modeling soot formation derived from solid fuels

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2018-11-01

Modeling Soot Formation Derived from SolidFuelsAlexander Jon JosephsonBrigham Young University

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Modeling Soot Formation Derived from Solid Fuels

Alexander Jon Josephson

A dissertation submitted to the faculty ofBrigham Young University

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

David O. Lignell, ChairThomas H. FletcherJeremy N. Thornock

Larry L. BaxterBradley R. Adams

Department of Chemical Engineering

Brigham Young University

Copyright © 2018 Alexander Jon Josephson

All Rights Reserved

ABSTRACT

Modeling Soot Formation Derived from Solid Fuels

Alexander Jon JosephsonDepartment of Chemical Engineering, BYU

Doctor of Philosophy

Soot formation from complex solid fuels, such as coal or biomass, is an under-studiedand little understood phenomena which has profound physical effects. Any time a solid fuel iscombusted, from coal-burning power plants to wildland fires, soot formation within the flamecan have a significant influence on combustion characteristics such as temperature, heat flux, andchemical profiles. If emitted from the flame, soot particles have long-last effects on human healthand the environment.

The work in this dissertation focuses on creating and implementing computational modelsto be used for predicting soot mechanisms in a combustion environment. Three models are dis-cussed in this work; the first is a previously developed model designed to predict soot yield in coalsystems. This model was implemented into a computational fluid dynamic software and resultsare presented. The second model is a detailed-physics based model developed here. Validation forthis model is presented along with some results of its implementation into the same software. Thethird model is a simplified version of the detailed model and is presented with some comparisoncase studies implemented on a variety of platforms and scenarios.

While the main focus of this work is the presentation of the three computational modelsand their implementations, a considerable bulk of this work will discuss some of the technical toolsused to accomplish this work. Some of these tools include an introduction to Bayesian statisticsused in parameter inference and the method of moments with methods to resolve the ’closure’problem.

Keywords: soot formation, particulate emissions, coal, biomass

ACKNOWLEDGMENTS

I am indeed grateful to my BYU advisor, David Lignell, and LANL mentor, Rod Linn, who

both have been encouraging, guiding, and willing to sit and listen, even when I’ve been completely

wrong.

I’ve had supportive parents who not only set a stellar example of wisdom and hard-work

in the early parts of my life, but have continued to show interest and support to all aspects of my

work throughout.

Without a patient and understanding wife, Rachel, this work wouldn’t be what it is. Not

only has she been influential, encouraging, and supportive but she’s been willing to put-off many

of the world’s comforts in order that I might pursue this work and degree. Though they don’t

realize it, both of my children, Gideon and Eleanor, have made sacrifices of worldly comforts for

this work as well.

Most importantly, I’m grateful to a merciful, patient, and loving Father in Heaven who has

made all things in my life possible.

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Chapter 1 Introduction and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Soot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Flame Impacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Health Impacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.3 Environmental Impacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Soot Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.1 Formation in Gaseous Fuels . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.2 Challenges to Soot Formation . . . . . . . . . . . . . . . . . . . . . . . . 141.2.3 Formation from Solid Fuels . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.3 Oxy-Fuel Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.4 Modeling Wildland Fires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Chapter 2 Computational Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.1 Resolution of Particle-Size Distributions . . . . . . . . . . . . . . . . . . . . . . . 30

2.1.1 Sectional Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.1.2 Derived-Distribution Method . . . . . . . . . . . . . . . . . . . . . . . . . 332.1.3 Method of Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.2 Bayesian Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.2.1 Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.2.2 Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.2.3 Marginal Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.2.4 Posterior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Chapter 3 Existing Model Implementation . . . . . . . . . . . . . . . . . . . . . . . . 523.1 The Brown Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2 Arches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.3 Simulation Set-Up and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3.1 Oxy-Fuel Combustor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.3.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.3.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Chapter 4 Modeling Soot Consumption . . . . . . . . . . . . . . . . . . . . . . . . . 704.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2.1 Oxidation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

iv

4.2.2 Oxidation Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.2.3 Gasification Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.2.4 Gasification Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.3 Bayesian Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.4.1 Oxidation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.4.2 Gasification Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.4.3 Rate-Informed Priors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.4.4 Rate Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Chapter 5 Detailed Modeling of Soot from Solid Fuels . . . . . . . . . . . . . . . . . 1035.1 Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.1.1 Precursors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.1.2 Soot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.2 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.2.1 Coal System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.2.2 Biomass System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Chapter 6 Simplified Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.1 Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.1.1 Precursor Inception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366.1.2 Thermal Cracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1416.1.3 Soot Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1436.1.4 Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1446.1.5 Surface Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1456.1.6 Coagulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1476.2.1 Coal Flat-Flame Burner . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1476.2.2 LES Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Chapter 7 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 1537.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1537.2 Possible Model Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1547.3 Future Development of a Surrogate Model for FIRETEC . . . . . . . . . . . . . . 155

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Appendix A Model Derivations for Developed Detailed Soot Model . . . . . . . . . . . 174A.0.1 Soot Nucleation from Sections 5.1.1 and 5.1.2 . . . . . . . . . . . . . . . . 174A.0.2 Precursor Deposition from Sections 5.1.1, 5.1.2, and 5.1.2 . . . . . . . . . 175A.0.3 Precursor Cracking from Section 5.1.1 . . . . . . . . . . . . . . . . . . . . 178

v

A.0.4 Soot Coagulation from Sections 5.1.2 and 5.1.2 . . . . . . . . . . . . . . . 181A.0.5 Surface Reactions from Sections 5.1.2 and 5.1.2 . . . . . . . . . . . . . . . 185A.0.6 Expansion of a grid function, Equation 5.35 . . . . . . . . . . . . . . . . . 187

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LIST OF TABLES

2.1 Resolved statistical moments of the experimental distribution of Figure 2.1. . . . . . . 372.2 Resolved weights and abcissas of the 6 resolved moments in Table 2.1. . . . . . . . . . 392.3 Experimental data for example gas-reactor. . . . . . . . . . . . . . . . . . . . . . . . . 422.4 Ranges over which a & b parameters were analyzed for the example gas-reactor. . . . . 422.5 Calibrated parameters from the Bayesian inference for the simple gas-reactor example. 50

3.1 Transport equation source terms in the Brown Model. . . . . . . . . . . . . . . . . . . 553.2 Proximate and ultimate analysis for Utah SUFCO and Skyline coals. . . . . . . . . . . 603.3 Flow rates for the two simulated experiments. . . . . . . . . . . . . . . . . . . . . . . 603.4 Comparisons the average soot volume fraction across the flame from optical measure-

ments and simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.1 Studies from which oxidation data were extracted for model development. . . . . . . . 744.2 Studies from which gasification data were extracted for model development. . . . . . . 804.3 Range over which model parameters were tested. . . . . . . . . . . . . . . . . . . . . 844.4 Calibrated parameters for soot oxidation, Equation 4.3. . . . . . . . . . . . . . . . . . 864.5 Calibrated parameters for H2O gasification of soot, Equation 4.12. . . . . . . . . . . . 894.6 Calibrated parameters for CO2 gasification of soot, Equation 4.11. . . . . . . . . . . . 91

5.1 Reactions and reaction rates used in precursor cracking scheme (rates in kmolem3s , con-

centrations in kmolem3 , and activation energies in J

mole K ). . . . . . . . . . . . . . . . . . 1085.2 Surface growth mechanism where ki = AT n exp

(−ERT

)[7]. . . . . . . . . . . . . . . . 111

5.3 Proximate and ultimate analyses for the six coals tested [121]. . . . . . . . . . . . . . 1205.4 Precursor species fractions as described in Section 5.1.1 for the coal experiments. . . . 1225.5 Proximate and ultimate analyses for the biomass fuels tested. . . . . . . . . . . . . . . 1305.6 Precursor species fractions as described in Section 5.1.1 for the biomass experiments. . 131

6.1 Sooting potential model for biomass with calibrated parameters for Equations 6.4and 6.5. Tg and P are the gas temperature (K) and log-pressure (log(atm)) respec-tively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.2 Computational expense comparison between the detailed model of Chapter 5 and thesimplified model of Chapter 6 and found in the OFC simulation of Section 6.2.2. . . . 151

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LIST OF FIGURES

1.1 Effects of soot processes in the climate system. . . . . . . . . . . . . . . . . . . . . . 61.2 Basic outline of the soot formation process. . . . . . . . . . . . . . . . . . . . . . . . 81.3 Illustration of the HACA mechanism [57]. . . . . . . . . . . . . . . . . . . . . . . . . 101.4 Illustration of the mechanism for aromatic deposition onto the surface of a soot parti-

cle [57]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Agglomeration of soot particles in a hypothetical box at different temperatures. . . . . 121.6 Overview of the soot formation process as found in complex solid fuel systems. . . . . 201.7 Comparison between pyrene, a common PAH soot precursor in gaseous systems, and

a theoretical tar molecule as constructed based on elemental composition, molecularweight, and aromatic content [10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.8 Diagram of a proposed oxy-coal reactor. As proposed by Buhre et al. [22]. . . . . . . . 23

2.1 Example of a soot particle-size distribution as collected from a pre-mixed flame exper-iment [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2 A graphical representation of the sectional method as applied to a soot PSD where 8sections are applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3 A mono-dispersed distribution with η = 11.22. . . . . . . . . . . . . . . . . . . . . . . 332.4 A lognormal distribution with η = 2.86 and σ = 0.43. . . . . . . . . . . . . . . . . . . 342.5 A bimodal, lognormal/power law, distribution with the following parameters: α =

3.35, k = 5.14, η = 2.85, σ = 0.42. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.6 This is a model-informed prior of the ab joint probability space as informed by the

basic linear model used in gas-reactor example. . . . . . . . . . . . . . . . . . . . . . 442.7 This is a Gaussian-likelihood of the ab joint probability space as computed using a

data from Table 2.3 and Equation 2.15 in the gas-reactor example. . . . . . . . . . . . 472.8 This is a posterior of the ab joint probability space as computed using the prior of

Figure 2.6 and likelihood of Figure 2.7 in the gas-reactor example. . . . . . . . . . . . 492.9 Marginalized PDFs for the a and b parameters as taken from the posterior in Figure 2.8. 492.10 Linear mode, Equation 2.15, fitted to data from Table 2.3 using Bayesian inference. . . 50

3.1 Diagram of the downward burner and draft portion of the oxy-fuel combustor at theUniversity of Utah. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.2 Results of the SUFCO coal simulations [158]. From left to right the figures depict: (a)temperature (max = 2500 K, min = 300 K), (b) carrier gas mixture fraction (max = 1,min = 0), (c) coal off-gas mixture fraction (max = 0.3, min = 0), and (d) CO molefraction (max = 0.7, min = 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.3 Results of the SUFCO coal simulations, showing the number densities of (a) 20 µm(max = 5E10, min = 0), (b) medium (max = 1E9, min = 0), and (c) large (max = 2.5E7,min = 1.0E2) sized particles within the reactor. . . . . . . . . . . . . . . . . . . . . . 63

3.4 Results of the SUFCO coal simulations, showing (a) the tar mass fraction (max = 0.03,min = 0), (b) soot particle number (max = 1E19, min = 1E12), and (c) soot volumefraction (max = 6 ppmv, min = 0 ppmv). . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.5 Results of the SUFCO coal simulations, showing (a) the CO2 mole fraction (max = 1,min = 0) and (b) O2 mole fraction (max = 1, min = 0). . . . . . . . . . . . . . . . . . 65

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3.6 Results of the SUFCO coal simulations with soot gasification, showing (a) the sootparticle number (max = 1E5, min = 5E16) and (b) soot volume fraction (max = 6ppmv, min = 0 ppmv). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.7 Results of the Skyline coal simulations [185], showing (a) the temperature (max = 2500 K,min = 300), (b) small particle number density (max = 4.4E10, min = 1.0E6, logarithmicscaling), and (c) large particle number density (max = 6.0E8, min = 1.0E1, logarithmicscaling). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.8 Results of the Skyline coal simulations, showing (a) the tar mole fraction (max = 0.001,min = 0), (b) soot particle number (max = 1E16, min = 0), and (c) soot volume fraction(max = 0.24 ppmv, min = 0 ppmv). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.9 Line of sight measurements of the soot volume fraction across the flame. Solid linesrepresent optical measurements while dotted line represent simulation results. Blue isat the root of the flame, green at the middle of the flame, and red is at the tip of the flame. 68

4.1 PDFs of each of the oxidation parameters in Equation 4.3. Contours indicate jointPDFs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.2 Comparison of predicted rates of soot oxidation by calibrated, with parameters in Table4.4, model and those rates collected from the literature. Those experiments that aremeasured only oxidation by O2, such as TGA, are filled symbols (R2 = 0.75). . . . . . 87

4.3 Comparison of oxidation rates as predicted by the NSC oxidation model [140] andthose rates collected from the literature (R2 = 0.65). . . . . . . . . . . . . . . . . . . 88

4.4 Comparison of oxidation rates as predicted by the NSC oxidation model combinedwith Neoh et al.[141] calculated collision efficiency for OH and those rates collectedfrom the literature (R2 = 0.71). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.5 PDFs of each of the H2O gasification parameters in Equation 4.12. . . . . . . . . . . 904.6 Comparison of predicted rates of soot gasification via H2O by calibrated model, pa-

rameters in Table 4.5, and those rates collected from the literature (R2 = 0.87 minusNeoh Data). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.7 PDFs of each of the CO2 gasification parameters in Equation 4.11. . . . . . . . . . . 924.8 Comparison of predicted rates of soot gasification via CO2 by calibrated model, pa-

rameters from Table 4.6, and those rates collected from the literature (R2 = 0.62). . . . 924.9 Comparison of predicted rates of soot gasification via CO2 by individually calibrated

models and those rates collected from the literature. . . . . . . . . . . . . . . . . . . 934.10 Model-informed priors for the CO2 gasification model. Derived with mode values at

ACO2=3.06E-17 and ECO2=5.56E3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.11 PDFs of each of the oxidation parameters in Equation 4.3 derived using the model-

informed priors of Figure 4.10. Contours indicate joint PDFs. . . . . . . . . . . . . . 954.12 Model-informed priors for the oxidation model. Derived with mode values at AO2=7.98E-

1, EO2=1.77E5, and AOH=1.89E-3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.13 PDFs of each of the oxidation parameters in Equation 4.3 derived using the model-

informed priors of Figure 4.12. Contours indicate joint PDFs. . . . . . . . . . . . . . 984.14 PDF of the calculated gasification rate in Higgins experiment where the flame data was

at 1200 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.15 Comparison of the model predicted oxidation rate with confidence bounds versus the

measured rate in Higgins’s experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . 100

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5.1 Basic outline of PAH thermal cracking. . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.2 Result of numerical study considering the evolution of precursors from Pittsburgh #8

coal at 1800 K as found in Section 5.2.1. Results were 0.004, 0.283, 0.503, and 0.210for Xphe, Xnapth, Xtol , and Xben respectively. . . . . . . . . . . . . . . . . . . . . . . . . 110

5.3 Diagram of the complete HACA mechanism illustrating growth of a benzene ring. . . 1115.4 Diagram of flat flame burner used by Ma [120]. Reproduced with permission. . . . . . 1205.5 Simulation results, continuous dotted lines, are compared to reported experimental

data, individual marks. Results are soot mass yield as a percent of original fuel mass(dry and ash free). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.6 Average particle collision diameter across the flame portion of the Pittsburgh # 8 coalexperiments as predicted by the simulation. . . . . . . . . . . . . . . . . . . . . . . . 125

5.7 Particle shape factor across the flame portion of the Utah Hiawatha coal experiments. . 1255.8 Soot mass yield with an additional ‘maximum sooting potential’ solid line representing

the mass yield of tars released into the system. . . . . . . . . . . . . . . . . . . . . . . 1265.9 Soot mass yield deposited on the soot filters of the coal-flame collection system. . . . . 1285.10 Results of biomass-derived soot simulations compared to reported experimental data.

Results are displayed as a mass percent of the parent fuel (dry and ash free). . . . . . . 1315.11 Blue bars represent experimentally measured particle-size distributions and red lines

represent simulation resolved moments fitted to a log-normal distribution. . . . . . . . 132

6.1 Comparison between results given by CPDbio versus the proposed sooting potentialempirical model. Different colors represent different biomass components: cellulose(blue), hemicellulose softwood/hardwood (green/yellow), and lignin softwood/hardwood(magenta/red). The left plot shows the comparison for tar mass yield (R2=0.811) andthe right plot shows the comparison for tar mass size (R2=0.856). . . . . . . . . . . . . 139

6.2 Comparison between results given by CPD versus the proposed sooting potential em-pirical model. The left plot shows the comparison for tar mass yield (R2=0.794) andthe right plot shows the comparison for tar mass size (R2=0.854). . . . . . . . . . . . . 140

6.3 Variation of time-averaged precursor ratios from numerical study as temperature (left)and initial number density (right) are varied. . . . . . . . . . . . . . . . . . . . . . . . 142

6.4 Comparison between empirical model and numerical study for predicting precursor-type fractions. The black straight 45°represents a perfect agreement between the two(R2=0.919). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.5 Particle number density and soot volume fraction simulation results from the coal flat-flame burner with entrained oxygen, comparing simplified model against the detailedmodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.6 Results of the comparative LES coal simulations. From left to right the figures depict:Soot volume fraction predicted by the detailed soot model (max (red) = 3.5 ppmv,min (blue) = 0 ppmv), soot volume fraction predicted by the simplified soot model(max = 3.5 ppmv, min = 0 ppmv), soot particle number density from detailed model(max = 1E21 #/m3, min = 0 #/m3), and soot particle number density from simplifiedmodel (max = 1E21 #/m3, min = 0 #/m3). . . . . . . . . . . . . . . . . . . . . . . . . 150

A.1 Visual evidence of iteration reorganization. . . . . . . . . . . . . . . . . . . . . . . . 182

x

NOMENCLATURE

Ai Pre-exponential factor for reaction iCa Collision frequency constantCmin # Number of carbon atoms per incipient soot particle〈d〉 Shape factordi m Diameter of species iE kW Energy outputEi

Jkmol Activation energy for reaction i

fv,S ppmv Soot volume fractionF kg

hr Fuel inputHC Hydrogen to carbon atomic ratioI Conditional factors of an eventki Rate constant for reaction iks

kgm2s Reaction rate per unit particle surface area

kBm2kgs2K Boltzmann constant

mi kg Mass of species iMi

kgi

m3 Particle size distribution generalized moment iNa

#kmol Avogadro’s number

NS#kg Number of soot particles per unit volume of gas

Ni#

m3 Number densidty of particles of size iOC Oxygen to carbon atomic ratioPi Pa Partial pressure of species iPr unitless Prandtl numberR J

kmolK Ideal gas constantr2 Residual errorR2 Coefficient of determinationSi m2 Surface areaSAv,i

m2

m3 Surface area per unit volume of iSc unitless Schmidt numberT K TemperatureTg K Gas temperature~u m

s Gas velocityV % Mass percent of volatile matterwi

#m3 Weight of QMoM

xi Vector of parameters iXi Mole fraction of species iyi Data from experiment iYi Mass fraction of species iα Fitted parameterβ

m3

s Collision frequencyχi

#m2 Number of sites per unit surface area

∆ kg Change of mass involved with a single reaction∆Hreac

Jkg Heat of pyrolysis

xi

∆HvapJkg Heat of vaporization

ε Van der Waals Enhancement Factorη m Assumed particle sizeγ Calibrated model outputλi Mean free path of species iµ

kgms Viscosity

µi, j kg Harmonic mean massπ 3.14159ρi

kgm3 Density of species i

σ Standard deviationFunctions and indicatorsδ () indicates a Dirac delta functionf () indications a generic function with dependencies inside the paranthesisg() indications a generic function with dependencies inside the paranthesisLi() indicates a Lagrangian interpolationP() or p() indicates probability of event within paranthesis∩ indicates the intersection of two eventsx|y indicates conditionality[i] indications a concentration of species i

xii

CHAPTER 1. INTRODUCTION AND OVERVIEW

Motivation and funding for the work in this dissertation come from two sources: the

Carbon-Capture Multidisciplinary Simulation Center (CCMSC) at the University of Utah and the

Earth and Environmental Sciences (EES) Division at Los Alamos National Laboratory. CCMSC

was investigating full-scale boiler simulations of oxy-coal power plants [177] and EES division

was performing wildland fire simulations [33]. These two areas of research have a common thread:

soot formation mechanisms from solid complex fuels.

This introductory chapter will present a quick review of soot formation mechanisms and

modeling approaches. Characteristics of the soot phenomena will be presented and motivations of

why an understanding of these mechanics are important will be included. There will then be a quick

discussion of oxy-fuel technologies and wildland fire sciences with emphasis on soot formation in

these environments.

1.1 Soot

Soot is a collection of carbonaceous particles found in nearly all combustion environments,

from the burning wax candle to a diesel engine, and is a result of incomplete combustion. The

yellow color of a flame is usually due to the incandescence of soot particles [108], and is evidence

of a ‘sooting’ flame. Non-sooting flames, such as a pre-mixed flame where fuel and oxidizer are

mixed together before ignition, will not have this yellow spectra, unless Na is burned, and have

very different flame characteristics due to the lack of soot.

Soot particles range widely in size. Observations have recorded particles in sizes from

<0.005 µm to >1500 µm in collision diameter [97, 69]. At the molecular level, these particles

are primarily a carbon structure loosely representative of graphite, that is a honeycomb of aromatic

rings in a plane; however, soot particles contain enough amorphous regions to significantly change

the overall structure such that soot characteristics are distinct from graphite [193]. Particle struc-

1

ture varies with source but some general characteristics are consistent. At a microscopic level,

soot is formed from small, roughly spherical, primary particles of a critical size. Critical size is

system dependent. These spherical primary particles aggregate together forming broader chain-

like structures, referred to as aggregates, but under a electron microscope the individual primary

particles are still distinct [38, 120, 190]. Aggregate shape and size may vary between fuel-types

and systems.

While predominately carbon, an elementary analysis of soot particles will show that other

elements are also present. Unsurprisingly, hydrogen is attached to the carbon-skeleton throughout

the particle. Oxygen is also commonly found in soot samples, particularly soot from solid-fuel

systems, which will be discussed later, and those samples collected post-combustion, where the

surface of the particle has been partially oxidized and contains a large variety of oxygen-based

functional groups [193]. As soot is a direct product of fuels, any inorganics found in the fuel will

usually be found in the soot as well, but in lesser amounts. Experimentalists have observed soot

particles with significant amounts of nitrogen, sulfur, phosphorus, potassium, and silica, along

with trace amounts of calcium, chlorine, sodium, zinc, and barium [190, 193]. It is not known how

these elements, especially the metals, are attached to the carbon skeleton, but it is suspected that

many are actually chemically bonded and not just ash contaminates loosely attached to the soot

samples [190, 49].

The formation of soot within a combustion system impacts internal flame characteristics,

and, if emitted from the flame, the particles can have heavy impacts on the environment and human

health.

1.1.1 Flame Impacts

An important attribute of combustion processes is the thermal radiation released by the

flame to the surrounding environment. Soot particles are known to have both high adsorption

capabilities and high emissivity [9] leading to heavy impacts on thermal radiation.

Unlike the surrounding gases, which emit photons only in discrete energy bands, soot par-

ticles strongly emit photons across a continuous energy spectrum. This is possible because of the

amorphous and non-homogeneous nature of soot which allows for a continuous energy spectrum

in the particle’s inter-molecular bonding, rotations, and vibrations. As a result, while gases may be

2

powerful photon adsorbers/emitters across a small portion of the energy spectrum, soot particles

may adsorb/emit photons across the entire energy spectrum allowing for penetration of emitted

photons through the surrounding gas at wavelengths not observed by combustion without soot. In

comparison to other particle species (fuel, char, etc), soot primary particles are very small but with

a high number density and tend to be the same temperature as the surrounding gas. In a flame,

this high temperature, high particle number density, and high surface area to volume ratio allows

greater emissivity than other larger and cooler particles [195]. In heavily sooting flames, the ra-

diative emissions from soot particles can account for upwards of 30% of the flame’s total thermal

radiation [51].

Models developed to predict radiative heat transfer due to soot usually modify a general

gas absorption coefficient based on the amount of soot present in the flame [86]. Unfortunately,

the impact of soot particles on this coefficient is broad and depends on the nature of the particle

surface. It has been found that as aggregates form, morphology and surface consistency can have

significant impacts on the radiative heat transfer [9] indicating that in complex fuel systems the

simplification of basing alterations in the gas absorption coefficient on the soot volume fraction

could prove to be inadequate. Such may be the case in the design of power-generating boilers,

where the heat flux to boiler walls is one of the most critical quantities of interest and variations in

particle radiative heat transfer directly alters that heat flux [92].

Increases in thermal radiation from soot lead to a greater heat loss in local areas where

soot occurs. These heat loss values lead to lower local temperatures. Although the total effect of

soot on local temperatures is difficult to measure, many sophisticated models have been developed

which couple soot and radiative heat loss. It has been observed that combustion simulations which

neglect soot formation tend to be much hotter than experimental observations. In some cases,

the differences in temperature between simulations which accounted for soot and those that didn’t

can be quite severe; Xu et al. observed differences as great as 236 C° [201] in regions where

soot concentrations were the highest in their simulations. This lowering of local temperatures

alters local gas chemistry as the balance of gas-phase chemical mechanisms are highly temperature

dependent [35, 166]. In particular, the concentration of radical species would be expected to lessen.

Just as the presence of soot particles affects local temperatures and chemistry, these in turn affect

3

soot formation processes [119]. This interplay between soot formation and local heat loss increases

the difficulty of successfully predicting soot particle quantities [6].

Overall flame chemistry is impacted by soot particles in other ways as well. As soot par-

ticles are primarily carbon, they act as a carbon sink in local chemistry profiles until the particles

reach the stoichiometric point, or flame front, where they are oxidized and release that carbon into

the surrounding gas. In the case of complex fuels such as wood or coal, soot particles can poten-

tially act as a nitrogen sink [49], greatly altering fuel-NOx emissions from these fuels [133, 146].

A proper accounting of soot formation is important for detailed simulations of any combustion

system.

1.1.2 Health Impacts

If soot particles escape the flaming portion of a combustion system, they are released into

the surrounding environment as an aerosol. These aerosols can be transported over great distances

spreading the effect of the combustion system over a large footprint. Of greatest concern in this

footprint are the human health effects generated by these aerosols. The health effects of soot parti-

cles is an area of increased interest and intense research. The full-implications of soot particles on

human health is not known, and it is often difficult to separate the effects of combustion-generated

aerosols, like soot, and ambient environmental aerosols, like dust [114]. However, given the char-

acteristics of soot particles and what is known of their evolution, at least some health impacts of

soot have been identified and investigated by researchers [97, 114, 26].

The largest concern for soot aerosols involves the interaction between these particles and

the human respiratory system. Epidemiologists typically will characterize aerosol particles by their

size [192], whether the particles are normal (>10 µm), fine (2.5 µm< 10 µm), or ultrafine (<2.5

µm), as the different sizes have varied effect on the respiratory system.

Exposure to normal particles has minimal effect on the respiratory system since the parti-

cles are typically filtered by nasal follicles and cause enough immediate irritation to be expelled

quickly through coughing or sneezing. Fine particles have tendencies to accumulate in upper respi-

ratory passages of the throat and nasal. This accumulation can cause problems, such as sore throat,

nasal infections, etc., but the effects tend to be short-term [98]. In severe cases, especially relevant

to firefighter safety, inhalation of large quantities of these normal and fine particles will saturate

4

the upper respiratory system and penetrate into the lungs causing blockage to the bronchus and

other lung airways potentially causing asphyxiation. Even when asphyxiation does not occur, this

inhalation of large quantities of particles often causes thermal and chemical burns throughout the

respiratory system as well as in ‘soft’ areas (eyes, ears, armpits, palms, etc) on the body’s exterior.

The ultrafine particles cause more long-term problems as they tend to penetrate directly to

the lungs and lodge in the alveoli of the lungs. As these particles undergo many transformations in

the atmosphere, discussed in the following section, they become carriers for organic compounds

which cause significant chemical damage to the surrounding lung cells [83, 106]. This continual

chemical and mechanical irritation to lung-cells leads to increased risk of asthma, bronchitis, and

other respiratory related diseased. Long-term exposure to combustion generated aerosols is known

to be carcinogenic, leading to increased cases of lung-cancer, and mutagenic, causing surrounding

lung-cell to mutate and hindering their functional capabilities [137, 114, 98, 26].

In addition to respiratory problems caused by the inhalation of soot particles, there is an

increasing body of research concerning the effects of aerosols on the circulatory system. It is

suspected that particles residing in the lung alveoli will break down to smaller polycyclic aromatic

hydrocarbons (PAHs) which dissolve in lipids and are absorbed through the lung walls directly into

the bloodstream [137]. Once in the blood stream, these PAHs are known to be carcinogenic, caus-

ing increased risk of heart diseases and cardiovascular cancers [157]. When inorganics, especially

metals, are attached to the particle carbon skeleton, they also can be carried into the bloodstream

where, even in trace amounts, a whole new set of medical problems may arise including blood

poisoning, white cell/red cell mutations, and others [114].

While not all health effects of combustion-generated aerosols have been discovered or re-

searched, it has become evident over the last several decades that the impacts of released soot

aerosols can be both diverse and long-lasting.

1.1.3 Environmental Impacts

In addition to human health effects, soot production also can have severe and negative im-

pacts on the environment. The impacts of soot aerosols on the environment have been summarized

in the flowchart of Figure 1.1. While this is not a comprehensive list of all potential impacts of

soot aerosols, it does include a number of the highest concerns among environmentalists.

5

EmissionSourceSourcesvaryextensively,canbefromopenfires(wildoragricultural),transportation,mining,residential,powergeneration,etc.

AtmosphericTransformationsParticlesmixandreactwithco-emittedgasesandaerosols(organicacids,sulfates,dust,etc.)Particleagglomeration,condensation,oxidation,anddilutionalloccur

Hydro-atmosphericInterfaceAerosolparticlescanactasanucleationsiteforwatermoleculestocondense.

AtmosphericRadiationParticlesabsorbphotonsintheatmospherecoolingthesurfacebutwarmingtheatmosphere

Snow/RainParticlesaredepositedonthesurfacebynucleationinrain/snow.Particleshaveusuallybeenacidifiedintheatmospherecreatingalightacidrain.

SurfaceEffectsDepositedparticlesincreasesurfaceadsorption.Whereparticleshavesettledonsnoworice,impactscanbesevere

Figure 1.1: Effects of soot processes in the climate system.

Not all soot particles are the same. The source of the soot production, both in terms of the

parent-fuel and the system in which the formation took place, heavily influences the environmental

impact of the soot produced. As an example, compare the soot formed in a natural-gas reactor

against that produced in a pine forest wild fire. In the first instance, natural-gas tends to be a

‘clean’ fuel, implying that the fuel has a tendency to completely combust emitting very few soot

particles, especially in a reactor designed for that purpose. Those particles that are emitted from

the natural gas flame will be almost completely carbon in aromatic rings. On the other hand, a

pine forest wild fire will emit a much larger number of soot particles as the ‘sooting potential’

of this fuel is much higher and the irregularity of the system will cause more opportunities for

particles to be emitted to the atmosphere. In addition, the particles themselves will likely have

more inorganics embedded in the structure of the particles along with more aliphatic carbon. The

inorganics and aliphatic carbon will cause particles emitted from the pine forest fire to be more

reactive in the atmosphere than the natural-gas particles causing further differences between the

particles and their impact on the environment [89].

Once particles are emitted from the combustion system, they undergo a variety of reac-

tions and transformations in the atmosphere. These reactions include continual particle-particle

6

agglomeration as particles collide and stick. The agglomeration of particles is more likely in hotter

environments, such as within a flame envelope, as the collision frequency increases, but does not

stop at lower temperatures and will continue even at very low atmospheric temperatures as long

as the particle number density allows for collisions to occur [54, 167]. Particles also react with

co-emitted gases and aerosols like ash or dust from the combustion system, this alters the surface

of the particle. Both organic and inorganic functional groups are formed as a result of these re-

actions [196]. In the upper stratosphere, partial oxidation occurs readily as particles encounter

increasing concentrations of OH and O3 [12, 83].

Within the flame, soot particles have a continuous radiative absorption band, and the al-

teration in the particles’ surface, due to the many reactions which have occurred, increases the

particles’ emissitivity [78, 89, 196]. Once lofted into the atmosphere, these particles will absorb

and reflect increasing amounts of sunlight from the earth, altering the planet’s albedo, which is a

measure of diffusive reflection of solar radiation, warming the atmosphere but cooling the planet

surface. It is this principle which is the basis of the nuclear winter; should enough particles be

lofted into the atmosphere at the same time, they would reflect/absorb enough light as to suffi-

ciently cool the surface to a perpetual winter.

While suspended in the upper stratosphere the presence of soot particles has the potential

to alter the hydro-atmospheric interface. Snow and rain occur as droplets of water form from

condensed water vapor. The presence of aerosols, such as soot, promotes the condensation of

this water vapor by providing nucleation sites on which ice forms readily [72]. As ice begins to

form around the particles, their increased temperature, a result of increased emissitivity, causes

warmer precipitation with heavier droplets [149]. During the atmospheric reactions, particularly

the reactions with sulfides, particles usually become slightly acidic, thus creating a light acid rain

when mixed into the hydro-atmospheric interface [167].

Once deposited on the surface, either through natural settling or deposition through precipi-

tation, the radiative emissitivity of soot particles continues to impact the local environment. Where

particles settle on dark surfaces, such as soil-rich surfaces or rock out-croppings, the emissitivity

effects are typically negligible and may even have positive effects as they deliver small amounts

of minerals and nutrients to the soil, that is after we have considered any light acid consequences.

Where particles deposit on light surfaces, snow or ice, the increased emissitivity of a darkened sur-

7

SootPrecursors

Nucleation Coagulation

Soot Nuclei Primary Soot Particles

Aggregation

Soot Aggregates

SootPrecursorsLight Gases

Light Gases

OxidationOxidation

OxidationOxidation

Surface Growth Surface Growth

Surface Growth

Figure 1.2: Basic outline of the soot formation process.

face will significantly warm surfaces [148, 77]. The effect of this warming on light surfaces tends

to be much longer lasting that any surface cooling effects the aerosol had in the atmosphere, to

the extent that it is estimated that soot emissions have the second greatest global climate warming

impact, after the cumulative impact of all CO2 emissions [142, 182].

1.2 Soot Formation

Soot formation generally refers to any mechanism that governs the evolution of soot in

a combustion environment before particle emission to the surrounding environment. The term

formation is a bit of a misnomer as particle consumption mechanisms are often lumped into the

‘formation’ of soot. In this section, a review of soot formation mechanisms for gaseous fuels is

presented, followed by a discussion of the difficulties associated with modeling soot formation,

and finally there will be a section discussing soot formation mechanisms from solid fuels.

1.2.1 Formation in Gaseous Fuels

The process of soot formation from gaseous and liquid fuels has been well studied and there

are sophisticated models describing theses mechanisms [17, 69, 183, 97, 82]. Most liquid fuels

may be characterized with gaseous fuels because with respect to soot formation processes they are

8

the same. The basic steps of soot formation are outlined in Figure 1.2 and include: nucleation,

coagulation, aggregation, and surface reactions.

Particle nucleation is the first step and often the rate determining step in soot formation

for gaseous fuels. The process of nucleation begins with the production of PAHs from the gas

phase in fuel-rich regions. These PAHs are a variety of species with multiple aromatic rings

bonded together in different configurations. Naphthalene would be the smallest PAH with only

two aromatic rings; other examples include anthracene, phenanthrene, triphenylene, pyrene, and

coronene. These PAHs are nonpolar and completely made up of carbon and hydrogen with delo-

calized electrons shared among the aromatic rings. Developing gas-phase chemical mechanisms

which accurately predict the species and concentrations of PAH is an area of detailed research

with no common consensus from the combustion community of the ‘best’ approach [24], but every

proposed PAH mechanism begins with the formation of a single aromatic ring.

The exact mechanism of the initial aromatic ring, which is essential to all proposed nucle-

ation models, is a matter of considerable debate [24]. Various mechanisms have been proposed

via reactions of C2H2, C3H2, C3H3, nC4H3, C4H4, C4H5 and nC4H5 [116, 57, 184]. Each of

these mechanisms involves reactions with highly reactive species (free radical, unstable ring, etc.)

that require the high-temperature environment of combustion to even be present in the gas phase.

Once an aromatic ring is produced, those rings will continue to grow according to various

gas phase mechanisms forming multi-ring aromatics or PAHs. As these molecules grow, they will

transition from a large molecule to a particle. Distinctions between a large PAH molecule and

the incipient soot particle are fuzzy. As a general rule, many researchers think of the incipient

soot particle as a diamer of pyrene molecules [135], establishing this line at a molecular size of

roughly 400 grams per mole, anything smaller is PAH while anything larger is soot. Nucleation,

the formation of an incipient soot particle, may occur either through the gradual growth of PAHs

to particle size or, more commonly, through the coalescence of two PAH molecules as they collide

and stick [128, 56]. When concentrations of PAH are high, nucleation is dominated by the coa-

lescence of PAH molecules and the eventual growth of PAH molecules, through various chemical

mechanisms, becomes negligible.

Once the incipient soot particle is formed, overall soot mass will continue to evolve in a

system through continued surface growth. Surface growth involves chemical reactions between a

9

Figure 1.3: Illustration of the HACA mechanism [57].

particle surface and the surrounding gas-phase [129]. While there may exist thousands of possible

reactions at this interface, two have been identified by researchers as critical to particle growth.

The first is the hydrogen-abstraction-carbon-addition (HACA) mechanism which begins with the

radicalizing of the particle surface by means of abstracting a hydrogen atom from its surface as

displayed at the start of Figure 1.3.

The radicalized particle surface then reacts with acetylene in the surrounding gas to form an

additional ring attached to the original surface. This overall mechanism is a propagation reaction,

meaning that the end product still contains a radical so that the reaction may continue indefinitely as

long as concentrations of acetylene are available in the local environment and no other termination

reaction occurs.

The second growth reaction of interest to researchers is the deposition of aromatics, typi-

cally PAHs, onto the surface of the particle, as seen in Figure 1.4.

In concept, proposed chemical mechanisms for aromatic deposition are similar to HACA.

