Modeling Soot Formation Derived from Solid Fuels

Transcription

1 Brigham Young University BYU ScholarsArchive All Theses and Dissertations Modeling Soot Formation Derived from Solid Fuels Alexander Jon Josephson Brigham Young University Follow this and additional works at: Part of the Chemical Engineering Commons BYU ScholarsArchive Citation Josephson, Alexander Jon, "Modeling Soot Formation Derived from Solid Fuels" (2018). All Theses and Dissertations This Dissertation is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in All Theses and Dissertations by an authorized administrator of BYU ScholarsArchive. For more information, please contact

2 Modeling Soot Formation Derived from Solid Fuels Alexander Jon Josephson A dissertation submitted to the faculty of Brigham Young University in partial fulfillment of the requirements for the degree of Doctor of Philosophy David O. Lignell, Chair Thomas H. Fletcher Jeremy N. Thornock Larry L. Baxter Bradley R. Adams Department of Chemical Engineering Brigham Young University Copyright 2018 Alexander Jon Josephson All Rights Reserved

3 ABSTRACT Modeling Soot Formation Derived from Solid Fuels Alexander Jon Josephson Department of Chemical Engineering, BYU Doctor of Philosophy Soot formation from complex solid fuels, such as coal or biomass, is an under-studied and little understood phenomena which has profound physical effects. Any time a solid fuel is combusted, from coal-burning power plants to wildland fires, soot formation within the flame can have a significant influence on combustion characteristics such as temperature, heat flux, and chemical profiles. If emitted from the flame, soot particles have long-last effects on human health and the environment. The work in this dissertation focuses on creating and implementing computational models to be used for predicting soot mechanisms in a combustion environment. Three models are discussed in this work; the first is a previously developed model designed to predict soot yield in coal systems. This model was implemented into a computational fluid dynamic software and results are presented. The second model is a detailed-physics based model developed here. Validation for this model is presented along with some results of its implementation into the same software. The third model is a simplified version of the detailed model and is presented with some comparison case studies implemented on a variety of platforms and scenarios. While the main focus of this work is the presentation of the three computational models and their implementations, a considerable bulk of this work will discuss some of the technical tools used to accomplish this work. Some of these tools include an introduction to Bayesian statistics used in parameter inference and the method of moments with methods to resolve the closure problem. Keywords: soot formation, particulate emissions, coal, biomass

4 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.

5 TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES NOMENCLATURE vii viii xi Chapter 1 Introduction and Overview Soot Flame Impacts Health Impacts Environmental Impacts Soot Formation Formation in Gaseous Fuels Challenges to Soot Formation Formation from Solid Fuels Oxy-Fuel Combustion Modeling Wildland Fires Chapter 2 Computational Tools Resolution of Particle-Size Distributions Sectional Methods Derived-Distribution Method Method of Moments Bayesian Inference Prior Likelihood Marginal Likelihood Posterior Chapter 3 Existing Model Implementation The Brown Model Arches Simulation Set-Up and Results Oxy-Fuel Combustor Simulations Results and Discussion Conclusions Chapter 4 Modeling Soot Consumption Introduction Methods Oxidation Model iv

6 4.2.2 Oxidation Data Gasification Model Gasification Data Bayesian Implementation Results Oxidation Model Gasification Model Rate-Informed Priors Rate Prediction Discussion Conclusions Chapter 5 Detailed Modeling of Soot from Solid Fuels Model Development Precursors Soot Validation Coal System Biomass System Conclusions Chapter 6 Simplified Modeling Model Development Precursor Inception Thermal Cracking Soot Nucleation Deposition Surface Reactions Coagulation Simulations Coal Flat-Flame Burner LES Simulation Conclusions Chapter 7 Conclusions and Future Work Conclusions Possible Model Improvements Future Development of a Surrogate Model for FIRETEC References Appendix A Model Derivations for Developed Detailed Soot Model A.0.1 Soot Nucleation from Sections and A.0.2 Precursor Deposition from Sections 5.1.1, 5.1.2, and A.0.3 Precursor Cracking from Section v

7 A.0.4 Soot Coagulation from Sections and A.0.5 Surface Reactions from Sections and A.0.6 Expansion of a grid function, Equation vi

