesel Engines. Aerosol Sci Technol 2007;41:485–96. doi:10.1080/02786820701225820.

[53] Dobbins RA, Fletcher RA, Chang H-C. The evolution of soot precursor particles in a diffusion flame. Combust Flame 1998;115:285–98. doi:10.1016/S0010-2180(98)00010-8.

[54] Dobbins RA, Fletcher RA, Lu W. Laser microprobe analysis of soot precursor particles and carbonaceous soot. Combust Flame 1995;100:301–9. doi:10.1016/0010-2180(94)00047-V.

[55] Dobbins RA, Govatzidakis GJ, Lu W, Schwartzman AF, Fletcher RA. Carbonization Rate of Soot Precursor Particles. Combust Sci Technol 1996;121:103–21. doi:10.1080/00102209608935589.

[56] Megaridis CM, Dobbins RA. Comparison of Soot Growth and Oxidation in Smoking and Non–Smoking Ethylene Diffusion Flames. Combust Sci Technol 1989;66:1–16. doi:10.1080/00102208908947136.

[57] Puri R, Richardson TF, Santoro RJ, Dobbins RA. Aerosol dynamic processes of soot aggregates in a laminar ethene diffusion flame. Combust Flame 1993;92:320–33. doi:10.1016/0010-2180(93)90043-3.

[58] Santoro RJ, Semerjian HG, Dobbins RA. Soot particle measurements in diffusion flames. Combust Flame 1983;51:203–18. doi:10.1016/0010-2180(83)90099-8.

[59] Commodo M, Sgro LA, Minutolo P, D’Anna A. Characterization of Combustion-Generated Carbonaceous Nanoparticles by Size-Dependent Ultraviolet Laser Photoionization. J Phys Chem A 2013;117:3980–9. doi:10.1021/jp401061d.

[60] D’Alessio A, D’Anna A, Gambi G, Minutolo P. The spectroscopic characterisation of UV absorbing nanoparticles in fuel rich soot forming flames. J Aerosol Sci 1998;29:397–409. doi:10.1016/S0021-8502(97)00457-6.

[61] De Falco G, Commodo M, Bonavolontà C, Pepe GP, Minutolo P, D’Anna A. Optical and electrical characterization of carbon nanoparticles produced in laminar premixed flames. Combust Flame 2014;161:3201–10. doi:10.1016/j.combustflame.2014.05.021.

[62] Öktem B, Tolocka MP, Zhao B, Wang H, Johnston M V. Chemical species associated with the early stage of soot growth in a laminar premixed ethylene–oxygen–argon flame. Combust Flame 2005;142:364–73. doi:10.1016/j.combustflame.2005.03.016.

[63] Zhao B, Uchikawa K, Wang H. A comparative study of nanoparticles in premixed flames by scanning mobility particle sizer, small angle neutron scattering, and transmission electron microscopy. Proc Combust Inst 2007;31:851–60. doi:10.1016/j.proci.2006.08.064.

168

[64] Wang H, Zhao B, Wyslouzil B, Streletzky K. Small-angle neutron scattering of soot formed in laminar premixed ethylene flames. Proc. Combust. Inst., vol. 29 II, 2002, p. 2749–57.

[65] Williams TC, Shaddix CR, Jensen KA, Suo-Anttila JM. Measurement of the dimensionless extinction coefficient of soot within laminar diffusion flames. Int J Heat Mass Transf 2007;50:1616–30. doi:10.1016/j.ijheatmasstransfer.2006.08.024.

[66] Blevins LG, Fletcher RA, Benner BA, Steel EB, Mulholland GW. The existence of young soot in the exhaust of inverse diffusion flames. Proc Combust Inst 2002;29:2325–33. doi:10.1016/S1540-7489(02)80283-8.

[67] D’Alessio A, D’Anna A, D’Orsi A, Minutolo P, Barbella R, Ciajolo A. Precursor formation and soot inception in premixed ethylene flames. Symp Combust 1992;24:973–80. doi:10.1016/S0082-0784(06)80115-3.

[68] Köylü ÜÖ, Faeth GM, Farias TL, Carvalho MG. Fractal and projected structure properties of soot aggregates. Combust Flame 1995;100:621–33. doi:10.1016/0010-2180(94)00147-K.

[69] Hu B, Köylü ÜÖ. Size and Morphology of Soot Particulates Sampled from a Turbulent Nonpremixed Acetylene Flame. Aerosol Sci Technol 2004;38:1009–18. doi:10.1080/027868290519111.

[70] Yazicioglu AG, Megaridis CM, Campbell A, Lee K-O, Choi MY. Measurement Of Fractal Properties Of Soot Agglomerates In Laminar Coflow Diffusion Flames Using Thermophoretic Sampling In Conjunction With Transmission Electron Microscopy And Image Processing. Combust Sci Technol 2001;171:71–87. doi:10.1080/00102200108907859.

[71] Sorensen CM, Feke GD. The Morphology of Macroscopic Soot. Aerosol Sci Technol 1996;25:328–37. doi:10.1080/02786829608965399.

[72] Abián M, Peribáñez E, Millera Á, Bilbao R, Alzueta MU. Impact of nitrogen oxides (NO, NO2, N2O) on the formation of soot. Combust Flame 2014;161:280–7. doi:10.1016/j.combustflame.2013.07.015.