It involves the radicalization of the particle surface, but instead of reacting with acetylene in the

surrounding gas-phase, reactions occur with other aromatics such as PAH. In many systems, this

source of surface growth is limited on a global scale as the lifetime of PAH is short compared to that

of soot; as a result soot and PAH are typically not found in the same region of the flame as PAH is

mostly consumed in the initial formation of soot. However, in regions of overlap between PAH and

soot, this mechanism becomes very influential. In more complex systems, such as multi-injection

10

Figure 1.4: Illustration of the mechanism for aromatic deposition onto the surface of a soot parti-cle [57].

nozzle systems, PAH and soot profiles overlap more and this mechanism becomes increasingly

globally important [40].

These two surface growth mechanisms account for most of the soot mass growth in com-

bustion systems, and models of varying sophistication have been developed to capture their influ-

ence [129, 116].

In addition to surface growth reactions, soot particles agglomerate as particles collide and

stick together. Soot agglomeration has two extremes: particle coagulation and aggregation. Co-

agulation occurs when two small particles collide and stick together but malleability of the small

particles along with the continual growth pattern forms a single larger spherical particle. After

a coagulation event individual molecules from the original colliding particles are indistinguish-

able [167]. In the case of small spherical particles we can think of coagulation as two small

spheres coming together, blending, and forming a larger spherical particle. Aggregation, on the

other hand, occurs when two particles collide and stick but retain the original structure of two in-

dividual particles now stuck together. In a combustion environment, true particle agglomeration is

neither perfect coagulation nor perfect aggregation, but rather somewhere in between with smaller

particles tending towards coagulation and larger particles tending towards aggregation behavior.

As a result of this shift away from coagulation towards aggregation as particles grow larger, chain

aggregates form.

Rates of particle agglomeration are founded on the collision and sticking of particles to-

gether. Frequencies of collisions between small particles is well understood with the Kinetic The-

ory of Gases; however, as particles grow larger they go from a free-molecular flow regime to a

11

Figure 1.5: Agglomeration of soot particles in a hypothetical box at different temperatures.

continuous one which alters the frequency of collision [54, 18]. Overall rates of particle agglom-

eration are dependent on a number of factors (temperature, pressure, etc.); however, the largest

factor tends to be the number density of particles. Figure 1.5 shows the result of a hypothetical

experiment where particles are injected into a box with fixed volume at different temperatures

and allowed to agglomerate together. Time is shown on the x-axis and number of particles on

the y-axis. Rates of agglomeration were computed using the relation developed by Seinfeld and

Pandis [167], a physics based model where the frequency of particle collision is computed as they

transition between a free-molecular regime to a continuous regime. As can be seen in the figure,

rates of agglomeration are initially high as the number of particles is also high and rapidly slows

down as the total number decreases. Note, the rate of particle agglomeration has a dependency

order on temperature which varies from 1/2 to 1 (larger particles having a weaker dependency than

smaller particles) while the dependency order on particle number density is consistently 2. This

indicates that temperature plays a secondary role to particle number density in determining overall

rates of agglomeration.

Soot particles will continue to grow as they are transported through the fuel-rich region of

any combustion environment, but once they enter an oxidizer-rich region, temperature-permitting,

they will be consumed through reactions between particle surfaces and the surrounding gas. This

transition between a fuel-rich region to oxidizer-rich region usually begins at the flame-front. These

12

consumption reactions can be categorized into two types of reactions: oxidation reactions and

gasification reactions.

Soot oxidation reactions are an area with a long history of research because in most systems

they account for the bulk of consumption reactions. Oxidation reactions occur when an oxidizing

agent, generally O2, OH, or O, collide with a particle surface where a redox reaction occurs pulling

carbon or hydrogen off the surface of the particle in a highly exothermic reaction. Products of

oxidizing reactions usually include: CO, CO2, and H2O.

Some of the first investigators of soot oxidation assumed that soot was consumed solely via

the reaction of an O2 molecule with the particle surface [109], and oxidation models were devel-

oped based on O2 concentrations. It was quickly determined that the presence of OH molecules

greatly influenced rates of soot consumption and hence was included in oxidation models [183]. In

more recent studies, emphasis has been placed on the influence of O radicals in flames [113], par-

ticularly in high temperature flames where the O radical concentration is relatively high [194, 55].

However, due to the coexistence and highly correlated concentrations of O with O2 and/or OH, it

is difficult to experimentally differentiate between oxidation via O versus oxidation by O2 and OH

without molecular modeling. As a result, many models do not explicitly consider oxidation by O,

rather, this effect is implicit in the rates used for O2 or OH.

Gasification reactions, on the other hand, are not as well studied or understood as oxidation

reactions. Gasification reactions occur when a high energy molecule collides with the particle

surface and transfers enough energy to break a bond within the particle structure [22]. As a result,

a broken fragment is released to the surrounding gas phase from the particle surface while the

original reactant gas may or may not be chemically altered [130]. The reaction itself tends to be

energetically neutral to slightly endothermic and the products of gasification tend to vary much

more widely than oxidation and can include: light gases (H2, CO, etc.), small hydrocarbons,

alcohols, and carbonyls [130].

In traditional combustion environments, soot consumption is dominated by oxidation kinet-

ics, while gasification occurs in small enough quantities to be considered negligible [51]; however,

this is not always the case. In the past, a fair amount of research explored using soot gasification

as a means of cleaning soot particles from engine-exhaust catalysts as the oxidation could be too

exothermic and damage the catalyst itself [144, 203]. Unsurprisingly, in recent research it has been

13

found that soot gasification can be significant in gasification systems, where soot may still form

within the unit, but the lack of oxidizing agents leads to gasification dominating particle consump-

tion [152]. In addition, it has been found in oxy-fuel environments the increased concentrations

of CO2 and H2O leads to increased rates of gasification, enough so that neglect of gasification

models can lead to significant uncertainty in predicting particle concentrations [1].

Current research on soot consumption has placed large emphasis on the evolution of par-

ticle surface reactivity. Researchers have developed mechanisms reflecting the many elementary

chemical reactions [65, 67] and mechanical changes [173] occurring at the particle surface during

consumption. There is also ongoing research investigating correlations between particle surface

reactivity and the particle inception environment [153, 190, 103]. These investigations into the

changing particle surface reactivity bear great promises into better fundamental understanding of

the mechanisms and processes of particle oxidation and gasification.

As soot particles evolve through a system, they may fragment into smaller particles. This

fragmentation process can be a mechanical fragmentation as the soot aggregate experiences stresses

through collisions with other particles or walls of a system [156]. But more commonly, this frag-

mentation is chemical. As an aggregate undergoes oxidation, or gasification, the surface of the

particle is consumed, but in sections where the aggregate particle thickness is small this surface

consumption can lead to fragmentation of the aggregate [206, 173, 136]. While the concept of

fragmentation is quiet simple, the breaking of a particle into multiple pieces, fragmentation has

proven surprisingly difficult to predict and model [68, 66, 43, 80, 162, 174].

1.2.2 Challenges to Soot Formation

The modeling of soot formation and evolution in combustion system pose large challenges

to researchers. Soot particles make an aerosol, a cloud of particles, surrounded by a highly reactive

gas. Characteristics of this aerosol, such as particle size distribution or particle concentration, are

very difficult to experimentally measure, particularly within the combustion environment, leaving

much uncertainty in available data [169]. With such large uncertainties, validation of formation

theories is difficult. This coupled with limited ability/understanding to effectually portray the

interactions between soot particles and their surroundings makes the field of soot formation both

interesting and unfinished.

14

Experiments

As with any model, the creation of soot formation models relies on data collected from

real-world experimental observations. Herein lies the greatest challenge to accurately model soot

formation. Where there is accurate experimental data to draw from, an accurate model is possible.

Without it, modelers lack both the direction and anchorage that good experiments can provide in

the theoretical world. There are two major ways in which soot particles are measured: the first

involves a physical collection probe and the second involves an optical measurement.

Physical collection probes vary greatly in design [2, 130, 134, 161, 38, 121], but all gen-

erally attempt to collect physical samples of particles either through a vacuum probe or through

particle deposition on a surface. While the physical collection of particles has given invaluable

data and insight into the nature of these particles, every contrived collection system also has its

deficiencies. The most common collection systems involve some sort of vacuum system to suck

up particles which are then separated and cooled. As stated in the previous section, particles will

continue to agglomerate and evolve, even at low temperatures [167]; as a result, once particles

are actually analyzed, they’re not the same as when they were first captured from the flame. The

particle collection system changed the nature of the particle forming environment and, hence, the

measured characteristics of the particle distribution. Most of these systems have a tendency to im-

pose an artificial particle concentration by forcing particles through some sort of tube or hose. This

actually increases the rate of agglomeration and as a result particles are significantly larger with a

different morphology than when first collected from the reaction location. In addition, the very act

of inserting a probe into a combustion environment will alter local temperatures, chemistry pro-

files, and flow dynamics of that environment thus also altering the soot formation process [2]. This

is not to say that the data collected from a physical probe is worthless. Indeed it is very valuable

data, but must be used with caution, understanding some of the effects that the physical probe have

on the measurements.

Optically collected data tend to be more reliable than physical collection data but also must

be used with caution. A variety of techniques have been developed for optically measuring soot

particles in a flame [81, 108, 169, 185, 189] and two of the most common are the application of

Mie scattering and two-color transmittance measurements.

15

The Mie scattering technique involves the use of a laser across the flame domain with wave-

lengths much larger than that of the size of soot particles, as the light hits the particles it is scattered

in a pattern and intensity as predicted by the Mie solution of Maxwell’s equations of electromag-

netism. By analyzing the scatter of light one may predict the size of a homogeneous solid sphere

and the concentration of these spheres across the width of the laser beam [81]. The technique is

well validated with a large array of circumstances for predicting particle size and concentration but

has a couple of problems. In early stages of a flame, where primary particles are newly created

and have yet to aggregate, Mie scattering works fairly well as the particles are roughly spherical

and more homogeneous but it is hard to quantify uncertainty in the measurements. At latter stages,

where particles have begun to aggregate, these particles are neither homogeneous nor spherical

and Mie scattering techniques do much worse. In addition, where solid fuel particles are used as a

fuel source, other particles, fuel or char, contribute to light scattering as well giving an additional

complexity and uncertainty to measurements [81].

Two-color transmittance measurements use two different colored lasers, typically a red and

green, and measure the transmittance, or ratio of light which passed through the flame without

being scattered or absorbed. This technique requires substantial calibration, but by comparing the

ratio of transmittance between the two colors one may deduce the amount of soot which passed

through the laser beam [189]. This is because the surface scattering effect of soot particles is unique

at these two wavelengths compared to other particles/gases which pass through the light [185].

This technique is only an indirect measurement of the surface area of soot particles, in order to

distinguish particle characteristics such as volume or mass certain assumptions have to be made.

There is also some worry that other species may be close enough to soot surfaces, in terms of light

transmittance, as to introduce false positives or negatives to experimental results [108]. Despite

these concerns, the two-color transmittance method is becoming increasingly popular and valuable

in the study of soot formation.

Modeling

In a computer simulation of a combustion system, the domain of the system is typically

broken into hundreds/thousands/millions of grid cells. Intensive thermodynamic properties (tem-

perature, pressure, chemical mole fractions, etc) are generally considered to be constant over any

16

given cell but not necessarily the same between cells, although sometimes a mapping is used

within a cell instead of constant profiles. Navier-Stokes transport equations [15], are numerically

discretized and applied to each cell individually and the array of cells are allowed to evolve using

those equations and a small time-step. From a modeler’s perspective, there are a number of unique

challenges to representing the soot formation phenomena effectively in this type of simulation.

The same issues with particle size and morphology that challenges experimentalists also

challenges modelers, but in a slightly different way. For experimentalists, the challenge is to quan-

tify and characterize the size and shape of soot particles in different environments. For modelers,

the challenge is to convey that size and shape in a way that is computationally feasible. In any

sample of soot particles, the size and shape varies greatly across the sample [2]. In a grid cell, the

particles are also a non-uniform distribution, but how does one portray that distribution? Chapter

2 will discuss in further detail different methods to portray the particle size distribution once a

characteristic has been chosen with which to define the particles’ size, such as effective collision

diameter or mass.

Shape, as opposed to size, is more difficult to represent, and although a variety of tech-

niques have been proposed to portray particle shape there is no universally used method [53, 11,

135, 154, 125]. As particles grow larger and larger, a spherical approximation becomes increas-

ingly inaccurate as particles agglomeration tends towards the aggregation extreme. Aggregation

usually increases the complexity of a system from a modeling stand point because the morphology

of a soot particle changes and the particle’s surface area is not longer discernible [147]. Typically,

soot aggregates are large enough to lie squarely in the continuum flow regime of particle-particle

collisions, but the differences in particle morphology can increase the complexity of aggregate-

aggregate collisions. In addition, aggregates have more available surface area for surface reactions

(growth and consumption) than a spherical particle of equivalent mass, allowing both faster sur-

face growth and faster consumption depending on the surrounding gases. While the morphology

of soot aggregates has been long studied, and there are physical parameters developed to describe

this morphology (evolving fractal dimensions, etc.), there remains much uncertainty on the char-

acterization of soot aggregates and associated formation.

Particle-gas interactions, including the previously discussed HACA, oxidation, and gasi-

fication reactions, are highly dependent on the local chemistry of the surrounding gas and these

17

chemistry profiles are difficult to model. In a high-temperature system, the chemistry mechanisms

of all feasible reactions becomes very complex. Even in simple fuel systems, thousands of differ-

ent reactions are occuring involving hundreds of unique species. Many have proposed detailed and

complex mechanisms [7, 196, 175] that do very well but inevitably are missing some reactions or

represent reaction rates imperfectly. To evolve every reaction at every point in time using thermo-

dynamic principles is computationally too expensive for most any simulation. Another approach

is to assume the chemistry is at instantaneous equilibrium for a given temperature. The equilib-

rium state is computed by minimizing Gibb’s free energy and tabulated before the simulation by

cell heat loss and mixture fraction (the ratio of mass originating from the fuel). The equilibrium

approach does well, assuming an accurate mechanism, at predicting bulk species concentrations

(O2, CO2, CH4, etc.) but is often found to severely underpredict the concentration of radical

species which are essential to so many of these particle reactions [194, 67, 122, 170]. Another

increasingly popular approach is to use a laminar flamelet model. The idea of a flamelet model is

that any flame, turbulent or laminar, can be characterized by series of thin laminar flamelets. Each

flamelet is locally one-dimensional, a mixture fraction dimension, in a transition from complete

oxidizer to complete fuel under a certain flame strain. A simulation cell need only be characterized

by where it exists on the laminar flamelet scale and the chemistry profile may be read directly

from experimental data or a precomputed flamelet data [35, 198]. There are other approaches to

representing local chemistry, but regardless of the approach there are advantages and shortcomings

to each which either add additional uncertainty or computational costs that should be considered.

The introduction of chemistry creates additional problems for modeling soot formation.

The time-scale of different reactions in the soot formation process vary widely, lending a tendency

to numerical stiffness. Some reactions, such as particle oxidation at the flame front, are happening

very fast, having a very small time-scale, whereas other reactions, such as the particle-particle

agglomeration at small number densities happen much slower, and have a much larger time-scale.

In simulation, if we use a large time step, on order of the larger time-scale, we will capture the

evolution of slower process correctly but the faster reaction can cause instabilities in the simulation.

An instability occurs when the rate of a reaction causes physically impossible results. For example,

suppose we have a concentrations of 1E-10 kg/m3 of particles which are being consumed at a rate

of 5E-10 kg/m3s. If we were take a step of 0.5 seconds we may say that there are now -1.5E-10

18

kg/m3. Obviously, it is impossible to have a negative concentration of particles in the domain and

such a result would ‘break’ a simulation. An initial solution may be to take timesteps on the scale

of the faster reaction; however this can lead to problems on the other end as numerical error, due

to a computer’s rounding precision or small quantities of model uncertainty, compound drastically

giving unrealistic rates for the slower reaction. This is what is meant by numerical stiffness and

techniques, such as implicit methods or partial equilibrium assumptions, have to be explored to

resolve these issues.

Computational expense is always a consideration for any simulation. On a case-by-case

basis, a balance of accuracy to expense must be evaluated and various models are adapted to fit

the balance. While most developed soot models include the major processes discussed in Sec-

tion 1.2.1 [36, 117, 164], not all do. Many models are simplified to reduce computational costs

while maintaining model predictability within a range of controllable environments [107, 111,

116]. Because of the expensive considerations as well as the before mentioned complications

to soot formation modeling, even the most sophisticated models often contain large quantities of

uncertainty and should be used with an understanding of these uncertainties [129].

1.2.3 Formation from Solid Fuels

Like soot formed from gaseous fuels, soot formed from solid fuels follows many of the

basic steps of the process portrayed in Figure 1.2. The primary difference comes in the source of

soot-precursors. Unlike the gaseous fuels, where the rate determining step is usually the formation

of PAHs from the gas-phase profiles, solid fuels tend to give off tars straight from the solid phase

which act as the primary soot precursor in most solid fuel systems. A brief outline of the soot

formation process for complex solid fuels is found in Figure 1.6.

As a solid fuel heats up, it undergoes primary pyrolysis or devolatilization, a thermo-

chemical decomposition of the parent fuel which results in the volatilization of minor components

within the fuel structure [179]. Details of primary pyrolysis are extensive, complex, and beyond

the scope of this work [28, 50, 99, 159, 180, 181, 199] and will only be summarized here in brief.

Complex solid fuels, such as coal or wood can be thought of as carbon clusters bonded together

through various molecular bridges and side-chains. Some of these bridges and chains are strong,

some are weaker. As the parent fuel heats up these bridges and chains begin to break and mutate,

19

Parent Solid Fuel

Char

Light Gases

Primary SootTar/PAH

Soot Aggregates

Devo

latil

izatio

n

CharOxidation

Oxidation PAH Production

NucleationAgglomeration

Oxidation/Gasification

Figure 1.6: Overview of the soot formation process as found in complex solid fuel systems.

releasing volatiles from the solid structure. Bridges and carbon structure of the fuel will continue

to transform releasing some volatiles and restructuring the solid until all side-chains and labile

bridges are gone [30]. At its conclusion, primary pyrolysis results in three major products as seen

in Figure 1.6: char, light gases, and tar.

Char is the solid structure remaining after devolatilization and is primarily carbon, with the

fuel inorganics eventually released as ash after char oxidation [176]. Almost entirely aromatic, it

is thus less reactive than the surrounding gases but will still react as oxidizing agents diffuse to

the surface. The primary pyrolysis process volatilizes large portions of the solid fuel leaving large

pores throughout the char structure [150]. This porosity plays an influential role in the oxidation of

char particles as the available surface area for oxidation increases significantly as oxidizing agents

diffuse into these pores [172].

Light gases from primary pyrolysis are all gases small enough in molecular weight as to

remain as gases even at standard temperature and pressure. These gases are predominately CO and

H2O, but other gases are found in abundance as well: CO2, H2, CH4, and other small hydrocar-

bons. The exact composition of these light gases is system dependent and will vary as any gas does

within the combustion environment as various temperature-dependent chemical mechanisms take

effect.

20

HN

CH3

CH3

CH3CH3

H3C

HO

HO

Figure 1.7: Comparison between pyrene, a common PAH soot precursor in gaseous systems, anda theoretical tar molecule as constructed based on elemental composition, molecular weight, andaromatic content [10].

Tar, like the light gases, is also a volatile released during primary pyrolysis. Unlike the

light gases, if tar were cooled to room temperature it would condense to a liquid-like substance.

Tar is made up of hundreds, if not thousands, of possible species of heavier hydrocarbons that tend

to be mostly aromatic, and these molecules serve as the primary soot precursors in most solid fuel

systems [204]. However, there are significant differences between tar released from solid fuels and

PAHs built from gas-phase mechanisms, and it is these differences that lead to differences in soot

formation between gaseous fuel systems and solid fuel systems.

An example of these differences comes from the molecular size distributions of gaseous

PAHs versus tars released from solid fuels. Gaseous PAHs tend to have a narrower distribution

of molecular sizes, ranging from naphthalene (128 g/mole) to circumcoronene (667 g/mole) with

a mode at pyrene (202 g/mole). Tar, has a much broader distribution ranging from 100 g/mole

to 3000 g/mole with a peak around 350-400 g/mole and a log-normal distribution [85]. These

distributions are much different, with tar not only being more variable in size, but also tending to

be larger than PAHs.

The yield of soot precursors, either tar or PAH, tends to be much different as well. A

gaseous system tend to yield less than 15% of the fuel mass as PAHs. This figure is fuel and

system dependent, with heavier fuels producing more PAH than lighter fuels [186], and hotter

systems tending to produce more than cooler systems. Solid fuel systems, can yield up to 40% of

the parent fuel’s mass as tar [121, 10], thus most solid-fuel systems tend to have a greater potential

for producing soot than gas-fuel systems [192].

21

PAHs built from gas-phase mechanisms are completely aromatic containing nearly all car-

bon with only some hydrogen on the outer rings. This leads to soot particles produced in gaseous

systems to be largely carbonaceous with only small amounts of hydrogen attached to the parti-

cle surface. Tars, on the other hand, tend to contain inorganics and aliphatic groups within the

molecule as seen in Figure 1.7 [10, 60]. For coal tars, the elemental composition and aromatic

percentage tend to reflect that of the parent coal [52]. For biomass tars, it is not as easy to predict

the aromatic percentage or elemental composition but it is known that tars produced tend to have a

lesser aromatic percentage than coal tars but also tend to reflect the elemental composition of the

parent biomass, but with much less oxygen [42].

Tars are more reactive and volatile than PAHs. The first reason has to do with pure concen-

trations soot precursors in a combustion system. In gaseous systems, PAHs must be built-up from

light gases through a variety of possible mechanisms discussed in Section 1.2.1. Each step in these

mechanisms is reversible, but the concentration of reactants is much greater than the concentration

of products, thus each reaction is thermodynamically pushed towards the formation of more PAH

to reach an equilibrium. Solid fuel systems, on the other hand, have a flood of precursors as a result

of parent-fuel devolatilization. This flood of precursors pushes any mechanisms towards equilib-

rium, or back towards more light gases. The differences in structure and elemental composition

also alter the reactivity of the precursors. PAHs have a greater aromaticity than tars and are thus

structurally more stable [42]. Also, the presence of inorganics, particularly oxygen and metals,

increases the reactivity of tar [19]. These differences of reactivity shift soot formation processes,

and an accounting of tar volatility is vital to accurately predict soot concentrations in solid-fuel

systems.

As tars act as the primary soot precursor in most systems, soot particles produced in solid

fuel systems also tend to contain inorganics and aliphatic branches embedded within the particle

structure [205, 190]. In many cases, variation in soot elemental composition may affect both

the reactivity of the particle itself or other aspects of the combustion environment. For example,

the embedding of metals (Na, K, etc) within soot particles produced from biomass can catalyze

oxidation reactions at the particle surface, increasing surface reactivity, [23, 190, 188, 191]; or

nitrogen may be stored within a coal system’s soot particles and only released when those particles

are oxidized later in the combustion system, thus altering the NOx formation process [161, 160].

22

Boiler

Gas – Gas Heater

Mill

Feed-water Heater

Oxygen Pre-

Heater

ElectrostaticPrecipitator Filter

Cooler

ASU

CompressorCO2

air

water

coal

Proposed Oxy-fuel System

Figure 1.8: Diagram of a proposed oxy-coal reactor. As proposed by Buhre et al. [22].

1.3 Oxy-Fuel Combustion

Oxy-fuel combustion was first proposed by Abraham et al. [3], as a method to achieve

CO2 purification and desulfurization in the flue gas, which are costly post-combustion recovery

processes. At the time, this new technology was largely overlooked; but with the increasing con-

cern of CO2 effects on climate [182], further investigation into carbon-capture technologies, such

as oxy-fuel combustion and others [105, 118] have become warranted.

The foundation of oxy-fuel combustion is the addition of air-separation units (ASU) at the

front-end of the combustion process as can be seen in Figure 1.8. At the ASU, O2 is separated

from N2, heated, and fed into the boiler as the oxidizing agent to combust the fuel. This ASU is

expensive in its power consumption and reduces the overall efficiency of the power plant introduc-

ing a parasitic load of about 22% [187]. Improvements and methods of application are an area of

extensive research [105].

The lack of atmospheric N2 in the boiler yields multiple benefits which could justify the

expense of the ASU:

23

• Without N2 in the boiler, sources of thermal NOx are eliminated during the combustion

process, leading to a significant decrease in the overall yield of NOx in the flue gas [87]

• CO2 is expensive to separate from N2 in the post combustion clean-up [143], but with the

prior removal of N2, CO2 separation from the flue gas becomes much more economical as

the flue gas is primarily composed of CO2 and H2O which can easily be condensed [44].

• Particles have a tendency to burn more completely because of higher temperatures and

greater access to O2, leading to greater boiler efficiency and less load in post combustion

clean-up processes [29].

While there are benefits to oxy-fuel combustion, the drastic change in the combustion en-

vironment leads to many differences in power plant operation. Besides the addition of an ASU,

the importance of flue-gas becomes emphasized. As the burning of fuel in pure O2 yields incred-

ibly high temperatures [16], it becomes necessary to regulate temperature with recycled flue gas.

This recycled flue gas not only lowers burn temperatures but also affects behavior of combustion.

The presence of high concentrations of tri-atomic molecules (CO2 and H2O) greatly increases the

thermal radiative properties of the gases [6, 92], increases effects of particle gasification [1], and

alters flame structure [45].

Post-combustion processes are greatly affected by oxy-fuel combustion. It has been pos-

tulated, that contamination of trace elements in the flue gas would increase, and consideration of

this increase may be burdensome for any post-combustion processes [64, 87]. In addition to ad-

justments in standard flue gas clean-up units, an additional unit for the treatment of CO2 must

be added. This unit cools and compresses a pure stream of CO2 for subsequent industrial use or

sequestration [118, 143].

In regard to soot formation, oxy-fuel combustion processes have potential to greatly al-

ter soot yields in comparison to conventional combustion processes. The effects of an oxy-fuel

environment on soot formation are threefold:

First, high concentrations of CO2 and H2O gasify soot particles. In conventional combus-

tion environments, particle gasification is usually considered to be negligible and the consumption

of soot particles is fully dominated by oxidation [51]. Gasification occurs as high energy molecules

collide with the surface a soot particle surface and transfer enough energy to break intra-particle

24

bonds and release a portion of the particle’s surface molecules as gas into the surrounding envi-

ronment. In conventional combustion systems, the bulk gas is overwhelmingly made up of mono-

atomic and diatomic molecules, N2 being the most abundant, and these molecules usually lack

intra-molecular energy to transfer to the soot particle surface upon collision. Oxy-fuel combustion

systems, on the other hand, contain high concentrations of tri-atomic molecules, particularly CO2and H2O, due to the high rate of flue-gas recycled back into the system for temperature control.

The extra atomic bonds of these tri-atomic molecules greatly increase the potential to contain intra-

molecular energy, through more vibrational, rotational, and electronic modes of energy [166]. This

increase intra-molecular energy increases the reactivity of these molecules for gasification, it also

increases the heat capacity of these molecules. An indicator to the effectiveness of a species as a

gasifying agent can be seen in its heat capacity. Thus the presence of high tri-atomic concentrations

increases particle gasification and, while still secondary to oxidation, becomes an increasingly im-

portant source of soot consumption [1].

Second, due to the increased concentrations of H2O and CO2, the radiative heat transfer of

the system gases increases [5]. Just as the greater heat capacity of the tri-atomic molecules indi-

cates for a potential for greater transfer of energy on impact with a particles surface, the increases

in vibrational, rotational, and electronic modes of energy allows for tri-atomic molecules to emit

photons across a broader range of the energy spectrum. This increases overall emissitivity of the

bulk gases in oxy-fuel conditions. Increases in overall emissitivity increase the local heat losses

in hot environments and thus lowers local temperatures in a reactor. Even though the radiative

effect of the oxy-fuel environment lowers local temperatures, oxy-fuel systems are capable of op-

erating at higher temperatures due to the lack of a N2 diluent in oxidizer feed [6]. As discussed

in Section 1.2.1, many soot formation mechanisms are temperature dependent and the balance of

these mechanisms will be altered by system operating temperature. For example, Zeng et al. [205]

noted a trend between soot yield and temperature. Starting their experiments at 800 K, they noted

that initially as system temperatures increased the soot yield declined, but as temperatures contin-

ued to increase soot yield reversed trend and inclined. This trend of initial decline followed by

inclining soot yield against increasing temperature may be explained by two competing mecha-

nisms. At low temperatures, tar mechanisms dominate soot formation; as temperatures increase,

tar thermal cracking rates increase and soot yields decline. However, as temperatures increase PAH

25

concentrations also increase [57], thus at very high temperatures, such as may be found in some

oxy-fuel systems, it is possible that the primary source of soot precursors, tar versus PAH, changes

significantly.

Third, the higher flame temperatures and the changes in O2 concentration can greatly af-

fect local chemistry profiles, those chemistry profiles play significant roles in soot formation as

well [57, 128]. The interdependence between temperature and local chemistry was discussed pre-

viously in Section 1.2.1. But the new balancing of chemistry with soot formation mechanisms

not only affects oxidation and gasification, as stated earlier, but also will have impacts on surface

growth mechanisms.

1.4 Modeling Wildland Fires

Wildland fires have become increasingly rampant and dangerous over the last few decades

for reasons both known and unknown. While critical to a healthy environment, wildland fires can

pose great danger to human life, health, property, and can have long-lasting environmental impacts.

The field of wildland fires is one of vast information and data, with large amounts of understanding

in many phenomena, and almost no understanding in others. For example: the fundamentals of

heat transfer are well known and developed; however, how a fire may spread via this heat transfer

from the ground (surface fire) into the canopies of towering trees (crowning fire) is both hard to

understand and even harder to predict.

Wildland fires vary immensely in scale of spread and intensity. The ‘ideal’ fire, healthy

to the ecosystem, remains on the ground, not crowning to the tree tops and consumes floor de-

bris and small vegetation, allowing room for new growth and boosting an ecosystem’s carbon

cycle. This type of fire spreads rapidly, as determined by current weather and climate, but over

a smaller domain (tens of hectacres) and is in lower temperature, typically ranging from 550 to

800 °C . Unfortunately, an increasing number of wildland fires are not ‘ideal’ and in some cases

create firestorms which can be quite severe. A firestorm occurs when the fire intensity becomes so

high that the mere convection drafts caused by the fire are violently destructive. Driven by self-

generated weather and climate, firestorms burn over a much larger domain (tens of thousands of

hectacres) and cause immense damage to both the short-term and long-term health of the ecosys-

tem. Temperatures within a firestorm have been postulated to reach as high as 1800 °C . A survey

26

of actual wildland fires shows a distribution of fire types spanning conditions from the ‘ideal’ fire,

to the firestorm, and everything in between; however, the vast majority are closer to the ‘ideal’

end of the scale. Bulk fire behavior, which determines where a fire falls on this scale, is largely

governed by three factors: atmospheric conditions, fuel characteristics, and topography.

Studying the effects of atmospheric conditions on wildland fires is difficult because of the

heavy coupling between combustion physics and atmospheric conditions. Many of the most impor-

tant characteristics of wildland fires (spread, intensity, etc.) are highly dependent on atmospheric

conditions, but the combustion characteristics also have a compounding effect on those conditions.

As an example, consider wind speeds. The most important of atmospheric conditions, wind speed

and direction, usually serve as the largest indicator of fire spread; however, these fires induce large

natural convection swells that alter those wind speeds and can even overcome wind direction if the

winds themselves are weak. Similar interactive effects occur with atmospheric humidity, precipi-

tation, and pressures, which are lesser, but also important, conditions to a fire.

A wildland fire has the potential to spread across many different fuel types, each of which

can be unique in its combustion characteristics. Every fuel has a unique flash point, or temperature

at which it begins to burn, distribution of pyrolysis products, and energy yield. Not only are

there large variations between broad biomass types (grass, bush, tree, etc.), but even at a finer

level the combustion characteristics can vary. For example, softwood trees, like pine or fir, tend

to have a much lower flame temperature than hardwood, like oak or maple. Not only do different

species react differently, but different parts of each species affect pyrolysis behavior. In a spreading

wildfire, needles and leaves ignite and burn much more readily by advection than branches, limbs,

or tree trunks, and different fire intensities will burn different portions of biomass. In addition,

temporary biomass attributes, such as moisture content, significantly influence behavior as well.

Living plant matter burns differently than dead plant matter, even when the dead matter has been

rehydrated to moisture levels equivalent to its living counterpart.

Topography also plays an influential role on fire behavior. Fire-slope behavior is unique as

flames have a tendency to attach to slopes, this tendency is known as the Coanda effect. Between

the Coanda effect and the buoyancy of emitted hot gases, any fire will readily travel uphill. Fires

will rarely spread downhill, unless directed by high winds or another equally powerful driving

force. Hence, the topography of a landscape will often determine both the path of a wildfire as

27

well as spread rates and travel distances. Fuel density, which may or may not be categorized as

topography, contributes to the potential of fire growth and intensity. Areas with a thick fuel density,

especially of dead and dry fuels, have a much greater potential of creating high intensity fires than

low density wet fuels.

Prior to the 20th century, an attitude of complete fire suppression was established in the

United States and most other locations throughout the world. In 1905 the U.S. Forest Service was

established with the primary task of suppressing all fires on the forest reserves it administered. This

attitude of complete fire suppression, in effect for many decades, did not allow naturally occurring

fires to clear wildland debris. After 150 years of fire suppression, most North American forests

have an unnaturally high density of dry and dead fuels. These forests have been additionally

subjected to increasing global temperatures, large outbreaks of tree-killing insects, and regular

periods of drought, all of which further kill and dry fuels. As a result, when wildland fires occur

today, whether through natural or human causes, those fires have a much higher potential to grow

in intensity beyond what is healthy in the ecosystem.

With respect to soot formation, wildland fire behavior poses an interesting series of circum-

stances to be investigated. Establishing the total sooting potential of these fires is difficult due to

the heterogeneous fuel source. As stated before, each fuel type pyrolyzes uniquely. That pyroly-

sis behavior determines concentrations and structure of tars produced, which tars are the primary

soot precursor in this system. The evolution of those tars is temperature dependent and typically

wildland fires are low temperature fires 550-1200 °C . These lower temperatures tend to favor the

nucleation of soot over the breakdown of tars, thus wildland fires tend to produce and emit more

soot and precursor molecules than industrial combustion environments. The larger quantity of

emissions lead to interesting dynamics of post combustion particle evolution which is, in and of

itself, a new field of study.

While there have been several attempts to construct a comprehensive computational fluid-

dynamic (CFD) software that predicts wild-land fire behavior, only a few have succeeded with

extensive validation. One such CFD software is FIRETEC. Developed by Rodmann Linn at Los

Alamos National Laboratory, FIRETEC is a wildfire behavior model based on conservation of

mass, momentum, species, and energy [34, 33]. It combines a three-dimensional transport model

that uses a compressible-gas fluid dynamics formulation with a physics-based combustion model.

28

Coupled with HIGRAD, an atmospheric software package, FIRETEC/HIGRAD does reasonably

well predicting fire spread patterns and rates over large land areas.

Work in this dissertation deals directly with the abilities of FIRETEC to predict soot emis-

sions from a wild-land fire and predicting soot formation processes in coal systems. At this point,

there are no physics-based models existing in the literature for predicting soot emissions from

wildfires, with the exception of some smoking-point models, which are semi-physics based, but

with heavy empiricism. There are only a few limited models, described further in Chapter 3, for

predicting soot in coal systems.

29

CHAPTER 2. COMPUTATIONAL TOOLS

This chapter includes two major sections: one for the resolution of particle-size distribu-

tions, and another as an introduction to Bayesian Statistics. Although these two different tools may

seem unrelated and disjointed, both were used extensively throughout the work of this dissertation

and thus are included here.

2.1 Resolution of Particle-Size Distributions

Soot particles and precursors within a system vary greatly in size as they form and evolve.

In most any real system, the particle number is too large to resolve the formation and evolution of

individual particles. As a result, an Eulerian approach is applied, looking at a group of particles

within an observed volume rather than individual particles. An observed group of particles are not

homogenous; rather they tend to vary greatly in size and shape. When considering soot particles

and precursors in a system, it is typical to characterize particles by their mass; hence a particle size

distribution (PSD) can be constructed for any group of observed particles where the distribution

is based on particle mass. An example of an observed PSD for soot particles collected from a

biomass-gasification system can be seen in Figure 2.1 [38].

The true challenge that these distributions pose to combustion models is how to represent

a PSD in numerical terms during simulation. There are a number of proposed methods used by

researchers to represent a PSD and in this introduction three will be discussed. The three meth-

ods discussed are not comprehensive of all methods developed or used but embodies the most

commonly used methods in the current community.

2.1.1 Sectional Methods

A common approach to depicting PSDs is known as the sectional method. In this method

the PSD is broken into a discrete distribution with limited sections, each of which represents the

30

0 10 20 30 40 50 60Particle Diameter (nm)

106

107

108

109

1010

1011

1012

Num

ber D

ensit

y of

Par

ticle

s (#/

m3 )

Data

Figure 2.1: Example of a soot particle-size distribution as collected from a pre-mixed flame exper-iment [2].


106

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1011

1012

Num

ber D

ensit

y of

Par

ticle

s (#/

m3 )

DataSectional Model

Figure 2.2: A graphical representation of the sectional method as applied to a soot PSD where 8sections are applied.

31

number of particles found within a given section’s range. This concept is depicted in Figure 2.2,

which is a depiction of the sectional method where 8 discrete sections are used to represent the

PSD found in Figure 2.1.

Sectional methods have the advantage of capturing the shape of an evolving PSD. As soot

particles evolve in a system, the size distribution also evolves. Sectional methods are able to

capture the evolving shape fairly well. However sectional methods do have their disadvantages.

Sectional methods often can require a large number of sections to be transported and re-

solved in order to accurately estimate a PSD. As more sections are added, the accuracy of the

method increases but so does the computational cost. This gives more flexibility to the researcher

to balance a simulation accuracy against economic cost to best fit the needs of his or her project.

However, to gain a good approximation of a real soot PSD it is common to need 20+ sections to

be resolved. This indicates a transport and resolution of 10+ parameters during simulation, which

is a very large computational cost for most combustion simulations.

Sectional methods introduce complications with interplay between sections during simula-

tion time. A given section represents a range of particle sizes. When particles in a given section

agglomerate or grow, they result in a size that is not represented exactly by the discrete sections.

Hence a repartitioning of particles among the existing sections is required. This can be done in

several ways, but a common approach is to do this such that particle mass and particle number

are preserved. Sufficient to say, the interplay between sections and within a section itself, due to

particle agglomeration and growth, leads to increased complications to the sectional method and

thus higher computational costs to resolve those issues.