8 LIST OF TABLES 2.1 Resolved statistical moments of the experimental distribution of Figure Resolved weights and abcissas of the 6 resolved moments in Table Experimental data for example gas-reactor Ranges over which a & b parameters were analyzed for the example gas-reactor Calibrated parameters from the Bayesian inference for the simple gas-reactor example Transport equation source terms in the Brown Model Proximate and ultimate analysis for Utah SUFCO and Skyline coals Flow rates for the two simulated experiments Comparisons the average soot volume fraction across the flame from optical measurements and simulations Studies from which oxidation data were extracted for model development Studies from which gasification data were extracted for model development Range over which model parameters were tested Calibrated parameters for soot oxidation, Equation Calibrated parameters for H 2 O gasification of soot, Equation Calibrated parameters for CO 2 gasification of soot, Equation Reactions and reaction rates used in precursor cracking scheme (rates in kmole m 3 s, concentrations in kmole J, and activation energies in m 3 mole K ) Surface growth mechanism where k i = AT n exp ( ) E RT [7] Proximate and ultimate analyses for the six coals tested [121] Precursor species fractions as described in Section for the coal experiments Proximate and ultimate analyses for the biomass fuels tested Precursor species fractions as described in Section for the biomass experiments Sooting potential model for biomass with calibrated parameters for Equations 6.4 and 6.5. T g and P are the gas temperature (K) and log-pressure (log(atm)) respectively Computational expense comparison between the detailed model of Chapter 5 and the simplified model of Chapter 6 and found in the OFC simulation of Section vii

9 LIST OF FIGURES 1.1 Effects of soot processes in the climate system Basic outline of the soot formation process Illustration of the HACA mechanism [57] Illustration of the mechanism for aromatic deposition onto the surface of a soot particle [57] Agglomeration of soot particles in a hypothetical box at different temperatures Overview of the soot formation process as found in complex solid fuel systems Comparison between pyrene, a common PAH soot precursor in gaseous systems, and a theoretical tar molecule as constructed based on elemental composition, molecular weight, and aromatic content [10] Diagram of a proposed oxy-coal reactor. As proposed by Buhre et al. [22] Example of a soot particle-size distribution as collected from a pre-mixed flame experiment [2] A graphical representation of the sectional method as applied to a soot PSD where 8 sections are applied A mono-dispersed distribution with η = A lognormal distribution with η = 2.86 and σ = A bimodal, lognormal/power law, distribution with the following parameters: α = 3.35, k = 5.14, η = 2.85, σ = This is a model-informed prior of the ab joint probability space as informed by the basic linear model used in gas-reactor example 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 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 Marginalized PDFs for the a and b parameters as taken from the posterior in Figure Linear mode, Equation 2.15, fitted to data from Table 2.3 using Bayesian inference Diagram of the downward burner and draft portion of the oxy-fuel combustor at the University of Utah 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) 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 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) Results of the SUFCO coal simulations, showing (a) the CO 2 mole fraction (max = 1, min = 0) and (b) O 2 mole fraction (max = 1, min = 0) viii

10 3.6 Results of the SUFCO coal simulations with soot gasification, showing (a) the soot particle number (max = 1E5, min = 5E16) and (b) soot volume fraction (max = 6 ppmv, min = 0 ppmv) 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, logarithmic scaling), and (c) large particle number density (max = 6.0E8, min = 1.0E1, logarithmic scaling) 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) Line of sight measurements of the soot volume fraction across the flame. Solid lines represent optical measurements while dotted line represent simulation results. Blue is at the root of the flame, green at the middle of the flame, and red is at the tip of the flame PDFs of each of the oxidation parameters in Equation 4.3. Contours indicate joint PDFs Comparison of predicted rates of soot oxidation by calibrated, with parameters in Table 4.4, model and those rates collected from the literature. Those experiments that are measured only oxidation by O 2, such as TGA, are filled symbols (R 2 = 0.75) Comparison of oxidation rates as predicted by the NSC oxidation model [140] and those rates collected from the literature (R 2 = 0.65) Comparison of oxidation rates as predicted by the NSC oxidation model combined with Neoh et al.[141] calculated collision efficiency for OH and those rates collected from the literature (R 2 = 0.71) PDFs of each of the H 2 O gasification parameters in Equation Comparison of predicted rates of soot gasification via H 2 O by calibrated model, parameters in Table 4.5, and those rates collected from the literature (R 2 = 0.87 minus Neoh Data) PDFs of each of the CO 2 gasification parameters in Equation Comparison of predicted rates of soot gasification via CO 2 by calibrated model, parameters from Table 4.6, and those rates collected from the literature (R 2 = 0.62) Comparison of predicted rates of soot gasification via CO 2 by individually calibrated models and those rates collected from the literature Model-informed priors for the CO 2 gasification model. Derived with mode values at A CO2 =3.06E-17 and E CO2 =5.56E PDFs of each of the oxidation parameters in Equation 4.3 derived using the modelinformed priors of Figure Contours indicate joint PDFs Model-informed priors for the oxidation model. Derived with mode values at A O2 =7.98E- 1, E O2 =1.77E5, and A OH =1.89E PDFs of each of the oxidation parameters in Equation 4.3 derived using the modelinformed priors of Figure Contours indicate joint PDFs PDF of the calculated gasification rate in Higgins experiment where the flame data was at 1200 K Comparison of the model predicted oxidation rate with confidence bounds versus the measured rate in Higgins s experiment ix