[73] Feitelberg AS, Longwell JP, Sarofim AF. Metal enhanced soot and PAH formation. Combust Flame 1993;92:241–53. doi:10.1016/0010-2180(93)90036-3.

[74] Maricq MM. A comparison of soot size and charge distributions from ethane, ethylene, acetylene, and benzene/ethylene premixed flames. Combust Flame 2006;144:730–43. doi:10.1016/j.combustflame.2005.09.007.

[75] Hughes KJ, Turányi T, Clague AR, Pilling MJ. Development and testing of a comprehensive chemical mechanism for the oxidation of methane. Int J Chem Kinet 2001;33:513–38. doi:10.1002/kin.1048.

169

[76] Sheen DA, You X, Wang H, Løvås T. Spectral uncertainty quantification, propagation and optimization of a detailed kinetic model for ethylene combustion. Proc Combust Inst 2009;32:535–42. doi:10.1016/j.proci.2008.05.042.

[77] Sarathy SM, Vranckx S, Yasunaga K, Mehl M, Oßwald P, Metcalfe WK, et al. A comprehensive chemical kinetic combustion model for the four butanol isomers. Combust Flame 2012;159:2028–55. doi:10.1016/j.combustflame.2011.12.017.

[78] Herbinet O, Pitz WJ, Westbrook CK. Detailed chemical kinetic mechanism for the oxidation of biodiesel fuels blend surrogate. Combust Flame 2010;157:893–908. doi:10.1016/j.combustflame.2009.10.013.

[79] Miller JA, Klippenstein SJ. The Recombination of Propargyl Radicals and Other Reactions on a C 6 H 6 Potential. J Phys Chem A 2003;107:7783–99. doi:10.1021/jp030375h.

[80] Miller JA, Melius CF. Kinetic and thermodynamic issues in the formation of aromatic compounds in flames of aliphatic fuels. Combust Flame 1992;91:21–39. doi:10.1016/0010-2180(92)90124-8.

[81] Law ME, Westmoreland PR, Cool TA, Wang J, Hansen N, Taatjes CA, et al. Benzene precursors and formation routes in a stoichiometric cyclohexane flame. Proc Combust Inst 2007;31:565–73. doi:10.1016/j.proci.2006.07.259.

[82] Westmoreland PR. The prehistory of soot: small rings from small molecules. In: Bockhorn H, D’Anna A, Sarofim AF, Wang H, editors. Combust. Gener. Fine Carbonaceous Part., Karlsruhe: KIT Scientific Publishing; 2009, p. 720.

[83] Frenklach M, Gardiner WC, Stein SE, Clary DW, Yuan T. Mechanism of Soot Formation in Acetylene-Oxygen Mixtures. Combust Sci Technol 1986;50:79–115. doi:10.1080/00102208608923927.

[84] Marinov NM, Pitz WJ, Westbrook CK, Castaldi MJ, Senkan SM. Modeling of aromatic and polycyclic aromatic hydrocarbon formation in premixed methane and ethane flames. Combust Sci Technol 1996;116-117:211–87.

[85] Violi A, D’Anna A, D’Alessio A. Modeling of particulate formation in combustion and pyrolysis. Chem Eng Sci 1999;54:3433–42. doi:10.1016/S0009-2509(98)00460-6.

[86] Appel J, Bockhorn H, Frenklach M. Kinetic modeling of soot formation with detailed chemistry and physics: laminar premixed flames of C2 hydrocarbons. Combust Flame 2000;121:122–36. doi:10.1016/S0010-2180(99)00135-2.

[87] Slavinskaya NA, Riedel U, Dworkin SB, Thomson MJ. Detailed numerical modeling of PAH formation and growth in non-premixed ethylene and ethane flames. Combust Flame 2012;159:979–95. doi:10.1016/j.combustflame.2011.10.005.

170

[88] Wang Y, Raj A, Chung SH. A PAH growth mechanism and synergistic effect on PAH formation in counterflow diffusion flames. Combust Flame 2013;160:1667–76. doi:10.1016/j.combustflame.2013.03.013.

[89] Izvekov S, Violi A. A Coarse-Grained Molecular Dynamics Study of Carbon Nanoparticle Aggregation. J Chem Theory Comput 2006;2:504–12. doi:10.1021/ct060030d.

[90] Frenklach M, Wang H. Detailed mechanism and modeling of soot particle formation. Springer-Verlag GmbH & Company KG; 1994.

[91] Frenklach M, Wang H. Detailed modeling of soot particle nucleation and growth. Symp Combust 1991;23:1559–66. doi:10.1016/S0082-0784(06)80426-1.

[92] D’Anna A, Violi A, D’Alessio A, Sarofim AF. A reaction pathway for nanoparticle formation in rich premixed flames. Combust Flame 2001;127:1995–2003. doi:10.1016/S0010-2180(01)00303-0.

[93] Richter H, Benish TG, Mazyar OA, Green WH, Howard JB. Formation of polycyclic aromatic hydrocarbons and their radicals in a nearly sooting premixed benzene flame. Proc Combust Inst 2000;28:2609–18. doi:10.1016/S0082-0784(00)80679-7.