A third aspect of sectional methods to review arises from another example. Imagine a sec-

tional method for a soot PSD applied to a simulation which consists of a long stretched flame giving

the soot particles a long residence time. At early residence times, these soot particles are newly

formed and small, meaning they are all clustered in the first section. As time passes, particles ag-

glomerate and grow becoming larger and larger, thus moving up to newer sections and spreading

out among all sections. Eventually, particles can grow too large to be accurately depicted by the

pre-established sectional sizes, thus voiding the accuracy of the soot model. While the obvious

answer would be to add more sections to the higher end of the particle spectrum, this of course

increases expense. We may also broaden the range of each section, but this decreases accuracy,

32


106

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1010

1011

1012

Num

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ensit

y of

Par

ticle

s (#/

m3 )

DataMono-Dispersed Model

Figure 2.3: A mono-dispersed distribution with η = 11.22.

especially at the early times. It is also possible to have an adapting sectional method which self

adjusts section sizes to accommodate the shifts and optimize the sections to most accurately repre-

sent the PSD. This adaptation may allow for transport of fewer sections, decreasing computational

costs, but the adaptation scheme itself requires a certain overhead computational cost, increasing

computational costs. Thus we see another deficiency of sectional methods, which while they are

rectifiable, not without great computational cost.

2.1.2 Derived-Distribution Method

Another method to represent a PSD is to approximate the PSD with another distribution

which is well defined and established with prescribed parameters. It is these prescribed parameters

that evolve with a soot formation model.

Mono-Dispersed Distribution

The first distribution that is commonly found in the literature is a simple mono-dispersed

distribution. In this distribution, it assumed that all observed particles, in a single time and loca-

tion, are of the same size. Evolution of the distribution through time and space only affects two

33


106

107

108

109

1010

1011

Num

ber D

ensit

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Par

ticle

s (#/

m3 )

DataLog-normal Model

Figure 2.4: A lognormal distribution with η = 2.86 and σ = 0.43.

parameters: a weight (the number of particles) and an abscissa (the size of the particles),

f (dp) = N0 ∗δ (dp−η). (2.1)

Where f (dp) is the number of particles of size dp, N0 is the total number of particles in the distri-

bution, and η is the assumed size of the observed particles.

This distribution is often overlooked and not considered a ‘truly characterized distribu-

tion’ because of its simplicity; however, it can be a very powerful tool as it is computationally

inexpensive, with only two parameters, and surprisingly accurate. As a result, this distribution is

commonly distribution found throughout the literature, particularly when computationally expen-

sive simulations are employed. A visual portrayal of the distribution can be seen in Figure 2.3,

where the vertical bar, which is located at dp=11.22, is capped at N0 portraying the total number

of particles in this distribution.

Scaled Lognormal Distribution

Perhaps the most useful of distributions to approximate a soot PSD would be a scaled log-

normal distribution, depicted in Figure 2.4. The lognormal distribution is based on the Gaussian,

34


106

107

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109

1010

1011

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ensit

y of

Par

ticle

s (#/

m3 )

DataPower-law + Log-normal Model

Figure 2.5: A bimodal, lognormal/power law, distribution with the following parameters: α = 3.35,k = 5.14, η = 2.85, σ = 0.42.

or normal, distribution but derived over a log scale rather than a linear scale. It is defined as:

f (dp) =N0

dpσ√

2πexp[−(ln(dp)−η)2

2σ2

]. (2.2)

A lognormal distribution has only three parameters to be defined, η is the mean of the natural log

of the size variable (ln(dp) ), σ is the standard deviation of the same, and N0 is the total number

of particles represented by the distribution. The distribution tends to have a off-center mode value

with a long tail extended to higher values. This shape is due to the logarithmic scale to which the

distribution was first derived. Should this same distribution be plotted with x-axis on a log-scale

then it would appear Gaussian in form. The lognormal distribution tends to capture the shape of

larger particles in a true soot distribution as seen in Figure 2.1, but it also tends to misrepresent the

large presence of small particles. For most purposes, this misrepresentation of small particles leads

to small amounts of error as it is the large particles that tend to dominate most attributes of soot

production for which there is interest: impact on thermal radiation, combustion efficiency, etc.

35

Power-law Lognormal Distribution

To reduce error and capture the shape of a particle distribution at small particle sizes, a

bimodal distribution is sometimes used which combines a lognormal distribution with a power-

law distribution. In this case, larger particles are mostly represented by the lognormal distribution

while smaller particles are captured by a power-law.

f (dp) = N0

(αd−k

p +1

dpσ√

2πexp[−(ln(dp)−η)2

2σ2

])(2.3)

While this distribution provides the best fit for the soot PSD as seen in Figure 2.5, it contains 5

parameters that must be resolved. The expense of 5 parameters along with each distribution eval-

uation can be burdensome for modeling and simulation, thus this distribution is rarely used and

should the finer details be required, most modelers turn to alternative methods for PSD represen-

tation.

2.1.3 Method of Moments

An increasingly common way to depict PSDs, or in fact any distribution, is by resolving

a distribution’s statistical moments. In practice, a derived distribution method is only a subset of

the Method of Moments (MoM) as the model parameters are types of PSD statistical moments;

however, what is referred to as MoM in literature usually deals directly with the non-centralized

statistical moments

Mr =∞

∑i=0

mri Ni. (2.4)

Here, Ni represents the number of particles with size mi. The first 6 moments of the experimental

distribution of Figure 2.1 are depicted in Table 2.1. Note that the values of the moments decrease

logarithmically. This is a common feature of soot particle size distributions.

There exist an infinite number of possible statistical moments all representative of a single

distribution. If we were to resolve the same number of moments as there are particles in a system

we could fully resolve a PSD through a series of linear equations; however, this number of resolved

terms is computationally/economically impractical. There are developed techniques to build a full

distribution from a finite set of resolved moments [93], and many of these techniques are quiet

36

Table 2.1: Resolved statistical moments of the experimental distribution of Figure 2.1.

Moment Value UnitsM0 3.47E11 #

m3

M1 3.68E-16 kgm3

M2 2.03E-42 kg2

#m3

M3 2.27E-68 kg3

#2m3

M4 3.79E-94 kg4

#3m3

M5 8.15E-120 kg5

#4m3

effective, but each has its limitations and there is no generally effective tool for all situations.

Fortunately, a full set of statistical moments or full distribution is rarely required to derive all the

information desired about a soot PSD. Normally, the first two moments (number density of the

particles and mass density of all the particles) is adequate to compute soot volume fraction or

average particle size, which is usually all that is desired from a system. It has become common

practice in the soot modeling community to transport and resolve statistical moments of the soot

distribution in simulations.

The major concern with the method of moments as applied to soot modeling arises from the

closure problem. To illustrate this an example is given here. When particles are oxidized, particle

mass is consumed and returned to the gas-phase. This oxidation affects the PSD moments

dMr

dt= π

(6

πρs

)ks

∆

r−1

∑l=0

(rl

)∆

r−lMl+2/3. (2.5)

Details and derivation of this equation will be provided later in this work. Suffice it to say thatdMrdt = g(Ml+2/3), indicating that the rate of moment changes during simulation is dependent on

a fractional moment of the previous iteration. While fractional moments can be computed if the

entire distribution is known, the entire distribution is almost never known and only a finite number

of integer moments has been chosen to be resolved. How do we resolve these fractional moments?

37

Quadrature Method of Moments

First applied to aerosol dynamics by McGraw[127], the quadrature method of moments

(QMoM) uses a quadrature approximation based on the resolved whole moments

Ml+2/3 =∞

∑i=0

ml+2/3i Ni ≈

rmax/2

∑i=1

ml+2/3i wi. (2.6)

This is an approximation and the higher the value of rmax, the number of resolved moments, the

more accurate the approximation. We directly calculate the values of the weights (wi) and abscis-

sas1 (mi) of the quadrature with the resolved whole moments

Mr =rmax/2

∑i=1

wimri . (2.7)

This creates a series of equations which may then be solved to find both the weights and abscissas

given the resolved integer moments. As an example take the six moments of Table 2.1, rmax=6,

and resolve the fractional moment M2/3. This leads to a series of equations

M0 = w1m01 +w2m0

2 +w3m03,

M1 = w1m11 +w2m1

2 +w3m13,

...

M5 = w1m51 +w2m5

2 +w3m53.

(2.8)

Solving this series of equations, usually through a numerical matrix, can be numerically expensive

and inaccurate. McGraw [127] proposed a solution to solving the weights and abscissas, using the

product-difference algorithm [73] to produce a tri-diagnol Jacobi matrix with eigenvalues equal

to the abcissas and the first element of the eigenvectors is equal to the normalized weights. For

further details refer to Appendix A in McGraw’s article Description of Aerosol Dynamics by the

Quadrature Method of Moments [127].

1Abscissa is a general term used in all quadrature method of moments. In the case of a particle size distributionabscissa is a size quantity.

38

Table 2.2: Resolved weights and abcissas of the 6 resolved moments in Table 2.1.

Variable Valuew1 2.959E10w2 3.164E11w3 7.827E8m1 7.062E-27m2 4.402E-28m3 2.499E-26

Regardless of the method used to reduce this series of equations, its solution leads to the

weights and abcissas shown in Table 2.2. These values are now substituted into Equation 2.6

M2/3 = m2/31 w1 +m2/3

2 w2 +m2/33 w3 = 2.987E-7. (2.9)

Compare this value as computed from the actual fractional moment of 2.724E-7 as defined by the

data and we have a 9.7% linear error by using the quadrature approximation. That is a very good

approximation as it is within the same order of magnitude as the true answer and is less than 10%

total error.

Variations of QMoM have been explored and expounded over the last several years. Di-

rect QMoM (DQMoM) is a mathematical simplification of QMoM in which weights and abscissas

are taken as the independent variables directly, instead of using the moments, thus eliminating the

numerical expense of moment inversion [135, 102]. Conditional QMoM (CQMoM) converts a

moment set into nodes which ease computational costs in comparison to QMoM and allows for

multi-dimensional distributions, such as with particle mass and surface area coordinates, to be si-

multaneously resolved [165]. CQMoM tends to be computationally more expensive than DQMoM,

which also can handle multi-dimensional distributions, but has the ability to capture certain par-

ticle interactions and realizations that DQMoM cannot. Extended QMoM (EQMoM) introduces

a Gaussian distribution solution to the quadrature approximation allowing more complex PSDs to

be represented with fewer weights and abscissas [? ]. While this is not a comprehensive list of

QMoM alterations explored and presented in the literature, it presents the most commonly used

approaches of the present day with regard to QMoM.

39

Method of Moments with Interpolative Closure

Another powerful closure method found in the literature, developed by Frenklach [53],

uses interpolative closure (MoMIC) between integral terms to determine fractional terms. The

interpolative closure is accomplished using a Lagrangian interpolation

logMp = Lp (logM0, logM1, ..., logMrmax) , (2.10)

Lp (logM0, logM1, ..., logMrmax) =rmax

∑i=0

logMi

rmax

∏j=0j 6=i

p− ji− j

. (2.11)

Note that the Lagrangian interpolation in Equation 2.10, is interpolating between logrithmic val-

ues of the moments. This is possible because of the logrithmic relation between PSD statistical

moments as mentioned previously and evident in the computed statistical moments in Table 2.1.

Displayed in the above equations in a closure of fractional moments, but interpolative closure is

used to compute any fractional term where the intergals are known, or can be computed, but the

fractional cannot be solved directly.

The moments of the example statistical distribution shown in Table 2.1, are interpolated

with Equation 2.10 to give a value for M2/3 of 2.724E-07. This value contains only a 2.8% error

with the actual fraction moment as defined by the data. For this data set, MoMIC did even better

than QMoM in evaluating the fractional moment, but both methods are proven viable for resolving

fractional moments. The detailed modeling portion of this work uses MoMIC for determining

fractional moments, but could be adapted to use QMoM or one of its variants without too much

difficulty.

2.2 Bayesian Inference

The following section discusses aspects of Bayesian statistics in the context of E.T. Jaynes’s

textbook Probability Theory: The Logic of Science [91]. For further discussion and clarification of

these principles refer to that work. For further introduction to the basic methodologies of Bayesian

inference refer to Andrew Gelman’s Bayesian Data Analysis [62].

40

Bayesian inference is rooted in Bayes’ Law, which is derived from an axiom of conditional

probability,

P(A∩B) = P(A)P(B|A) = P(A|B)P(B). (2.12)

In words, the probability of events A and B both occurring is equal to the probability of A occurring

times the probability of B occurring given that A occurs and is also equal to the probability of B

occurring times the probability of A occurring given that B occurs. This definition is rearranged

algebraically,

P(A|B) = P(A)P(B|A)P(B)

, (2.13)

which is Bayes Law. This is a discrete form of Bayes Law, but the law holds true in a continuous

regime as well,

f (x|y, I) = f (x|I) f (y|x, I)f (y|I)

. (2.14)

It is in this context that Bayesian inference is applied. The vector x of Equation 2.14

represents a set of parameters describing a model. The vector y represents data relevant to the

model. The I variable indicates an inclusion of all conditional factors not represented by x or y

(i.e., environmental conditions).

Each term in Equation 2.14 has a distinct name and meaning. The names and meanings

will be elaborated on in the following sections. Each section will begin with the theory of Bayesian

statistics then be followed with an example from a basic illustration. This illustration uses a natural-

gas reactor, to demonstrate the power and use of Bayesian Inference. Energy output of the gas

reactor is modeled using as simple linear equation,

E = aF +b, (2.15)

where E represents the energy output from the reactor in kilowatts and F is the fuel input in kg per

hour. a and b are model parameters to be calibrated using the experimental data. In this example,

experimental data, energy output and fuel input, make up vector y, while model parameters, a and

b, make up vector x. The vector I would be inclusive of any other conditions not represented in our

simple model (pressure, complex chemistry, reactor fouling, etc.).

41

Table 2.3: Experimental data for example gas-reactor.

Experiment Fuel Input (kg) Energy Output (J)1 84 1.24E92 79 1.38E93 30 0.54E94 45 0.69E95 64 1.07E96 91 1.40E97 95 1.09E98 33 0.55E99 77 1.56E9

10 58 0.94E911 75 1.13E912 79 0.92E913 62 0.93E914 68 1.17E915 96 1.44E9

Table 2.4: Ranges over which a & b parameters were analyzed for the example gas-reactor.

Parameter Low Range High Rangea -3.0E7 5.0E7b -4.0E9 5.0E9

After a series of experiments, with data given in Table 2.3, the model, represented by

Equation 2.15, can be calibrated to predict energy output using Equation 2.14.

2.2.1 Prior

f (x|I) is the prior, and represents an initial degree of belief for the hypothesized x vector

of parameters. The prior is a multi-dimensional probability density function (PDF) describing the

plausibility of the x vector, but before any of the currently analyzed data are considered. This PDF

may be a result of engineering intuition, model form, or previously collected and analyzed data;

regardless, the prior represents any previous belief in the nature of x and may be overcome with a

substantial amount of data to the contrary. A prior may be constructed in a variety of ways. Forms

and formats for deriving effective priors is a large area of research and debate in the Bayesian

community.

42

The initial step to forming an effective prior is to chose an effective domain over which the

Bayesian inference is to be carried out. To this end, the model form and engineering intuition can

help to narrow the domain to be tested. In our simple linear case, Equation 2.15, the a parameter

should be a positive value due to the intuition that more fuel should increase energy output. Thus

a limit is set on the domain available for the a parameter. However, domains for both a and b are

infinitely large and thus some testing has to be done to find reasonable ranges for the parameter to

be evaluated. In this example, domains over which the a and b parameters were to be evaluated are

shown in Table 2.4.

The next step to forming an effective prior is to have an effective shape to the prior. This

is where most of the debate over priors is concerned, as Bayesian inference is not only used to

calibrate parameters but also to quantify uncertainty in those parameters. Parameter uncertainty is

dependent on the prior used, especially in systems with sparse data sets.

The goal of establishing a prior is to incorporate any previously known system information

about the parameters before evaluation of the data. To this end, the simplest evaluation is to assume

we know nothing of the parameters other than their possible range. A uniform prior, represents

this assumption. In a uniform prior, any possible combination of parameter values, any x vector,

has an equal probability, thus showing no preference towards any particular parameter values.

For the gas-reactor, we may wish to evaluate a combination of 100 different values of

the a parameter and 100 values of the b parameter linearly spaced across the domains shown in

Table 2.4. In this case, there are 1002 different combinations of parameters a and b, each with

the same weighting as they are linearly spaced. Thus a uniform prior would place a probability of

1/1002 for each unique parameter combination.

The use of a uniform prior is the most basic of evaluations but it does not necessarily

incorporate all previously known information. The form of the model chosen to represent a system

can, and in fact usually does, contains an inherent correlation between model parameters which

can be incorporated into a prior. Note that when Equation 2.15 is rearranged to solve for the b

model parameter,

b = E−aF, (2.16)

that b parameter contains an a dependence, as long as F is none-zero. As a result, any adjustment

in parameter a in model calibration should result in an adjustment to parameter b as well to com-

43

Figure 2.6: This is a model-informed prior of the ab joint probability space as informed by thebasic linear model used in gas-reactor example.

pensate. The relationship between parameters a and b can be translated into a prior through the

following procedure.

First, recognize that confidence levels are reflected in probability contours on a PDF. These

confidence levels are equivalent to residual errors while comparing model outputs to data,

r2 = ∑i[yi− f (xi)]

2 , (2.17)

where r2 is that residual error, yi are data, and f (xi) are model outputs. In forming a model-

informed prior, y and x are not taken as individual data but rather as generic variables. If we

obtained y by a set of expected x values and compared that to varied the values of x we would

could map a response surface of residual errors, r2, according to the variations in x. Contours on

this response surface are commensurate to contour lines on the x PDF; however, while we know

the value of the r2 contours, we do not know the value of the corresponding contours on the x

PDF, only their location. Assigning a value to these PDF contours requires additional insight but

essentially reflects a researcher’s confidence in the proposed model form.

In the gas-reactor example, where y is E and x is a vector of a and b, we first create a

generic data set in the range where we expect E data to be taken. To obtain the yi values we solve

for F using expected E values, those found in the middle of our analyzed domain (1.0E7 for a

and 5.0E8 for b). To obtain the f (xi) values we then vary ab joint and solve for F again with the

44

expected E values. Figure 2.6 shows the response surface produced by subtracting these obtained

y and x sets. By assigning values to the contours in the figure we now have a model-informed prior.

The best situation for a prior occurs when a previous analysis of the system has been carried

out with data independent of the current analysis. When this is the case, the most effective prior

would be the posterior of the previous analysis. The posterior is defined and expounded upon in

Section 2.2.4.

2.2.2 Likelihood

f (y|x, I) from Equation 2.14 is the likelihood and represents the compatibility of the given

data with a hypothesized x vector of parameters. This is computed by first computing a γ value,

the model output, with the hypothesized parameters of x. This γ is then compared against the

measured data y. The difference between the two values can be assigned a probability in a variety

of ways, but it is this probability which is the likelihood value.

The complication to computing a likelihood arises from the variety of ways in comparing

y, the experimental data, and γ , the model predicted data. When experimental data has defined

uncertainty, then the comparison becomes straightforward. Simply plug data and evaluated pa-

rameter values into the proposed model and compare the resulting value against the experiments’

quantified uncertainty.

Unfortunately, experimental data uncertainty is not always quantified or reported. In these

cases, γ is compared to y using another established distribution. Perhaps most common and readily

accepted is the normal or Gaussian distribution,

f (y|x, I) = p(y|γ,σ , I) =1√

2σ2πexp(−(y− γ)2

2σ2

). (2.18)

A Gaussian distribution can be used in the majority of cases to described the shape of uncertainty

in experimental data. Typically, the only exceptions occur when constraints limit the physical

possibility of data and thus uncertainty distributions will be skewed or discontinuous.

Unfortunately, by introducing a generic distribution to quantify uncertainty in the exper-

imental data we have also introduced an undefined parameter. The σ variable of Equation 2.18

represents a standard deviation of data as described in a Gaussian distribution; however, no stan-

45

dard deviation actually exists for a single data point. This σ is often referred to as a ‘nuisance

parameter’ as it has no physical meaning but rather is an internal parameter of the statistical anal-

ysis that has been introduced to fully compute a likelihood.

The introduction of a nuisance parameter alters the formation of the prior as well. Because

the likelihood is no longer a function of just the model parameters, we must expand the x vector to

include those nuisance parameters as well. Fortunately, to compute a new prior, the σ parameter

is independent of the model parameters and maybe evaluated separately,

f (x|I) = f (x∗|I) f (σ), (2.19)

where x∗, is a vector of only model parameters not including σ . A separate independent prior,

f (σ), may now be used for nuisance parameters. While there is much discussion on the form

which that prior should take, most research points to the use of Jeffrey’s prior for σ [48]. Jeffrey’s

prior,

f (σ) ∝1σ, (2.20)

gives preference to smaller values of σ , thus favoring model parameters which give γ quantities

closer in value to the reported data.

Thus far, the computation of the likelihood only considering one point of data. In the case

of multiple data points, the overall likelihood is multiplicative of individual comparisons,

f (y|x, I) = f (y0|x) · f (y1|x) · ... f (yn|x). (2.21)

In the use of the Gaussian distribution, Equation 2.18 becomes:

f (y|x, I) =nz

∏z=1

p(yz|γ,σ , I) =σ−nz

√2π

exp

(−12σ2

nz

∑z=1

(yz−uz)2

), (2.22)

where nz represents the number data points to be analyzed. In this case, a separate γz value is

computed based on the input conditions associated with each data point (fuel mass). It is important

to note, this method of calculation is only valid if all data points are independent of one another.

46

-2E7 0 2E7 4E7a (J/kg)

-1E9

0

1E9

2E9

b (J)

Figure 2.7: This is a Gaussian-likelihood of the ab joint probability space as computed using a datafrom Table 2.3 and Equation 2.15 in the gas-reactor example.

The likelihood for the ab joint probability space as computed using the Gaussian distribu-

tion and data from Table 2.3 is shown in Figure 2.7. As shown in the figure, the shape, not the

spacing, of the contours found in Figures 2.6 and 2.7 are roughly equivalent.

2.2.3 Marginal Likelihood

f (y|I) from Equation 2.14 is the marginal likelihood, also known as the model evidence. It

is related to the likelihood function, but with the model variables marginalized out,

f (y|I) =∫

f (y|x, I) f (x|I)dx, (2.23)

in effect removing any dependency on model variables. This term is sometimes called the model

evidence because of extensive research done which helps to justify model form based on the above

relation [20]. Computation of the true marginal likelihood is difficult and there is disagreement in

the Bayesian community on how this is accomplished.

For purposes of this work, it is sufficient to say that because the marginal likelihood has

no dependence on model variables, it is constant across all parameter evaluations. The marginal

likelihood acts as a normalization constant for the PDF produced from the multiplication of the

prior and likelihood. Thus, in this work we compute the marginal likelihood to be the normalization

constant for the prior/likelihood product across the evaluated parameter domain.

47

In the case of the gas-burner, we can evaluate the marginal likelihood across the evaluated

domain of a and b,

f (E|I) =∫ ahigh

alow

∫ bhigh

blow

f (E|a,b, I) f (a|I) f (b|I)db da. (2.24)

2.2.4 Posterior

f (x|y, I) is known as the posterior and is our desired output. It is the pdf of the calibrated

model parameters, x vector, given the data, and contains a mode which is ‘the best’ parameter

values for our model. The posterior pdf represents the degree of belief we have for the x vector

having accounted for experimental data, environmental conditions, and any prior information. The

posterior is a PDF describing the plausibility of the x vector across a domain of different model

parameter values. The resulting PDF will be in z dimensions, where z is the number of elements in

the vector x. Where z> 1, the posterior may be marginalized for each individual element producing

z PDFs unique to each of the elements of the x vector.

Marginalization of parameters simply involves the integration of the posterior across the

domain of an evaluated parameter. For example, if we had a model with 3 parameters and we

wanted to remove the third dimension from the posterior PDF,

f (x1,x2) =∫

f (x1,x2,x3)dx3, (2.25)

would result in a two dimensional PDF. Should we want to have only a one dimensional PDF for

the x1, we would simply integrate over the x2 parameter. Any nuisance parameters introduced

in the analysis, such as the σ of Equation 2.18, should be marginalized from the final presented

posterior.

Defining credible intervals over the full PDF is difficult as it is multi-dimensional. How-

ever, when the PDF is marginalized to a single parameter, it becomes easy to establish credible

intervals for that parameter. On the other hand, when the full PDF is marginalized to two parame-

ters, correlations between the parameters become easy to see. When the two dimensional PDF is

circular it indicates that the two parameters are independent of each other. As parameters are more

and more correlated, patterns will arise in the two dimensional PDF reflecting that correlation.

48

-2E7 0 2E7 4E7a (J/kg)

-1E9

0

1E9

2E9

b (J)

Figure 2.8: This is a posterior of the ab joint probability space as computed using the prior ofFigure 2.6 and likelihood of Figure 2.7 in the gas-reactor example.

-5E7 -2.5E7 0 2.5E7 5E7a (J/kg)

0

1E-8

2E-8

3E-8

4E-8a Marginal Probability

-2E9 0 2E9 4E9b (J)

0

2E-10

4E-10

6E-10b Marginal Probability

Figure 2.9: Marginalized PDFs for the a and b parameters as taken from the posterior in Figure 2.8.

Returning to our gas-reactor example, the computed posterior is seen in Figure 2.8. This

posterior is a result of multiplying the model-informed prior with the likelihood seen in Figures 2.6

and 2.7, respectively. This posterior is marginalized and depicted in Figure 2.9. As expected, from

the shape of the two dimensional posterior, both marginalized PDF can be characterized as normal,

or very close to normal*.

From this computed posterior, there are multiple ways to calibrate the linear model. The

‘best fit’, or the fit with the least error between model outputs and data, comes from the mode

of the full-dimensioned posterior. An alternative way to calibrate model parameters would be to

take the mode of each marginal PDF. This calibration may be referred to as the ‘safe fit’ because

although the total error between model predictions and experiments may not be minimized, these

parameters contain the highest degree of confidence when considering the entirety of the analyzed

49

Table 2.5: Calibrated parameters from the Bayesian inference for the simple gas-reactor example.

Parameter Unit Calibrated Value 95% Credible Intervala J/kg 1.18E7 -1.19E7 < a < 3.44E7b J 2.35E8 -1.43E9 < b < 1.99E9

Figure 2.10: Linear mode, Equation 2.15, fitted to data from Table 2.3 using Bayesian inference.

parameter space. When a posterior PDF result is multi-modal, it is possible to have the ‘best-fit’

peak be very sharp, indicating that while that calibration yields the lowest error, the confidence

in that solution is not very high. A multi-modal posterior PDF should not be possible for simple,

single-equation models, but when a multi-layered complex model is analyzed as a whole, multi-

modal posteriors are possible, even probable. Where the posterior PDF is mono-modal, as seen in

Figure 2.8, these two methods of calibration, the ‘best-fit’ and ‘safe-fit’, should result in similar if

not identical parameter calibrations.

The final calibrated parameters for the gas-reactor example can be seen in Table 2.5. Since

the posterior is mono-modal we can safety compute a 95% credible interval by simply integrating

the marginal PDFs of Figure 2.9 to 95% of the whole PDF, centered on the mode values. A visual

representation of the calibrated linear model fitted to the data is shown in Figure 2.10.

This method of discretizing a parameter space and analyzing each possible combination of

parameters for a prior, likelihood, and posterior will be referred to as the ‘brute-force’ method of

50

Bayesian inference. The ‘brute-force’ method yields a fully comprehensive posterior PDF of the

entire analyzed parameter space, but this full PDF comes at a cost. The computational cost of a

full Bayesian inference analysis scales by a power equal to the number of parameters used in the

model plus any nuisance parameters. In the consideration of multi-layered complex models, with

dozens or more parameters, it is usually not computationally feasible to analyze every parameter

combination. Instead, methods have been developed to streamline the process beginning with

a single parameter combination and using the results of a single parameter analysis to inform

the choice of another parameter combination to be analyzed. Perhaps the most robust of these

methods are known as Monte-Carlo Markov-Chain (MCMC) methods. MCMC methods are a

class of algorithms for sampling the probability space based on the use of a Markov chain that

evolves a posterior distribution through parameter sampling until an equilibrium is obtained. These

algorithms are an intense field of research and results have become very robust and hold much

promise for parameter calibration in simple and complex models [76, 75]. Use of these methods

can considerably lower computational costs by finding and exploring areas of the parameter space

higher in probability, areas of interest, and leaving low probability areas, areas of little interest,

unexplored. Further discussion for the advantages and disadvantages of the Bayesian methods,

along with a comparison to a simple-least-squares calibration is found later in Section 4.5.

In the work of this dissertation, Bayesian inference was used to calibrate model parameters

for some of the sub-models representing the particle physics described in Section 1.1, particularly

in Chapter 4. These sub-models were then used as a part of overall developed soot formation

models. These sub-models also then had associated uncertainties which could then be incorporated

in overall in soot prediction uncertainties, although that work was not carried out to its fullest extent

in this disseration.

51

CHAPTER 3. EXISTING MODEL IMPLEMENTATION

The overall purpose of the work in this dissertation is to create predictive models for soot

formation in solid fuel systems which balance the needs of accuracy and computational cost for

simulations. As a result, a variety of different models have been explored and developed. How-

ever, before new models were developed, existing models were explored and implemented into

simulations to prove whether they were adequate.

As discussed in Chapter 1, there are a large number of models developed for predicting

soot formation from gaseous fuels, as the bulk of soot research has focused on these systems over

the last several decades. Although the mechanisms from these models can be highly useful in

the development of a solid fuel model, none are adequate for extemporaneous use in solid fuel

systems. This is because all of these models are developed with highly condensed PAHs, built up

from gas-phase mechanisms, acting as the primary soot precursor. In nearly all solid fuel systems,

tar released from the parent fuel during primary pyrolysis acts as the primary soot precursor, not

the PAHs evolved from the gas-phase.

Recognizing this limitation of gaseous fuel models, we turn to existing solid fuel models.

Among these solid fuel models, the most common type of model is the ‘smoke-point’ model. The

fundamental concept of a smoke-point model is based on a simple, easily repeatable experiment.

This experiment involves the creation of a laminar diffusion flame; by adjusting fuel flow rates, an

experimenter can change the flame temperature. Temperature of the flame is increased until smoke,

or soot, first begins to escape the flame sheet. This is the smoke point, and by use of an oxidation

model one can find the thickness of the flame sheet, temperature, and oxygen concentration, and

thus the amount of oxidation soot particles had to endure to escape the flame. Knowing this oxi-

dation quantity tells one the amount of soot present before oxidation started. Thus a smoke-point

model contains an oxidation sub-model and a formation sub-model which is then calibrated to the

two points, the fuel-burn point, the minimum temperature at which fuel began to burn (no soot),

52

and the smoke-point, the minimum temperature at which enough soot was produced to escape the

flame.

Another similar model is an equivalence ration model as developed by Adams and Smith [4].

This model scales yield of soot to equivalence ratio, or the air/fuel ratio over the stoichiometric

air/fuel ratio. From experimental data, one can determine the ratio at which soot first begins to

form and the minimum ratio at which no burn-out occurs (all the soot formed is emitted from the

flame). With these two experimental points, one may use a simple empirical correlation with the

locally solved equivalence ratio to predict the local yield of soot.

Smoke-point models, equivalence ratio models, and others similar to it, are empirical mod-

els simply mapping inputs to outputs based on the results of experimental data. While useful in

systems where experiments have been done, it is difficult to extrapolate the use of these models

to any other type of scenario. It is much better to have a physics-based model which has been

developed and validated with a back-and-forth process with experiments.

Searching the literature for physics-based models for soot formation from solid fuels yields

very few results [28, 139, 21] and those models are still very limited in scope. Chen et al. [28] ap-

plied a soot model in the CFD software FLASHCHAIN, designed to predict the primary and sec-

ondary pyrolysis behavior of coal systems, although the details of this soot model are not readily

available to the public. Muto et al. [139] proposed a simple physics-based model which transported

terms for soot particle number density and soot mass density and involved sub-models for particle

inception from PAH, conversion of coal tar to soot, particle coagulation, surface growth, and oxi-

dation. Most valuable among the literature was a model developed by Brown and Fletcher [21] to

predict soot formation in coal systems. This model was implemented into simulations as a starting

point for further research and development.

3.1 The Brown Model

The Brown model was developed for predicting soot formation in coal systems. It is largely

based on much of the work done previously by Ma [120], and resolves the time-evolution of three

variables: mass fraction of soot (YS), mass fraction of tar (YT ), and the number of soot particles

per kilogram of gas (NS). These three variables are used to describe the soot and tar PSDs as

mono-dispersed distributions.

53

The model applies equations for conservation of mass for soot and tar and conservation of

particle number for soot,

~∇ · (ρg~uYS) = ~∇ ·(

µ

Sc~∇YS

)+SYS , (3.1)

~∇ · (ρg~uYT ) = ~∇ ·(

µ

Sc~∇YT

)+SYT , (3.2)

~∇ · (ρg~uNS) = ~∇ ·(

µ

Sc~∇NS

)+SNS . (3.3)

µ is the turbulent viscosity, and Sc is the turbulent Schmidt number. The above conservation

equations may be discretized and applied to simulations to resolve transport phenomena effects,

either through convection or diffusion, on these variables.

Source terms, the last term of Equations 3.1, 3.2, and 3.3, represent rates of creation or

destruction. Each source term is computed by a series of sub-models representing soot and tar

formation processes,

SYS = rFS− rOS, (3.4)

SYT = rFT − rFS− rGT − rOT , (3.5)

SNS = (Na/mCCmin)rFS− rAN . (3.6)

These processes include tar formation (rFT ), tar oxidation (rOT ), tar gasification (rGT ), soot forma-

tion (rFS), soot oxidation (rOS), and soot aggregation (rAN). Above, Cmin is the number of carbon

atoms in the incipient soot particle and mC is the molecular mass of a carbon atom. Submodels

define a rate of reaction for each process and were taken from work done previously and published

in the literature. Rate parameters for each of the submodels are given in Table 3.1.

Rates of tar formation,

rFT = SPtar, (3.7)

need to be defined by other means beyond the scope of this model. In implementing this model,

we used the coal percolation model for devolatilization (CPD) [50] to predict the primary pyrolysis

54

Table 3.1: Transport equation source terms in the Brown Model.

term A E (kJ/mole) sourcerFT N/A N/A Source term for tarrOT 6.77×105 (m3/kg · s) 52.3 Shaw et al. [171]rGT 9.77×1010 (1/s) 286.9 Ma [120]rFS 5.02×108 (1/s) 198.9 Ma [120]rOS 1.09×105 (kg ·K1/2/m2 · atm · s) 164.5 Lee et al. [109]rAN N/A N/A Fairweather et al. [46]

behavior of parent fuels. CPD will predict the total yield of volatiles and tar from devolatilization

as a mass percentage of the parent fuel. From this, we compute a percentage of the total volatiles

which are tars. Now when any devolatilization model is used in simulation we can set a percentage

of the volatiles to be tar and use that as SPtar. This is not the only way, and the Brown model

does not specify a method for determining this source term. Another effective method would be to

tabulate experimental volatile data and during simulation directly read this source term from that

tabulated data.

Rates of tar oxidation,

rOT = (ρgYT ) · (ρgYO2) ·AOT e−EOT /RT , (3.8)

were take from the work of Shaw et al. [171], who performed experiments measuring the oxidation

rates of coal volatiles from 14 different coal types. Results from these experiments were used to

calibrate Equation 3.8 across a broad temperature and coal-rank range. Note that the rates measured

by Shaw et al. were for oxidation across coal volatiles, not just tar.

Rates of tar gasification,

rGT = (ρgYT ) ·AGT e−EGT /RT , (3.9)

were taken from the work of Ma et al. [121], who performed experiments measuring the yield of

soot from coal flames on a flat flame burner. More details on these experiments will be shared

in Chapter 5 as these experiments have become invaluable in the validation of soot formation

models. In the experimental analysis, Ma proposed a simple empirical model for predicting soot

55

yield calibrated to experimental results. The simple model included a yield of tar from primary

pyrolysis as predicted by CPD, a tar gasification term, Equation 3.9, to account for the fact that

the soot mass yield was less than the tar mass yield from primary pyrolysis, and a soot formation

model,

rFS = (ρgYT ) ·AFSe−EFS/RT , (3.10)

also used in the Brown model. The constants from Equations 3.9 and 3.10 were tuned to fit exper-

imental data.

Rates of soot oxidation,

rOS = SAv,SpO2

T 1/2 AOSe−EOS/RT , (3.11)

were taken from the work of Lee et al. [109]. This work was one of the earliest studies on soot

oxidation. In this work, Lee et al. measured soot oxidation rates as a function of input O2 con-

centrations and particle surface area across a laminar diffusion flame. Measured rates were used

to calibrate Equation 3.11. To use this model we need an available surface area density of soot

particles which can be obtained from the resolved soot particle and mass densities,

SAv,S = (NSρg)πd2p = (NSρg)π

(6YS

πNSρs

)2/3

, (3.12)

assuming all particles to be spherical and the density of soot, ρs to be 1900 kg/m3.

Rates of soot aggregation,

rAN = 2Ca

(6mC

πρs

)1/6(6kBTρs

)1/2(ρgYS

mC

)1/6

(ρgNS)11/6, (3.13)

were taken from the work of Fairweather et al. [46] who developed this aggregation term for

predicting evolving soot radiative properties, tied directly to the available particle surface area, in

a turbulent jet flame with an intercepting cross-flow wind.

The combination of the resolution of these three resolved variables along with submodels

governing their source terms makes for an effective model for predicting soot concentrations in

56

coal systems. This model, and its implementation, was used as a starting point for future model

development.

3.2 Arches

The Brown model, and other further developed models described in later chapters, were im-

plemented into Arches, which is built within the Uintah computational framework [145]. Arches is

a finite-volume large eddy simulation (LES) computational fluid-dynamics (CFD) software pack-

age under development at the University of Utah and used extensively by CCMSC to simulate

oxy-coal boilers.

A basic simulation using CFD is one where the simulated domain is meshed into hun-

dreds/thousands/millions of cells. Navier-Stokes transport equations and equations of species con-

servation are applied simultaneously across all cells and evolved in time to create a simulation.