11 5.1 Basic outline of PAH thermal cracking Result of numerical study considering the evolution of precursors from Pittsburgh #8 coal at 1800 K as found in Section Results were 0.004, 0.283, 0.503, and for X phe, X napth, X tol, and X ben respectively Diagram of the complete HACA mechanism illustrating growth of a benzene ring Diagram of flat flame burner used by Ma [120]. Reproduced with permission 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) Average particle collision diameter across the flame portion of the Pittsburgh # 8 coal experiments as predicted by the simulation Particle shape factor across the flame portion of the Utah Hiawatha coal experiments Soot mass yield with an additional maximum sooting potential solid line representing the mass yield of tars released into the system Soot mass yield deposited on the soot filters of the coal-flame collection system 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) Blue bars represent experimentally measured particle-size distributions and red lines represent simulation resolved moments fitted to a log-normal distribution Comparison between results given by CPDbio versus the proposed sooting potential empirical 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 (R 2 =0.811) and the right plot shows the comparison for tar mass size (R 2 =0.856) Comparison between results given by CPD versus the proposed sooting potential empirical model. The left plot shows the comparison for tar mass yield (R 2 =0.794) and the right plot shows the comparison for tar mass size (R 2 =0.854) Variation of time-averaged precursor ratios from numerical study as temperature (left) and initial number density (right) are varied Comparison between empirical model and numerical study for predicting precursortype fractions. The black straight 45 represents a perfect agreement between the two (R 2 =0.919) Particle number density and soot volume fraction simulation results from the coal flatflame burner with entrained oxygen, comparing simplified model against the detailed model 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 #/m 3, min = 0 #/m 3 ), and soot particle number density from simplified model (max = 1E21 #/m 3, min = 0 #/m 3 ) A.1 Visual evidence of iteration reorganization x

12 NOMENCLATURE A i Pre-exponential factor for reaction i C a Collision frequency constant C min # Number of carbon atoms per incipient soot particle d Shape factor d i m Diameter of species i E kw Energy output J E i kmol Activation energy for reaction i f v,s ppmv Soot volume fraction kg F hr Fuel input H C Hydrogen to carbon atomic ratio I Conditional factors of an event k i Rate constant for reaction i kg Reaction rate per unit particle surface area k s m 2 s m 2 kg s 2 K k B Boltzmann constant m i kg Mass of species i kg M i i m 3 # N a kmol Particle size distribution generalized moment i Avogadro s number N S # kg Number of soot particles per unit volume of gas # N i Number densidty of particles of size i m 3 O C Oxygen to carbon atomic ratio P i Pa Partial pressure of species i Pr unitless Prandtl number J R kmolk Ideal gas constant r 2 Residual error R 2 Coefficient of determination S i m 2 Surface area m SA 2 v,i Surface area per unit volume of i m 3 Sc unitless Schmidt number T K Temperature T g K Gas temperature u m s Gas velocity V % Mass percent of volatile matter # w i m 3 x i X i y i Y i Weight of QMoM Vector of parameters i Mole fraction of species i Data from experiment i Mass fraction of species i α Fitted parameter m β 3 s Collision frequency # χ i Number of sites per unit surface area m 2 kg Change of mass involved with a single reaction Heat of pyrolysis H reac J kg xi

13 J kg H vap Heat of vaporization ε Van der Waals Enhancement Factor η m Assumed particle size γ Calibrated model output λ i Mean free path of species i kg µ ms Viscosity µ i, j kg Harmonic mean mass π kg ρ i Density of species i m 3 σ Standard deviation Functions and indicators δ() indicates a Dirac delta function f () indications a generic function with dependencies inside the paranthesis g() indications a generic function with dependencies inside the paranthesis L i () indicates a Lagrangian interpolation P() or p() indicates probability of event within paranthesis indicates the intersection of two events x y indicates conditionality [i] indications a concentration of species i xii

14 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

15 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 chainlike 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 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 particles 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

16 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 radiative 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