[94] Herdman JD, Miller JH. Intermolecular potential calculations for polynuclear aromatic hydrocarbon clusters. J Phys Chem A 2008;112:6249–56. doi:10.1021/jp800483h.

[95] Violi A, Venkatnathan A. Combustion-generated nanoparticles produced in a benzene flame: a multiscale approach. J Chem Phys 2006;125:054302. doi:10.1063/1.2234481.

[96] Chung SH, Violi A. Insights on the nanoparticle formation process in counterflow diffusion flames. Carbon N Y 2007;45:2400–10. doi:10.1016/j.carbon.2007.07.003.

[97] Woods IT, Haynes BS. Soot surface growth at active sites. Combust Flame 1991;85:523–5. doi:10.1016/0010-2180(91)90156-6.

[98] Dasch CJ. The decay of soot surface growth reactivity and its importance in total soot formation. Combust Flame 1985;61:219–25. doi:10.1016/0010-2180(85)90103-8.

[99] Cain JP, Camacho J, Phares DJ, Wang H, Laskin A. Evidence of aliphatics in nascent soot particles in premixed ethylene flames. Proc Combust Inst 2011;33:533–40. doi:10.1016/j.proci.2010.06.164.

[100] Wang H, Du DX, Sung CJ, Law CK. Experiments and numerical simulation on soot formation in opposed-jet ethylene diffusion flames. Symp Combust 1996;26:2359–68. doi:10.1016/S0082-0784(96)80065-8.

[101] Singh J, Patterson RIA, Kraft M, Wang H. Numerical simulation and sensitivity analysis of detailed soot particle size distribution in laminar premixed ethylene flames. Combust Flame 2006;145:117–27. doi:10.1016/j.combustflame.2005.11.003.

171

[102] Sabbah H, Biennier L, Klippenstein SJ, Sims IR, Rowe BR. Exploring the role of PAHs in the formation of soot: Pyrene dimerization. J Phys Chem Lett 2010;1:2962–7. doi:10.1021/jz101033t.

[103] Obolensky OI, Semenikhina V V., Solov’yov A V., Greiner W. Interplay of electrostatic and van der Waals forces in coronene dimer. Int J Quantum Chem 2007;107:1335–43. doi:10.1002/qua.21253.

[104] Teini PD, Karwat DMA, Atreya A. Observations of nascent soot: Molecular deposition and particle morphology. Combust Flame 2011;158:2045–55. doi:10.1016/j.combustflame.2011.03.005.

[105] Happold J, Grotheer H-H, Aigner M. Soot precursors consisting of stacked pericondensed PAHs. In: Bockhorn H, D’Anna A, Sarofim AF, Wang H, editors. Combust. Gener. Fine Carbonaceous Part., Stuttgart: KIT Scientific Publishing; 2009, p. 277–88.

[106] Friedlander SK. Smoke, dust, and haze : fundamentals of aerosol dynamics. Second. New York: Oxford University Press; 2000.

[107] Neoh KG, Howard JB, Sarofim AF. Soot oxidation in flames. In: Siegla DC, Smith GW, editors. Part. Carbon, Boston, MA: Springer US; 1981, p. 261–82. doi:10.1007/978-1-4757-6137-5.

[108] Lighty JS, Romano V, Sarofim AF. Soot Oxidation. Combust. Gener. Fine Carbonaceous Part., KIT Scientific Publishing; 2009, p. 523–36.

[109] Vander Wal RL, Tomasek AJ. Soot oxidation. Combust Flame 2003;134:1–9. doi:10.1016/S0010-2180(03)00084-1.

[110] Neoh KG, Howard JB, Sarofim AF. Effect of oxidation on the physical structure of soot. Symp Combust 1985;20:951–7. doi:10.1016/S0082-0784(85)80584-1.

[111] Xu F. Soot surface oxidation in hydrocarbon/air diffusion flames at atmospheric pressure. Combust Flame 2003;132:43–57. doi:10.1016/S0010-2180(02)00459-5.

[112] Fairweather M, Jones WP, Lindstedt RP. Predictions of radiative transfer from a turbulent reacting jet in a cross-wind. Combust Flame 1992;89:45–63. doi:10.1016/0010-2180(92)90077-3.

[113] Blanquart G, Pitsch H. Analyzing the effects of temperature on soot formation with a joint volume-surface-hydrogen model. Combust Flame 2009;156:1614–26. doi:10.1016/j.combustflame.2009.04.010.

[114] Singh J, Balthasar M, Kraft M, Wagner W. Stochastic modeling of soot particle size and age distributions in laminar premixed flames. Proc Combust Inst 2005;30:1457–65. doi:10.1016/j.proci.2004.08.120.

172

[115] Appel J, Bockhorn H, Wulkow M. A detailed numerical study of the evolution of soot particle size distributions in laminar premixed flames. Chemosphere 2001;42:635–45. doi:10.1016/S0045-6535(00)00237-X.

[116] Vlasov PA, Warnatz J. Detailed kinetic modeling of soot formation in hydrocarbon pyrolysis behind shock waves. Proc Combust Inst 2002;29:2335–41. doi:10.1016/S1540-7489(02)80284-X.

[117] Sirignano M, Kent J, D’Anna A. Modeling Formation and Oxidation of Soot in Nonpremixed Flames. Energy & Fuels 2013;27:2303–15. doi:10.1021/ef400057r.