This description is a direct numerical simulation, and accuracy of the simulation depends heavily

on cell size. For turbulent flows, such as a combustion reactor, the cell size must be smaller than the

Kolmogorov eddies, the smallest eddies, in order for a simulation to accurately model a turbulent

flow. LES is a scheme that allows for a simulation to use larger cell sizes while maintaining the

effect of smaller eddies, thus significantly lowering the computational cost of simulations. This is

accomplished by applying a spatial filter to the Navier-Stokes equations. All turbulent flow scales

larger than the filter size, the large eddies, are resolved through the discretized Navier-Stokes equa-

tions. All turbulent flow scales smaller than the filter size, the small eddies, are unresolved subgrid

fluctuations, and are modeled using a variety of turbulent flow models. Hence, LES derives its

name from the fact that large eddies are numerically resolved while small eddies are modeled.

This is only an overall view of how LES works, for more details one may refer to Frohlich et al.’s

Direct and Large-eddy Simulation X [59].

Arches was originally developed to simulate large pool-fires, that is a fire over a large pool

of liquid fuel, by solving filtered Navier Stokes equations at low Mach number using a pressure

projection scheme and user-defined boundary conditions. Since its original development, Arches

has been expanded with extensive particle physics to simulate coal-fuel flames. Now particu-

late fuels are traced in Arches simulations using an Eulerian particle transport method and the

size-distribution of particles is represented using DQMoM with either two or three weights and

57

abscissas Resolved variables describing the particles include the raw coal mass, three velocity

components, char mass, particle weight and enthalpy.

A variety of particle physics submodels are available in Arches. These include transport

models for drag forces, thermophoresis, and thermal radiation, as well as source models for fuel

swelling, devolatilization, char reactions, and ash-wall depositions. Chemistry profiles are tabu-

lated by mixture fraction and enthalpy before simulation with an assumed equilibrium gas compo-

sition.

The combination of these features, and other continually evolving features, has made

Arches an effective CFD software package for large-scale simulations and was used extensively in

this current work.

3.3 Simulation Set-Up and Results

Initial simulations to test the implementation of the Brown soot model were of an oxy-fuel

combustor (OFC) at the University of Utah’s Industrial Combustion and Gasification Research

Facility. This unit was chosen for simulations, as it is a smaller lab-scale unit with reasonable

domains for short and accurate simulations, while at the same time being large enough for fully-

developed turbulent flow. In addition, oxy-coal experiments were conducted previously measuring

soot volume fraction in this unit [185] providing an opportunity for model validation.

3.3.1 Oxy-Fuel Combustor

The OFC is a downward-fired 100 kilowatt lab-scale combustor unit. Figure 3.1 shows a

diagram of the burner and down-draft portion of the OFC, it does not show the full heat-exchanger

portion which would extend to the right of the diagram. The burner of this unit is swirl-stabilized

with a primary inlet and a secondary annulus inlet surrounding. Through the primary inlet, coal

particles are fed with a carrier gas, while through the secondary inlet an oxidizer is fed. The

oxidizer can be O2, a O2/CO2 mixture, or air, while the primary carrier gas is usually CO2 or N2.

The walls of the main burner chamber are 0.6 m in diameter and 1.2 m long with heated

walls as to minimize boundary layer effects. Quartz windows are inlaid in the walls for visual

observation and optical diagnostics in the main burner chamber. Flue gases pass from the burner

58

Furnace�---LES domain and Wall BC

Dimension of Test Rigs Items Units Value_real Value_LES-1 Value_LES-2

Burner structure (double channel)

ID_in m 0.0172 0.018 0.018 OD_in m 0.0215 0.024 0.024 ID_out m 0.0342 0.036 0.036 OD_out m 0.0425 6.00E-01 6.00E-01

Furnace Dimension for

LES

Diameter m 6.00E-01 6.00E-01 6.00E-01 Height m 1.2 1.2 1.2

Shrink Height m 0.3 0.3 0.3 outlet_Diameter m 0.132 0.132 0.132

outlet_length m 0.18 0.18 0.18

1

2

3

4 5

6

7

8

Heating parts

Glass windows

3mm:

17,423,308 cells

�mm: 3 times larger

Figure 3.1: Diagram of the downward burner and draft portion of the oxy-fuel combustor at theUniversity of Utah.

zone to the radiation zone through a slight narrowing of the combustion chamber. All along the

main burner and radiation zones are a series of sample ports through which varies probes and

measurement instruments are installed. A purge gas, typically of CO2 is blown over radiometers

in these ports to protect the surfaces from the high heat flux and ash build-up. After the radiation

zone, flue gases are sent through a series of heat exchangers before clean-up and ventilation.

3.3.2 Simulations

Two simulations were performed, one replicating experiments performed by Rezaei et al. [158]

and the other replicating experiments performed by Stimpson et al. [185].

The first simulation, those replicating the Rezaei et al. experiments, were chose because

these experiments had the reactor running at full capacity, or close to it. This high firing rate

produces larger quantities of soot with which simulations began to show the effect of soot and

the soot model upon the environment. These experiments were performed using a Utah SUFCO

59

Table 3.2: Proximate and ultimate analysis for Utah SUFCO and Skyline coals.

Coal Type Moisture Volatiles Ash C H N S O

SUFCO High-Vol Bit. 10.5 34.0 11.0 65.0 4.7 1.1 0.6 7.1Skyline High-Vol Bit. 10.0 39.2 10.0 63.0 4.6 1.2 0.6 10.6

Table 3.3: Flow rates for the two simulated experiments.

SUFCO SkylineBituminous Coal Bituminous Coal

Primary InletCoal: 13.81 (kg/hr) Coal: 6.80 (kg/hr)CO2: 5.40 (kg/hr) CO2: 10.8 (kg/hr)O2: 1.04 (kg/hr) O2: 2.08 (kg/hr)

T: 300 (K) T: 300 (K)Secondary

O2: 7.48 (kg/hr) O2 3.04 (kg/hr)T: 489 (K) T: 489 (K)

PurgeCO2: 3.08 (kg/hr) CO2: 3.08 (kg/hr)

T: 300 (K) T: 300 (K)3 radiometer inlets 3 radiometer inlets

high-volatile bituminous coal with proximate and ultimate analysis also shown in Table 3.2. Firing

rates are shown in Table 3.3 and can be seen to be significantly higher than the other simulation.

The second simulation, those replicating the Stimpson et al. experiments, were chosen

because Stimpson et al. optically measured soot volume fractions using a two color laser method in

a line of sight across the reactor at three different flame heights. These experiments were performed

using a Utah Skyline high-volatile bituminous coal with proximate and ultimate analysis as seen in

Table 3.2. Flow rates for the experiment can be seen in Table 3.3. At this firing rate, the reactor is

running at about half capacity as to create a flame that is optically thin enough for the two-colored

lasers to penetrate. As a result of the low firing rate, this experiment produces a low soot yield with

minimal impact but still one that is measurable.

Simulation space was limited to the main burner section of the OFC in both cases. By

limiting the simulation space we were able to reduce the computational cost of simulations signif-

60

(a) (b) (c) (d)

max

50%

75%

25%

min

Figure 3.2: Results of the SUFCO coal simulations [158]. From left to right the figures depict: (a)temperature (max = 2500 K, min = 300 K), (b) carrier gas mixture fraction (max = 1, min = 0), (c)coal off-gas mixture fraction (max = 0.3, min = 0), and (d) CO mole fraction (max = 0.7, min = 0).

icantly. Because the radiant and heat exchanger zones were not included in simulation, we were

not able to tell the total heat flux to boiler walls, which is usually the primary quantity of interest in

boiler simulations. However, these simulations did cover the entire flame area which was sufficient

as these simulations were primarily interested in flame structure and soot mechanisms, which all

occurred within the flame for these experiments.

Simulations meshed the main burner zone into cubed grid cells 4 mm on a side. This

amounted to 9.5 million cells across the entire domain. Simulations required approximately 25,000

CPU hours, and were run on the Fulton Supercomputer at BYU. The following results are all

shown after approximately 10 seconds of simulation time taken from a 2D plane passing through

the centerline of the reactor. The time of 10 seconds was chosen as it was observed, after numerous

simulations, that the soot profile did not vary much from time-step to time-step over the previous

and proceeding 2 seconds around this time period and thus we assumed we had obtained a steady-

state with regard to soot formation.

61

3.3.3 Results and Discussion

Figure 3.2 shows flame structure results of the SUFCO coal simulations. The image (a)

depicts instantaneous local temperatures. In this image we can see the overall outline of the flame

structure in the yellow and red portions of this image. As expected, the inlet streams are much

cooler than the overall reactor temperature, on the right hand side of the image we see the purge

streams around the three radiometers. Max temperatures peak around 2500 K and minimum tem-

peratures of the purge and primary inlet stream are 300 K. Overall reactor temperature, the green,

is around 1250 K, indicating a hot reactor [178].

Image (b) shows the mixture fraction of primary inlet carrier gases, mostly CO2. This

image is a decent indication of the extent of mixing in the reactor, as the other sources of gases are

coal-off gases and oxidizer. The ‘green’ color of the bulk of the reactor indicates a good mixing

between the gases. The second image from the right shows the mixture fraction of coal volatiles.

As coal particles undergo primary pyrolysis and devolatilize, these volatile gases quickly mix with

the surrounding gases and oxidize. The lack of large pockets of highly concentrated volatile gases

indicates a good rate of mixing due to high turbulence in this reactor set-up.

Image (c) shows local mass fraction of CO. CO results from the partial oxidation of carbon

by oxidizing agents, locations of high CO concentrations indicate regions of the highest reactivity.

In other words, we can see the flame structure from these concentrations. If the concentration

profile were reminiscent of a hollow tube, it would be an indication that a fully developed diffusive

flame was present where oxidizing agents must diffuse through a flame front to an inner core

‘pure’ fuel. This is not the case, indicating a flame more characteristic of a turbulence-driven

pseudo-premixed flame where the oxidizing agents are partially mixed with fuel particles during

primary pyrolysis and the combustion of volatiles is driven more by the mixing of the turbulence

than by oxidizer diffusion. This image also indicates a ‘lifted’ flame, where there is a notable

separation between the mouth of the burner and the reaction zone.

Figure 3.3 shows the number density component of non-reacted coal particles. The three

images are representative of the three weights used in the QMoM approximation of the particle

size distribution. In general, image (a) is representative mostly of small particles (20 µm), image

(c) is representative mostly of large particles (240 µm), and image (b) representative of particles

in between (120 µm). Although these three images are very similar, there are some subtle differ-

62

(a) (b) (c)

max

50%

75%

25%

min

Figure 3.3: Results of the SUFCO coal simulations, showing the number densities of (a) 20 µm(max = 5E10, min = 0), (b) medium (max = 1E9, min = 0), and (c) large (max = 2.5E7, min = 1.0E2)sized particles within the reactor.

ences. One such such difference is that larger particles penetrate further into the reactor than small

particles due to having more momentum. Dispersion of particles is due both to the turbulent flow

and the consumption of particles through devolatilization and oxidation. Small particles seem to

have the largest radial dispersion while large particle have very little radial dispersion until deep

in the reactor. In comparing this image against Figure 3.2, it is evident that small and medium

particles are largely dispersed within the flaming portion of the reactor, where large particles have

the ability to penetrate through the initial flame to a greater extent, thus extending the flame itself.

Figure 3.4 shows the results of the Brown model implementation in this simulation. Image

(a) shows instantaneous local mass fractions of tar. It is interesting to note the low concentration

of tar in the reactor at any given time, this is an indication of the high reactivity of tar. Tar is

an inherently unstable molecule in a combustion system and reacts quickly upon being released

as a volatile from the parent fuel. The tar can either thermally crack, or gasify as in the Brown

model, or it can nucleate into soot particles. The low concentrations of tar in the system indicate

the speed with which these reactions take place. The concentrations of tar are also not necessarily

continuous because location is determined by the pyrolysis of fuel particles. Large particles take

longer to pyrolyze than smaller particles, and since the QMoM weights tend to depict the particle

63

(a) (b) (c)

max

50%

75%

25%

min

Figure 3.4: Results of the SUFCO coal simulations, showing (a) the tar mass fraction (max = 0.03,min = 0), (b) soot particle number (max = 1E19, min = 1E12), and (c) soot volume fraction(max = 6 ppmv, min = 0 ppmv).

distribution as being closer to discreet than continuous we can get a non-continuous cloud of tar

concentrations.

Image (b) portrays the soot particle number, NS, from the Brown model. Image (c) shows

the soot volume fraction, fv,S computed from the modeled YS,

fv,S = (ρg/ρs)YS. (3.14)

Which is shown in units of ppmv, and is a common method to portray soot concentrations.

Of these two images, (c) is much more important as soot volume fraction is used as the

main indicator of the thermal radiation influence of soot particles. There is a problem with these

two images: the soot permeates through the entire reactor. From visual inspection it was known

that these experiments produced a ‘clean’ flame where no soot was observed to escape from the

flame. So how could the simulations be so far off?

Returning to the Brown model, we can search for deficiencies in the model which may

explain the concerning results. The most obvious deficiency in the Brown model is a lack of

surface growth terms in the particles’ surface reaction. However, the effect of these surface growth

terms would be small in these experiments because the reactor is turbulence-driven flame where

64

(a) (b)

max

50%

75%

25%

min

Figure 3.5: Results of the SUFCO coal simulations, showing (a) the CO2 mole fraction (max = 1,min = 0) and (b) O2 mole fraction (max = 1, min = 0).

there is not a zone in the flame where acetylene could be in high enough concentrations to cause

significant growth of particles. It is possible that the oxidation sub-model taken from Lee et al.

could be subpar for this system, but even with an updated oxidation model [141] results looked

much the same.

Abian et al. [1] compared soot formation in conventional versus oxy-fueled environments.

In that work it was noted that the higher concentration of tri-atomic molecules, CO2 and H2O,

in an oxy-fuel environment promoted the gasification of soot particles. Figure 3.5 shows the

mole fractions of CO2 and O2; note the high concentration of CO2 and low concentration of

O2 throughout the reactor. As the Brown model was originally designed for conventional air-fired

systems, soot gasification was not considered as it is fairly negligible in those systems [51]. As a

result, in simulations when soot particles penetrated through the thin O2 layer, they stopped being

consumed in the simulation, where in reality these particles were continually consumed through

interactions with CO2. To rectify this problem, a gasification sub-model was added to the Brown

model to reflect the consumption of soot via CO2 gasification. This sub-model was based on the

work of Qin et al. [153] who experimented on the gasification rates of biomass-derived soot in high

concentration CO2 in a thermogravimetric analysis set-up.

65

(a) (b)

max

50%

75%

25%

min

Figure 3.6: Results of the SUFCO coal simulations with soot gasification, showing (a) the sootparticle number (max = 1E5, min = 5E16) and (b) soot volume fraction (max = 6 ppmv, min = 0ppmv).

Figure 3.6 shows the resulting soot profiles of simulations with a soot gasification term

added. These profiles are what would be expected based on experimental observations and showed

the importance of gasification in oxy-fuel environments. In these experiments flue gas was not

recycled, but in a recycled flue gas system, H2O concentrations would also increase dramatically

and a gasification term as a result of soot/H2O interactions would be necessary. It should be

noted here, that though gasification was significant, as seen in the figures, it was still secondary to

oxidation for soot consumption.

Figure 3.7 shows the temperature and particle number density profiles of the Skyline coal

simulations. Image (a) depicts temperatures, which give a good indication of flame structure. In

this simulation, the flame is much smaller than the SUFCO experiment and tended to burn cooler,

with an average reactor temperature of 1100 K, due to the lower firing rate. Image (b) shows

the number density of small particles and image (c) shows the same for large particles. The bulk

of particles are small and disperse fairly close to the burner compared to large particles which

penetrate deeply into the reactor, and showed an almost puffing behavior.

Figure 3.8 shows a soot profile resulting from the Skyline coal simulation. As expected,

the tar profile, image (a), shows very small instantaneous concentrations of tar. There is a strange

66

(a) (b) (c)

max

50%

75%

25%

min

Figure 3.7: Results of the Skyline coal simulations [185], showing (a) the temperature(max = 2500 K, min = 300), (b) small particle number density (max = 4.4E10, min = 1.0E6, log-arithmic scaling), and (c) large particle number density (max = 6.0E8, min = 1.0E1, logarithmicscaling).

(a) (b) (c)

max

50%

75%

25%

min

Figure 3.8: Results of the Skyline coal simulations, showing (a) the tar mole fraction (max = 0.001,min = 0), (b) soot particle number (max = 1E16, min = 0), and (c) soot volume fraction (max = 0.24ppmv, min = 0 ppmv).

67

0.0 0.1 0.2 0.3 0.4 0.5 0.6Distance Along Line of Sight (m)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Soot

Vol

ume

Frac

tion

(ppm

v)

Simulation ValidationFlame RootFlame MiddleFlame TipExperiments

Figure 3.9: Line of sight measurements of the soot volume fraction across the flame. Solid linesrepresent optical measurements while dotted line represent simulation results. Blue is at the rootof the flame, green at the middle of the flame, and red is at the tip of the flame.

cluster of tar particles half-way down the reactor. This is due to the almost puffing behavior noted in

the large particles. As these particles penetrate deep into the reactor, devolatilization occurs lower,

and since tar is a direct result of devolatilization, we see some of this tar appear at a downward

location where those large particles have penetrated. It is not known whether this behavior was

experimentally seen, but it occurs in small time intervals throughout simulation and may be a

result of a small error in the simulation set-up. The puff of tar translates directly to soot particles

as well, thus we see, in image (b), an increased number of soot particles deep in the reactor. These

particles seem to be quickly consumed, though translating to only a slight increase in soot volume

fraction as shown in image (c).

The true advantage of this simulation is to validate the soot model. Stimpson et al. took line-

of-sight optical measurements of the soot volume fraction using a two-color transmittance method.

In the experiment, an the optical device was aligned along the reactor’s quartz window at the base,

middle, and tip of the visible flame. Numerically, a similar measurement was taken along the

simulation and averaged over multiple radial measurements resulting in the results of Figure 3.9.

This figure shows the experimental data with solid lines and simulation data with dotted lines. Blue

lines represent data at the flame base, green lines at the flame middle, and red lines at the flame tip.

As can be seen in the figure, the optical measurements only give an average soot volume fraction

68

Table 3.4: Comparisons the average soot volume fraction across the flame from optical measure-ments and simulations.

Flame LocationAverage Soot Volume Fraction (ppmv)

Relative ErrorExperimental Simulation

Root 0.0944 0.0804 14.8%Middle 0.0532 0.0456 14.3%

Tip 0.0542 0.0350 35.6%

across the flame while simulations give localized soot volume fractions. When an average soot

volume fraction is taken across the length of the flame and compared against experimental data the

agreement is decent, as seen in Table 3.4 which shows the error between the two in relation to the

experimental data. There is room for improvement, but the results are promising.

3.4 Conclusions

As can be seen in the above simulations, the Brown soot model yields decent results in

predicting soot in coal flames. However, this model does have deficiencies that this work and the

development of more-detailed models will help to address.

For oxy-fuel combustion it has been found that soot gasification is not a negligible mecha-

nism. A more detailed soot consumption model needs to explored. This consumption model needs

to include soot gasification from interactions with CO2 at a minimum and preferably with H2O as

well should a reactor be designed with a ‘wet’ flue gas recycle. It would be preferable to update

the oxidation mechanisms as well since the model developed by Lee et al. is out-of-date.

The Brown model does not include any surface growth mechanisms. In environments such

as the OFC, with a turbulence-driven flame, surface growth is minimal and not as important as

in diffusion (non-pre-mixed) flames where surface-growth species, such as acetylene, are more

prevalent [7]. In addition, this model describes the PSD of soot particles as a mono-dispersed

distribution, which is not a bad approximation, but not great either. The development of a detailed

soot model will also explore different methods of representing the PSD evolution.

Lastly, this model is designed only for predicting soot formation in coal systems. The work

of this dissertation provides an expansion to biomass systems as well.

69

CHAPTER 4. MODELING SOOT CONSUMPTION

4.1 Introduction

In the previous chapter, the importance of accurately modeling soot consumption was

shown as the Brown soot model did not have a gasification term in a system where soot gasifi-

cation was a significant source of consumption. This chapter includes work done in developing an

improved soot consumption model with considerations of oxidation and gasification.

Some of the first investigators of soot oxidation assumed that soot was consumed solely

via reaction of an O2 molecule with the particle surface [109], and oxidation models were devel-

oped based on the O2 concentration. It was quickly determined that the presence of OH molecules

greatly influenced rates of soot consumption and hence was included in oxidation models [183].

In more recent studies, emphasis has been placed on the influence of O radicals in flames [113],

particularly in high temperature flames where the O radical concentration is relatively high [194].

However, due to the coexistence of O with O2 and/or OH, it is difficult to experimentally differen-

tiate between oxidation via O versus oxidation by O2 and OH without molecular modeling. As a

result, many models do not explicitly consider oxidation by O, rather, this effect is implicit in the

rates used for O2 or OH.

In recent years there has been an increased interest in oxygen-enriched combustion (oxy-

fuel combustion) as a means of enabling carbon capture [187, 22]. Oxy-fuel combustion often

involves higher temperatures and higher H2O and CO2 concentrations due to flue gas recycling.

Most soot models have historically ignored gasification reactions, which tend to be small compared

to oxidation reactions in common combustion systems [51]. However, this may not be true in oxy-

fuel combustion, where the higher H2O and CO2 concentrations interact with particle surfaces and

lead to increased soot consumption [1].

Current research on soot consumption has placed large emphasis on the evolution of par-

ticle surface reactivity. Researchers have developed mechanisms reflecting the many elementary

70

chemical reactions [65, 67] and mechanical changes [173] occurring at the particle surface during

consumption. There is also ongoing research investigating correlations between particle surface

reactivity and the particle inception environment [153, 190].

Many experiments have been performed to investigate soot oxidation in premixed and non-

premixed flames. Fewer studies have been performed of soot gasification. In this paper, we analyze

data from 19 experiments to develop soot oxidation and gasification models to predict soot con-

sumption behavior over wide ranges of temperature and composition. To do this, we use Bayesian

inference to fit reaction model parameters to specified model forms. This method allows the model

to be easily extended to account for additional data sets, varying model forms, or more generic

problems.

4.2 Methods

This section describes the oxidation and gasification models along with the data sets used.

4.2.1 Oxidation Model

Although the process of soot oxidation is complicated, this study uses data collected from

experiments over the last several decades to fit a simple global model for use in simulation. This

model is based on irreversible, global oxidation reactions including:

C+O2 −−→ CO2, (4.1)

C+OH−−→ CO+H. (4.2)

This global model is both computationally inexpensive and simple, but still reasonably accurate.

This model is designed for use in large-scale simulations sensitive to computational cost. As this

consumption model will only require basic information for evaluation (local temperature, species

concentrations, and particle size), it reduces the number of transported and computed terms, which

can be costly in large-scale simulations.

This model is not a complete mechanism for soot oxidation and should be used with cau-

tion when considered for simulations outside of flames and may not be appropriate for detailed

71

simulations with fully resolved physics. A full mechanism for soot oxidation may contain hun-

dreds of possible reactions as soot particles react with various oxidizing species [61, 55]. Since

these reactions occur at the particle surface, considerations for gaseous species concentrations,

mass transport, and surface chemistry would all need to be included.

Due to the relatively small size of soot particles, soot oxidation models usually assume

particles are in the free molecular regime and transport limitations of oxidizing molecules from

the bulk gases within the flame to the particle surface are ignored. Transport effects may, however,

become important for large soot aggregates, especially in systems, such as coal combustion, for

which relatively high soot concentrations may be expected. Besides external transport, a com-

plete mechanism would need to consider particle surface and internal structure properties, such

as porosity, in a manner similar to char oxidation models [172]. Internal transport of O2 during

soot oxidation has also been studied recently and supports the relative minimal influence of porous

surface area to total soot oxidation rates [65, 68].

As the soot particle oxidizes, the surface chemistry changes and further affects later ox-

idation reactions [95]. When oxidation first begins, aliphatic branches first react with oxidizing

agents due to the weaker bonds holding these atoms to the particle surface. Once these branches

are all consumed, aromatic structures begin to break up, and depending on the size of the aromatic

cluster, will have varying activation energies. This means that the oxidation consumption reactions

are not uniform throughout the process of consumption but will vary in rate as the particle surface

chemistry changes. This level of detail, while important to note, is not normally considered in soot

modeling and is not used in the models presented here.

The following simple global model is proposed:

rox =1

T 0.5

(AO2PO2 exp

[−EO2

RT

]+AOHPOH

). (4.3)

Here, rox (kg/m2s) denotes the oxidation rate, T is temperature, A is an Arrhenius pre-exponential

factor, P is partial pressure, R is the gas constant, and E is an activation energy. This global model

is a modified Arrhenius equation with dependence on temperature and concentrations of O2 and

OH. Similar in form to previously developed models [74], it contains three fitted parameters: AO2 ,

EO2 , and AOH . Equation 4.3 assumes the following:

72

1. Oxidizing Agents Oxidation is assumed to occur by O2 and OH only. Contributions of

O are coupled with the oxidation by OH and not modeled separately. This is adequate for

the majority of flames. In flames, it was found that the OH and O account for most of

the consumption of soot [113], while in the TGA experiments nearly all consumption is

attributed to O2 and O [96]. In turbulent flames, higher mixing rates may allow for greater

interaction between O2 and soot than is found in laminar flames. As noted above, O rates

are not explicitly modeled but rather are absorbed with OH rates in flames and O2 rates in

TGA experiments, and so O oxidation is not explicitly considered here. Concentrations of

O2 and OH are taken from an equilibrated GRI mechanism according to temperature and

pressure. While equilibrated OH values may vary widely from actual OH concentrations,

equilibrium provided a standard which could be applied across all experimental data where

concentrations were not reported or attainable and equilibrium is a commonly used technique

for species predictions in large-scale simulations.

2. Transport Surface concentrations of oxidizing species are taken as the local concentrations

in the surrounding environment. Transport effects are then implicit in the pre-exponential

factor for the rate expressions.

3. Surface Chemistry This model does not attempt to capture changes in surface chemistry.

Conversion-dependent changes in rate coefficients are approximated with an effective acti-

vation energy. This effective activation energy is what is used for the O2 reaction, while the

effective activation energy for the OH reaction is considered to be negligibly small because

OH is such an effective oxidizer [74].

4.2.2 Oxidation Data

Experiments measuring soot oxidation have been carried out in many forms, and the liter-

ature contains many different studies. In this work, data were taken from 13 different sources and

typically fall under two different types of studies: those soot experiments performed with flames

and those in a non-flame environment. Most of the flame environments use a laminar flame; the

non-flame experiments mostly took place through thermogravimetric analysis (TGA), where soot

particles were exposed to an oxidizing environment at elevated temperatures. Table 4.1 summarizes

73

Table 4.1: Studies from which oxidation data were extracted for model development.

Study DataPoints

OxidizingAgent

Experiment Temperature (K)

Fenimore andJones, 1967 [47]

3 O2 & OH Pre-mixed EthyleneFlame

1530-1710

Neohet al., 1981 [141]

14 O2 & OH Laminar MethaneDiffusion Flame

1768-1850

Ghiassiet al., 2016a [65]

54 O2 & OH Premixed Varied-FuelFlame

1265-1570

Kimet al., 2004 [100]

2 O2 & OH Laminar EthyleneDiffusion Flame

1735-1740

Kimet al., 2008 [101]

3 O2 & OH Laminar EthyleneDiffusion Flame

1892-1916

Garoet al., 1990 [61]


1809-1851

Puriet al., 1994 [151]


1236-1774

Xuet al., 2003 [200]

15 O2 & OH Laminar MixedHydrocarbon Diffusion

Flames

1775-1900

Leeet al., 1962 [109]

29 O2 & OH Laminar MixedHydrocarbon Diffusion

Flame

1315-1660

Chanet al., 1987 [27]

14 O2 TGA 780-1210

Higginset al., 2002 [84]

28 O2 Tandem DifferentialMobility Analyzer

773-1348

Kalogirou andSama-

ras, 2010 [96]

6 O2 TGA 823-973

Sharmaet al., 2012 [170]

18 O2 TGA 823-923

74

the different experiments used for this study including the experimental method and the number of

data points.

Each of these experiments was performed differently and results were presented in different

ways.

As experimental uncertainties were not reported in the literature, a full analysis considering

both model and experimental uncertainties is not presented here. Quantified experimental errors

would improve the results presented in terms of the credible intervals (the Bayesian analog of

confidence intervals), and aid in ascribing variability to data and model forms.

All data needed to be converted to a common format for use in the proposed model. This

conversion of data, referred to as the instrumental model, involved making some assumptions about

the data or experimental conditions, thus introducing additional uncertainty. The instrument model

extracted a rate (measured in kgm−2 s−1), temperature (K), and species partial pressures (Pa) from

each data set to be used in the Bayesian analysis. A brief description of the experiments along with

some aspects of the instrumental model used are discussed below.

Fenimore and Jones [47] created soot with a fuel-rich ethylene premixed flame and the

soot was then fed to a second burner fired with a fuel-lean premixed flame. Oxidation rates were

taken from this second flame using quench probe measurements. Local gas temperatures were

reported and used to find local species concentrations assuming an equilibrium state of the GRI 3.0

mechanism in Cantera, a suite of object-oriented software tools for problems involving chemical

kinetics, thermodynamics, and transport processes [71].

Kim et al. [100, 101], Neoh et al. [141], and Xu et al. [200], all measured oxidation rates in

laminar diffusion flames. Local temperatures and concentrations of oxidizing agents were reported

along the flame. Rates were measured and converted to collision efficiencies for the different

oxidizing species, and these efficiencies were reported. For our study, these collision efficiencies

were converted back to rates through the following equation:

rox =ηimcr,iCiv

4, (4.4)

where ηi is the collision efficiency of species i, mcr,i is the mass of carbon removed due to the

oxidation by species i per mole of species i, Ci is the molar concentration of species i, and v is the

mean molecular velocity. Data from each of these experiments are assumed to be independent and

75

were all used to calibrate the Arrhenius pre-exponential factors and effective activation energies in

Equation 4.3 using Bayesian statistics.

Ghiassi et al. [65] used a two stage burner where a liquid fuel mixture was injected into

a premixed-fuel-rich region where soot particles were formed and then passed into a second fuel-

lean region where oxidation occurred. Particles were collected in the second region and analyzed

using a scanning mobility particle sizer. Rates of oxidation were extracted from the change in

particle size distribution in the fuel-lean region. Local temperatures and O2 concentrations were

measured while OH concentrations were modeled and reported by the experimenters.

Garo et al. [61] and Puri et al. [151] both measured oxidation rates of soot using laser-

induced fluorescence in a methane-air laminar diffusion flame. Temperatures, species partial pres-

sures, and oxidation rates were all reported. Reported values of O2 and OH were not used. Instead,

the calculated equilibrium values were used to preserve consistency between these data and other

collected data. Reported rate values were converted to units of kgm−2 s−1 for evaluation.

Chan et al. [27] and Lee et al. [109] each measured oxidation rates using a quench probe

in laminar diffusion flames burning propane and natural gas, respectively. Chan et al. performed

additional experiments using a TGA technique. For the flame, local gas temperatures were reported

along with oxidation rates. Those temperatures were used to find local concentrations of O2 and

OH along the flame front (stoichiometric point), assuming an equilibrium state of the GRI 3.0

mechanism in Cantera. The TGA temperature and rates were also reported along with a partial

pressure of O2 in the experimental setup. The reported rate values were converted to units of

kgm−2 s−1 for evaluation.

Higgins et al. [84], used a tandem differential mobility analyzer technique in which mono-

dispersed particles, collected from an ethylene diffusion flame, were subjected to an elevated tem-

perature in air and the change in particle diameter was measured. Particle diameter, temperature,

and residence time were reported. Rates were extracted by the experimenters from these data by

the following equation:

rox =ρs(d1−d2)

2t, (4.5)

where the density of the soot particles (ρs) was assumed to be 1850 kgm−3. The above equation

reflects the change of mass per surface area over a residence time of which the soot particle was

76

exposed to oxidizer. Partial pressures were again calculated using equilibrium of the GRI 3.0

mechanism.

Kalogirou and Samaras [96] and Sharma et al. [170], both used TGA techniques to record

oxidation rates of soot collected from a diesel engine. Reported data were temperature, O2 con-

centrations, and calculated rate constant (k) values of a single-step Arrhenius equation:

rox,rep = kXnO2. (4.6)

Kalogirou assumed a n=0.75 order dependence of O2, while Sharma assumed a 1.0 order depen-

dence and used the partial pressure of O2 rather than the molar fraction as displayed above. In

both cases, the Arrhenius equation gave rate data in units of s−1. These rates were converted to our

desired rates by:

rox =rox,repρsd1

6, (4.7)

where the soot density was again assumed to be 1850 kgm−3 and the initial particle diameter was

assumed to be 50 nm [74, 67]. This equation is a reflection of the mass of soot consumed per unit

of particle surface area.

4.2.3 Gasification Model

Gasification differs significantly from oxidation. Gasification generally has an endothermic

heat of reaction and products vary much more broadly. Examples of global gasification reactions

include

C+CO2 −−→ 2CO, (4.8)

C+H2O−−→ CO+H2, (4.9)

and these are the reactions used in this work. As with oxidation reactions, these global reactions

are considered irreversible. In the rate models presented below, the global nature of these reactions

is reflected in non-unity reaction orders. Tri-atomic species are particularly important in gasifica-

tion due to large amounts of potential energy contained within bond vibrations and rotations [22].

Sometimes the collision of these molecules with a soot particle results in the transfer of enough

77

energy to break bonds within the soot particle, similar to thermal pyrolysis. As a result of these

collisions and reactions, gasification tends to produce a larger variety of product species than oxi-

dation. Products of oxidation are usually limited to: CO, CO2, and H2O. Gasification reactions,

on the other hand, will often include these species along with H2, small hydrocarbons, alcohols,

carbonyls, and other species as products [130].

In oxy-fuel systems, the increased concentrations of CO2 and H2O are of interest. CO2is the most commonly considered gasification agent. H2O is often considered to be an oxidizer;

however, data in the literature has shown that the products of soot/H2O reactions are more indica-

tive of gasification than oxidation [130]. Other species are able to gasify as well, such as NO2, and

some research has been done on these reactions and rates [183, 32].

Like oxidation, gasification tends to be a complex surface reaction, dependent on many of

the same variables discussed above: transport effects, surface chemistry, and various gasification

agents [88]. As stated previously, gasification occurs via surface reactions with many different

possible species, especially high-internal-energy molecules with energy to transfer upon collision.

The model developed in this study only considers gasification by CO2 and H2O since these two

species are thought to be the only gasifying agents in high enough concentrations to have a notable

effect in either air-fired or oxy-fired boiler environments.

Although soot consumption via oxidation has long been an area of research, gasification of

soot has been much less studied. While gasification has long been discussed as a possible method

for removing soot build-up on filters in diesel engines, relatively little experimentation has been

done and gasification rates are not well known. In recent years there has been increased interest in

solid-fuel gasification for use in combined turbine cycles. During this gasification process, soot has

the potential to form and researchers have begun exploring soot models for these systems. Due to

the absence of oxygen in these systems, the only source of soot consumption is gasification. As a

result, there have been a few recent studies that consider gasification of soot, particularly biomass-

derived soots. These experiments, along with a few others found in the literature, are used to form

the proposed model of this study.

78

This model consists of two additive rate terms for gasification by CO2 and H2O:

rgs = rCO2 + rH2O, (4.10)

rCO2 = ACO2P0.5CO2

T 2 exp(−ECO2

RT

), (4.11)

rH2O =AH2OPn

H2O

T 1/2 exp(−EH2O

RT

). (4.12)

Rates in these equations are defined in units of (kg/m2s). Equation 4.11 represents gasification

due to attack by CO2 with a modified Arrhenius equation dependent on temperature and the par-

tial pressure of CO2. The CO2 order of reaction was extracted from Ref. 95. The temperature

dependence order was set after a series of statistical fittings to limit the number of adjustable pa-

rameters. Equation 4.11 contains two adjustable parameters: the Arrhenius pre-exponential factor

and activation energy, that are fit empirically to data with Bayesian statistics , as described below.

Equation 4.12 represents gasification by H2O. Like Equation 4.11, Equation 4.12 also

contains temperature and partial pressure dependencies, two similar adjustable parameters, and

a third adjustable parameter n for the H2O order of reaction. These two equations are analyzed

separately because researchers have studied gasification by CO2 and H2O independently.

4.2.4 Gasification Data

Table 4.2 summarizes the gasification data used here. The data are limited but represent

the experimentation done with regard to soot gasification found in the literature. More data are

desirable to obtain a more robust model, and one purpose of this study is to present a method that

can easily incorporate additional data as they become available.

Like the oxidation experiments, each of the gasification experiments was performed differ-

ently and results were presented in various ways. As for the oxidation experiments, uncertainties

for gasification were not included in the literature, however they are believed to be larger than

the uncertainties found in the oxidation experiments since the magnitude of gasification rates are

smaller than those for oxidation and thus small measurement errors yield higher relative errors.

These larger uncertainties are reflected in larger uncertainties in the model as well. In order to use

these data in the proposed model, each data point had to be converted to an instrumental model.

79

Table 4.2: Studies from which gasification data were extracted for model development.

Study # of Data Points Gasifying Agent Temperatures (K)Abian et al., 2012 [1] 14 CO2 1132-1650

Kajitani et al., 2010 [95] 6 CO2 1123-1223

Qin et al., 2013 [153] 3 CO2 305-1261

Otto et al., 1980 [144] 2 H2O & CO2 1066-1160

Arnal et al., 2012 [8] 6 H2O 1273

Chhiti et al., 2013 [31] 28 H2O 1373-1673

Neoh et al., 1981 [141] 14 H2O 1777-1815

Xu et al., 2003 [200] 15 H2O 1770-1840

The following is a brief description of each experiment along with some aspects of the instrumental

model used.

Abian et al. [1] produced soot particles in an ethylene diffusion flame. These particles

were collected and placed in a TGA under a N2/CO2 environment. The partial pressure of CO2was set and temperature was calculated given the elapsed time and a constant heating rate. Rates

of consumption were measured as the particles were heated and these rates were reported as a

conversion of the original mass over time. This reported rate was converted to kgm−2 s−1 using the

original sample mass along with an assumed initial particle diameter of 50 nm. Soot samples were

prepared under different environments by varying feed rates into the original ethylene diffusion

flame; however, it was found that the gasification rate minimally depended on the environment in

which the soot was produced. For the purposes of this model, that dependence was accounted for

by taking an average rate across all samples collected in different environments.

Kajitani et al. [95] and Qin et al.[153] also used a TGA to measure the reactivity of soot

collected from biomass derived soots. Both reported partial pressures of CO2 within the TGA as

well as conversion of soot particles as the experiment progressed. Rates were extracted using the

given particle heating rates along with an assumed initial particle diameter of 50 nm. Of particular

note is the observation made by Qin et al. that soot particles have a significant difference in

reactivity compared to char particles. Kajitani et al. remarked that the surface chemistry of soot

seemed to change throughout the experiment but minimally affected rates of gasification.