17 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 particles 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 potentially act as a nitrogen sink [49], greatly altering fuel-no x emissions from these fuels [133, 146]. A proper accounting of soot formation is important for detailed simulations of any combustion system 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 particles 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 characteristics 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 particles 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 respiratory 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

18 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, causing 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 researched, it has become evident over the last several decades that the impacts of released soot aerosols can be both diverse and long-lasting Environmental Impacts In addition to human health effects, soot production also can have severe and negative impacts 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

19 Atmospheric Radiation Particles absorb photons in the atmosphere cooling the surface but warming the atmosphere Hydro-atmospheric Interface Aerosol particles can act as a nucleation site for water molecules to condense. Atmospheric Transformations Particles mix and react with coemitted gases and aerosols (organic acids, sulfates, dust, etc.) Particle agglomeration, condensation, oxidation, and dilution all occur Snow/Rain Particles are deposited on the surface by nucleation in rain/snow. Particles have usually been acidified in the atmosphere creating a light acid rain. Surface Effects Deposited particles increase surface adsorption. Where particles have settled on snow or ice, impacts can be severe Emission Source Sources vary extensively, can be from open fires (wild or agricultural), transportation, mining, residential, power generation, etc. 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 reactions and transformations in the atmosphere. These reactions include continual particle-particle 6

20 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 reactions [196]. In the upper stratosphere, partial oxidation occurs readily as particles encounter increasing concentrations of OH and O 3 [12, 83]. Within the flame, soot particles have a continuous radiative absorption band, and the alteration 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 sufficiently 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 precipitation, 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

21 Light Gases Surface Growth Surface Growth Surface Growth Soot Precursors Nucleation Soot Nuclei Coagulation Primary Soot Particles Aggregation Soot Aggregates Oxidation Oxidation Oxidation Oxidation Light Soot Gases Precursors 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 CO 2 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 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

22 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 delocalized 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 nucleation models, is a matter of considerable debate [24]. Various mechanisms have been proposed via reactions of C 2 H 2, C 3 H 2, C 3 H 3, nc 4 H 3, C 4 H 4, C 4 H 5 and nc 4 H 5 [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 coalescence 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

23 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, typically 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

24 Figure 1.4: Illustration of the mechanism for aromatic deposition onto the surface of a soot particle [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 combustion systems, and models of varying sophistication have been developed to capture their influence [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. Coagulation 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 indistinguishable [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 individual 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 together. Frequencies of collisions between small particles is well understood with the Kinetic Theory of Gases; however, as particles grow larger they go from a free-molecular flow regime to a 11

25 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 agglomeration 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

26 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 O 2, 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, CO 2, and H 2 O. Some of the first investigators of soot oxidation assumed that soot was consumed solely via the reaction of an O 2 molecule with the particle surface [109], and oxidation models were developed based on O 2 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], particularly 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 O 2 and/or OH, it is difficult to experimentally differentiate between oxidation via O versus oxidation by O 2 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 O 2 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 (H 2, CO, etc.), small hydrocarbons, alcohols, and carbonyls [130]. In traditional combustion environments, soot consumption is dominated by oxidation kinetics, 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

27 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 consumption [152]. In addition, it has been found in oxy-fuel environments the increased concentrations of CO 2 and H 2 O 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 particle 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 fragmentation 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] 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

28 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 generally 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 impose 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 profiles, 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

29 The Mie scattering technique involves the use of a laser across the flame domain with wavelengths 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 electromagnetism. 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 (temperature, pressure, chemical mole fractions, etc) are generally considered to be constant over any 16

30 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 quantify 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 techniques 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 increasingly 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 aggregateaggregate 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 surface 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 characterization of soot aggregates and associated formation. Particle-gas interactions, including the previously discussed HACA, oxidation, and gasification reactions, are highly dependent on the local chemistry of the surrounding gas and these 17

31 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 different 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 thermodynamic 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 equilibrium 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 (O 2, CO 2, CH 4, 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/m 3 of particles which are being consumed at a rate of 5E-10 kg/m 3 s. If we were take a step of 0.5 seconds we may say that there are now -1.5E-10 18

32 kg/m 3. 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 Section [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] 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 thermochemical 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

33 Parent Solid Fuel Devolatilization Char Oxidation Char Light Gases Oxidation/Gasification Oxidation PAH Production Tar/PAH Nucleation Primary Soot Agglomeration Soot Aggregates 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 H 2 O, but other gases are found in abundance as well: CO 2, H 2, CH 4, and other small hydrocarbons. 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