[118] Zhang Q, Thomson MJ, Guo H, Liu F, Smallwood GJ. A numerical study of soot aggregate formation in a laminar coflow diffusion flame. Combust Flame 2009;156:697–705. doi:10.1016/j.combustflame.2008.10.022.

[119] Violi A, Sarofim AF, Voth GA. Kinetic monte carlo–molecular dynamics approach to model soot inception. Combust Sci Technol 2004;176:991–1005. doi:10.1080/00102200490428594.

[120] Schuetz CA, Frenklach M. Nucleation of soot: Molecular dynamics simulations of pyrene dimerization. Proc Combust Inst 2002;29:2307–13. doi:10.1016/S1540-7489(02)80281-4.

[121] Park SH, Rogak SN, Bushe WK, Wen JZ, Thomson MJ. An aerosol model to predict size and structure of soot particles. Combust Theory Model 2005;9:499–513. doi:10.1080/13647830500195005.

[122] Wen JZ, Thomson MJ, Lightstone MF, Rogak SN. Detailed Kinetic Modeling of Carbonaceous Nanoparticle Inception and Surface Growth during the Pyrolysis of C 6 H 6 behind Shock Waves. Energy & Fuels 2006;20:547–59. doi:10.1021/ef050081q.

[123] Zhang Q, Thomson MJ, Guo H, Liu F, Smallwood GJ. Modeling of Oxidation-Driven Soot Aggregate Fragmentation in a Laminar Coflow Diffusion Flame. Combust Sci Technol 2010;182:491–504. doi:10.1080/00102200903463050.

[124] Dworkin SB. Serial and distributed-memory parallel computation of sooting, steady and time-dependent, laminar flames using a modified vorticity-velocity formulation. Yale University, 2009.

[125] Von Kármán T. On laminar and turbulent friction. Washington: National Advisory Committee on Aeronautics; 1946.

[126] Kee RJ, Miller JA, Evans GH, Dixon-Lewis G. A computational model of the structure and extinction of strained, opposed flow, premixed methane-air flames. Symp Combust 1989;22:1479–94. doi:10.1016/S0082-0784(89)80158-4.

[127] Turns S. An introduction to combustion: concepts and applications/. McGraw-Hill Ser Mech Eng 2000.

173

[128] Viskanta, R, Mengüç MP. Radiation heat transfer in combustion systems. Prog Energy Combust Sci 1987;13:97–160. doi:10.1016/0360-1285(87)90008-6.

[129] Sparrow EM, Cess RD. Radiation heat transfer. Belmont, CA, United States: Brooks/Cole Publishing; 1966.

[130] Liu F, Guo H, Smallwood GJ. Effects of radiation model on the modeling of a laminar coflow methane/air diffusion flame. Combust Flame 2004;138:136–54. doi:10.1016/j.combustflame.2004.04.007.

[131] Ju Y, Guo H, Maruta K, Liu F. On the extinction limit and flammability limit of non-adiabatic stretched methane–air premixed flames. J Fluid Mech 1997;342:315–34. doi:10.1017/S0022112097005636.

[132] Hall RJ. Radiative dissipation in planar gas-soot mixtures. J Quant Spectrosc Radiat Transf 1994;51:635–44. doi:10.1016/0022-4073(94)90117-1.

[133] Hubbard GL, Tien CL. INFRARED MEAN ABSORPTION COEFFICIENTS OF LUMINOUS FLAMES AND SMOKE. J Heat Transfer 1978;100:235–9.

[134] Smyth KC, Shaddix CR. The elusive history of m�= 1.57 – 0.56i for the refractive index of soot. Combust Flame 1996;107:314–20. doi:10.1016/S0010-2180(96)00170-8.

[135] Kaplan C, Baek S, Oran E, Ellzey J. Dynamics of a strongly radiating unsteady ethylene jet diffusion flame☆. Combust Flame 1994;96:1–21. doi:10.1016/0010-2180(94)90154-6.

[136] Thurgood CP, Pollard A, Becker HA. TN quadrature set for the discrete ordinates method. J Heat Transfer 1995;117:1068–70.

[137] Liu F, Smallwood GJ, Gülder ÖL. Application of the statistical narrow-band correlated-k method to low-resolution spectral intensity and radiative heat transfer calculations — effects of the quadrature scheme. Int J Heat Mass Transf 2000;43:3119–35. doi:10.1016/S0017-9310(99)00343-9.

[138] Liu F, Guo H, Smallwood GJ, Gülder ÖL. Effects of gas and soot radiation on soot formation in a coflow laminar ethylene diffusion flame. J Quant Spectrosc Radiat Transf 2002;73:409–21. doi:10.1016/S0022-4073(01)00205-9.

[139] Liu F, Smallwood GJ, Gülder ÖL. Application of the statistical narrow-band correlated-k method to non-grey gas radiation in CO2–H2O mixtures: approximate treatments of overlapping bands. J Quant Spectrosc Radiat Transf 2001;68:401–17. doi:10.1016/S0022-4073(00)00033-9.

[140] Slavinskaya NA, Frank P. A modelling study of aromatic soot precursors formation in laminar methane and ethene flames. Combust Flame 2009;156:1705–22. doi:10.1016/j.combustflame.2009.04.013.