80

Otto et al. [144] were the first to experiment on soot gasification by collecting diesel soot

on filters and exposing that soot to exhaust gas from four CVS-CH cycles. TGA experiments were

carried out first with H2O as the gasifying agent and then repeated with CO2. Rates (µg/m−2s),

partial pressures of the gasifying agents, and temperatures were reported. Otto et al. noted that

data collected for CO2 gasification should be used with caution due to low accuracy.

Arnal et al. [8] used a flow reactor to study the water vapor reactivity of Printex-U, a

commercial carbon black considered as a surrogate for diesel soot. Temperatures and the changing

concentrations of CO, CO2, and H2 were reported. Assuming the only source of carbon in the

system came from the Printex-U, we determined a rate of soot consumption as the CO and CO2concentrations increased. Once again an initial particle diameter of 50 nm was assumed.

Chhiti et al. [31] explored soot gasification by H2O in bio-oil gasification using a lab-scale

Entrained Flow Reactor, and reported the soot yield and temperature over time. Soot particles were

added to the reactor and first pyrolyzed in an inert environment over a given amount of time. This

was repeated in an environment containing a reported partial pressure of H2O. The gasification

rate was determined assuming a constant number of particles that lost mass uniformly from all

particles.

The experiments of Neoh et al. [141] and Xu et al. [200] included H2O reactions, and these

were described in the previous section.

Data from each of these experiments are assumed to be independent and are all used to

calibrate the parameters in the gasification model, Equations 4.10-4.12.

4.3 Bayesian Implementation

In this study, Bayesian inference, detailed in Chapter 2 is used to determine the probability

of a set of parameters describing the oxidation and gasification models based on the collected data.

Here, an example is detailed showing the steps taken to calibrate parameters in the H2O portion of

the gasification model found in Equation 4.12. As noted in Chapter 2, the effective dimensionality

of the system needed for evaluation is np + 1, as we use a Gaussian distribution to determine

the likelihood function which introduces one additional nuisance parameter (σ ) which is given a

Jeffrey’s prior. We discretize the parameter space domain using a structured np + 1 dimensional

grid stored as an np +1 dimensional array.

81

1. The parameter values in each dimension were initially determined over a very broad range

within the physically possible space. This range was refined to smaller ranges with multiple

iterations of these steps to where there was some detectable probability in order to better

detail the posterior presented in this work. The gasification by H2O, Equation 4.12, contains

np=3 adjustable parameters: AH2O, n, and EH2O. The final ranges over which these and all

other parameters were tested are shown in Table 4.3.

2. The selected ranges are discretized into a series of potential parameter values to be tested in

different combinations. 150 points were used for all parameters. Logarithmic spacing was

used for all parameters except ECO2 , EH2O, and n, which had linear spacing.

3. A prior needs to be established. In this study, a uniform prior was used for the model param-

eters, meaning that all combinations of model parameters had uniform probability. Jeffrey’s

prior was used for the σz values. The uniform prior for the model parameters was subsumed

in the posterior’s normalization constant and not explicitly considered.

4. For the current experiment, at a given point in the np + 1 dimensional grid (corresponding

to a given value of x) modeled rates are computed for each experimental data point. A

combination of parameters is selected to be tested against every data point. From these

parameters and in computing the modeled rates, the secondary data collected from literature

(partial pressures and temperature) are used that correspond to each experimental data point.

For H2O gasification, the modeled rates are computed using using Equation 4.12.

5. These modeled rates are compared to the rates given by the data using Equation 2.18 to

calculate a likelihood that this combination of parameters describes a data point. For a given

grid point (a given value of x), the likelihood for all points in a given experiment is the

product of the likelihoods for the individual data points.

6. This likelihood value is multiplied by the Jeffrey’s prior for the σz, and the uniform prior

(done implicitly) for the rate model parameters. The product is a posterior value at the given

grid point x for the given experiment.

82

7. The previous three steps are repeated for each point in the np + 1 dimensional grid. The

result for H2O gasification is a four dimensional array holding the (unnormalized) posterior

PDF for the given experiment.

8. This posterior is then marginalized to remove the σz dimension by numerically integrating

over all points that shared the same Arrhenius pre-exponential factor, activation energy, and

reaction order. That is, by integrating along grid lines in the σz direction. The resultant three-

dimensional unnormalized PDF is the discretized posterior. This posterior can be easily

normalized to yield a true PDF so that its (numerical) integral is one [91, 62].

9. Steps 4-8 are now repeated for the second (and subsequent) experimental data sets. The final

posterior f (x|y, I) is then the product of the posterior terms for the individual experiments.

Equivalently, the posterior from step 8 for the previous experiment can be used as the prior

of the model parameters for the current experiment since the likelihood is multiplied by the

prior in step 6. In this case, a final multiplication of the posterior terms for the individual

experiments is not needed since the product is built up sequentially. This interpretation is

consistent with the Bayesian approach of making use of prior information as it becomes

available. The order in which the experiments are processed does not affect the final poste-

rior, nor does it matter if all the data in the experiments are evaluated in one step or several,

as long as each data point is only evaluated once.

10. A final one dimensional PDF for each individual parameter is produced by marginalizing

the multi-dimensional PDF to each parameter. This is done similarly to the marginalization

in step 8 above. For a given single parameter of interest (PoI), the np− 1 dimensional grid

at each value of the discrete PoI is numerically integrated and the result is normalized so

that the PDF integrates to one. For H2O gasification, with EH2O as the PoI, we have the

numerical equivalent of

f (EH2O|y) =∫∫

f (x|y)dAH2Odn. (4.13)

83

Table 4.3: Range over which model parameters were tested.

Equation Parameter Range

4.3AO2 1E-2 to 1E2

EO2 1E5 to 2.51E5

AOH 3.16E-4 to 1E-2

4.11ACO2 1E-18 to 1E-15

ECO2 0 to 3E4

4.12AH2O 1E2 to 3.16E7

EH2O 1E5 to 5E5

n 0 to 1

4.4 Results

This section contains results of the Bayesian analysis as applied to the aforementioned data

sets. It is important to note that these results are not to be considered absolute but, due to the nature

of Bayesian statistics, can and should be updated as more experimental data become available. This

is especially important for soot gasification where few data are currently available in the literature.

4.4.1 Oxidation Model

Results for the parameter calibration of Equation 4.3 can be seen in Figure 4.1. The three

diagonal figures are the resultant marginal PDFs of each of the adjustable parameters. Each PDF

is approximately lognormal in appearance. It is interesting to note that the curve for AO2 is much

more broad than AOH : the marginal PDF of AO2 spans over 2 full orders of magnitude, while that

for AOH spans less than one order of magnitude. This is due to the relative importance of these

two parameters and the influence of slight variations on the overall rate. In the flame experiments,

oxidation by OH is the predominant mechanism of oxidation and tends to influence overall rates

more than oxidation by O2. As a result, the flame experiments defined AOH the OH Arrhenius

constant more distinctly than AO2 . EO2 has a sharp peak compared to either AO2 or AOH . This

peak is due to the TGA experiments, which were dominated by O2 oxidation. Slight variations

in EO2 had a stronger impact on overall rate than AO2 the O2 Arrhenius constant variations and

84

1e-2 1e-1 1e0 1e2 1e-2

AO

2

0

0.2

0.4

0.6

Marg

inal P

oste

rior

1e-2 1e-1 1e0 1e2 1e-2

AO

2

1.5e5

1.7e5

2.0e5

EO

2

1.5e5 1.7e5 2.0e5

EO

2

0

0.5

1

Marg

inal P

oste

rior ×10-4

1e-2 1e-1 1e0 1e2 1e-2

AO

2

1e-3

2e-3

5e-3

AO

H

1.5e5 1.7e5 2.0e5

EO

2

1e-3

2e-3

5e-3

AO

H

1e-3 2e-3 5e-3

AOH

0

500

1000

1500

Marg

inal P

oste

rior

Figure 4.1: PDFs of each of the oxidation parameters in Equation 4.3. Contours indicate jointPDFs.

was therefore more defined. The mode of each of the marginal PDFs is reported in Table 4.4

as the calibrated parameters for Equation 4.3; credible intervals are also shown. The value of

AOH =1.89E-3 kgK1/2

Pam2 scorresponds to a collision efficiency of 0.15, which is consistent with

previous literature values (see Ref. 67 for a discussion).

The off-diagonal plots of Figure 4.1 are contour plots showing the two-variable PDFs be-

tween the three different parameters. The top of these three plots shows a heavy correlation be-

tween AO2 and EO2 . A correlation is to be expected because these two parameters are used in

combination to describe the oxidation reaction as occurs by the O2 molecule. There is a positive

correlation between EO2 and AO2 , which is consistent with an increase in AO2 being offset by an

increase in EO2 for a given rate. The shape of the correlation is consistent with the model form.

In contrast to the EO2/AO2 PDF, the AOH/AO2 and AOH/EO2 PDFs show little correlation between

85

Table 4.4: Calibrated parameters for soot oxidation, Equation 4.3.

Variable Value90% Credible Interval

UnitsLower Bound Upper Bound

AO2 7.98E-1 1.94E-1 5.15E0 kgK1/2

Pam2 sEO2 1.77E5 1.57E5 1.94E5 J

mol

AOH 1.89E-3 1.06E-3 3.14E-3 kgK1/2

Pam2 s

their respective parameter pairs. The correlation that is present is slightly negative so that increases

in AO2 and EO2 result in decreases in AOH . These low correlations are due to the nature of the ex-

periments from which data was derived. Oxidation in TGA experiments were due entirely to the

O2 mechanism, whereas oxidation in flame experiments were dominated via the OH mechanism.

Figure 4.2 shows the agreement between rate data collected from the literature and the rates

predicted by the calibrated model for soot oxidation by O2 and OH. This figure displays a parity

plot of model calculated rates and literature reported rates. The solid line indicates perfect agree-

ment between the model and the data, so the degree of scatter about this line is a measure of the

error in the model and scatter in the measured data. The R2 statistic (coefficient of determination),

using log10 rates, is 0.75 for this comparison. As can be seen in the figure, there is reasonable

agreement between the data and the model with large deviations occurring in only a few data sets.

For reference, the data span eight orders of magnitude.

For comparison, Figure 4.3 shows another parity plot between the collected rates and the

rates predicted by the Nagle/Strickland-Constable (NSC) model [140]. Here, R2 = 0.65. The NSC

model represents the oxidation of graphite by O2. As can be seen in the figure, the NSC model

tends to over-predict oxidation of soot particles for TGA experiments and under-predict oxidation

for flame experiments where OH is significant, indicating a significant difference between soot

and graphite surface chemistries. Another common model uses a combination of the NSC O2 and

Neoh OH oxidation models (using a collision efficiency of 0.13, as found by Neoh et al.[141]).

Figure 4.4 shows the agreement between the collected data and data predicted by this combined

model. Here, R2 = 0.71. While this combined model does better than the NSC model alone at

predicting soot oxidation, the calibrated model is slightly more accurate (R2 value of 0.75 vs 0.71).

86

10-10

10-8

10-6

10-4

10-2

100

Measured Rates (kg/m2*s)

10-10

10-8

10-6

10-4

10-2

100

Ca

lcu

late

d R

ate

s (

kg

/m2*s

)

Fenimore

Neoh

Ghiassi

Kim

Garo

Puri

Xu

Lee

Chan

Higgins

Kalogirou

Sharma

Figure 4.2: Comparison of predicted rates of soot oxidation by calibrated, with parameters in Table4.4, model and those rates collected from the literature. Those experiments that are measured onlyoxidation by O2, such as TGA, are filled symbols (R2 = 0.75).

The improvement is modest, however, and indicates the NSC/Neoh combined model is nearly

optimal over a wide range of reported oxidation rates. This is an unexpected but important result.

While it is not the authors’ expectation that the proposed model replace the well-established

NSC/Neoh combined model on the basis of our results, the use of Bayesian statistics for calibra-

tion allows for the quantification of parameter uncertainty as shown in Figure 4.1, such a joint-

parameter PDF is not available for parameters in the NSC/Neoh model. The similarity between the

NSC/Neoh and the calibrated oxidation models lends confidence to our application of Bayesian

statistics to the calibration of the soot gasification models, for which there are no strongly estab-

lished models in the literature.

4.4.2 Gasification Model

H2O Gasification

Results for the parameter calibration of H2O gasification are presented in Figure 4.5 and

Table 4.5. As in the above discussion, this figure contains the parameter marginal PDFs on the

87

10-10

10-8

10-6

10-4

10-2

100


10-10

10-8

10-6

10-4

10-2

100

Ca

lcu

late

d R

ate

s (

kg

/m2*s

)

Fenimore

Neoh

Ghiassi

Kim

Garo

Puri

Xu

Lee

Chan

Higgins

Kalogirou

Sharma

Figure 4.3: Comparison of oxidation rates as predicted by the NSC oxidation model [140] andthose rates collected from the literature (R2 = 0.65).

10-10

10-8

10-6

10-4

10-2

100


10-10

10-8

10-6

10-4

10-2

100

Ca

lcu

late

d R

ate

s (

kg

/m2*s

)

Fenimore

Neoh

Ghiassi

Kim

Garo

Puri

Xu

Lee

Chan

Higgins

Kalogirou

Sharma

Figure 4.4: Comparison of oxidation rates as predicted by the NSC oxidation model combinedwith Neoh et al.[141] calculated collision efficiency for OH and those rates collected from theliterature (R2 = 0.71).

88

Table 4.5: Calibrated parameters for H2O gasification of soot, Equation 4.12.



AH2O 6.27E4 8.31E3 2.47E7 kgK1/2

Pan m2 sEH2O 2.95E5 2.66E5 3.26E5 J

moln 0.13 0.02 0.46 —–

diagonal plots and contour plots showing the two parameter join-PDFs between parameters on the

off-diagonal plots. Modes of the marginal PDFs are given in Table 4.5. As expected, the marginal

PDFs show fairly clear distributions that could be characterized as approximately lognormal (nor-

mal for n). The PDF for the reaction order was only taken out to zero because a negative reaction

order was not considered in the form of this global model.

There exists an almost linear correlation between EH2O and the log of AH2O, indicating a

close linking between these two parameters, as was noted for the oxidation reaction above. How-

ever, there is a much different correlation between the reaction order n and either EH2O or AH2O,

with nearly round contours until the reaction order n drops to low levels. This shape of contour

implies that the H2O reaction order is fairly independent of the other two parameters, except at

low values of n, where there appears to be a positive correlation between n and EH2O or AH2O.

This indicates that the rates are mostly governed by AH2O and EH2O, unless the reaction order is

sufficiently low (on the order of 0.5 or less), where the other parameters must be adjusted to com-

pensate. Figure 4.6 shows the agreement between data collected from the literature and calibrated

model prediction using a parity plot like that shown in the previous section. The rate data measured

and predicted span ten orders of magnitude. The agreement between the calibrated model and the

data is relatively good, with most predictions within an order of magnitude of the data. Note that

individual data sets show consistent bias with respect to the model. For example, the model tends

to consistently over-predict the Chhiti data. Considering only a single data set normally would

allow better agreement than when considering all sets together.

89

1e2 1e4 1e5 1e7

AH

2O

0

2

4

Marg

inal P

oste

rior ×10-6

1e2 1e4 1e5 1e7

AH

2O

2.5e5

3e5

3.5e5

EH

2O

2.5e5 3e5 3.5e5

EH

2O

0

2

4

Marg

inal P

oste

rior ×10-5

1e2 1e4 1e5 1e7

AH

2O

0

0.25

0.5

n

1e5 4e5 7e5

EH

2O

0

0.25

0.5

n

0 0.25 0.5

n

0

2

4

6

Marg

inal P

oste

rior

Figure 4.5: PDFs of each of the H2O gasification parameters in Equation 4.12.

CO2 Gasification

Results for the parameter calibration of CO2 gasification are shown in Figure 4.7. The

two diagonal plots are the marginal PDFs for the two adjustable parameters in Equation 4.11. The

modes of these two PDFs are given in the Table 4.6. The PDF for the activation energy was cut off

at zero, and negative activation energies were not considered. The PDF value at an ECO2 value of

0 implies that a straight ACO2 with no exponential activation energy term,

rCO2 = ACO2P0.5CO2

T 2, (4.14)

could be used to describe the data, but not as well as the current proposed model. The authors

expect that more data would support the form of this model and the activation energy PDF would

90

10-15 10-10 10-5 100


10-15

10-10

10-5

100

Ca

lcu

late

d R

ate

s (

kg

/m2*s

)

Arnal

Chhiti

Neoh

Otto

Xu

Figure 4.6: Comparison of predicted rates of soot gasification via H2O by calibrated model, pa-rameters in Table 4.5, and those rates collected from the literature (R2 = 0.87 minus Neoh Data).

Table 4.6: Calibrated parameters for CO2 gasification of soot, Equation 4.11.



ACO2 3.06E-17 1.17E-17 1.57E-16 kgPa1/2 K2 m2 s

ECO2 5.56E3 6.04E2 1.95E4 Jmol

become more narrow within the positive range. The full PDF of these parameters is shown in

the contour plot in Figure 4.7. As can be seen in this plot, ECO2 and the log of ACO2 are highly

correlated in a linear relationship, as expected by the model form.

Figure 4.8 shows the parity plot of the data and the calibrated model for the CO2 gasifica-

tion rates. A large amount of scatter is seen in this plot and the model is much less accurate than

for the oxidation and H2O gasification rates. This discrepancy is due to the combined effects of in-

consistencies between experiments, and the inability of the model form chosen to reproduce these

data sets as accurately. The data in the four sets span approximately four orders of magnitude. The

model captures the measured rates within an order of magnitude for most of the data points.

91

8e-18 3e-17 1e-16

ACO

2

0

2

4

Ma

rgin

al P

oste

rio

r ×1016

8e-18 3e-17 1e-16

ACO

2

0

1e4

2e4

EC

O2

0 1e4 2e4

ECO

2

0

0.5

1

1.5

Ma

rgin

al P

oste

rio

r ×10-4

Figure 4.7: PDFs of each of the CO2 gasification parameters in Equation 4.11.

10-12

10-11

10-10

10-9

10-8

10-7

10-6


10-12

10-11

10-10

10-9

10-8

10-7

10-6

Ca

lcu

late

d R

ate

s (

kg

/m2*s

)

Abian

Kajitani

Otto

Qin

Figure 4.8: Comparison of predicted rates of soot gasification via CO2 by calibrated model, pa-rameters from Table 4.6, and those rates collected from the literature (R2 = 0.62).

92

10-12 10-10 10-8 10-6


10-12

10-10

10-8

10-6

Ca

lcu

late

d R

ate

s (

kg

/m2*s

)

Kajitani

10-12 10-10 10-8 10-6


10-12

10-10

10-8

10-6

Ca

lcu

late

d R

ate

s (

kg

/m2*s

)Qin

10-12 10-10 10-8 10-6


10-12

10-10

10-8

10-6

Ca

lcu

late

d R

ate

s (

kg

/m2*s

)

Abian

10-12 10-10 10-8 10-6


10-12

10-10

10-8

10-6

Ca

lcu

late

d R

ate

s (

kg

/m2*s

)

Otto

Figure 4.9: Comparison of predicted rates of soot gasification via CO2 by individually calibratedmodels and those rates collected from the literature.

Figure 4.9 shows the same parity plots as above, but here the gasification model has been

individually calibrated to each data set instead of all the data sets combined. As can be seen in the

figure, the proposed model fits three of the four data sets, with some difficulty in fitting the data

measured by Kajitani et al. [95] This indicates that the form of the model used was reasonable but

there may be differences between data sets that could be explored more thoroughly.

Gasification rates tend to be much smaller than oxidation rates—small enough that simple

thermal pyrolysis of soot samples may not be considered negligible in these experiments. As

a result, some of the experiments may appear to gasify faster than others due to differences in

pyrolysis. In addition, the structure of the soot particle surface may have a much larger impact

on gasification than on oxidation. Two of these experiments were carried out with the expressed

purpose of exploring changes in the rate as the surface chemistry changed over time [95, 1]. The

model used here does not account for such changes. Despite these and other factors, the model

93

1e-18 1e-17 1e-16 1e-15ACO2

0

0.5e15

1.0e15

1.5e15

2.0e15

2.5e15

Mar

gina

l Prio

r

1e-18 1e-17 1e-16 1e-15AH2O

0

1.0e4

2.0e4

3.0e4

E CO

2

0 1.0e4 2.0e4 3.0e4ECO2

1.0e-5

2.5e-5

5.0e-5

6.5e-5

Mar

gina

l Prio

r

Figure 4.10: Model-informed priors for the CO2 gasification model. Derived with mode values atACO2=3.06E-17 and ECO2=5.56E3.

form chosen was the best of those tested. As more experimentation is carried out and more data

become available in the literature, a more accurate model should be compiled and calibrated using

the techniques discussed in this study.

4.4.3 Rate-Informed Priors

The previous section’s work was repeated but instead of using uniform priors for each anal-

ysis, model-informed priors were used as described in Section 2.2.1. In order to establish these

model-informed priors, an initial calibrated-parameter vector is used from which contours radiate

out. The shape of the contours is model-defined. In this exercise we used the calibrated param-

eters from the previous sections to create the model-informed priors and then reran the Bayesian

inference using those priors instead of the uniform priors. The analysis in this section will display

results from the CO2 gasification first, as there are only two calibrated parameters and correlations

are easier to see and understand, then the oxidation model, a three parameter model, will be shown.

94

8e-18 3e-17 1e-16ACO

2

0

2

4

Mar

gina

l Pos

terio

r #1016

8e-18 3e-17 1e-16ACO

2

0

1e4

2e4

E CO

2

0 1e4 2e4ECO

2

0

0.5

1

1.5

Mar

gina

l Pos

terio

r #10-4

Figure 4.11: PDFs of each of the oxidation parameters in Equation 4.3 derived using the model-informed priors of Figure 4.10. Contours indicate joint PDFs.

Figure 4.10 is the model-informed prior for Equation 4.11, the CO2 portion of the gasifi-

cation model. The ACO2 and ECO2 correlation, seen in the bottom-left plot, appears linear if plotted

on a semi-xlog plot as ellipses radiating from the mode. This is due to the relation between ACO2

and ECO2 found in the model where ACO2 is outside the exponential function while ECO2 is within

it. Although these ellipses radiate out from the PDF mode, they are not centered on the mode. This

is because as ACO2 gets smaller and ECO2 gets larger, the overall model consistently predicts rates

closer and closer to zero, there is a maximum residual error that can be obtained, in this direction,

where the rate equals zero. On the other hand, as ACO2 gets larger and ECO2 gets smaller, residual

error will consistently get larger and larger, well beyond the residual error which would result if

the overall predicted rate was zero. Residual error is inversely proportional to probability; there-

fore, smaller values of ACO2 are more probable than larger values, as reflected in the top plot, a

marginalized PDF of the ACO2 prior. On the same thread, large values of ECO2 are more probable

than small values as seen in the bottom-right plot, a marginalized PDF of the ECO2 prior. This

example shows how marginalization can often wash out the finer details of the overall PDF.

When using this model-informed prior in the Bayesian calibration the result can be seen

in Figure 4.11. This figure is very similar to Figure 4.7 indicating the method we used to assign

95

confidences to the model-informed priors’ contours did not reflect a high confidence in the prior.

Rather, the analysis still heavily favors the data collected from the literature rather than the model

form. However, there is one notable difference between the figures brought about by the model-

informed prior. Note in the bottom-left plot, a small kink in the ellipse contours toward the higher

values of ACO2 . This deformity is consistent with the model-informed prior, which clearly favored

higher values of ACO2 , and confirms an earlier suspicion that this model is not perfectly consistent

with the data and it is possible/probable that another yet untested model form would do better in

describing the CO2 gasification data.

Figure 4.12 shows the model-informed prior for Equation 4.3 for soot oxidation. Unfortu-

nately, because this is a three parameter system, some of the finer details of the full 3-dimensional

prior have been washed out by marginalization. For example, the overall prior mode is given in

the figure caption; however, this is not the mode of the individual plots as the overall trend of the

parameter probabilities washed out the prior peak. There is still a lot of information to be derived

from this figure even with the washed-out details. As with the CO2 prior, the AO2 /EO2 correlation

is linear when plotted on a semi-xlog plot, consistent with the expectation discussed before. The

relations between AOH and the other parameters is more telling, showing that as AOH increases in

value, the value of the other parameters becomes less predominate in some sort of uncharacterized

exponential relation. This is because we have assumed EOH to equal zero, therefore AOH quickly

becomes the most important parameter in determining overall rates. The overall probability of AOH

favors smaller values for the same reason stated for the ACO2 parameter in the above paragraphs.

Figure 4.13 is also very similar to Figure 4.1, for the same reason stated before for the

CO2 gasification model. However, in that analysis we saw a slight deformity arise with the use

of a model-informed prior. In this analysis, we see no such deformity indicating the form of

the proposed model was excellent for describing the data collected from the literature, further

validating the analysis performed. In fact, the only difference between the analysis with the model-

informed prior and with the uniform priors is a slight, almost unperceivable, narrowing of the

contours and marginal parameter PDFs, indicating we could easily assign a higher confidence

to the contours in the model-informed prior and thus increasing the confidence in our analysis.

Calibrated ‘optimal’ parameters do not change using these rate-informed priors.

96

1e-2 1e-1 1e0 1e1 1e2AO2

0.010

0.012

0.014

0.016M

argi

nal P

rior

1e-2 1e-1 1e0 1e1 1e2AO2

1.0e5

1.5e5

2.0e5

2.5e5

E O2

1.0e5 1.5e5 2.0e5 2.5e5EO2

0

1e-5

2e-5

3e-5

4e-5M

argi

nal P

rior

1e-2 1e-1 1e0 1e1 1e2AO2

4e-4

1e-3

3e-3

1e-2

A OH

1.0e5 1.5e5 2.0e5 2.5e5EO2

4e-4

1e-3

3e-3

1e-2

A OH

10 3 10 2

AOH

60

80

100

120

140M

argi

nal P

rior

Figure 4.12: Model-informed priors for the oxidation model. Derived with mode values atAO2=7.98E-1, EO2=1.77E5, and AOH=1.89E-3.

As model form for the H2O gasification model were similar enough to the two models

above that no additional insight was expected to be gained from this additional analysis.

4.4.4 Rate Prediction

The results of Sections 4.4.1 and 4.4.2 can be used to predict soot consumption rates along

with a quantified uncertainty for those predictions. This is illustrated in this section using, for

instance, the Higgins et al. [84] data for soot oxidation.

97

1e-2 1e-1 1e0 1e2 1e-2AO

2

0

0.2

0.4

0.6

Mar

gina

l Pos

terio

r

1e-2 1e-1 1e0 1e2 1e-2AO

2

1.5e5

1.7e5

2.0e5

E O2

1.5e5 1.7e5 2.0e5EO

2

0

0.5

1

Mar

gina

l Pos

terio

r #10-4

1e-2 1e-1 1e0 1e2 1e-2AO

2

1e-3

2e-3

5e-3

A OH

1.5e5 1.7e5 2.0e5EO

2

1e-3

2e-3

5e-3

A OH

1e-3 2e-3 5e-3AOH

0

500

1000

1500

Mar

gina

l Pos

terio

r

Figure 4.13: PDFs of each of the oxidation parameters in Equation 4.3 derived using the model-informed priors of Figure 4.12. Contours indicate joint PDFs.

In Figure 4.14, a PDF of the soot oxidation rate in Equation 4.3 is shown for a single data

point measured by Higgins where the flame has a temperature of 1225 K and partial pressures of

PO2 = 21,300 Pa and POH = 6.22E-7 Pa. This PDF is obtained from the full joint PDF calculated

for the oxidation parameters and displayed in Figure 4.1. Each combination of parameters tested

results in a calculated rate; the associated probability with that combination of parameters is equal

the probability of the calculated rate. Just as with the marginal PDFs displayed in Figure 4.1, a

normalization constant is computed and used to determine the final PDF of Figure 4.14.

The vertical line in the figure indicates the measured rate reported by Higgins and falls near

the center of the calculated PDF. This PDF was calculated using discrete bins. The width of the

98

10-7

10-6

10-5

10-4

10-3

Rate (kg/m2 s)

0

0.5

1

1.5

2

2.5

3

3.5

Pro

ba

bili

ty

×104

Figure 4.14: PDF of the calculated gasification rate in Higgins experiment where the flame datawas at 1200 K.

calculated PDF indicates the uncertainty in this calculation. As more data are analyzed from the

literature this PDF will narrow and the uncertainty will shrink.

Figure 4.15 shows the comparison of multiple data points measured by Higgins compared

to the model predicted rates. There were two independent measurements taken out at each temper-

ature by the experimenters, and all measurements are shown in this plot. This figure also shows

a 90% credible interval evaluated from the calculated PDF at each point. Like Figure 4.14, Fig-

ure 4.15 indicates that with the current analysis there is a moderate degree of uncertainty in the

oxidation model, but all the reported rates lie close to the center of the calculated uncertainty

bounds.

99

1050 1100 1150 1200 1250 1300 1350

Temperature of the Flame (K)

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Ra

te o

f S

oo

t O

xid

atio

n (

kg

/m2 s

)

Model Predicted

Measured Rate

90% Credible Interval

Figure 4.15: Comparison of the model predicted oxidation rate with confidence bounds versus themeasured rate in Higgins’s experiment.

4.5 Discussion

The previous section demonstrated the use of Bayesian statistics to calibrate global models

for soot consumption. This method of model calibration has a few advantages and disadvantages

over more traditional model calibration techniques, such as minimization of summed square error.

The first clear advantage of using a Bayesian calibration method, compared to that of a

least-summed-squares, is the production of a full PDF for the parameter-space from which un-

certainty quantification can be easily extracted. Other methods of extracting uncertainty from

calibrated parameters assume a fixed PDF for the parameter space and test from that distribution

using either Student’s t-test or an f-test.

This full PDF comes at a cost. The computational cost of a full Bayesian analysis scales by

a power equal to the number of parameters used in the models plus any nuisance parameters. In the

case of the soot consumption model calibrated in this study, when the parameter space of the oxi-

dation model was doubled the number of computations required was increased sixteen-fold. There

are methods to reduce the computational costs of a Bayesian analysis such as the use of Markov

chain Monte Carlo (MCMC) methods. MCMC methods are a class of algorithms for sampling

100

from the probability space based on the use of a Markov chain that evolves a posterior distribution

through sampling until an equilibrium is obtained. These algorithms are an intense field of research

and results have become very robust and hold much promise for parameter calibration in simple

and complex models [76, 75]. Even with such improvements, least-summed-squares usually re-

quires only a fraction of the computation cost. However, for the present study, computational costs

did not limit the technique.

In principle, the final result of a least-summed-squares calibration and a Bayesian calibra-

tion should yield the same results [91]. Both methods are based on the use of the Gaussian Distri-

bution found in Equation 2.18. Because σ is a nuisance parameter, to maximize the probability of

this distribution the numerator of the exponential should be minimized:

max(p(yz,i|µz(x),σz)) = min((yz,i−µz(x))2) , (4.15)

which is the basis of least-summed-squares.

In the case of Bayesian calibration, this distribution is used as the likelihood function. Once

the probability space is calculated the mode is used as the calibrated parameter set. In this study, the

modes of the marginal parameter PDFs were used instead of the absolute mode of the probability

space, but these tend to be the same for simple, single-peaked topologies, as occur in Figs. 4.1,

4.5, and 4.7. If the probability surface topology is more complex, e.g., with multiple peaks of

high probability, the mode of the probability space will differ from the mode of the parameter-

marginal PDFs. This is an indication that there is likely disagreement between data sets and the

proposed model, and is clearly indicated by the Bayesian processes, in contrast to a least-squared-

sum analysis that would not necessarily reveal this discrepancy.

The Bayesian analysis presented is a calibration technique for parameters of a given model.

This analysis is not strictly a model optimization because the form of the model does not change

during the analysis, only the parameter values [90]. In this study, different forms of a soot con-

sumption model were analyzed including a collision-efficiency model, simple Arrhenius equations,

and modified Arrhenius equations, with varied temperature and concentration dependencies.

101

4.6 Conclusions

Global models for soot particle oxidation and gasification were presented with parameters

calibrated using Bayesian methods. Besides providing the model parameters, this method also

gives full joint parameter PDFs and uncertainties, which provide more detailed information, with

fewer assumptions, than are available by other methods such as by minimizing least sum square er-

rors. PDFs of the calibration were presented along with parity plots displaying agreement between

model predicted rates and those collected from the literature. The oxidation model shows good

results and was robust enough for use in large scale simulation. The gasification model showed

reasonable results for H2O gasification, but only marginal results for CO2 gasification when con-

sidering all data sets. Individual data sets could be fit with much more accuracy. The R2 values

for the oxidation, H2O, and CO2 gasification models are 0.75, 0.87, and 0.62, respectively. As

new data become available, these could easily be incorporated into the model to reduce uncer-

tainty in the calibrated model parameters. This is especially true for the performance of the CO2gasification. Further research into model forms including additional soot physics could reduce

possible model bias and possibly improve consistency among experiments. While the oxidation

model was an improvement over the NSC O2 + Neoh OH combined oxidation model R2 = 0.71,

the improvement is modest.

The calibrated oxidation model can be used to calculate rates along with their uncertainties.

An example was given using the Higgins et al.[84] experiments. Results were compared to the data

and it was found that all reported data fell within determined credible intervals of the model.

102

CHAPTER 5. DETAILED MODELING OF SOOT FROM SOLID FUELS

This chapter presents a developed physics-based detailed model for predicting soot forma-

tion from complex-solid fuels along with two validation cases, one using coal and the other using

biomass. Results of the proposed model are compared against measured soot concentrations.

5.1 Model Development

Soot formation is dependent on the presence of soot precursors and the transformation of

soot particles throughout a system. The proposed model describes PSDs and their time-evolution

for both soot precursors and soot particles; however, the method used to represent each PSD will

be different. We use the abbreviation of PSD to describe the distribution of soot precursors for

convenience despite the size of precursors being too small to be considered particles.

The precursor PSD is represented using a sectional method. In the sectional method, a

series of pseudo-chemical species are used to represent all precursors that are within a section of

the full PSD. Each section is a subset of the PSD with different size ranges. The combination of

all sections represents the entirety of the precursor PSD,

NPAHtotal =

nbins

∑i=0

NPAHi , (5.1)

where NPAHi is the number density of precursor molecules within a given section. Upper and lower

bounds of each section were determined by molecular weight in this work, but can be determined

by other indicators, such as collision diameter. NPAHi refers to all precursors within a given section,

not just PAHs formed from light gases.

As the molecular weight range of the precursor PSD remains roughly fixed and sufficiently

narrow (150-3500 g/mole), a sectional approach for representing the PSD is both accurate and

computationally affordable. On the other hand, the soot PSD range is not fixed and highly de-

pendent on system configuration, sometimes growing to very broad ranges. Thus using a sectional

103

approach to represent the PSD becomes increasingly difficult; the presented model uses the method

of moments to represent the soot PSD. The method of moments involves the use of a set of statis-

tical moments that describe a PSD,

Mr =∞

∑i=1

mri Ni, (5.2)

where Mr is the resolved rth moment, mi is the molecular weight of particle i, and Ni is the number

density of particles i. In theory, every discrete distribution can be described by a finite set of

moments. However, in most cases a true soot PSD would require a set of moments well beyond

computational possibility and so only a few moments are used; the more moments resolved, the

more accurate the representation of the true PSD. Validation cases presented in this study were

limited to the resolution of 6 integer moments for the soot PSD [53].

Interpolative closure, as developed by Frenklach [53], was used to resolve all fractional

moments needed by the model. Interpolative closure uses a Lagrangian interpolation between

resolved whole moments to determine fractional moments that arise in the submodels used to

describe the time evolution of the PSD moments. The Lagrangian interpolation is given by

logMp = Lp (logM0, logM1, ..., logMn) , (5.3)

Lp (logM0, logM1, ..., logMn) =n

∑i=0

logMi

n

∏j=0j 6=i

p− ji− j

. (5.4)

Details for the time-resolution of each precursor section or soot moment used in this model are

given below. For further details on model derivations and justifications, refer to Appendix A.

5.1.1 Precursors

As mentioned above, the precursor PSD is represented by the sectional method. The rate

of formation of each section’s number density is determined by a series of submodels, written as

dNPAHidt

= r f ormi− rnucli− rdepoi− rcracki + rgrowthi− rconsumei, (5.5)

104

where the r expressions represent the formation, soot nucleation, deposition, thermal cracking,

surface growth, and consumption of each precursor section.

Precursor Formation

Precursors are formed in two ways: release from the parent fuel during primary pyrolysis,

or molecular build-up from light gases,

r f ormi = Rpyreneδ (mpyrene−mi)+Rpyrolysisi. (5.6)

PAH formation from light gas precursors, Rpyrene, is modeled using a gas-phase chemistry

mechanism developed by Appel, Bockhorn, and Frenklach [7] (ABF mechanism), which details

the production of pyrene, a common species used to model soot nucleation. The ABF mechanism

can be implemented in Cantera, a suite of software tools for problems involving chemical kinetics,

thermodynamics, and/or transport processes [71], or another similar software, and used to deter-

mine the production rate of pyrene in the gas-phase. The molecular weight of pyrene is 202.25

kg/kmol and contributes to the formation in only one PSD section; hence the delta function in the

first term of Equation 5.6.

Precursors released from the parent fuel, Rpyrolysisi in Equation 5.6, are evolved directly

into sections of the precursor PSD according to their molecular weight. Release rates need to be

determined by methods outside the scope of this model but may either be modeled or taken from

experimental data.

Soot Nucleation

Soot nucleation is modeled as the coalescence of two precursors to form a soot particle.

This process removes the two precursors from the precursor PSD and adds a soot particle to the

soot PSD represented by the soot moments. In terms of the precursor PSD, nucleation was given

by Frenklach and Wang [58] as

rnucli =nbins

∑j=1

βPAHi, j NPAH

i NPAHj . (5.7)

105

where β PAHi, j represents the frequency of collision between the two sectional species; in the free-

molecular collision regime, β PAHi, j is given by

βPAHi, j = 2.2

√πkBT2µi, j

(di +d j

)2, (5.8)

µi, j =mim j

mi +m j, (5.9)

di =Chm1/2i , (5.10)

Ch = dA

√2

3mC, (5.11)

where kB is Boltzmann’s constant, T is temperature, µi, j is the reduced mass of species i and j, di is

the collision diameter of species i, dA is the diameter of a single aromatic ring (0.28 nm), mC is the

mass of a single carbon atom (12.01 amu), and 2.2 is the van der Waals enhancement factor, which

accounts for the attraction of van der Waals forces as well as a collision efficiency [79, 131, 58].