174

[141] Dworkin SB, Zhang Q, Thomson MJ, Slavinskaya NA, Riedel U. Application of an enhanced PAH growth model to soot formation in a laminar coflow ethylene/air diffusion flame. Combust Flame 2011;158:1682–95. doi:10.1016/j.combustflame.2011.01.013.

[142] Zsély IG, Zádor J, Turányi T. Uncertainty analysis of updated hydrogen and carbon monoxide oxidation mechanisms. Proc Combust Inst 2005;30:1273–81. doi:10.1016/j.proci.2004.08.172.

[143] Wang H, You X, Joshi A V., Davis SG, Egolfopoulos F, Law CK. USC Mech Version II. High-Temperature Combustion Reaction Model of H2/CO/C1-C4 Compounds 2007. http://ignis.usc.edu/USC_Mech_II.htm.

[144] Smoluchowski M. Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen. Zeitschrift Fuer Phys Chemie 1917;92:129–68.

[145] Gelbard F, Tambour Y, Seinfeld JH. Sectional representations for simulating aerosol dynamics. J Colloid Interface Sci 1980;76:541–56. doi:10.1016/0021-9797(80)90394-X.

[146] Jin Jwang Wu, Flagan RC. A discrete-sectional solution to the aerosol dynamic equation. J Colloid Interface Sci 1988;123:339–52.

[147] Zhang Q, Guo H, Liu F, Smallwood GJ, Thomson MJ. Modeling of soot aggregate formation and size distribution in a laminar ethylene/air coflow diffusion flame with detailed PAH chemistry and an advanced sectional aerosol dynamics model. Proc Combust Inst 2009;32:761–8. doi:10.1016/j.proci.2008.06.109.

[148] Homann KH. Formation of large molecules, particulates and ions in premixed hydrocarbon flames; Progress and unresolved questions. Symp Combust 1985;20:857–70.

[149] Bockhorn H, editor. Soot Formation in Combustion. vol. 59. Berlin, Heidelberg: Springer Berlin Heidelberg; 1994. doi:10.1007/978-3-642-85167-4.

[150] Guo H, Liu F, Smallwood GJ, Gülder ÖL. Numerical study on the influence of hydrogen addition on soot formation in a laminar ethylene–air diffusion flame. Combust Flame 2006;145:324–38. doi:10.1016/j.combustflame.2005.10.016.

[151] Wen JZ, Thomson MJ, Park SH, Rogak SN, Lightstone MF. Study of soot growth in a plug flow reactor using a moving sectional model. Proc Combust Inst 2005;30:1477–84. doi:10.1016/j.proci.2004.08.178.

[152] Rogak SN, Flagan RC. Coagulation of aerosol agglomerates in the transition regime. J Colloid Interface Sci 1992;151:203–24. doi:10.1016/0021-9797(92)90252-H.

[153] Zhang Q. Detailed Modeling of Soot Formation/Oxidation in Laminar Coflow Diffusion Flames. University of Toronto, 2009.

175

[154] Coffee T, Heimerl J. Transport algorithms for premixed, laminar steady-state flames. Combust Flame 1981;43:273–89. doi:10.1016/0010-2180(81)90027-4.

[155] Bird RB, Stewart WE, Lightfoot EN. Transport Phenomena. New York: John Wiley and Sons; 2006.

[156] Mathur S, Tondon PK, Saxena SC. Thermal conductivity of binary, ternary and quaternary mixtures of rare gases. Mol Phys 1967;12:569–79. doi:10.1080/00268976700100731.

[157] Sorensen CM, Wang GM. Note on the Correction for Diffusion and Drag in the Slip Regime. Aerosol Sci Technol 2000;33:353–6. doi:10.1080/02786820050121549.

[158] Naumann K-H. COSIMA—a computer program simulating the dynamics of fractal aerosols. J Aerosol Sci 2003;34:1371–97. doi:10.1016/S0021-8502(03)00367-7.

[159] Talbot L, Cheng RK, Schefer RW, Willis DR. Thermophoresis of Particles in a Heated Boundary Layer. vol. 101. 1980.

[160] Li Z, Wang H. Drag force, diffusion coefficient, and electric mobility of small particles. II. Application. Phys Rev E 2003;68:061207. doi:10.1103/PhysRevE.68.061207.

[161] Li Z, Wang H. Thermophoretic force and velocity of nanoparticles in the free molecule regime. Phys Rev E 2004;70:021205. doi:10.1103/PhysRevE.70.021205.

[162] Patankar S. Numerical Heat Transfer and Fluid Flow. CRC Press; 1980.

[163] Zheng X, Liu C, Liao C, Liu Z, McCormick S. Multigrid Method for Modeling Multi-Dimensional Combustion with Detailed Chemistry. Seventh Copp. Mt. Conf. Multigrid Method, Denver, Colorado: 1996.

[164] Grcar JF. The twopnt program for boundary value problems: Version 3.10 of March 1992. Livermore, CA (United States): 1992.

[165] Toselli A, Widlund OB. Domain Decomposition Methods — Algorithms and Theory. vol. 34. Berlin/Heidelberg: Springer-Verlag; 2005. doi:10.1007/b137868.

[166] Gropp W, Lusk E, Thakur R. Using MPI-2: Advanced Features of the Message-Passing Interface. MIT Press; 1999.

[167] Kee RJ, Rupley FM, Miller JA. CHEMKIN-II: A FORTRAN Chemical Kinetics Package for the Analysis of Gas-Phase Chemical Kinetics. Livermore, CA, USA: 1989.

[168] Kee RJ, Dixon-lewis G, Warnatz J, Coltrin ME, Miller JA. A Fortran Computer Code Package For The Evaluation Of Gas-Phase, Multicomponent Transport Properties. Livermore, CA (United States): 1986.

176

[169] Lutz AE, Kee RJ, Grcar JF, Rupley FM. OPPDIF: A Fortran program for computing opposed-flow diffusion flames. Albuquerque, NM, and Livermore, CA (United States): 1997. doi:10.2172/568983.

[170] Curtis A, Powell M, Reid J. On the estimation of sparse Jacobian matrices. J Inst Math Appl 1974.

[171] Kee RJ, Grcar JF, Smooke MD, Miller JA, Meeks E. PREMIX: A FORTRAN Program for Modeling Steady Laminar One-Dimensional Premixed Flames. San Diego, CA, United State: 2000.

[172] Mueller ME, Blanquart G, Pitsch H. A joint volume-surface model of soot aggregation with the method of moments. Proc Combust Inst 2009;32:785–92. doi:10.1016/j.proci.2008.06.207.

[173] Mueller ME, Blanquart G, Pitsch H. Hybrid Method of Moments for modeling soot formation and growth. Combust Flame 2009;156:1143–55. doi:10.1016/j.combustflame.2009.01.025.

[174] Ulrich GD, Subramanian NS. III. Coalescence as a Rate-Controlling Process. Combust Sci Technol 1977;17:119–26. doi:10.1080/00102207708946822.

[175] Sander M, West RH, Celnik MS, Kraft M. A Detailed Model for the Sintering of Polydispersed Nanoparticle Agglomerates. Aerosol Sci Technol 2009;43:978–89. doi:10.1080/02786820903092416.

[176] Sander M, Patterson RIA, Braumann A, Raj A, Kraft M. Developing the PAH-PP soot particle model using process informatics and uncertainty propagation. Proc Combust Inst 2011;33:675–83. doi:10.1016/j.proci.2010.06.156.

[177] D’Anna A, Sirignano M, Kent J. A model of particle nucleation in premixed ethylene flames. Combust Flame 2010;157:2106–15. doi:10.1016/j.combustflame.2010.04.019.

[178] Koch W, Friedlander SK. The effect of particle coalescence on the surface area of a coagulating aerosol. J Aerosol Sci 1989;20:891–4. doi:10.1016/0021-8502(89)90719-2.

[179] Park SH, Rogak SN. A One-Dimensional Model for Coagulation, Sintering, and Surface Growth of Aerosol Agglomerates. Aerosol Sci Technol 2003;37:947–60. doi:10.1080/02786820300899.

[180] Frenkel J. Viscous flow of crystalline bodies under the action of surface tension. J Phys 1945;9:385.

[181] Dobrushin R, Kotecký R, Shlosman S. Wulff construction: a global shape from local interaction. 1992.

177

[182] Friedlander SK, Wu M. Linear rate law for the decay of the excess surface area of a coalescing solid particle. Phys Rev B 1994;49:3622–4. doi:10.1103/PhysRevB.49.3622.

[183] Zachariah MR, Carrier MJ. Molecular dynamics computation of gas-phase nanoparticle sintering: a comparison with phenomenological models. J Aerosol Sci 1999;30:1139–51. doi:10.1016/S0021-8502(98)00782-4.

[184] Barone AC. Morphological characterization of the early process of soot formation by atomic force microscopy. Combust Flame 2003;132:181–7. doi:10.1016/S0010-2180(02)00434-0.

[185] Reilly PT., Gieray R., Whitten W., Ramsey J. Direct observation of the evolution of the soot carbonization process in an acetylene diffusion flame via real-time aerosol mass spectrometry. Combust Flame 2000;122:90–104. doi:10.1016/S0010-2180(00)00105-X.

[186] Vander Wal RL. Soot Precursor Carbonization: Visualization Using LIF and LII and Comparison Using Bright and Dark Field TEM. Combust Flame 1998;112:607–16. doi:10.1016/S0010-2180(97)00171-5.

[187] Fenimore CP, Jones GW. Coagulation of soot to smoke in hydrocarbon flames. Combust Flame 1969;13:303–10.

[188] Howard JB, Wersborg BL, Williams GC. Coagulation of carbon particles in premixed flames. Faraday Symp Chem Soc 1973;7:109. doi:10.1039/fs9730700109.

[189] Boedeker LR, Dobbs GM. Soot distribution and cars temperature measuremnts in axisymmetric laminar diffusion flames with several fuels. Symp Combust 1988;21:1097–105. doi:10.1016/S0082-0784(88)80340-0.

[190] Smyth KC. http://www.bfrl.nist.gov. Build Fire Res Lab Natl Inst Stand Technol 1999. http://www.nist.gov/el/fire_research/flamereduc/diffusion_flamedata.cfm.

[191] International Sooting Flame Workshop 2014. http://www.adelaide.edu.au/cet/isfworkshop/.