The effect of nucleation on the soot PSD moments is expressed later in Section 5.1.2. Other

mechanisms for soot nucleation have been proposed in the literature [115, 197, 126] and may be

adapted to augment the currently proposed sub-model

Precursor Deposition

Soot growth via precursor deposition is modeled with the following precursor-soot collision

rate,

rdepoi =−∞

∑j=1

βi, jNsootj NPAH

i , (5.12)

where βi, j is a collision frequency that includes the collision efficiency. Balthasar and Frenklach

[11] expressed this model in terms of the precursor sizes and soot moments (derivation details are

found in A.0.2)

rdepoi = 2.2

√πkBT

2

(C2

hm1/2i Msoot

0 +2ChCaCsMsoot1/3 +C2

s C2am−1/2

i Msoot2/3

)NPAH

i . (5.13)

106

Here, Cs and Ca are the spherical soot collision diameter and the particle shape deviation from

spherical

Cs =

(6

πρs

)1/3

, (5.14)

Ca = (3−3〈d〉)+(3〈d〉−2)C〈d〉, (5.15)

where 〈d〉 is a shape factor related to the surface area of soot particles, detailed further in Section

5.1.2. C〈d〉 is a proportionality constant determined by a Monte-Carlo fitting to be 1.9125 [11].

Precursor Thermal Cracking

Thermal cracking is the chemical break-up of larger molecules, such as precursors, into

lighter gases and is heavily influenced both by the chemistry of the molecule and temperature

[41, 168]. In gaseous fuels, PAH molecules are made up of various aromatic rings, which are fairly

sTable and have only a small probability of thermally cracking. As more rings are added, forming

soot particles, the molecule becomes more sTable due to van der Waals forces, and eventually

thermal cracking becomes negligible [39]. For complex solid fuels, precursors are mostly volatile

tars released during primary pyrolysis. These tars are not completely made up of aromatic rings

but rather contain aliphatic and non-carbon components, reflective of the parent fuel [37]. These

inorganics and aliphatic groups make tars much more receptive to thermal cracking than gaseous-

fuel PAHs [123].

Thermal cracking of the precursor PSD is represented using a model developed by Marias

et al. [124]. In this model, tars are characterized as four basic types: phenol, toluene, naphthalene,

and benzene. While the precursors are not actually phenol, toluene, naphthalene, or benzene, these

four species are used as surrogates. In mathematical terms we may say 1 mole of precursors is

taken as 1 mole of a mixture of phenol, toluene, naphthalene, and benzene. Each of these types

undergo different reactions, as mapped in Figure 5.1. These reactions either convert one type to

another with the difference of mass being released into the gas phase, or crack completely into

lighter gases. The rates of each of these reactions are given in Table 5.1.

107

Precursor

Napthalene

LightGases Benzene

ToluenePhenol

R2

R1

R5

R4R3

Figure 5.1: Basic outline of PAH thermal cracking.

Table 5.1: Reactions and reaction rates used in precursor cracking scheme (rates in kmolem3s , concen-

trations in kmolem3 , and activation energies in J

mole K ).

Reaction Rates

C6H6O−−→ CO+0.4C10H8+0.15C6H6 R1 = k1[C6H6O]

+0.1CH4+0.75H2 k1 = 1.00×107 exp(−1.0×105

RT

)C6H6O+3H2O−−→ 2CO+CO2+3CH4 R2 = k2[C6H6O]

k2 = 1.00×108 exp(−1.0×105

RT

)C10H8+4H2O−−→ C6H6+4CO+5H2 R3 = k3[C10H8][H2]

0.4

k3 = 1.58×1012 exp(−3.24×105

RT

)C7H8+H2 −−→ C6H6+CH4 R4 = k4[C7H8][H2]

0.5

k4 = 1.04×1012 exp(−2.47×105

RT

)C6H6+5H2O−−→ 5CO+6H2+CH4 R5 = k5[C6H6]

k5 = 4.40×108 exp(−2.2×105

RT

)

108

The Marias et al. model is translated into the number density change of precursor sections

by multiplying the rates of reaction by the fraction of molecular weight cracked into light gas,

(5.16)rcracki =

(31.194

k1Xphe + k2Xphe +50

128k3Xnapth [H2]

0.4 +1492

k4Xtol [H2]0.5 + k5Xben

)NPAH

i ,

where kn values are given in Table 5.1. Details for this equation’s derivation are given in A.0.3.

[H2] is the concentration of H2 measured in kmolem3 . Xphe, Xnapth, Xtol , and Xben are the mole fractions

of surrogate precursors. The difficulty in using this sub-model lies in specifying the Xphe, Xnapth,

Xtol , and Xben values. In this study, the fractions are taken as constant and the values are determined

through a numerical study.

This numerical study was performed uniquely for each fuel/system considered. We evolve

a representative group of precursors using the cracking scheme detailed in Table 5.1, at constant

temperature and H2 concentrations, until 98% of the precursors are fully converted to light gases.

The time averaged mole fractions of the precursors are computed and used as constant values for

Xphe, Xnapth, Xtol , and Xben in subsequent soot simulations. Temperature, H2, and total initial pre-

cursor concentrations are set equal to peak system values as these values are a close representation

of the conditions where thermal cracking occurs.

The initial precursor components are estimated as follows. We start with equal parts phe-

nol, toluene, and naphthalene. But we want to maintain an initial aromatic/aliphatic carbon ratio

reflective of the actual system. This is done by adding methyl groups to the toluene precursor com-

ponents, thus during the numerical study the toluene components are really polymethylbenzenes.

To also maintain the given initial oxygen mass fraction, phenol groups are added to the phenol

precursor components, thus during the numerical study the phenol components are really polyphe-

nolicbenzenes. If the parent fuel is coal, the initial elemental composition and aromatic carbon

fraction are the same as the parent coal. For biomass, the elemental compositions and aromatic

carbon content were taken from Dufour et al. [42], which were 42.6% oxygen, 50.7% carbon, and

5.9% hydrogen; with 50% of the carbon as aromatic.

With an initialization of precursors with aromatic carbon ratios and oxygen mass fractions

consistent with what would be found in the system precursors, we evolve these precursors in time

according to the thermal cracking reactions. The precise reactions of Table 5.1 cannot be used in

this exercise because the ‘toluene’ precursor component is not exactly toluene and the ‘phenol’

precursor component is not exactly phenol. The reactions in Table 5.1 need to be modified slightly

109

0.0 0.1 0.2 0.3 0.4Time (ms)

0.0

0.2

0.4

0.6

0.8

1.0

Conc

entra

tion

(#/m

3 ) [1

019]

0.0 0.1 0.2 0.3 0.4Time (ms)

0.0

0.2

0.4

0.6

0.8

1.0

Type

Mol

e Fr

actio

n

Phenol Toluene Naphthalene Benzene

Figure 5.2: Result of numerical study considering the evolution of precursors from Pittsburgh #8coal at 1800 K as found in Section 5.2.1. Results were 0.004, 0.283, 0.503, and 0.210 for Xphe,Xnapth, Xtol , and Xben respectively.

to accommodate these differences. Reaction 4 is changed so that one methyl group is removed from

the ‘toluene’ component per reaction (i.e., a trimethylbenzene would become a dimethylbenzene.)

This means that only one reaction in every n reactions would produce benzene, where n is the

number of methyl groups added to the toluene components to adjust the initial aromatic/aliphatic

carbon ratio. Similar adjustments are made to reactions 1 and 2, where a single instance of reaction

1 or 2 only removes one OH group from the component until a true phenol is present. Then

reactions 1 and 2 occur as shown in the table. Reactions 3 and 5 are unchanged.

Figure 5.2 shows the results of this numerical study as performed for Pittsburgh #8 coal at

1800 K, which is discussed later in Section 5.2.1.

Precursor Growth

Particles are able to either increase or decrease in mass through interactions with the sur-

rounding gas phase. Increases in mass are modeled using the hydrogen abstraction and carbon

addition (HACA) mechanism.

Details of the HACA mechanism have been carefully studied and validated [7, 56, 129,

128]. Concentrations of radical species are higher in a combustion environment, and these radi-

cal species, particularly H·, react with the particle surface abstracting a hydrogen atom, leaving

a radical surface site. This radical site then reacts with acetylene in the surrounding gas, adding

110

Figure 5.3: Diagram of the complete HACA mechanism illustrating growth of a benzene ring.

Table 5.2: Surface growth mechanism where ki = AT n exp(−E

RT

)[7].

No. Reaction A ( m3

kmol·s·Kn ) n E ( Jmole )

1 C−H+H· −−→ C·+H2 4.2×1010 54,392

1R C−H+H·←−− C·+H2 3.9×109 46,024

2 C−H+OH· −−→ C·+H2O 1.0×107 0.734 5,932

2R C−H+OH·←−− C·+H2O 3.68×105 1.139 7,093

3 C·+H· −−→ C−H 2.0×1010

4 C·+C2H2 −−→ C−H+H· 8.0×104 1.56 15,762

the acetylene’s carbon to the surface. Another acetylene molecule is attached in the same way,

completing an additional aromatic ring on the surface of the original particle and releasing another

H· into the surrounding gas. HACA is a self-sustaining chain reaction due to the number of rad-

ical species remaining constant throughout the mechanism. Figure 5.3 illustrates the addition of

aromatic rings through the HACA mechanism. Kinetic rates for HACA are given in Table 5.2.

Each reaction rate given in Table 5.2 assumes a first order dependence on the gaseous

species. The overall reaction rate (kg/m2s) takes the form

RHACA = 2mCk4[C2H2]αχC·. (5.17)

111

χC· represents a number density of sites on the particle surface which have been radicalized. The α

parameter is the fraction of those surface sites kinetically available for reaction. Early implemen-

tations of HACA used an α value of 1 due to a lack of data. Appel et al. [7], derived an empirical

correlation for calculating α ,

α = tanh(

alog µ1

+b), (5.18)

where µ1 =M1M0

, and a and b are given as

a = 12.65−0.00563T, (5.19)

b =−1.38+0.00068T. (5.20)

The χC· value is computed using steady-state assumptions of the HACA mechanism in Table 5.2

χC· = 2χC−Hk1[H]+ k2[OH]

k−1[H2]+ k−2[H2O]+ k3[H]+ k4[C2H2]. (5.21)

χC−H is the number density of sites on the particle surface available for reaction, estimated to be

2.3×1019 sites/m2 [7].

The addition of mass to particles is accomplished by converting the mass added through

HACA into an equivalent number of particles added to a PSD section

rgrowthi =RHACASPAH

i NPAHi

mi. (5.22)

The surface area, SPAHi , of a precursor molecule is [186]

SPAHi = 5×10−20 ·NPAH

C,i , (5.23)

NPAHC,i =

mi

mC. (5.24)

112

Precursor Consumption

We model the consumption of precursors via oxidation and gasification. Oxidation of a par-

ticle surface is an exothermic reaction between surface carbon/hydrogen atoms and oxidizing gases

(O2 and OH here), leading to products of combustion: CO2, H2O, or CO [141, 74]. Gasification,

on the other hand, is a less exothermic, possibly endothermic, reaction between a particle surface

and gaseous molecules, such as H2O or CO2, and results in a more diverse array of gaseous prod-

ucts which may include: products of combustion, small hydrocarbons, alcohols, carbonyls, and

other species [31, 110].

The proposed model uses the global consumption submodel developed in Chapter 4 [94].

Oxidation and gasification rates (kg/m2s) are given in Equations 4.3 and 4.10. Both rates are mass

consumption per unit surface area of the particles (kg/m2s).

Similar to the growth term in Equation 5.22, the consumption of particle number is ac-

complished by converting the mass consumed into an equivalent number of particles from a PSD

section,

rconsumei =

(Roxidation +Rgasi f ication

)SPAH

i NPAHi

mi. (5.25)

5.1.2 Soot

As mentioned above, the soot PSD is represented using the method of moments. Moment

rates are determined by a series of submodels,

dMr

dt= Nur +Grr +Dpr +Cgr, (5.26)

where the terms on the right-hand side of the Equation represent nucleation, net surface growth (or

consumption), precursor deposition, and particle coagulation.

Soot Nucleation

Nucleation of soot particles is accomplished through the coalescence of two precursor

molecules. Section 5.1.1 describes the process of this coagulation and its effect on the precur-

113

sor PSD. The expression for its effect on the soot PSD is similar [58],

Nur =nbins

∑i=1

nbins

∑j=i

βi, j(mi +m j)rNPAH

i NPAHj , (5.27)

where βi, j again represents the frequency of collision between precursor species i and j, it is

computed using Equation 5.8.

Soot Coagulation

Coagulation of soot particles is computed based on the collision frequency between soot

particles [53]

Cgr =12

r−1

∑k=1

(rk

)(∞

∑i=1

∞

∑j=1

mki mr−k

j βi, jNiN j

). (5.28)

(rk

)is the binomial coefficient. Note, that coagulation does not effect the first PSD moment, thus

Cg1 = 0.

The βi, j term, representative again of particle collision frequency, is dependent on the flow

regime (continuum or free-molecular). Model details and derivations are provided in A.0.4.

βi, j in the continuum flow regime is

βCi, j = KC

(m−1/3

i +m−1/3j +K′C

[m−2/3

i +m−2/3j

])(m1/3

i +m1/3j ), (5.29)

leading to coagulation source terms in the continuum regime for r = 0 and r ≥ 2,

Cgc0 =−Kc[M2

0 +M−1/3M1/3 +K′c(M−1/3M0 +M−2/3M1/3)], (5.30)

(5.31)Cgcr =

12

Kc

r−1

∑k=1

(rk

)[2MkMr−k + Mk+1/3Mr−k−1/3 + Mk−1/3Mr−k+1/3

+ K′c(Mk−1/3Mr−k + MkMr−k−1/3 + Mk+1/3Mr−k−2/3 + Mk−2/3Mr−k+1/3)],

where the Kc = 2kBT/3µ and K′c = 2.514λ f /(CsCa), and µ is the gas viscosity. Cs and Ca are

evaluated using Equations 5.14 and 5.15. Fractional moments are computed using Lagrangian

interpolation among logarithms of integer moments using Equation 5.3.

114

Coagulation in the free molecular regime is more difficult as the βi, j expression is

βf

i, j = K f

(m1/3

i +m1/3j

)2(

1mi

+1

m j

)1/2

, (5.32)

and results in a non-expandable form of summations in Equation 5.28. Therefore, a grid function

is established and evaluated using Lagrangian interpolation [53],

Cg f0 =−1

2K f f (0,0)1/2 , (5.33)

Cg fr =

12

K f

r−1

∑k=1

(rk

)f (k,r−k)1/2 , (5.34)

where the K f = εC2aC2

s√

πkBT/2 and ε is the Van der Waals efficiency factor, taken as 2.2. The

grid function f (x,y)k is

f (x,y)k =∞

∑i=1

∞

∑j=1

(1mi

+1

m j

)k

mxi my

j

(m1/3

i +m1/3j

)2NiN j. (5.35)

Fractional values of k needed to evaluate Equations 5.33 and 5.34 are computed using Lagrangian

interpolation among the grid function evaluated at integer values of k [53]. An example of how to

resolve these grid functions is given in A.0.6.

A harmonic mean of the coagulation source terms in the continuum and free-molecular

regimes is used to determine the final coagulation rate

Cgr =Cgc

rCg fr

Cgcr +Cg f

r. (5.36)

Soot Surface Growth and Consumption

Just as the precursor PSD was affected by the growth or consumption of precursors through

the interactions between a precursor’s surface and the surrounding gas phase, the soot PSD also

changes through the mechanisms previously discussed: HACA growth, oxidation, gasification, and

precursor deposition. Details for the HACA, oxidation, and gasification were previously discussed

115

in Sections 5.1.1 and 5.1.1. The rate of change of the number density of particle i is given by

dNi

dt=

ks

∆m(Ni−1Si−1−NiSi). (5.37)

ks is the rate of a surface reaction (HACA, oxidation, or gasification) and is equal to RHACA, -

Roxidation, or -Rgasi f ication given in Equations 5.17, 4.3, and 4.3. ∆m is the mass change to the

particle due to a single reaction. For HACA, ∆m = 2mC, while for oxidation/gasification ∆m = mC.

Applying moments, the net soot growth/consumption moment source term for r ≥ 1 is derived to

be

(5.38)Grr = πC2s

ks

∆mm2/3−〈d〉

0

r−1

∑k=0

(rk

)(∆m)r−kMk+〈d〉.

For r = 0, Gr0 = 0. Precursor deposition was discussed in Section 5.1.1 and the moment source

term for r ≥ 1 is

Dpr = 2.2

√πkBT

2

r−1

∑k=0

(rk

)(C2

hMPAHr−k+1/2Msoot

k +2ChCaCsMPAHr−k Msoot

k+1/3+C2s C2

aMPAHr−k−1/2Msoot

k+2/3

),

(5.39)

where the Ch, Cs, and Ca constants were given in Equations 5.11, 5.14, and 5.15. For r = 0, we

have Dp0 = 0. The precursor PSD moment is calculated across all resolved sections

MPAHj =

nbins

∑i=1

m ji NPAH

i . (5.40)

Soot Aggregation

Modeling soot aggregation deals directly the morphology of soot particles. As particles

grow in size, particle morphology shifts from roughly spherical to aggregate chains. This behavior

is modeled using the approach of Balthasar and Frenklach [11], in which an additional statistical

moment is introduced which is related to the particle surface area. This moment, M〈d〉, is defined

through the total particle surface area density, S,

S = S0

∞

∑i=1

(mi

m0

)〈d〉Ni =

S0

m〈d〉0

M〈d〉, (5.41)

116

where S0 and m0 refer to the surface area and mass of an incipient soot particle upon nucleation.

〈d〉 is a shape factor, which can vary from 2/3, where the particles have the minimum possible

surface area (spherical), to 1 , where particles have the maximum possible surface area (a chain of

non-overlapping incipient particles). 〈d〉 is estimated using M0, M1, and M〈d〉,

〈d〉=log µ〈d〉log µ1

, (5.42)

where µ〈d〉 =M〈d〉M0

and µ1 = M1M0

. While the introduction of 〈d〉 does not completely resolve the

particle morphology, it can provide a particle collision diameter and surface area available for

gas-surface reactions.

M〈d〉, the surface moment, is solved similar to other moments, with submodels for particle

nucleation, precursor deposition, and net surface growth/consumption,

dM〈d〉dt

= Nu〈d〉+Dp〈d〉+Gr〈d〉. (5.43)

The nucleation source, assuming spherical primary particles, is

Nu〈d〉 = m2/30 Nu0. (5.44)

The deposition source term is determined by Lagrangian interpolation of the Dpi terms for

the resolved integer moments

Dp〈d〉 = L〈d〉 (logDp1, logDp2, logDp3) . (5.45)

Surface growth and consumption terms require the use of another grid function gk. The

source term is

Gr〈d〉 = πC2s

ks

∆mm2/3−〈d〉

0(g〈d〉−M2〈d〉

), (5.46)

with details and derivations given in A.0.5. As in Equation 5.37, ks is the rate of a surface reaction

(HACA, oxidation, or gasification) and is equal to RHACA, -Roxidation, or -Rgasi f ication. Similar to

f (x,y)l in Equation 5.35, gk is computed at integer values and used to interpolate to g〈d〉. The grid

117

function gk needed in Equation 5.46 is

gk =k

∑i=0

(ki

)∆mk−iMi+〈d〉, (5.47)

where ∆m represents the mass of carbon change resulting from a single reaction (∆m = 2mC for

HACA, and ∆m = mC for oxidation and gasification).

In using this aggregation model, Balthasar and Frenklach [11] note that “constituent par-

ticles of the evolving aggregate are assumed to have point contacts with each other and, conse-

quently, coagulation is assumed not to contribute to the change in the total surface area.” Initially,

this would imply that coagulation would not affect M〈d〉. However, a problem arises in coagulation

dominated regions where M1 and M〈d〉 remain stationary, but M0 decreases. The decreasing num-

ber of particles pushes M0 toward M1 and 〈d〉 (computed from Equation 5.42) decreases below its

lower bound of 2/3.

To resolve this issue, we recognize M〈d〉 not as an absolute surface area moment, but rather

on a scale between M0 and M1. Therefore, as particle coagulation affects one end of that scale,

M0, it must effect M〈d〉 as well. As the proposed submodel for particle coagulation in Section

5.1.2 is not designed to resolve fractional moments such as M〈d〉, the equations are modified and

Lagrangian interpolation is incorporated again using a grid function. Like the above coagula-

tion scheme, submodels resolve the coagulation rate for both the free-molecular and continuum

regimes. The continuum regime moment source term,

(5.48)Cgc〈d〉 = Kc

(12

h〈d〉 −(2M0M〈d〉 + M1/3M〈d〉−1/3 + M−1/3M〈d〉+1/3

+ K′C[M0M〈d〉−1/3 + M−2/3M〈d〉+1/3 + M〈d〉−2/3M1/3 + M−1/3M〈d〉

])),

uses a grid function hk in order to interpolate to h〈d〉 using Lagrange interpolation as before

(5.49)hk =

∞

∑i=0

∞

∑j=0

(mi + m j

)k(

2 + m−1/3i m1/3

j + m1/3i m−1/3

j

+ K′C[m−1/3

i + m1/3i m−2/3

j + m−2/3i m1/3

j + m−1/3j

])NiN j.

Coagulation in the free-molecular regime,

(5.50)Cg f〈d〉 = K f

(12

f (0,0)〈d〉+1/2 − f (〈d〉,0)1/2

),

118

uses the grid function given in Equation 5.35. Details and derivations are given in A.0.4 and a

example of how to resolve grid functions is given in A.0.6.

Once the Cg〈d〉 is computed for both regimes, the results are weighted according to Equa-

tion 5.36 above. This solution leads to an increased computational expense and the addition of the

Cg〈d〉 term can be numerically stiff, but it is also accurate.

5.2 Validation

The proposed soot model has been implemented in several forms and the code have been

verified. For validation of the proposed soot model, comparisons between model predicted and

experimentally measured soot profiles were carried out for two different systems. The first sys-

tem is a coal-fired laminar flat flame burner [121]. The second system is a biomass-fed gasifier

[190]. Adequate data was published for both experiments to successfully reproduce the systems

for simulation, allowing for model validation.

5.2.1 Coal System

Ma et al. [121, 120] collected soot from a coal-fired laminar flat flame burner, as depicted in

Figure 5.4. In this system, a Hencken flat-flame burner establishes a pre-mixed, fuel-lean laminar

flame with in-flows of CH4, H2, and dilution N2. Coal particles were steadily added to the center

of the flame with an N2 carrier gas. Proximate and ultimate analyses for three of the tested coals

are summarized in Table 5.3.

The Hencken burner used is made up of a honey-comb mesh with small-diameter tubes

inserted through the mesh-pores. Gases rapidly mix over the honeycomb and create a laminar

flame sheet with a nearly uniform temperature profile [25]. This particular burner was a square 5

cm on a side. Ma measured the spatial variation of temperature with a thermocouple at different

heights and radial locations and found that within the inner 3 cm of the flame, temperatures varied

radially by less than 40 K (about 2%) after the initial mixing layer (the first 2 cm above the burner.)

As particles entered the flame, primary pyrolysis occurred and particles devolatilized, re-

sulting in precursors and lighter volatiles escaping into the gas phase, leaving a char particle behind.

Volatile gases and char were collected by a nitrogen suction probe suspended at varying heights

119

Oxidizer

Fuel

Quench Nitrogen

Cooling Water

Coal Particles from the Feeder

Vacuum Pumps

Flowmeters

Char Collector

Cooling Water

Water Bath

Cyclone/

Virtual Impactor

Char-Leg Filter

Soot-Leg Filter

Suction Probe

Flat- Flame Burner

Water Traps

F

Quartz Tower

To Air

F

Char Stream

Soot Cloud

Figure 3.10. Particle collection and separation system.

41

Figure 5.4: Diagram of flat flame burner used by Ma [120]. Reproduced with permission.

Table 5.3: Proximate and ultimate analyses for the six coals tested [121].

Coal Type Moisture Volatiles Ash C H N S O

Utah Hiawatha High-Volatile B Bituminous

7.58 38.78 9.14 80.53 5.96 1.33 0.47 11.71

Pittsburgh #8 High-Volatile A Bituminous

1.87 37.10 4.11 84.70 5.40 1.71 0.92 7.26

Illinois #6 High-Volatile A Bituminous

6.94 38.69 15.13 76.65 4.93 1.47 6.93 10.01

above the burner. This suction probe dilutes incoming gas with cool nitrogen through jets at the

probe tip and through the porous walls of the probe, reducing the temperature of the collected

sample to approximately 700 K at the mouth of the probe. Additional diluent nitrogen permeates

the length of the probe walls to reduce sticking of particles on the inside of the probe.

From the probe, samples enter a virtual impactor where the momentum of heavier particles

(char) carries them into a horizontal cyclone with a cut-off diameter of 5 µm. Particles with a larger

120

diameter were collected in a char trap on the bottom of the cyclone, whereas smaller diameter

particles passed through a soot filter at the top of the cyclone. In the virtual impactor, gases and

small particles (soot) bend into a side arm. On this side arm is a soot filter through which gases

pass. Gases from both the cyclone and the virtual impactor side arm pass through a water bath for

cooling, water traps, flow meters, and other analysis equipment.

Data reported by Ma et al. included thermocouple readings along the flame centerline,

with particle residence times at the same locations. Also reported were char, soot, and volatile

yields from the suction probe collected along the flame centerline at varying heights. These soot

yields were collected from two sources. The first source was from the two soot filters previously

described, and these particles range in size from approximately 0.5-5.0 µm in diameter, as smaller

particles would likely pass through the filter and larger particles ended up in the char collector.

These larger particles were the second source of soot particles as they were separated from char

using a sieve with 38 µm openings.

Coal Simulations

As this system is both laminar and approximately one-dimensional, per the burner design,

simulations replicating the environment for soot formation were computationally inexpensive and

allowed for validation of the proposed soot model.

Simulations were carried out in one dimension for 120 mm along the gas flow direction.

Ma [120] reported experimentally measured particle residence times at four locations for each coal

type. These measurements were used to estimate instantaneous particle velocities. These particle

velocity profiles, reported gas temperatures, and fuel properties (Table 5.3) were used with the

Coal Percolation for Devolatilization (CPD) model [50] to predict particle devolatilization and the

release of precursors during primary pyrolysis. As stated above, the soot model depends on an

accurate prediction of soot precursors released from the parent fuel during primary pyrolysis. CPD

can be modified to output a sectional size distribution of precursors during primary pyrolysis with

section number and size dependent on coal type. These same sections were carried over to the

precursor sectional model.

These simulations resolved the precursor PSD with 9 sections and the soot PSD with 6

statistical moments and a shape factor. Sections of the precursor PSD and moments of the soot

121

Table 5.4: Precursor species fractions as described in Section 5.1.1 for the coal experiments.

Temp (K) CoalMole Fraction


1650 Utah Hiawatha 0.008 0.424 0.508 0.067

1650 Pittsburgh #8 0.008 0.427 0.501 0.064

1650 Illinois #6 0.006 0.408 0.502 0.084

1800 Utah Hiawatha 0.004 0.277 0.503 0.216

1800 Pittsburgh #8 0.004 0.283 0.503 0.210

1800 Illinois #6 0.003 0.245 0.505 0.247

1900 Utah Hiawatha 0.003 0.198 0.505 0.294

1900 Pittsburgh #8 0.003 0.213 0.504 0.280

1900 Illinois #6 0.002 0.164 0.508 0.326

PSD are transported in the z-direction by advection via the following the balance equations

d(uzNPAHi )

dz=

dNPAHidt

, (5.51)

d(uzMr)

dz=

dMr

dt, (5.52)

assuming negligible axial diffusion relative to advection, and no significant pressure differential.

Velocities, uz, were interpolated among experimentally measured values and dz was kept constant

at 1.2E-5 m, resulting in 10,000 steps per simulation.

Calculation of soot surface reaction rates for both PSDs requires species concentrations of

C2H2, H, H2, O2, OH, CO2, and H2O. Chemical equilibrium at the local experimental tempera-

ture was assumed for these gas phase species using the ABF mechanism discussed in Section 5.1.1.

The production rate of pyrene was computed from this gas state using the rate from the ABF mech-

anism, and any produced pyrene was added to the precursor PDF as described in Section 5.1.1. A

soot cloud of 3 cm diameter was observed experimentally, and in simulation it was assumed that

soot particle and chemical species concentrations were uniformly distributed across this cloud.

122

0 50 1000

10

20

30

Yiel

d (%

)

Utah Hiawatha

0 50 100z (mm)

Pittsburgh #8

0 50 100

Illinois #6

1650 K exp1650 K sim



Figure 5.5: Simulation results, continuous dotted lines, are compared to reported experimentaldata, individual marks. Results are soot mass yield as a percent of original fuel mass (dry and ashfree).

As described in Section 5.1.1 for the thermal cracking submodel, precursors were charac-

terized as phenol, toluene, naphthalene, and benzene types. The mole fractions of these types are

given in Table 5.4. The component fractions appear to vary more strongly with temperature than

with coal type. For all species and temperatures, naphthalene fractions remain fairly constant. At

higher temperatures toluene and phenol are exchanged for benzene. The precursor type fractions

are arguably the only ‘tunable’ parameters for this simulation, but even these were not tuned to ex-

perimental data but rather computed as the expected time-evolution of the precursors in the system.

This detailed model otherwise contained no parameters tuned to fit the experimental data.

Coal Results

Ma reported soot collected from both filters and sieved from the char trap. These data are

compared against the results of our simulations in Figure 5.5. The plots in this Figure display the

yield of soot, as a mass percent of the parent coal, collected at different heights above the burner

(which correlate to different particle residence times). The markers represent reported experimen-

tal results and the lines represent the simulations. Results are shown for three temperatures for

each of the three coals. As can be seen in the figure, there is good agreement between experiments

and simulations with regard to soot formation trends and locations. There is some disagreement

between the magnitude of soot yield, but even this level of disagreement has is less than many soot

123

prediction models [69]. The curve shapes found in the Figure are indicative of reaction mecha-

nisms but are consistent across all experiments. The total yield of soot is directly linked to the

volatile yield of the parent fuel, as all three of these coals are high-volatile coals, all three have

significant amounts of soot formed in their systems.

The location of soot formation is largely driven by the devolatilization rate of parent fuel.

As the fuel devolatilizes, precursors are released into the system and immediately begin to nucle-

ate or crack. The short time of soot mass build-up, occurring between 15 and 35 mm above the

burner, seems to indicate that the life-span of these precursors in the flame is very short. In each

of the cases, soot started to form approximately 15 mm above the burner. The higher temperature

systems tend to form soot more quickly, but form less soot overall, compared to the lower temper-

ature systems. This is because the higher temperatures force higher collision frequencies among

precursors, thus increasing soot nucleation rates. These increases are offset by increased thermal

cracking reaction rates, causing more precursor consumption and leading to an overall smaller soot

yield.

Around 35 mm above the burner, all the precursors have been consumed and the soot yield

levels off. Initially there is a slight, almost imperceptible drop in yield due to oxidation. This drop

is most easily seen in the 1650 K Pittsburgh #8 experiment, but is present in all curves. Within

the parent coal particles was a small amount of oxygen which becomes OH, and it is this OH that

begins to oxidize the soot. However, the OH is also consumed in oxidizing the soot particles, and

is itself fully consumed before too long. C2H2, which causes surface growth, also is only present

in small amounts and is fully consumed by the soot particles very quickly. Surface growth and

consumption effects, like oxidation, are very small and are largely masked by soot nucleation in

the initial mass build-up.

Note in Figure 5.5 that the yield of the soot mass levels off around 25 to 35 mm above the

burner for all cases. This is because in these low-temperature pre-mixed flames there is little to no

pyrene or acetylene present in the chemistry of the system. This translates to very little particle

mass increase due to gaseous growth of particles once the precursors released during primary

pyrolysis are consumed. However, although no mass increase is occurring after the initial soot

formation, this does not indicate that all mechanisms have stopped. Figure 5.6 shows the average

particle collision diameter within the flame. The average particle size is continually increasing

124

0 25 50 75 100z (mm)

5

10

15

20

Parti

cle D

iam

eter

(nm

)

1650 K1800 K1900 K

Figure 5.6: Average particle collision diameter across the flame portion of the Pittsburgh # 8 coalexperiments as predicted by the simulation.

0 25 50 75 100z (mm)

0.66

0.68

0.70

0.72

0.74

0.76

Parti

cle S

hape

Fac

tor

1650 K1800 K1900 K

Figure 5.7: Particle shape factor across the flame portion of the Utah Hiawatha coal experiments.

across the system as particles coagulate, changing the available particle surface area available

for oxidation/gasification at the flame layer. This seems to indicate that particle size is strongly

dependent on residence time and not only on mass yield.

Figure 5.7 shows that as the particle collision diameter grows the particles also become less

spherical. Recall the description of the shape factor parameter 〈d〉 (as described in Section 5.1.2)

indicates that at 〈d〉= 2/3 the particles are spherical but as 〈d〉 increases the particles become less

spherical and have more surface area. Initially, as particle concentrations are very small, the profile

is noisy as numerical errors dominate the computation of the shape factor. However, as particle

concentrations increase there is an initial steep growth of the particle shape factor which quickly

drops again. This trend is clearly evident in the 1650 K experiment but is present to a lesser extent

in the other two experiments as well. This quick drop is the result of a slight amount of oxidation,

125

0 50 1000

10

20

30

40

Yiel

d (%

)

Utah Hiawatha

0 50 100z (mm)

Pittsburgh #8

0 50 100

Illinois #6


1800 K exp1800 K simMaximum Sooting Potential


Figure 5.8: Soot mass yield with an additional ‘maximum sooting potential’ solid line representingthe mass yield of tars released into the system.

which tends to round-out particles. There are not many oxidizing agents in this pre-mixed flame,

but there are some, mostly OH, which quickly attack particle surfaces, consuming both agent and

particle. The overall impact of this oxidation is hard to see in Figure 5.5 but is much more evident

in Figure 5.7. After this initial oxidation we see the shape factor climb steeply once again until

around 35 mm, at which point the precursors are fully consumed as described earlier. Once the

precursors are consumed, the shape factor continues to climb but at a lesser rate. This steady climb

is an indication of continued particle agglomeration throughout the flame, also seen in Figure 5.6.

The combination of these two figures indicates that not only are particles growing in size, but are

becoming more chain-like throughout the agglomeration-dominated region 35-50 mm above the

burner.

In coal systems, tar is the dominant source of precursors and thus the dominant source of

soot mass. An additional simulation of the burner without coal was done with soot precursors only

coming from pyrene as described above. This simulation yielded soot mass less than 2% of the coal

system. This shows an important quantity then is the amount of tar that is converted to soot. This

value will be system dependent, but Figure 5.8 reproduces Figure 5.5 with a maximum sooting

potential line included. These lines are an indication of the soot yield that would be observed

if all tar molecules were converted to soot. As can be seen in the figure, not all tar molecules

were nucleated to soot particles, the rest thermally cracked, oxidized, or were gasified. In the

case of Utah Hiawatha: 61%, 56%, and 53% of the tar mass was converted to soot, dependent on

126

temperature. For Pittsburgh #8: 78%, 73%, and 70% mass was converted. And for Illinois #6:

74%, 70%, and 69% mass was converted.

Experimental uncertainties were not reported, nor has a full uncertainty quantification for

this model been done, so the precise discrepancy between the simulations and experiments is not

known. Sources of error within the experiment nearly all lead to decreased collection of soot.

The soot cloud was visually estimated by Ma to be around 3 cm while the opening to the suction

probe was only 2.5 cm. This suction probe did have a vacuum applied to it which helped to

collect most of the flame’s soot cloud, but it is possible that some soot particles were not collected

within the system. Additionally, small amounts of soot were known to deposit on the walls of

the soot collection system, thus leading to reduced mass in measurements. Within the suction

probe itself, nitrogen permeated the length of the probe walls to prevent particles sticking to the

walls, but this permeating nitrogen was not consistent through the virtual impactor, injection tube,

side arm, or cyclone. The soot filter pore size was 1 µm, but this filter is effective at capturing

smaller particles as well; there were certainly particles that passed through the collection filters as

a 1.0 µm collision diameter is a fairly large soot aggregate [130]. The cumulative effect of these

uncertainties is difficult to quantify, but these uncertainties would result in the actual soot produced

in the system being more than that reported. The simulation results consistently ‘over-predicted’

the measured soot concentrations within the system, and this is consistent with the sources of error.

(The one exception to this is the 1650 K experiment with Illinois #6.) These results help to validate

both the experiments and the proposed soot model for coal systems.

Particle Agglomeration

A problem occurs when comparing Figure 5.6 against the experimental setup. The maxi-

mum particle size predicted by this simulation is on the order of 20 nm diameter within the flame,

while the soot filters at the back-end of the collection system had pore sizes on the order of 1 µm.

So how do particles grow to that size? The answer, we believe, is two-fold:

First, when modeling, we assumed particles were uniformly distributed across the observed

soot cloud (3 cm diameter). In reality, this will not be a uniform distribution as soot particles are

largely concentrated in the centerline of the flame and concentrations would decline towards the

wings of the soot cloud, this type of distribution is commonly observed around pyrolyzing coal

127

0 50 100z (mm)

0

10

20

30

Yiel

d (%

)

Utah Hiawatha

0 50 100z (mm)

Pittsburgh #8

0 50 100

Illinois #6




Figure 5.9: Soot mass yield deposited on the soot filters of the coal-flame collection system.

particles [51]. High concentration of soot particles towards the reactor centerline would result in

higher rates of agglomeration than that predicted during the simulation as the frequency of particle-

particle collisions would also increase.

Second, once soot particles are extracted from the flame via the nitrogen suction probe

they are still an aerosol, though diluted. Even at the lower temperature, particles will continue to

agglomerate as seen in the study represented by Figure 1.5. By extending out the simulation, we

were able to perform a small validation on the soot agglomeration portion of this proposed model.

Temperatures of the collection system were taken to be 700 K, as described by Ma. Dilution

of the aerosol was also accounted for as particles were diluted by N2 permeating the walls of

the suction probe and particles traveled through sections of the collection systems with varying

cross-sections. Once particles reached the soot filters in simulation, we assumed them to have a

log-normal distribution characterized by the first three resolved moments and said that any particles

over 5 µm in diameter were not captured on the filter but would rather have passed into the char

trap.