[192] Santoro RJ, Yeh TT, Horvath JJ, Semerjian HG. The Transport and Growth of Soot Particles in Laminar Diffusion Flames. Combust Sci Technol 1987;53:89–115. doi:10.1080/00102208708947022.

[193] Eaves NA, Dworkin SB, Thomson MJ. The importance of reversibility in modeling soot nucleation and condensation processes. Proc Combust Inst 2014. doi:10.1016/j.proci.2014.05.036.

[194] Veshkini A, Dworkin SB, Thomson MJ. A soot particle surface reactivity model applied to a wide range of laminar ethylene/air flames. Combust Flame 2014;161:3191–200. doi:10.1016/j.combustflame.2014.05.024.

178

[195] Eaves NA, Veshkini A, Riese C, Zhang Q, Dworkin SB, Thomson MJ. A numerical study of high pressure, laminar, sooting, ethane–air coflow diffusion flames. Combust Flame 2012;159:3179–90. doi:10.1016/j.combustflame.2012.03.017.

[196] Lahaye J. Particulate carbon from the gas phase. Carbon N Y 1992;30:309–14.

[197] Lewis IC. Chemistry of carbonization. Carbon N Y 1982;20:519–29.

[198] Harris SJ, Weiner AM. Chemical Kinetics of Soot Particle Growth. Annu Rev Phys Chem 1985;36:31–52. doi:10.1146/annurev.physchem.36.1.31.

[199] Frenklach M, Wang H. Detailed modeling of soot particle nucleation and growth. Symp Combust 1991;23:1559–66. doi:10.1016/S0082-0784(06)80426-1.

[200] Haynes BS, Wagner HG. The Surface Growth Phenomenon in Soot Formation. Zeitschrift Für Phys Chemie 1982;133:201–13. doi:10.1524/zpch.1982.133.2.201.

[201] Harris SJ, Weiner AM. Determination of the Rate Constant for Soot Surface Growth. Combust Sci Technol 1983;32:267–75. doi:10.1080/00102208308923661.

[202] Kronholm DF, Howard JB. Analysis of soot surface growth pathways using published plug-flow reactor data with new particle size distribution measurements and published premixed flame data. Proc Combust Inst 2000;28:2555–61. doi:10.1016/S0082-0784(00)80672-4.

[203] Frenklach M. On surface growth mechanism of soot particles. Symp Combust 1996;26:2285–93. doi:10.1016/S0082-0784(96)80056-7.

[204] Frenklach M, Moriarty NW, Brown NJ. Hydrogen migration in polyaromatic growth. Symp Combust 1998;27:1655–61. doi:10.1016/S0082-0784(98)80004-0.

[205] Colket MB, Hall RJ. Soot Formation in Combustion. vol. 59. Berlin, Heidelberg: Springer Berlin Heidelberg; 1994. doi:10.1007/978-3-642-85167-4.

[206] Mauss F, Bockhorn H. Soot Formation in Premixed Hydrocarbon Flames: Prediction of Temperature and Pressure Dependence. Zeitschrift Für Phys Chemie 1995;188:45–60. doi:10.1524/zpch.1995.188.Part_1_2.045.

[207] Mauss F, Schäfer T, Bockhorn H. Inception and growth of soot particles in dependence on the surrounding gas phase. Combust Flame 1994;99:697–705. doi:10.1016/0010-2180(94)90064-7.

[208] Faccinetto A, Desgroux P, Ziskind M, Therssen E, Focsa C. High-sensitivity detection of polycyclic aromatic hydrocarbons adsorbed onto soot particles using laser desorption/laser ionization/time-of-flight mass spectrometry: An approach to studying the soot inception process in low-pressure flames. Combust Flame 2011;158:227–39. doi:10.1016/j.combustflame.2010.08.012.

179

[209] Kazakov A, Wang H, Frenklach M. Detailed modeling of soot formation in laminar premixed ethylene flames at a pressure of 10 bar. Combust Flame 1995;100:111–20. doi:10.1016/0010-2180(94)00086-8.

[210] Chernov V, Zhang Q, Thomson MJ, Dworkin SB. Numerical investigation of soot formation mechanisms in partially-premixed ethylene–air co-flow flames. Combust Flame 2012;159:2789–98. doi:10.1016/j.combustflame.2012.02.023.

[211] Herdman JD, Connelly BC, Smooke MD, Long MB, Miller JH. A comparison of Raman signatures and laser-induced incandescence with direct numerical simulation of soot growth in non-premixed ethylene/air flames. Carbon N Y 2011;49:5298–311. doi:10.1016/j.carbon.2011.07.050.

[212] Shaddix CR, Smyth KC. Laser-induced incandescence measurements of soot production in steady and flickering methane, propane, and ethylene diffusion flames. Combust Flame 1996;107:418–52. doi:10.1016/S0010-2180(96)00107-1.

[213] Whitesides R, Kollias AC, Domin D, Lester WA, Frenklach M. Graphene layer growth: Collision of migrating five-member rings. Proc Combust Inst 2007;31:539–46. doi:10.1016/j.proci.2006.07.034.

[214] Whitesides R, Domin D, Salomón-Ferrer R, Lester WA, Frenklach M. Embedded-ring migration on graphene zigzag edge. Proc Combust Inst 2009;32:577–83. doi:10.1016/j.proci.2008.06.096.