Figure 5.9 shows the result of these extended simulations. This Figure is showing the ex-

perimental soot mass yield of soot collected on the filters and the simulated soot mass yield of

all particles smaller than 5 µm at the location of the soot filters. With a quick glance, it would

seem that the extended simulation did not do very well in capturing the soot dynamics of the col-

lection system as experimental data shows a clear decrease in collected yield while simulations

still predict a constant yield. There are many reasons for this, the most important being that even

128

the extended simulation does not capture the additional agglomeration happening within the flame

due to the particle concentration distribution discussed previously. This additional flame agglom-

eration is very likely the cause of the shape of the concentration profile in the experimental data.

Higher concentrations of particles near the flame centerline increases particle agglomeration caus-

ing enough size differences along centerline soot particles to create a noTable size difference at

different points within the flame. In simulation, we assumed a uniform average concentration

across the soot cloud, therefore centerline concentrations were significantly diluted and therefore

agglomeration rates. As a result, in simulation, not enough agglomeration occurred within the

flame to make a noTable difference in particle size within the flame itself.

Another cause for differences in the shape of yield profiles occurs from the collection sys-

tem itself. There are complex flow dynamics occurring in the collections system (gas dilution, ex-

pansion, mixing, re-circulation, etc.) which cannot be captured by the one dimensional assumption

made in these simulations also resulting in significant predictive errors. So while these extended

one-dimensional simulations are insufficient for total predictive capabilities, they are still informa-

tive. Note the reduced simulated yields in comparison to the full yields of Figure 5.5, this reduction

indicates that simulated particles are on the same size order as particles collected in experiments

(micron-order diameters). Thus, while the one-dimensional simulation is a crude representation

of the flow dynamics in the collection system, final predicted particle sizes are now on the same

order as those experimentally observed. And with that similar residence time and temperature the

detailed model yields particles of similar size as those in the actual experiment.

5.2.2 Biomass System

Trubetskaya et al. [190], collected soot from a fast-pyrolysis drop-tube reactor which gasi-

fied three types of biomass at two different temperatures, 1250 °C and 1400 °C. Biomass was fed

into the reactor at a rate of ∼0.2 g min−1, where it was rapidly heated and pyrolyzed as it fell

through the reactor. Reaction products were passed through a cyclone where larger particles (char

and fly ash) were separated and fine particles (soot) were captured on a filter attached to the outlet

of the cyclone [190, 70]. Proximate and ultimate analysis of the three biomass types are given in

Table 5.5.

129

Table 5.5: Proximate and ultimate analyses for the biomass fuels tested.

Biomass Type Moisture Volatiles Ash C H N S+Cl O

Pinewood (Softwood) 5.1 86.6 0.3 53.1 6.5 .0.06 0.02 40.3

Beechwood (Hardwood) 4.5 79.4 1.4 50.7 5.9 0.13 0.04 43.3

Wheat Straw 5.5 77.5 4.1 46.6 6.1 0.6 0.2 46.5

Collected particles were analyzed in a number of ways: elemental analysis, ash composi-

tional analysis, FTIR spectroscopy, X-ray diffraction, thermogravimetric analysis, N2 adsorption

analysis, transmission electron microscopy (TEM), electron energy-loss spectroscopy, particle size

distribution analysis, and graphitic structure. For purposes of validation, we focus here on the re-

ported soot yield data and the particle size distribution analysis. Soot yield data were obtained for

both an organic fraction and an inorganic fraction (through a standard ash test) of soot collected

from the exhaust gas. However, in all cases soot was overwhelmingly organic, and inorganic frac-

tions were only detecTable in Wheat Straw soot and Beechwood soot at the higher temperature.

The particle size distributions were estimated manually from TEM images. For every experiment,

50 particles were separated for the size analysis and every particle was assumed to spherical.

Biomass Simulations

In the simulations, we assumed that all soot was completely organic. Concentrations of

precursors released during the primary-pyrolysis of the biomass were estimated using CPD-bio, an

adaptation of CPD for estimating the behavior of biomass devolatilization using the same structure

principles derived for CPD [112]. Particle temperatures, velocities, and residence times were com-

puted using the devolatilization model provided in the supplemental material of the original study

[190]. These temperature profiles were then used in CPD-bio to predict tar yields segregated into

a sectional precursor PSD. These simulations resolved the precursor PSD with 10 sections and the

soot PSD with 6 statistical moments along with the shape factor.

Precursors were again characterized into different types and the results are shown in Ta-

ble 5.6. Some trends we observed for coal seem to be consistent for biomass as well. There does

not appear to be much difference in precursor type fractions between biomass species but there

130

Table 5.6: Precursor species fractions as described in Section 5.1.1 for the biomass experiments..

Temp (°C) BiomassFraction


1250 Pinewood 0.157 0.415 0.424 0.004

1250 Beechwood 0.156 0.415 0.425 0.004

1250 Wheat Straw 0.152 0.417 0.427 0.004

1400 Pinewood 0.089 0.444 0.459 0.007

1400 Beechwood 0.088 0.445 0.460 0.007

1400 Wheat Straw 0.085 0.446 0.462 0.007

1250 °C 1400 °C0

2

4

6

8

10

Yiel

d (%

)

Pinewood

1250 °C 1400 °C

Beechwood

1250 °C 1400 °C

Wheat Straw

Experiment Simulation

Figure 5.10: Results of biomass-derived soot simulations compared to reported experimental data.Results are displayed as a mass percent of the parent fuel (dry and ash free).

does seem to be a heavy correlation between the type fractions and temperature. Although there

does not appear to be much variation between different biomass species, there is a significant dif-

ference between precursor type fractions for the biomass in Table 5.6 and type fractions for coal in

Table 5.4.

Simulations assumed that chemical species and soot concentrations were uniform across

the diameter of the reactor (2 cm) and chemical equilibrium using the ABF mechanism was as-

sumed for gaseous species. We treated the soot formation simulation as a plug-flow reactor with

Equations 5.51 and 5.52 solved for both precursor PSD sections and soot PSD moments.

131

0

10

20

30

40

50

Perc

enta

ge (%

)

Pinewood 1250 °C Pinewood 1400 °C

0

10

20

30

40

50

Perc

enta

ge (%

)

Beechwood 1250 °C Beechwood 1400 °C


0

10

20

30

40

50

Perc

enta

ge (%

)

Wheat Straw 1250 °C


Wheat Straw 1400 °C

Figure 5.11: Blue bars represent experimentally measured particle-size distributions and red linesrepresent simulation resolved moments fitted to a log-normal distribution.

Biomass Results

Figure 5.10 shows simulation results compared to the experimental data. As can be seen

in the figure, there is good agreement between simulations and experiments with the simulation

results all lying within or very close to the reported error bounds of the experiments; the only

exception is the 1250 °C experiment for the Beechwood fuel. The model also captures the trends

of the experiments, where higher temperatures generally led to higher rates of precursor thermal

cracking, which led to lower soot yields, as seen in the Pinewood and Beechwood experiments.

Soot yields from wheat straw, on the other hand, went up as the system responded to differences in

the chemistry of the wheat straw, which was also captured by the model. In general, the softwood

produces more soot than either the hardwood or the straw. This trend is seen in both experiments

and in simulations.

132

The proposed detailed model does not resolve a full particle distribution but rather only

moments of the distribution. In order to compare the experimentally analyzed distributions against

the computed statistical moments, the resolved moments were fitted to a log-normal distribution.

With this assumption, a PSD could be reconstructed for each set of conditions and compared

directly to available experimental data as seen in Figure 5.11. In the experiments, 50 particles

were analyzed for each set of conditions via visual analysis, and the results are shown as the

blue bars in the figure. The red lines represent the first three simulation moments set to a log-

normal distribution. While there exist discrepancies between experimental data and simulation

results, the two are highly complementary, with the exception of the 1250 °C Pinewood experiment.

This experiment’s difference may be due to the log-normal assumption used to reconstruct the

distributions. This particular system had a much longer residence time than the others resulting in

a flatter experimental distribution.

5.3 Conclusions

A physics-based model for predicting soot formation from solid-complex fuels was pro-

posed. This model has a number of advantages for predictability in a wide variety of flames. Re-

searchers should be comforTable extrapolating the use of this model without parameter calibration

specific to their situation.

That being said, the model does not include every possible mechanism that can affect soot

formation. For example, it is known from reported research [23, 190], that the presence of in-

organics, Na, K, S, etc., in the soot particle structure can have catalytic effects on the chemical

interactions between particle surface and surrounding gases. The exact effects of these inorganics

are not fully quantified or developed into a model form yet and thus not included here. While it is

believed that catalytic effects are small, they are a source of error that researchers should be aware

of, especially for biomass fuels which have a tendency to have more inorganics present.

In the model’s current formulation, oxidation and gasification consume particle mass, which

affects the higher moments of the soot PSD; however, it does not affect the zeroth moment, particle

number density. As a result, when particles are fully consumed, simulation results may indicate a

number of particles still present in the system where there is little or no mass. In addition, particles

have a tendency to fragment [156, 206], whether through a mechanical breakage of an aggregate

133

or through chemical consumption. Currently, this model does not account for any particle frag-

mentation.

Section 5.1.1 refers to the use of a submodel developed by Marias et al. [124] for predicting

thermal cracking rates of soot precursors. This submodel requires a precursor characterization, and

in this study we used time-averaged values for those precursor types determined by a numerical

study described above. A numerical study done for every fuel type under unique conditions is un-

desirable and work is ongoing to improve aspects of this sub-model’s implementation. In addition,

the total sensitivity of these type-fractions to overall soot yields is not completely quantified and

also an area of ongoing model improvement.

The numerical economy of the Method of Moments applied in this model allows for de-

tailed resolution of the soot PSD to be coupled with the resolution of other physics in reactive

flows. However, even with these advantages the computational expense of the proposed model

may be too high for use in large-scale simulations. This is because the full-detailed model pre-

sented contains multiple sections to be resolved for the precursor PSD and at minimum 4 moments

to be resolved for the soot PSD with a large number of processes affecting each term. However, the

detailed model presented is useful in calibrating simpler models for use in larger CFD simulations.

In conclusion, this proposed soot model shows promising results for predicting soot par-

ticle formation in a large variety of systems, but researchers using the model should be aware of

implementation details and limits to tailor its use in their own systems.

134

CHAPTER 6. SIMPLIFIED MODELING

In the previous chapter, a developed detailed model was presented for the formation of soot

in solid fuel systems and was validated against experiments. That detailed model can be computa-

tionally expensive and thus is often not appropriate for large-scale simulations. This chapter will

present a simplified model which is both easier to implement in simulation and computationally

more economic in terms of CPU hours, memory allocation, and stored drive space. In addition

to the simplified model, this chapter also presents some simulation results comparing the detailed

model against the simplified model.

6.1 Model Development

The proposed simplified model solves only three quantities: the number density of soot-

precursor molecules (Ntar), the number density of soot particles (Nsoot) and the mass density of

soot particles (Msoot), that’s the mass of soot particle per volume of gas. These three terms may

be subject to transport phenomena such as diffusion or convection schemes in ways specific to the

simulation scenario, for an example see Equations 3.2, 3.1, and 3.3 from Chapter 3. The generation

and consumption rates for each term are defined here.

The primary simplification to the detailed model deals with the representation of the pre-

cursor and soot PSDs. In the detailed model, the precursor PSD was represented with a sectional

method, while the soot PSD was represented with MoM. Both of these methods require the resolu-

tion of many terms with increased complexity as source terms interacted with each other, such as

the closure of fractional moments. In this simplified model, both PSDs are represented as mono-

dispersed PSDs with the weight of precursors fixed and the weight of soot particles resolved along

with number densities for both precursor molecules and soot particles.

135

Rates of generation/consumption for each of these terms is defined by many of the same

submodels found in the detailed model,

dNtar

dt= rPI−2rSN− rPD− rTC +2508NtarrPS, (6.1)

dNsoot

dt= rSN− rSC, (6.2)

dMsoot

dt= rSN +mtarrPD +π

(6msoot

πρs

)2/3

NsootrSS. (6.3)

These equations include terms for precursor inception (rPI), precursor deposition (rPD), thermal

cracking (rTC), soot nucleation (rSN), soot coagulation (rSC), and surface reactions (rPS and rSS).

6.1.1 Precursor Inception

In the previously presented detailed model, the formation of precursors was computed as a

summation of two sources: PAH build-up from light gases and the release of tar volatiles during

primary pyrolysis. This simplified model dismisses the PAH build-up from light gases as a neg-

ligible source of precursors [51]; however, should researchers determine that a particular system

for which this model is applied contains a significant build-up of PAH, amalgamation of a PAH

mechanism, such as the ABF mechanism [7], should be simple.

Precursor inception from the release of tar volatiles is modeled using a ’sooting potential’

model unique to fuel type and pyrolysis conditions. This model predicts the fraction of volatiles,

resulting from primary pyrolysis, which may be considered as soot precursors along with their

average molecular size.

The coal percolation model for devolatilization (CPD) [50] along with its biomass adapta-

tion (CPDbio) [112] were used as gold standards to which the sooting potential model was cali-

brated. CPD is a network devolatilization model designed to predict products of primary pyrolysis

for solid fuels. CPD-CP has a submodel combination for predicting particle temperature profiles

if a user specifies the surrounding gas temperatures, pressure and particle velocities; this particle

temperature profile, along with CNMR parameters of a fuel, are fed into the CPD portion of the

136

code to predict pyrolysis behavior. To calibrate the sooting potential model, CPD was executed

thousands of times varying input parameters to create a comprehensive data set to which parame-

ters could be tuned. During the calibration of the sooting potential model, it was quickly found that

fuel particle velocities had a minimal effect on total tar yield and tar size and so particle velocities

were kept a constant velocity (2.5E-5 (m/s)) for the data creation.

When using CPDbio, predicting products of biomass primary pyrolysis is accomplished

by first predicting the devolatilization behavior of five biomass components: cellulose, galacto-

gluco-mannose (softwood hemicellulose), xylose (hardwood hemicellulose), softwood lignin (with

higher concentrations of guaiacyl constituents), and hardwood lignin (with higher concentrations

of syringyl constituents). Each component is determined independently and summed together,

weighted by the respective mass percentage of each component in a given biomass, to predict the

overall devolatilization behavior of the given biomass species.

CPDbio was executed 1000 times for each biomass component over a wide range of pres-

sures and gas temperatures, 0.1<P (atm)<100 and 800<T (K)<3000, using a Latin hypercube

sampling method. This generated 1000 data points to which rational empirical models of the

forms

ytar =a+bTg + cP+dT 2

g + eP2 + f TgP+gT 3g +hT 2

g P+ iTgP2 + jP3

k+ lTg +mP+nT 2g +oP2 + pTgP+qT 3

g + rT 2g P+ sTgP2 + tP3 , (6.4)

mtar =a+bTg + cP+dT 2

g + eP2 + f TgP+gT 3g +hT 2

g P+ iTgP2 + jP3

k+ lTg +mP+nT 2g +oP2 + pTgP+qT 3

g + rT 2g P+ sTgP2 + tP3 , (6.5)

were fitted. In Equations 6.4 and 6.5, Tg represents the gas temperature (K) and P represents

the logarithm of the pressure (atm). Calibration was accomplished using a series of least-squares

fittings for all 20 parameters. Insignificant parameters (those with an influence less than 5% on

final yields and sizes) with were eliminated and the proposed models refitted leaving the equations

shown in Table 6.1. Like CPDbio, the sooting potential model predicts behavior for five different

biomass components. To find the total biomass devolatilization behavior, simply sum together

those components weighted by the mass fraction of the given component in the biomass

(6.6)ytar = ytar,cellycell + ytar,hw/hcyhw/hc + ytar,sw/hcysw/hc + ytar,hw/ligyhw/lig + ytar,sw/ligysw/lig,

mtar = mtar,cellycell + mtar,hw/hcyhw/hc + mtar,sw/hcysw/hc + mtar,hw/ligyhw/lig + mtar,sw/ligysw/lig.

(6.7)

137

Table 6.1: Sooting potential model for biomass with calibrated parameters for Equations 6.4and 6.5. Tg and P are the gas temperature (K) and log-pressure (log(atm)) respectively.

Component Model

Celluloseytar,cell =

-1.57E5+290.6Tg−0.022T 2g +8.00TgP+3.60E-5T 3

g −0.036E-2T 2g P

−2.03E5+382.9Tg+11.2TgP+4.53E-5T 3g −0.042T 2

g P

mtar,cell =-3.06E4+242.2Tg+1.05E4P−1.84E3P2−83.1TgP+461.8P3

0.635Tg−0.145TgP−0.021TgP2−2.78P3

HardwoodHemicellulose

ytar,hw/hc =-5.21E5+3.12E3Tg−0.382T 2

g −1.08E3TgP+0.207T 2g P

5.75E3Tg−2.65E3TgP−1.45E-4T 3g +0.518T 2

g P

mtar,hw/hc =236.7TgP2−5.92E4P3

0.608TgP2−109.4P3

SoftwoodHemicellulose

ytar,sw/hc =7.05E4+144.4Tg−1.29E-5P+0.233T 2

g −7.41E-5T 3g

3.69E5+91.0Tg−3.22E5P+0.725T 2g −2.08E-4T 3

g

mtar,sw/hc =−6.41E4P2+50.0TgP+26.0TgP2+1.56E4P3

-1.65E3P2+0.126TgP+0.072TgP2+41.3P3

HardwoodLignin

ytar,hw/lig =9.04E4−76.2Tg−3.43E4P+6.03E-3T 2

g +36.6TgP+7.69E-6T 3g −0.011T 2

g P1.37E5−117.5Tg−3.66E4P+0.012T 2

g +39.3TgP+1.00E-5T 3g −0.012T 2

g P

mtar,hw/lig =4.78E6−8.40E3Tg+7.36T 2

g +3.39E6P2−573.1TgP−1.23E-3T 3g +340.1TgP2−4.85E5P3

8.13Tg+1.47E4P2−2.64TgP2+997.9P3

SoftwoodLignin

ytar,sw/lig =9.15E5−609.8Tg−3.00E5P+0.070T 2

g +231.8TgP+1.32E-T 3g −0.046T 2

g P1.43E6−1.10E3Tg−3.02E5P+0.22T 2

g +235.3TgP−0.047T 2g P

mtar,sw/lig =9.15E5−609.8Tg−3.00E5P+0.0703T 2

g +231.8TgP+1.318E-5T 3g −0.046T 2

g P1.43E6−1.103E3Tg−3.02E5P+0.219T 2

g +235.3TgP−0.0468T 2g P

Note that this sooting potential model neglects the behavior of extractives in biomass in part be-

cause extractives can vary so greatly that an individual characterization would need to be done

for every species, which is not possible in a general model such as this. Fortunately, extractives

typically make up a small fraction of most biomass species (approximately 1-5%) [13].

Figure 6.1 shows the effectiveness of this empirical sooting potential model against CPDbio.

Both of these plots are parity plots where results of the sooting potential model is plotted against

the x-axis while results of CPDbio are plotted against the y-axis. The black 45° line represents a

perfect match between the two models. As can be seen in the figures, generally the sooting poten-

tial model follows the trends of CPDbio with good agreement (R2=0.811 and 0.856 for soot mass

138

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0CPD Predicted

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Mod

el P

redi

cted

Tar Mass Yield Predictionshardwood_hemicellulosesoftwood_hemicellulosehardwood_ligninsoftwood_lignincellulose

100 200 300 400 500 600 700CPD Predicted

100

200

300

400

500

600

700

Mod

el P

redi

cted

Tar Size Predictions (g/mole)

Figure 6.1: Comparison between results given by CPDbio versus the proposed sooting potentialempirical model. Different colors represent different biomass components: cellulose (blue), hemi-cellulose softwood/hardwood (green/yellow), and lignin softwood/hardwood (magenta/red). Theleft plot shows the comparison for tar mass yield (R2=0.811) and the right plot shows the compar-ison for tar mass size (R2=0.856).

yield and molecular size respectively) but there is room for improvement should a better model

form be discovered that is as computationally inexpensive as this proposed one.

To create a sooting potential model for coal fuels we needed to add an extra compo-

nent of varying 13C NMR parameters. These parameters may be obtained through a correla-

tion developed by Genetti et al. [63] which links these parameters to the elemental composition

and volatile matter content of the parent coal. Through this correlation and the use of CPD-

CP, we again developed a database of 1000 data points resulting from varying O/C atomic ratio

(0.01< OC <0.35), H/C atomic ratio (0.3< HC <1.1), volatile matter content (2<Vol (%)<80),

pressure (0.1<P (atm)<100), and gas temperature (800<T (K)<3000). This database was used in

a similar way to calibrate surrogate models of a form similar to Equations 6.4 and 6.5; as before,

negligible parameters were eliminated leaving

(6.8)ytar =

-124.2 + 35.7P + 93.5OC − 223.9O2C + 284.8HC − 107.3H2

C

+ 5.48V + 0.014V 2 − 58.2PCH − 0.521PV − 5.32HCV

-303.8 + 52.4P + 1.55E3OC − 2.46E3O2C + 656.9HC − 266.3H2

C + 15.9V

+ 0.025V 2 − 90.0PHC − 462.5OCHC + 4.80OCV − 17.8HCV

139

0.0 0.2 0.4 0.6 0.8 1.0CPD Predicted

0.0

0.2

0.4

0.6

0.8

1.0

Mod

el P

redi

cted

Tar Mass Yield Predictions

200 300 400 500 600 700 800CPD Predicted

200

300

400

500

600

700

800

Mod

el P

redi

cted

Tar Size Predictions

Figure 6.2: Comparison between results given by CPD versus the proposed sooting potential em-pirical model. The left plot shows the comparison for tar mass yield (R2=0.794) and the right plotshows the comparison for tar mass size (R2=0.854).

and

(6.9)mtar =

3.12E5 + 16.4Tg + 4.34E5OC − 8.48E5HC + 6.38E5H2C

− 361.3V − 0.221TgV − 6.39E5OCHC + 1.91E3HCV

753.6 + 0.042Tg + 83.9OC − 1.77E3HC + 1.20E3H2C + 5.09E-3TgP

− 0.024TgHC − 5.27E-TgV + 0.513PV − 361.0OCHC3.83HCV

.

In these equations P is the logarithm of the pressure measured in atmospheres. OC and HC are

the atomic ratio of oxygen and hydrogen to carbon respectivily. V is the mass percent of volatile

matter in the parent coal. Tg is the gas temperature. Unlike biomass, these surrogate models are

absolute for predicting the tar mass yield and average molecular weight as a result of pyrolysis and

do not need to be recombined from components.

Figure 6.2 shows the effectiveness of this empirical sooting potential model against CPD.

Generally the sooting potential model follows the trends of CPD with good agreement (R2=0.794

and 0.854 for soot mass yield and molecular size respectively). It is interesting to note that in

calibrating these surrogate models all terms to gas temperature dropped out of Equation 6.8 as

negligible and pressure only places a minor role in determining tar size (Equation 6.9). These

characteristics show potential for further investigation in creating a more physics-based sooting

potential model.

140

Using the sooting potential model, either for biomass or for coal, we may predict the rate

of precursor inception as a fraction of the rate volatiles are released during primary pyrolysis rv

(kg/m3s),

rT I =ytar rv

mtar, (6.10)

of which there are many developed models [202, 159, 104, 50].

6.1.2 Thermal Cracking

Thermal cracking of precursors into light gas is modeled in the same way as the detailed

model (Section 5.16) using the submodel developed by Marias et al. [124]. Like before, the crack-

ing of precursor molecules results in mass lost from precursors to light gases as these precursors

undergo transformations. The simplified model assumes that all particles are of the same fixed

size. As a result, to account for the changes of mass due to thermal cracking we convert the mass

loss to an equivalent change in number of precursors,

(6.11)rcracki =

(31.194

k1Xphe + k2Xphe +50

128k3Xnapth [H2]

0.4 +1492

k4Xtol [H2]0.5 + k5Xben

)NPAH .

Justification for this model was given previously in the preceding chapter. The difficulty in this

submodel is designating values for Xphe, Xnapth, Xtol , and Xben.

In the previous chapter, a numerical study was detailed for determining these values. In

deriving the simplified soot model, this numerical study was executed over a wide range of inputs,

temperature, oxygen mass fraction, aromatic/aliphatic carbon ratio, H2 concentration, and initial

precursor number density, to determine both parameter sensitivity and to derive a simple empiri-

cal model for deciding mole fraction quantities. This series of studies revealed that the two most

important parameters in determining mole fraction quantities were temperature and initial precur-

sor number density. The other three parameters, oxygen mass fraction, aromatic/aliphatic carbon

ration, and H2 concentration, all had negligible effects on the time-average precursor ratios.

Figure 6.3 shows the results of varying temperature (left) and initial number density (right)

over a wide range, 500<T (K)<3000 and 1E10<Ntar (#/m3)<1E25. Observe that at low tempera-

tures and high number densities the fractions all collapse to 1/3, the initialization of the numerical

study. This collapse is because at these conditions thermal cracking becomes negligible in compar-

ison to soot nucleation mechanisms. On the other hand, at high temperatures the thermal cracking

141

500 1000 1500 2000 2500 3000Temperature (K)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Mol

e Fr

actio

n

Temperature Variation

1016 1018 1020 1022 1024

Concentration (#/m^3)0.0

0.1

0.2

0.3

0.4

0.5

Concentration Variation

Phenol Toluene Naphthylene BenzeneFigure 6.3: Variation of time-averaged precursor ratios from numerical study as temperature (left)and initial number density (right) are varied.

dominates soot nucleation, and as a result phenol and toluene-type precursors disappear quickly,

being converted to a benzene-type. It is evident with the varying temperature plot that in terms

of reactivity, phenol>toluene>naphthalene/benzene, which is expected because phenol has the

presence of oxygen, and aromatics are molecularly more stable than aliphatics.

Using the results of Figure 6.3, an empirical model was proposed of form,

xi =tanh

(a+bT + cC+dT 2 + eC2 + f TC

)m

+g+hT + iC+ jT 2 + kC2 + lTC, (6.12)

where T is the temperature and C is the logarithm (base 10) of the initial precursor concentration.

This model form is over-defined, with 12 tunable parameters. Using optimization software, these

12 parameters were tuned for each precursor type against results from the numerical study where

temperature and concentrations were varied with a Latin hypercube sampling. Once these parame-

ters were tuned, negligible ones (those with an influence on the final xi value of less than 5%) were

discarded, and the remaining parameters were tuned again. This procedure was done iteratively

until only significant parameters remained. Results of this parameter calibration yielded empirical

models for each of the xi parameters,

(6.13)Xphe =16

tanh(5.73− 0.00384 T − 0.159 C)− 0.218 + 0.0277 C,

Xnapth =12

tanh(−1.98 + 6.18E-4 T + 0.124 C − 0.00285 C2 + 4.14E-7 T 2 − 4.97E-5 TC

)− 0.576 + 0.000233 T − 1.69E-7 T 2,

(6.14)

142

0.0 0.2 0.4 0.6 0.8 1.0Numerical Study

0.0

0.2

0.4

0.6

0.8

1.0

Mod

el P

redi

cted

Precursor Type Fraction PredictionPhenolTolueneNaphthaleneBenzene

Figure 6.4: Comparison between empirical model and numerical study for predicting precursor-type fractions. The black straight 45°represents a perfect agreement between the two (R2=0.919).

(6.15)Xtol =13

tanh(17.3− 0.00869 T − 1.08 C + 0.0199 C2 + 0.000365 T C

)+ 0.000265 T − 0.000111 C2 − 9.32E-6 TC,

(6.16)Xben = 1− Xphe − Xnapth − Xtol,

which can be used to predict these type fractions with ease and during model implementation

instead of requiring a previous numerical study. These empirical models produce decent results in

comparison to the numerical study, as seen in Figure 6.4.

6.1.3 Soot Nucleation

Soot nucleation occurs through the coalescence of two precursor molecules to form an

incipient soot particle

rSN = εβPN2tar. (6.17)

Here βPAH represents a frequency of collision between precursors and ε is a steric factor, the Van

der Waals enhancement factor, with a value of 2.2 [54]. From kinetic collision theory, we can

143

compute the frequency of collision between two molecules in the free-molecular regime

βP = d2PAH

√8πkBT

mtar. (6.18)

dPAH , the effective diameter of the precursor, can be computed using a geometric relationship

assuming that the precursor is highly condensed [58]

dPAH = dA

√2mtar

3mC. (6.19)

Given these definitions, Equation 6.17 can be expanded and then simplified

rSN =4εd2

AN2tar√

2πkBT mtar

3mC. (6.20)

Note that the above equation represents the number of incipent soot particles created through the

nucleation process. Two precursors are consumed for every one soot particle created; therefore, to

obtain the total number of precursors consumed from soot nucleation multiply this term by 2 as

seen in Equation 6.1.

6.1.4 Deposition

When a precursor collides with a soot particle, there is a likely chance that the precursor

will stick to the surface of the soot particle, thus growing the particle’s surface. This is the process

of precursor deposition and is modeled as

rPD = εβpsNtarNsoot , (6.21)

using a frequency of collision, βps, between precursors and particles. ε is the Van der Waals

enhancement factor. We compute the frequency of collision assuming a free-molecular flow regime

βps = (dsoot +dPAH)2√

πkBT2mtar

. (6.22)

144

dPAH is the effective diameter of the precursors and is computed using Equation 6.19 and dsoot is

the effective diameter of the soot particles,

dsoot =

(6msoot

πρs

)1/3

, (6.23)

where msoot is the mass of individual soot particles defined as msoot =MsootNsoot

. Substituting the

collision frequency and effective diameters back into Equation 6.21 yields

(6.24)rPD

= ε√

kBT

[m1/2

tar

(2π

)1/6(3msoot

ρs

)2/3

+dA

(3

mC

)1/2

π1/3(

6msoot

ρs

)2/3

+d2A

(2πmtar

9

)1/2]

NtarNsoot .

This term represents the rate of precursors depositing on the surface of soot particles. To obtain the

mass accumulation which results, the second term in Equation 6.3, we simply multiple the number

rate of precursor deposition by the mass of the precursors being deposited.

6.1.5 Surface Reactions

This model considers three types of surface reactions: surface growth through the es-

tablished hydrogen-abstraction-carbon-addition mechanism (HACA), consumption through oxi-

dation, and gasification.

HACA is a literature-established growth mechanism [7, 56, 129, 128] described previously

in Section 5.1.1. The overall reaction rate is given in Equation 5.17. Use of this equation requires

the first two moments of a PSD. For soot this is not a problem as Nsoot and Msoot are the zeroth and

first moments, respectively, of the soot PSD and can be used directly. For precursors, the zeroth

moment of the distribution is resolved directly, Ntar, and the second can be computed using the

assumed molecular size,

Mtar = mtarNtar. (6.25)

Oxidation and gasification rates are resolved using the work of Chapter 4, given in Equations 4.3

and 4.10.

The total effect of all three surface reactions are the sum of the individual processes,

rSS or rPS = RHACA−Roxidation−Rgasi f ication. (6.26)

145

This is a rate per unit of available surface area. For precursors, the surface area is computed

through an empirical correlation developed by Tielens [186] and results in the coefficient of the

last term of Equation 6.1. This mass change of precursors is then converted to an equivalent

number of particles produced or consumed as we assume all particles are a constant size. The

surface area of soot particles is assumed to be spherical and the resulting area is seen in the last

term of Equation 6.3.

6.1.6 Coagulation

Particle-particle coagulation only affects the number density term of soot particles as to-

tal soot mass is conserved throughout the process. The basic concept of coagulation is that two

spherical particles collide, stick, and mold forming one larger particle that is still roughly spherical

rCS = βSN2soot . (6.27)

Computing the frequency of collisions among soot particles is more difficult than among precursors

or between precursors and particles. This is because soot particles can grow to very large sizes,

large enough that the flow and transport of soot particles can no longer be modeled with free-

molecular flow regime assumptions, but rather as particles grow in size they increasingly show

characteristics of a continuum flow regime. To capture this potential in flow regime we model

coagulation in both a free-molecular and continuum flow regime and use the Knudsen number

Kn =2λsoot

dsoot(6.28)

a ratio of particle mean free path to particle diameter to determine which flow regime we are in

and which solution to use.

In the free-molecular flow regime, the frequency of particle collisions is computed in a way

similar to those discussed before with the soot nucleation and precursor deposition submodels,

βf

S = εd2soot

√8πkBTmsoot

(6.29)

where dsoot is computed from Equation 6.23.

146

In the continuum flow regime, the frequency of particle collision was modeled by Seinfeld

and Pandis [167] as

βcS =

8kBT3µ

(1+1.257Kn) , (6.30)

where µ is the gas viscosity.

Should we be firmly in the free-molecular flow regime, Kn < 0.1, then we use βf

S in Equa-

tion 6.27 to model the coagulation rate. If we are firmly in the continuum flow regime, Kn > 10,

then we use β cS in Equation 6.27. If we are in the transition regime, 0.1 < Kn < 10, then we use a

weighted combination

βtS =

β cS

1+Kn+

βf

S1+1/Kn

(6.31)

in Equation 6.27.

6.2 Simulations

This simplified model is proposed as a replacement to the detailed model of Chapter 5

for systems that are too complex or computationally expensive for the latter. Thus it is important

to realize the comparability of these two models. For this purpose, two simulations have been

performed to juxtapose these two models.

6.2.1 Coal Flat-Flame Burner

Details of this system were given in Section 5.2.1. The validation simulations for the

detailed model were repeated but with one exception, small amounts of oxidizers, 2.63E-7 Pa

of OH and 2.17E-2 Pa of O2, were numerically entrained in the flow to allow small amounts

of oxidation to occur. In the experiment, coal particles are introduced into a fuel-rich flow and

soot/char are collected by the suction probe before encountering an oxygen-rich region. Thus little

to no oxidation occurs to soot particles in the experimental set-up. In this model comparison an

exploration of all mechanisms, including oxidation, is desirable, thus these oxidizing species were

numerically entrained, and kept constant, in the flow to compare the effects of partial oxidation on

the soot profiles.

147

0 25 50 75 100z (mm)

1015

1016

1017

1018

1019

Num

ber D

ensit

y (#

/m3 )

Particle Number Density

0 25 50 75 100z (mm)

0

1

2

3

PPBv

Soot Volume Fraction

Detailed ModelSimplified Model

Figure 6.5: Particle number density and soot volume fraction simulation results from the coalflat-flame burner with entrained oxygen, comparing simplified model against the detailed model.

Figure 6.5 shows the results of these simulations where the proposed simplified model is

directly compared against the previously developed detailed model. The left plot compares the

resulting particle number density between the two models. Here the simplified model tends to pre-

dict a higher number density. This is probably because the simplified model assumes a molecular

size of tar to be 350 g/mole, whereas the detailed model resolves the precursor distribution over

five sections, three of which have mass higher than 350. Smaller tar molecules tend to crack away

faster than the larger particles, as a result larger molecules tend to make a higher percentage of soot

particles in the detailed model, but in both cases total mass of precursors going to soot is similar.

Therefore, the total number of particles predicted by the simplified model is more than the detailed

model.

The right plot compares soot volume fraction predictions between the detailed and sim-

plified models. Because the initial mass of particles resulting from soot nucleation is roughly

equivalent in the two simulations, the simple model predicts a greater availability of total particle

surface area, due to the larger number density, (i.e. smaller particles but more of them). A larger

number of small particles leads to increased surface area at which oxidation can take place. This

may be the cause of the lower overall soot volume fraction in Figure 6.5. Although the models

do not perfectly agree, the curves shown by the detailed and simplified models follow the same

trends very closely and predict similar particle profiles in this system. Given the large difference

in computational cost, and the difficulty and uncertainty in soot modeling, the agreement between

the detailed and simplified models is considered quite good.

148

6.2.2 LES Simulation

The above coal flat-flame burner provided a good comparison between the two models but

the system configuration is simple. To provide a more complex comparison of these two models,

LES were perfomed of the OFC described in Section 3.3.1. These simulations were carried out to

10 seconds of simulation time using the LES software, Arches, described in Section 3.2.

The fuel was a Skyline coal with Proximate and Ultimate analysis shown earlier in Ta-

ble 3.2; inlet flow rates were double those shown in Table 3.3. Fuel density was 1300 kg/m3 and

the dry/ash free fuel enthalpy was taken as -1.161E6 J/kg. The fuel is represented by a particle

distribution resolved using DQMoM with 3 quadrature nodes at 20, 120, and 240 µm in diame-

ter. Initially, the total weight of the fuel is divided up as 42.1% small particles, 30.6% medium

particles, and 27.3% large particles. Internal coordinates of the fuel particle distribution resolved

include 3 coordinate-velocities, temperature, number density, particle diameter, raw fuel mass,

char mass, and particle enthalpy.

Fuel pyrolysis is modeled using a first-order weighted yield model (FOWY)

dVdt

= Adevol exp(−Edevol

RT

)(V∞−V ) , (6.32)

where V is the volatile yield of the parent fuel. This model was calibrated against CPD assuming

a maximum temperature of 2300 K and a heating rate of 1E5 K/s yielding values of 1.972E7 (1/s),

1.133E4 (J/mole), and 0.664 for Adevol , Edevol , and V∞ respectively.

Char oxidation is modeled using a global reaction rate

dMchar

dt= AcharPn

O2exp(−Echar

RT

), (6.33)

where Mchar is the consumption rate of carbon per m2 of available surface area for oxidation.

Values for Achar, Echar, and n were taken from work done by Murphy and Shaddix [138] and were

4.128 (kg/m2 atmn), 45.5E6 (J/mol), and 0.18 respectively.

Thermal radiation was resolved using discrete ordinates with 8 ordinates [132]. Absorption

coefficients were computed for the grey gases and soot aerosol cloud using the Hottel et al. [86]

model.

149

Soot Volume Fraction Soot Particle Number Density

Detailed Model Simplified Model Detailed Model Simplified Model

max

50%

75%

25%

min

Figure 6.6: Results of the comparative LES coal simulations. From left to right the figuresdepict: Soot volume fraction predicted by the detailed soot model (max (red) = 3.5 ppmv,min (blue) = 0 ppmv), soot volume fraction predicted by the simplified soot model(max = 3.5 ppmv, min = 0 ppmv), soot particle number density from detailed model(max = 1E21 #/m3, min = 0 #/m3), and soot particle number density from simplified model(max = 1E21 #/m3, min = 0 #/m3).

Under the above conditions, two simulations were carried out to provide a more complete

comparison between the two proposed soot models.