[215] Smooke MD, Hall RJ, Colket MB, Fielding J, Long MB, McEnally CS, et al. Investigation of the transition from lightly sooting towards heavily sooting co-flow ethylene diffusion flames. Combust Theory Model 2004;8:593–606. doi:10.1088/1364-7830/8/3/009.

[216] Miller JH. The kinetics of polynuclear aromatic hydrocarbon agglomeration in flames. Symp Combust 1991;23:91–8. doi:10.1016/S0082-0784(06)80246-8.

[217] Miller JH. Aromatic excimers: evidence for polynuclear aromatic hydrocarbon condensation in flames. Proc Combust Inst 2005;30:1381–8. doi:10.1016/j.proci.2004.08.192.

[218] Zhang Q, Guo H, Liu F, Smallwood GJ, Thomson MJ. Implementation of an advanced fixed sectional aerosol dynamics model with soot aggregate formation in a laminar methane/air coflow diffusion flame. Combust Theory Model 2008;12:621–41. doi:10.1080/13647830801966153.

[219] D’Alessio A, Barone AC, Cau R, D’Anna A, Minutolo P. Surface deposition and coagulation efficiency of combustion generated nanoparticles in the size range from 1 to 10nm. Proc Combust Inst 2005;30:2595–603. doi:10.1016/j.proci.2004.08.267.

180

[220] Israelachvili JN. Intermolecular and surface forces / Jacob N. Israelachvili. London ; San Diego: Academic Press; 1991.

[221] D’Anna A, Commodo M, Sirignano M, Minutolo P, Pagliara R. Particle formation in opposed-flow diffusion flames of ethylene: An experimental and numerical study. Proc Combust Inst 2009;32:793–801. doi:10.1016/j.proci.2008.06.200.

[222] Lindstedt RP, Waldheim BBO. Modeling of soot particle size distributions in premixed stagnation flow flames. Proc Combust Inst 2013;34:1861–8. doi:10.1016/j.proci.2012.05.047.

[223] Totton TS, Misquitta AJ, Kraft M. A quantitative study of the clustering of polycyclic aromatic hydrocarbons at high temperatures. Phys Chem Chem Phys 2012;14:4081–94. doi:10.1039/c2cp23008a.

[224] Rapacioli M, Calvo F, Spiegelman F, Joblin C, Wales DJ. Stacked clusters of polycyclic aromatic hydrocarbon molecules. J Phys Chem A 2005;109:2487–97. doi:10.1021/jp046745z.

[225] Rapacioli M, Calvo F, Joblin C, Parneix P, Spiegelman F. Vibrations and thermodynamics of clusters of polycyclic aromatic hydrocarbon molecules: the role of internal modes. J Phys Chem A 2007;111:2999–3009. doi:10.1021/jp068821z.

[226] Zhao B, Yang Z, Johnston M V., Wang H, Wexler AS, Balthasar M, et al. Measurement and numerical simulation of soot particle size distribution functions in a laminar premixed ethylene-oxygen-argon flame. Combust Flame 2003;133:173–88. doi:10.1016/S0010-2180(02)00574-6.

[227] Zhao B, Yang Z, Li Z, Johnston M V., Wang H. Particle size distribution function of incipient soot in laminar premixed ethylene flames: effect of flame temperature. Proc Combust Inst 2005;30:1441–8. doi:10.1016/j.proci.2004.08.104.

[228] McQuarrie DA. Statistical Mechanics. University Science Books; 2000.

[229] Tan X. Towards a comprehensive electronic database of polycyclic aromatic hydrocarbons and its application in constraining the identities of possible carriers of the diffuse interstellar bands. Spectrochim Acta A Mol Biomol Spectrosc 2009;71:2005–11. doi:10.1016/j.saa.2008.07.038.

[230] Miller JH. Aromatic excimers: evidence for polynuclear aromatic hydrocarbon condensation in flames. Proc Combust Inst 2005;30:1381–8. doi:10.1016/j.proci.2004.08.192.

[231] Raj A, Sander M, Janardhanan V, Kraft M. A study on the coagulation of polycyclic aromatic hydrocarbon clusters to determine their collision efficiency. Combust Flame 2010;157:523–34. doi:10.1016/j.combustflame.2009.10.003.

181

[232] Chernov V, Thomson MJ, Dworkin SB, Slavinskaya NA, Riedel U. Soot formation with C1 and C2 fuels using an improved chemical mechanism for PAH growth. Combust Flame 2014;161:592–601. doi:10.1016/j.combustflame.2013.09.017.

[233] Castaldi MJ, Marinov NM, Melius CF, Huang J, Senkan SM, Pitz WJ, et al. Experimental and modeling investigation of aromatic and polycyclic aromatic hydrocarbon formation in a premixed ethylene flame. Symp. Combust., vol. 1, Combustion Inst; 1996, p. 693–702.

[234] Laskin A, Lifshitz A. Thermal decomposition of indene. Experimental results and kinetic modeling. Symp Combust 1998.

[235] Eaves NA, Thomson MJ, Dworkin SB. The Effect of Conjugate Heat Transfer on Soot Formation Modeling at Elevated Pressures. Combust Sci Technol 2013;185:1799–819. doi:10.1080/00102202.2013.839554.