The first simulation, using the detailed model of Chapter 5 as a soot model, yielded an

average soot volume fraction of 0.1436 ppmv across the entire domain and 0.927 ppmv along the

reactor centerline. A 2-dimensional cross-section passing through the reactor centerline is shown

in Figure 6.6. The figure on the far left depicts an instantaneous profile of the soot volume fraction

as predicted by the detailed model, while the third figure from the left depicts the particle number

density.

The second simulation, using the simplified model presented in this chapter as a soot model,

yielded an average soot volume fraction of 0.0770 ppmv across the entire domain and 0.735 ppmv

along the reactor centerline. Figure 6.6 shows the same 2-dimensional cross-section of the reactor

150

Table 6.2: Computational expense comparison between the detailed model of Chapter 5 and thesimplified model of Chapter 6 and found in the OFC simulation of Section 6.2.2.

Detailed Model Simplified Model

Lines of code to implement model in Arches 1233 613

Simulation CPU hours 6300 820

Average centerline soot volume fraction (ppmv) 0.935 0.735

with the second from left showing soot volume fraction and fourth from the left showing particle

number density.

The predicted profiles for the two figures are very similar, but with a few key differences.

The main difference is the detailed model predicts a larger amount of soot produced. As before,

with the flat-flame burner, the number density of the detailed model is closer to the burner with

similar soot volume fraction, implying a smaller particle surface density and therefore consumption

reactions are more limited. The difference between these two models was less pronounced in the

LES simulation than the flat flame coal burner but the trends are the same.

Another difference came from the rate of particle mixing. It seemed the detailed model cap-

tured slightly more dispersion of soot particles than the simplified model did, shown in the slightly

darker peripheries of the number density plots in Figure 6.6. This slight increase in dispersion

may be a result of a number of things, but most likely is that the diminished effect of consumption

reactions allowed particles to disperse more since a small number of particle penetrated the flame’s

reaction zone in the detailed model, but not in the simplified model.

While the two models predict differences in the soot particle distribution across the reactor,

the predictions are both comparable, and provide a quantitative validation the proposed simplified

model. The clear advantages of the simplified model are shown in Table 6.2. This table shows

that soot volume fraction predictions between the two models are comparable but the complexity

of model implementation, reflected in part by the number of lines of code, is much different.

Implementing the simplified model is much simpler, requiring half the lines of code. At the same

time, computational cost is drastically reduced, simulation CPU hours were reduced by nearly

8 times. In the simplified model simulation, most of the computational cost was absorbed by a

resolution of thermal radiation and the cost of computing the simplified model was trivial. In the

151

detailed model simulation, on the other hand, computing the detailed model was non-trivial, and

even surpassed the computational cost of solving thermal radiation.

6.3 Conclusions

The detailed model presented in Chapter 5 has proven accurate in predicting soot produced

from solid complex fuel. But the resolution of these physics comes at a high computational cost,

and when implemented in large simulations becomes infeasible. The proposed simplified model

of this chapter significantly cuts computational costs by making two key assumptions: first that

all particles are spherical and second that distributions in a simulation cell are mono-dispersed.

These two assumptions, along with the application of a few surrogate models, greatly simplify the

mathematics of the model and reduce computational costs while maintaining a promising level of

accuracy in predictions.

152

CHAPTER 7. CONCLUSIONS AND FUTURE WORK

7.1 Conclusions

Work in this dissertation has proposed three models for predicting soot formation from

solid complex fuels.

The first model, the Brown model [21], was a previously developd model for predicting

soot in coal reactors. In this work, this model was first reproduced and then implemented into a

CFD software. Simulations showed that while this model was excellent for predicting soot pro-

files in traditional air-fired coal boilers it was inadequate to predict profiles in an oxy-coal boiler.

Upon further investigation it was found that soot gasified with high concentrations of CO2; thus

a soot gasification model and new soot oxidation model were derived with uncertainty attached

to the model parameters using Bayesian statistics. With a modified gasification term and updated

oxidation term, the Brown model was able to predict soot profiles with high confidence within a

oxy-coal boiler. When compared to optical experimental data, relative error between simulation

and experiments was 14-35% for soot volume fraction.

The second model developed, the detailed model, is a more physics-based model using the

method of moments with interpolative closure [53] for the soot portion of the model and other

chemistry fundamentals for the precursor portion of the model. Unlike the Brown model, the

detailed model is capable of high fidelity predictions of soot for both coal and biomass in any

combustion system. This model was validated against coal soot data collected by Ma et al. [121]

on a flat flame-burner and showed good agreement for soot profiles throughout the flame both in

location and quantity of yielded soot. Validation was also carried out against biomass gasification

experiments and simulated soot yields were within the error bounds of reported experimental re-

sults. Particle size distributions were also compared and showed a high level of agreement between

model predictions and experimental results.

153

The third model developed, the mono-dispersed model, was a simplification of the second

model with significant computational cost reductions. The coal flat flame burner set-up was used

again to compare the detailed and simplified model. In these simulations the initial mass yield

of soot particles is nearly identical but with the simplified model predicting a higher number of

smaller particles. As oxidation occurs this difference in particle number and size results in the

simplified model predicting a higher rate of consumption than the detailed model. However, dif-

ferences are small and showing great promise for the simplified model. In comparative LES, both

models perform similar enough to almost be indistinguishable. The simplified model does show

slightly less particle dispersion than the detailed model most likely due to the higher rate of con-

sumption. In computational cost, the simulation using the simplified model took 1/8 of the CPU

hours, thus showing the significant cost reductions while maintaining high fidelity.

The following guidelines are given to determine when each model should be used.

1. Simulations of smaller systems, biomass or coal, that require a high degree of accuracy but

are small enough to justify high resolution should use the detailed model.

2. Systems where strange particle size distributions are expected (due to abnormal mixing) use

the detailed model.

3. Simulations of biomass systems where computational cost is important should use the mono-

dispersed model.

4. Simulations of traditional coal systems use the Brown Model with gasification term added if

oxy-coal.

5. Simulations of non-traditional coal systems (gasification, two-stage burner, etc) use the

mono-dispersed model.

6. If in doubt use detailed model and try to reduce to mono-dispersed model if possible.

7.2 Possible Model Improvements

While the presented developed models may represent a current state-of-the-art in modeling

soot formation from solid complex fuels, there are certainly areas of potential improvement.

154

Each submodel which is based on collision rate (soot nucleation, precursor deposition, and

soot coagulation) uses a Van der Waals enhancement factor. This factor is meant to combine the

effects of collision efficiency and Van der Waals attractive forces. This factor is derived only for

particle-particle interactions within a limited range [79]. This factor was then applied liberally to

particles across all domains and precursor molecules as well. While the enhancement factor may

very well be applicable this liberally it is unknown as yet if there is a better method to account for

these forces.

This work applied a surrogate molecule approach to the chemistry of tar. In reality, tars

released by coal or biomass during primary pyrolysis include hundreds of possible species, most

of which are unknown but is an active area of research [163]. As more becomes known about

the speciation of tar, updated mechanisms should be applied to the detailed model to maintain the

model as a comprehensive model. In addition, as additional gas-phase mechanisms are explored

and updated, such as surface growth through a propagyl reaction [155], these should also be added

to the detailed model.

In this work, a particle shape factor was applied to capture morphology. This particle shape

factor has been previously presented in the literature but is not fully explored. As an example, this

work suggested an improvement of including a coagulation term to its evolution. More research

and development should be carried out exploring the potential of this particle shape factor, and if

necessary, particle morphology schemes might be applied to these developed models to reduce the

numerical stiffness often introduced when resolving particles’ shape moment.

7.3 Future Development of a Surrogate Model for FIRETEC

The presentation of the second and third models is adequate for the needs of CCMSC in

the full-scale boiler simulations. However, neither model can be used by the EES division in wild-

land fire simulations. This is because FIRETEC, discussed in Section 1.4, does not resolve fluid

dynamics on the combustion scale (approx. 1 cm) but on a much larger scale (approx. 1 m). All

three of the presented models require a resolution at the combustion level for use in CFD.

Future work includes the creation of a soot emissions model for FIRETEC. This will be

accomplished using the simplified model but modified with a modial model application. Within a

155

cell of FIRETEC there exists regions of pyrolyzing fuel, fuel-rich flame, flame-reaction zones, and

fuel-lean quenched zones. In each of these zones soot particles are undergoing different processes.

FIRETEC resolves a fuel consumption rate at each timestep. Using the ‘sooting potential’

model developed in Chapter 6 we can predict what mass fraction of that consumed fuel is converted

into tars and the size of those tars. From this produced tar we can determine a fraction of it which

converts to soot particles by computing rates of thermal cracking and soot nucleation using the

submodels of the simplified model. These computations give a rate of soot inception within a

FIRETEC cell, and the size of the incipient particles.

Determining a volume of fuel-rich flame in a cell is accomplished by comparing an ex-

pected oxygen concentration, that which would occur by advection, and actual oxygen concentra-

tion, a variable resolved in FIRETEC. In this fuel-rich region particles coagulate and grow through

the submodels in the simplified model for particle coagulation and surface growth. A surrogate

model is developed to determine the particle-size distribution already within the flame-region,

needed to determine rates coagulation and surface growth.

The size of the reaction zone is determined by first computing the flame surface area using

a correlation developed by Zhou and Mahalingam [207] which computes flame surface density

as governed by overall mixture fractions, stoichiometry, and turbulence. Flame surface area is

multiplied by reaction zone thickness, computed by a correlation developed by Bilger [14]. Within

this reaction zone, particles are oxidized according the derived oxidation model of Chapter 4.

Any particles not found in the fuel-rich zone or the reaction zone are assumed to be in the

fuel-lean quenched zone. Here only particle aggregation occurs modeled as the continuum regime

coagulation using the simplified model. By dividing a single cell into different zones, an effective

modial model, we are able to apply the developed models of soot formation to FIRETEC despite

not fully-resolving the combustion kinetics within a simulation. This simplified-modial soot model

represents a fourth and final model for predicting soot form solid complex fuels.

156

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173

APPENDIX A. MODEL DERIVATIONS FOR DEVELOPED DETAILED SOOT MODEL

This appendix is added to give greater details of some of the submodel derivations in the

detailed model of Chapter 5. This study uses the Method of Moments with Interpolative Closure

(MoMIC), a commonly used model in the soot formation literature, but its derivation is not readily

available. As a result, one may find multiple inconsistent variations of MoMIC among different

research groups. One purpose of this Appendix is to provide a complete derivation for MoMIC

which new researchers may use as a reference and as an aid to understanding the model.

A.0.1 Soot Nucleation from Sections 5.1.1 and 5.1.2

Soot nucleation is based on the coalescence of precursors

(A.1)dNPAH

idt

= −nbins

∑j=1

βPAHi, j NPAH

i NPAHj ,

where βi, j represents the frequency that precursors NPAHi and NPAH

j collide and stick together. The

frequency factor β PAHi, j is computed from collision theory

(A.2)βi, j = (dPAHi + dPAH

j )2

√πkBT2µi, j

,

where kB is Boltzmann’s constant, T is the temperature, and µi, j is the reduced mass of precursors

i and j(A.3)µ =

mim j

mi + m j.

dPAHi represents the collision diameter of precursor i, which we compute using a geometric rela-

tionship for the most condensed PAH species of size mi [58]

(A.4)dPAH

i = dA

√2mPAH

i3mC

,

= Ch

√mPAH

i ,

174

with dA being the diameter of a single aromatic ring, 1.395√

3 A.

Equation A.2 only describes the frequency of collision between two non-interacting spher-

ical molecules. A van der Waals enhancement factor ε = 2.2 [56, 111] is applied to Equation A.2,

resulting in Equation 5.8.

Equation A.1 expresses the change in precursor sections’ number densities due to soot

nucleation. To evaluate the effect on the soot PSD we sum across all sections of the precursor PSD

and divide by two to discount the double-counting of nucleation occurrences

(A.5)dNdt

=12

nbins

∑i=1

nbins

∑j=1

βPAHi, j NPAH

i NPAHj .

Here N would indicate a soot particle the total number of soot particles. From here we can convolve

the above equation with the moment definition, Equation 5.2, to obtain

(A.6)∞

∑k =1

mrkdNdt

=∞

∑k=1

mrk12

nbins

∑i=1

nbins

∑j=1

βPAHi, j NPAH

i NPAHj .

Substitute mk = mPAHi +mPAH

j ,

(A.7)∞

∑k =1

mrkdNdt

=∞

∑k=1

12

nbins

∑i=1

nbins

∑j=1

(mi + m j)rβ

PAHi, j NPAH

i NPAHj ,

(A.8)∞

∑k =1

mrkdNk

dt=

12

nbins

∑i=1

nbins

∑j=1

(mi + m j)rβ

PAHi, j NPAH

i NPAHj ,

(A.9)dMr

dt=

12

nbins

∑i=1

nbins

∑j=1

(mi + m j)rβ

PAHi, j NPAH

i NPAHj .

Which is equivalent to Equation 5.27.

A.0.2 Precursor Deposition from Sections 5.1.1, 5.1.2, and 5.1.2

The submodel for the precursor deposition, from the perspective of the precursors, begins

the same as the nucleation submodel

(A.10)dNPAH

j

dt= −

∞

∑i=1

βi, jNiNPAHj .

This equation represents the change of the number NPAHj of precursors as they collide and stick

with Ni particles. The frequency factor βi, j is computed similar to Equation A.2, with a few small

175

differences. We assume the mass mi of the soot particle is much larger than the precursor molecule

m j. Therefore the reduced mass is µi, j = m j. We also substitute a soot particle diameter as this

reaction occurs with the collision of a soot particle and precursor instead of two precursors

(A.11)βi, j = 2.2(di + dPAHj )2

√πkBT2m j

.

The diameter of the soot particle is a function of the particle mass and shape factor,

(A.12)di = CaCsm1/3i ,

where Cs, defined in Equation 5.14, is the coefficient related to the diameter of a spherical particle,

and Ca, defined in Equation 5.15, is a coefficient relating to the particle shape deviation from

spherical. Substitute this diameter definition back into Equation A.11

(A.13)βi, j = 2.2(CaCsm1/3i +Chm1/2

j )2

√πkBT2m j

,

(A.14)βi, j = 2.2

√πkBT

2(C2

aC2s m−1/2

j m2/3i + 2CaCsChm1/3

i +C2hm j).

Substitute β back into Equation A.10

(A.15)dNPAH

j

dt= −2.2

√πkBT

2

∞

∑i=1

(C2aC2

s m−1/2j m2/3

i + 2CaCsChm1/3i +C2

hm1/2j )NiNPAH

j ,

and apply the definition of moments from Equation 5.2

(A.16)dNPAH

j

dt= −2.2

√πkBT

2(C2

aC2s m−1/2

j M2/3 + 2CaCsChM1/3 +C2hm1/2

j M0)NPAHj ,

which is the same as Equation 5.13.

To derive the effects of precursor deposition on the soot moments, we start by defining the

change in the number of particles of a given size, mi. The number of particles, Ni, increases as

smaller particles, of size mi−mPAHj , grow to mi through the deposition process. Ni decreases as

those particles grow larger also through deposition

(A.17)dNi

dt=

nbins

∑j=1

(βi− j, jNi− jNPAH

j − βi, jNiNPAHj

).

We convolve Equation A.17 using the moment definition, Equation 5.2, to obtain

(A.18)∞

∑i =1

mridNi

dt=

∞

∑i=1

mri

nbins

∑j=1

(βi− j, jNi− jNPAH

j − βi, jNiNPAHj

),

176

(A.19)dMr

dt=

∞

∑i=1

nbins

∑j=1

mri βi− j, jNi− jNPAH

j︸︷︷︸Term1

−∞

∑i=1

nbins

∑j=1

mri βi, jNiNPAH

j︸︷︷︸Term2

.

Now we will treat each term individually. Discretize each PSD as a series of sections defined by

the minimal possible size, mC. When each is discretized, we can say that mri = irmr

C (also note

mrj = jrmr

C). Substitute this definition into the first term of A.19

(A.20)Term1 =∞

∑i=1

nbins

∑j=1

mrCirβi− j, jNi− jNPAH

j .

Now we define k = i− j and switch the order of the summations

(A.21)Term1 =nbins

∑j=1

∞

∑k=1− j

mrC(k + j)r

βk, jNkNPAHj .

There are no particles of negative or zero size, therefore we may set all portions of the summation

where k <= 0 to be equal to zero

(A.22)Term1 = 0 +nbins

∑j=1

∞

∑k=1

mrC(k + j)r

βk, jNkNPAHj .

Substituting the binomial expansion of (k+ j)r yields

(A.23)Term1 =nbins

∑j=1

∞

∑k=1

mrC

r

∑l=0

(rl

)jr−lkl

βk, jNkNPAHj ,

(A.24)Term1 =nbins

∑j=1

r

∑l=0

(rl

)∞

∑k=1

jrmrCklml

C

jlmlC

βk, jNkNPAHj .

(A.25)Term1 =nbins

∑j=1

r

∑l=0

(rl

)∞

∑k=1

mr−lj ml

kβk, jNkNPAHj ,

Returning to Equation A.19, we substitute in the resolved value for the first term

(A.26)dMr

dt=

nbins

∑j=1

r

∑l=0

(rl

)∞

∑k=1

mr−lj ml

kβk, jNkNPAHj −

∞

∑i=1

nbins

∑j=1

mri βi, jNiNPAH

j .

Note, that when r = l the first and second terms are equivalent. Therefore,

(A.27)dMr

dt=

nbins

∑j=1

r−1

∑l=0

(rl

)∞

∑i=1

mr−lj ml

iβi, jNiNPAHj .

177

Now substitute the βi, j from Equation A.14, and simplify by using the definition of moments,

Equations 5.2 and 5.40,

dMr

dt=

nbins

∑j=1

r−1

∑l=0

(rl

)∞

∑i=1

mr−lj ml

i2.2

√πkBT

2(C2

aC2s m−1/2

j m2/3i + 2CaCsChm1/3

i +C2hm1/2

j )NkNPAHj ,

(A.28)

dMr

dt= 2.2

√πkBT

2

r−1

∑l=0

(rl

)(C2

aC2s MPAH

r−l−1/2Ml+2/3 + 2CaCsChMPAHr−l−1/2Ml+1/3 +C2

hMPAHr−l+1/2Ml),

(A.29)

to obtain Equation 5.39.

We can carry out this derivation for the soot surface moment but we will find that the

fractional moments, Md , leads to complications in the first term of Equation A.19. We can use

a grid function to resolve that first term, but given the nature of this submodel it is simpler and

computationally less expensive to resolve the submodel using Lagrangian Interpolation between

the already resolved full moment terms as seen in Equation 5.45. In addition, this interpolation is

just as accurate as applying a grid function to the first term of Equation A.19.

A.0.3 Precursor Cracking from Section 5.1.1

The entire principle behind the cracking scheme of this model is to take the Marias et al.

model, that seen in Table 5.1, and apply it in way consistent with the sectional method of precursor

evolution. We define the characterization of the precursor species

NPAHi = Ni,Phenol +Ni,Naphthylene +Ni,Toluene +Ni,Benzene, (A.30)

Ni,Phenol = xpheNPAHi , (A.31)

Ni,Naphthalene = xnapthNPAHi , (A.32)

Ni,Toluene = xtolNPAHi , (A.33)

178

Ni,Benzene = xbenNPAHi , (A.34)

where xi represents a mole fraction of a precursor section that may be characterized by phenol,

naphthalene, toluene, or benzene. The critical portion of this model is the use of phenol, naphtha-

lene, toluene, and benzene directly as surrogates to represent those molecules

Ni,Phenol ≈ NC6H6O, (A.35)

Ni,Naphthalene ≈ NC10H8, (A.36)

Ni,Toluene ≈ NC7H8, (A.37)

Ni,Benzene ≈ NC6H6. (A.38)

The above equations may seem odd as the two species do not have the same mass, but it is important

that we recognize that this approximation holds up with respect to how the species crack, as in that

a single cracking reaction would result in a similar proportion of mass loss from the surrogate

molecule and the actual precursor molecule and the rate of cracking reactions are approximately

the same. Now working with just the surrogate in mind, we want to know the rate of mass cracked

to gas for each surrogate species. Generalizing rates from Table 5.1, we may compute a rate of

mass production for gas from the surrogate species

Mgas = ∑miνRi, (A.39)

where ν is the stoichiometric coefficient associated with each gaseous species, positive if the

species is a product and negative if it is a reactant. This equation only includes light gas species, not

any of the surrogate species; therefore, the first equation of Table 5.1 would only consider species

CO, CH4, and H2. Recognizing now that the mass of gas produced is equal to the mass loss of

surrogate species, we may convert that mass loss to an equivalent number of surrogate molecules

179

consumed(A.40)Mgas = Msurrogate,

(A.41)Nsurrogate

Na'

Msurrogate

msurrogate.

The division of Avogadro’s number is to convert kmoles to a number of molecules. Putting all this

together for each reaction of the table,

(A.42)dNC6H6O

Nadt=

(mCO + 0.1mCH4 + 0.75mH2)k1 [C6H6O]

mC6H6O,

(A.43)dNC6H6O

Nadt=

(3mCO + mCO2 + 3mCH4 − 3mH2O)k2 [C6H6O]

mC6H6O,

(A.44)dNC10H8

Nadt=

(4mCO + 5mH2 − 4mH2O)k3 [C10H8] [H2]0.4

mC10H8

,

(A.45)dNC7H8

Nadt=

(mCH4 − mH2)k4 [C7H8] [H2]0.5

mC7H8

,

(A.46)dNC6H6

Nadt=

(5mCO + mCH4 + 6mH2 − 5mH2O)k5 [C6H6]

mC6H6

.

Combine Equations A.40 and A.41 as they both show the change of phenol. Now we take the

approximation we set at the beginning of this derivation, xiNPAHi ≈Ni where i is a surrogate species

and substitute into both sides of the above equations. Note that [i] = NiNa

,

(A.47)

d(xpheNPAHi )

Nadt=

(mCO + 0.1mCH4 + 0.75mH2)k1(xphiNPAH

i )Na

mC6H6O

+(3mCO + mCO2 + 3mCH4 − 3mH2O)k2

(xphiNPAHi )

Na

mC6H6O,

(A.48)d(xnapthNPAH

i )

Nadt=

(4mCO + 5mH2 − 4mH2O)k3(xnapthNPAH

i )Na

[H2]0.4

mC10H8

,

(A.49)d(xtolNPAH

i )

Nadt=

(mCH4 − mH2)k4(xtolNPAH

i )Na

[H2]0.5

mC7H8

,

180

(A.50)d(xbenNPAH

i )

Nadt=

(5mCO + 1mCH4 + 6mH2 − 5mH2O)k5(xbenNPAH

i )Na

mC6H6

.

Conveniently, Avogadro’s number would cancel out on both sides of each equation. Returning to

Equation A.30, we substitute some values and take the derivative of both sides

(A.51)dNPAH

idt

=d(xpheNPAH

i )

dt+

d(xnapthNPAHi )

dt+

d(xtolNPAHi )

dt+

d(xbenNPAHi )

Nadt,

substitute in our derived equations above, Equations A.47-A.50, and the known molecular weights

and we get Equation 5.16 from the model.

A.0.4 Soot Coagulation from Sections 5.1.2 and 5.1.2

Similar to the collision between soot particles and precursors, the start of this model begins

with two terms, the first representing the production of a given sized particle through the collision

and sticking of two particles of lesser size, and the second representing the consumption of a given

sized particle as it collides and sticks with another particle

(A.52)dNi

dt=

12

i−1

∑j=1

β j,i− jNi− jN j −∞

∑j=1

βi, jNiN j.

Convolve this equation with the moment definition, Equation 5.2, to obtain

(A.53)∞

∑i =1

mridNi

dt=

∞

∑i=1

mri

(12

i−1

∑j=1

β j,i− jNi− jN j −∞

∑j=1

βi, jNiN j

),

(A.54)dMr

dt=

12

∞

∑i=1

i−1

∑j=1

mri β j,i− jN jNi− j −

∞

∑i=1

∞

∑j=1

mri βi, jNiN j.

If we iterate across the j parameter space first and then across the i space second, we can

reorganize our iterization limits. Refer to Figure A.1 for a visual representation of this summation

reorganization

(A.55)dMr

dt=

12

∞

∑j=1

∞

∑i= j+1

mri β j,i− jN jNi− j −

∞

∑i=1

∞

∑j=1

mri βi, jNiN j.

Substitute, k = i− j, 2

(A.56)dMr

dt=

12

∞

∑j=1

∞

∑k+ j= j+1

mrk+ jβ j,kN jNk −

∞

∑i=1

∞

∑j=1

mri βi, jNiN j,

181

ii

jj

Figure A.1: Visual evidence of iteration reorganization.

(A.57)dMr

dt=

12

∞

∑j=1

∞

∑k=1

(mk + m j

)rβ j,kN jNk −

∞

∑i=1

∞

∑j=1

mri βi, jNiN j.

Equation A.57 is the base equation common for all particle coagulation. From here we’ll de-

rive four different submodels5: whole moment resolution for both continuum and free-molecular

continuum flow regimes, and fractional moment resolution (for the surface moment) resolution in

both regimes as well. First we will resolve the whole moment submodels. Through a binomial

expansion we know that

(A.58)(mi + m j)r =

r

∑k=0

(rk

)mk

i mr−kj .

Therefore(A.59)

dMr

dt=

12

∞

∑i=1

∞

∑j=1

r

∑k=0

(rk

)mk

i mr−kj β j,iN jNi −

∞

∑i=1

∞

∑j=1

mri βi, jNiN j.

Where k = r, the first term and second term are equivalent and cancel each other out

(A.60)dMr

dt=

12

∞

∑i=1

∞

∑j=1

r−1

∑k=1

(rk

)mk

i mr−kj β j,iNiN j.

Now we resolve the frequency of coagulation parameter for the continuum regime, βi, j, as defined

by Seinfeld and Pandis [167]

(A.61)βCi, j = KC

Ci

m1/3i

+C j

m1/3j

(m1/3i + m1/3

j

),

(A.62)KC =2kBT3η

,

(A.63)Ci = 1 + 1.257Kni,

182

(A.64)Kni =2λ f

di,

(A.65)di = CaCsm1/3i ,

where kB is Boltzmann’s constant, T is the temperature, η is the gas viscosity, and λ f is the

gas mean free path, Ca and Cs were defined earlier in Equations 5.15 and 5.14 respectively. We

substitute all these definitions back into Equation A.61 and then that back into Equation A.60,

(A.66)βCi, j = KC

1 + 1.257 2λ f

CaCsm1/3i

m1/3i

+

1 + 1.257 2λ f

CaCsm1/3j

m1/3j

(m1/3i + m1/3

j

),

(A.67)K′C =2.514λ f

CaCs,

(A.68)βCi, j = KC

(m−1/3

i + m−1/3j + K′C

[m−2/3

i + m−2/3j

])(m1/3

i + m1/3j ),

dMr

dt=

12

∞

∑i=1

∞

∑j=1

r−1

∑k=0

(rk

)mk

i mr−kj KC

(m−1/3

i +m−1/3j +K′C

[m−2/3

i +m−2/3j

])(m1/3

i +m1/3j )NiN j,

(A.69)

(A.70)

dMr

dt=

KC

2

∞

∑i=1

∞

∑j=1

r−1

∑k=0

(rk

)(mk

i mr−kj + mk+1/3

i mr−k−1/3j + mk−1/3

i mr−k+1/3j + mk

i mr−kj

+ K′C[mk−1/3

i mr−kj + mk+1/3

i mr−k−2/3j + mk−2/3

i mr−k+1/3j + mk

i mr−k−1/3j

])NiN j,

(A.71)dMr

dt=

KC

2

r−1

∑k=0

(rk

)(MkMr−k + Mk+1/3Mr−k−1/3 + Mk−1/3Mr−k+1/3 + MkMr−k

+ K′C[Mk−1/3Mr−k + Mk+1/3Mr−k−2/3 + Mk−2/3Mr−k+1/3 + MkMr−k−1/3

]).

This is equivalent to Equation 5.31, coagulation in the continuum flow regime. Sinc the binomial

expansion of Equation A.58 does not hold when r = 0, we treat this case separately by deriving a

simplified equation starting from Equation A.57

(A.72)dM0

dt= −1

2

∞

∑j=1

∞

∑k=1

β j,kN jNk,

183

substitute for βi,k from Equation A.61,

(A.73)dM0

dt= −1

2

∞

∑j=1

∞

∑k=1

KC

(m−1/3

i + m−1/3j + K′C

[m−2/3

i + m−2/3j

])(m1/3

i + m1/3j )N jNk,

(A.74)dM0

dt= −KC

(M2

0 + M1/3M−1/3 + K′C[M0M−1/3 + M1/3M−2/3

]).

Equivalent to Equation 5.30 coagulation in the continuum flow regime. Now we will resolve the

frequency of coagulation parameter, βi, j, in the free-molecular flow regime, again as defined by

Seinfeld and Pandis [167]

(A.75)βFi, j = (di + d j)

2

√πkBT2µi, j

,

(A.76)µi, j =mim j

mi + m j.

Substitute definitions into Equation A.75 and then back into Equation A.60,

(A.77)βFi, j = (CaCsm

1/3i +CaCsm

1/3j )2

√πkBT

2 mim jmi+m j

,

(A.78)βFi, j = C2

aC2s

√πkBT

2(m1/3

i + m1/3j )2

(1mi

+1

m j

)1/2

,

(A.79)K f = C2aC2

s

√πkBT

2,

(A.80)dMr

dt=

12

∞

∑i=1

∞

∑j=1

r−1

∑k=0

(rk

)K f mk

i mr−kj (m1/3

i + m1/3j )2

(1mi

+1

m j

)1/2

NiN j.

This is equivalent to Equation 5.34 in the paper. As the term (m1/3i +m1/3

j )2 cannot be expanded

because of the fractional power, we use a grid function with Lagrangian interpolation as described

in the paper. An example of grid function resolution is shown later in Section A.0.6.

Resolving the surface moment’s coagulation submodel is more difficult because of the

fractional nature of the moment. The first term from Equation A.57 cannot be expanded because

of the fractional exponential. Like before we use a grid function to resolve the first term of the

equation after substituting the βi, j values for each regime, Equations A.61 and A.75, into the term.

184

The grid functions to be resolved are h〈d〉 from Equation 5.49 for the continuum flow regime and

f 0,0〈d〉 from Equation 5.35 for the free-molecular flow regime. Grid functions for these equations

will not be expanded here and are left up to the reader, but an example of a grid function expansion

is found in the following Section, A.0.6.

The second term can be expanded by substituting the βi, j values for each regime, Equations

A.61 and A.75, into that term and resolving each into moment expressions. The end result can be

seen in Equations 5.48 and 5.50 for the continuum and free-molecular flow regimes respectively.

A.0.5 Surface Reactions from Sections 5.1.2 and 5.1.2

This section refers to surface growth, via HACA, or surface consumption, via oxida-

tion/gasification, as the derivation is the same for all these submodels. In each case, the model

derivation is the same, differences only arise in the rate of reaction and the sign of the reaction.

The starting point for our derivation deals once again with the number of molecules of a given size

changing as molecules grow/shrink to that size and others grow/shrink beyond that size.

(A.81)dNi

dt=

ks

∆m(Ni−1Si−1 − NiSi) ,

ks (kg/m2 s) is the reaction rate per particle surface area and is unique to whichever process we are

considering (HACA, oxidation, gasification), ∆m is the change of mass due to a single reaction.

Thus ks∆m represents the number of reactions occurring per second and unit surface area of particles.

Convolve the definition of a moment, Equation 5.2, with Equation A.81

(A.82)∞

∑i =1

mridNi

dt=

∞

∑i=1

mri ks

∆m(Ni−1Si−1 − NiSi) ,

(A.83)dMr

dt=

ks

∆m

(∞

∑i=0

mri+1SiNi −

∞

∑i=1

mri SiNi

).

If we define the iterations of the sum to be in units of ∆m, then mi+1 = mi +∆m.

(A.84)dMr

dt=

ks

∆m

(∞

∑i=0

(mi + ∆m)rSiNi −∞

∑i=1

mri SiNi

),

(A.85)dMr

dt=

ks

∆m

((m0 + ∆m)rS0N0 +

∞

∑i=1


∑i=0

mri SiNi

),

185

(A.86)dMr

dt=

ks

∆m

(∞

∑i=1


∑i=1

mri SiNi

),

From our definition of 〈d〉, Equation 5.41, we can determine S, representing the surface area of all

the particles, and thus Si, the surface area of particle i,

S =∞

∑i=1

SiNi = S0

∞

∑i=1

(mi

m0

)〈d〉Ni, (A.87)

(A.88)Si = S0

(mi

m0

)〈d〉,

(A.89)S0 = πm2/30 C2

s .

Substitute this surface area into Equation A.86

(A.90)dMr

dt=

ks

∆m

(∞

∑i=1

(mi + ∆m)rπm2/3

0 C2s

(mi

m0

)〈d〉Ni −

∞

∑i=1

mri πm2/3

0 C2s

(mi

m0

)〈d〉Ni

).

Substitute the binomial expansion of (mi +∆m)r into Equation A.90

(A.91)dMr

dt=

ks

∆m

(∞

∑i=1

r

∑k=0

(rk

)∆mr−kmk

i πm2/30 C2

s

(mi

m0

)〈d〉Ni−

∞

∑i=1

mri πm2/3

0 C2s

(mi

m0

)〈d〉Ni

).

When k=r, the first and second terms cancel out

(A.92)dMr

dt=

ksπC2s m2/3−〈d〉

0∆m

∞

∑i=1

r−1

∑k=0

(rk

)∆mr−kmk+〈d〉

i Ni,

(A.93)dMr

dt=

ksπC2s m2/3−〈d〉

0∆m

r−1

∑k=0

(rk

)∆mr−kMk+〈d〉,

which is equivalent to Equation 5.37.

To resolve the surface moment submodel we go back to Equation A.90,

(A.94)dMd

dt=

ksπC2s m2/3−d

0∆m

∞

∑i=1

(mi + ∆m)〈d〉m〈d〉i Ni︸︷︷︸Term1

−∞

∑i=0

m2〈d〉i Ni︸︷︷︸

Term2

,

in this equation, the second term resolves easily, but the first term cannot be expanded because of

the fractional exponent. Once again we use a grid function, Equation 5.47, to resolve this term.

With the introduction of the grid term, g〈d〉, and the resolution of the second term we get the

equivalence of Equation 5.37.

186

A.0.6 Expansion of a grid function, Equation 5.35

We will expand the grid function for the soot coagulation among whole moments for the

free-molecular flow regime as an example of how this grid function is used. This particular grid

function is in reference to Equation 5.35 but the expansion process is the same for all grid functions

used throughout this work.

Where l = 0,

(A.95)f k,r−k0 =

∞

∑i=1

∞

∑j=1

(1mi

+1

m j

)0

mki mr−k

j (m1/3i + m1/3

j )2NiN j,

(A.96)f k,r−k0 =

∞

∑i=1

∞

∑j=1

(mk+2/3i mr−k

j + 2mk+1/3i mr−k+1/3

j + mki mr−k+2/3

j )NiN j,

(A.97)f k,r−k0 = (Mk+2/3Mr−k + 2Mk+1/3Mr−k+1/3 + MkMr−k+2/3).

Where l = 1,

(A.98)f k,r−k1 =

∞

∑i=1

∞

∑j=1

(1mi

+1

m j

)1

mki mr−k

j (m1/3i + m1/3

j )2NiN j,

(A.99)f k,r−k1 =

∞

∑i=1

∞

∑j=1

(mk−1/3i mr−k

j + 2mk−2/3i mr−k+1/3

j + mk−1i mr−k+2/3

j + mki mr−k−1/3

j

+ 2mk+1/3i mr−k−2/3

j + mk+2/3i mr−k−1

j )NiN j,

(A.100)f k,r−k1 = Mk−1/3Mr−k + 2Mk−2/3Mr−k+1/3 + Mk−1Mr−k+2/3

+ MkMr−k−1/3 + 2Mk+1/3Mr−k−2/3 + Mk+2/3Mr−k−1.

Where l = 2,

(A.101)f k,r−k2 =

∞

∑i=1

∞

∑j=1

(1mi

+1

m j

)2

mki mr−k

j (m1/3i + m1/3

j )2NiN j,

f k,r−k2 =

∞

∑i=1

∞

∑j=1

(mk−4/3i my

j + 2mk−5/3i m j+1/3

j + mk−2i mr−k+2/3

j + 2mk−1/3i mr−k−1

j

+ 4mk−2/3i mr−k−2/3

j + 2mk−1i mr−k−1/3

j + mk+2/3i mr−k−2

j + 2mk+1/3i mr−k−5/3

j

+ mki mr−k−4/3

j )NiN j,

(A.102)

187

f k,r−k2 = Mk−4/3My + 2Mk−5/3M j+1/3 + Mk−2Mr−k+2/3 + 2Mk−1/3Mr−k−1 + 4Mk−2/3Mr−k−2/3

+ 2Mk−1Mr−k−1/3 + Mk+2/3Mr−k−2 + 2Mk+1/3Mr−k−5/3 + MkMr−k−4/3.

(A.103)

Where l = 3,

(A.104)f k,r−k3 =

∞

∑i=1

∞

∑j=1

(1mi

+1

m j

)3

mki mr−k

j (m1/3i + m1/3

j )2NiN j,

(A.105)

f k,r−k3 =

∞

∑i=1

∞

∑j=1

(mk−2/3i mr−k

j + 2m8/3i mr−k+1/3

j + mk−3i mr−k+2/3

j + 3mk−4/3i mr−k−1

j

+ 6mk−5/3i mr−k−2/3

j + 3mk−2i mr−k−1/3

j + 3mk−1/3i mr−k−2

j + 6mk−2/3i mr−k−5/3

j

+ 3mk−1i m4/3

j + mk+2/3i mr−k−3

j + 2mik+1/3mr−k−8/3j + mk

i mr−k−7/3j )NiN j,

f k,r−k3 = Mk−2/3Mr−k + 2M8/3Mr−k+1/3 + Mk−3Mr−k+2/3 + 3Mk−4/3Mr−k−1

+ 6Mk−5/3Mr−k−2/3 + 3Mk−2Mr−k−1/3 + 3Mk−1/3Mr−k−2 + 6Mk−2/3Mr−k−5/3

+ 3Mk−1M4/3 + Mk+2/3Mr−k−3 + 2mik+1/3Mr−k−8/3 + MkMr−k−7/3.

(A.106)

These four values, f k,r−k0 , f k,r−k

1 , f k,r−k2 , and f k,r−k

3 are used with a Lagrangian interpolation

scheme, Equations 5.3, along with their inter values, 0,1,2, and 3, to find f k,r−k1/2 .

188

modeling soot formation derived from solid fuels

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