34 CH 3 CH 3 HN CH 3 HO H 3 C CH 3 Figure 1.7: Comparison between pyrene, a common PAH soot precursor in gaseous systems, and a theoretical tar molecule as constructed based on elemental composition, molecular weight, and aromatic content [10]. HO 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 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

35 PAHs built from gas-phase mechanisms are completely aromatic containing nearly all carbon 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 particle 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 concentrations 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 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 equilibrium, 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

36 Oxygen Pre- Heater Boiler Feed-water Heater ASU air Gas Gas Heater Electrostatic Precipitator Filter Cooler water Mill CO 2 Compressor Proposed Oxy-fuel System coal 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 CO 2 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 concern of CO 2 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, O 2 is separated from N 2, 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 introducing a parasitic load of about 22% [187]. Improvements and methods of application are an area of extensive research [105]. The lack of atmospheric N 2 in the boiler yields multiple benefits which could justify the expense of the ASU: 23

37 Without N 2 in the boiler, sources of thermal NO x are eliminated during the combustion process, leading to a significant decrease in the overall yield of NO x in the flue gas [87] CO 2 is expensive to separate from N 2 in the post combustion clean-up [143], but with the prior removal of N 2, CO 2 separation from the flue gas becomes much more economical as the flue gas is primarily composed of CO 2 and H 2 O which can easily be condensed [44]. Particles have a tendency to burn more completely because of higher temperatures and greater access to O 2, 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 environment 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 O 2 yields incredibly 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 (CO 2 and H 2 O) 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 postulated, 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 adjustments in standard flue gas clean-up units, an additional unit for the treatment of CO 2 must be added. This unit cools and compresses a pure stream of CO 2 for subsequent industrial use or sequestration [118, 143]. In regard to soot formation, oxy-fuel combustion processes have potential to greatly alter soot yields in comparison to conventional combustion processes. The effects of an oxy-fuel environment on soot formation are threefold: First, high concentrations of CO 2 and H 2 O gasify soot particles. In conventional combustion 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

38 bonds and release a portion of the particle s surface molecules as gas into the surrounding environment. In conventional combustion systems, the bulk gas is overwhelmingly made up of monoatomic and diatomic molecules, N 2 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 CO 2 and H 2 O, 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 intramolecular 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 important source of soot consumption [1]. Second, due to the increased concentrations of H 2 O and CO 2, the radiative heat transfer of the system gases increases [5]. Just as the greater heat capacity of the tri-atomic molecules indicates 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 operating at higher temperatures due to the lack of a N 2 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 continued 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 mechanisms. 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

39 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 O 2 concentration can greatly affect 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 previously in Section 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 debris 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 selfgenerated 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 ecosystem. Temperatures within a firestorm have been postulated to reach as high as 1800 C. A survey 26

40 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 important 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, precipitation, 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

41 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 circumstances 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 pyrolysis 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 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 fluiddynamic (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

42 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 emissions 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

43 CHAPTER 2. COMPUTATIONAL TOOLS This chapter includes two major sections: one for the resolution of particle-size distributions, 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 methods discussed are not comprehensive of all methods developed or used but embodies the most commonly used methods in the current community 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

44 10 12 Data Number Density of Particles (#/m 3 ) Particle Diameter (nm) Figure 2.1: Example of a soot particle-size distribution as collected from a pre-mixed flame experiment [2]. Number Density of Particles (#/m 3 ) Data Sectional Model Particle Diameter (nm) Figure 2.2: A graphical representation of the sectional method as applied to a soot PSD where 8 sections are applied. 31

45 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 resolved 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 simulation 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 sectional 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 agglomerate 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

46 Number Density of Particles (#/m 3 ) Data Mono-Dispersed Model Particle Diameter (nm) Figure 2.3: A mono-dispersed distribution with η = 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 represent 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 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 location, are of the same size. Evolution of the distribution through time and space only affects two 33

47 Number Density of Particles (#/m 3 ) Data Log-normal Model Particle Diameter (nm) Figure 2.4: A lognormal distribution with η = 2.86 and σ = parameters: a weight (the number of particles) and an abscissa (the size of the particles), f (d p ) = N 0 δ(d p η). (2.1) Where f (d p ) is the number of particles of size d p, N 0 is the total number of particles in the distribution, and η is the assumed size of the observed particles. This distribution is often overlooked and not considered a truly characterized distribution 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 expensive simulations are employed. A visual portrayal of the distribution can be seen in Figure 2.3, where the vertical bar, which is located at d p =11.22, is capped at N 0 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 lognormal distribution, depicted in Figure 2.4. The lognormal distribution is based on the Gaussian, 34

48 Number Density of Particles (#/m 3 ) Data Power-law + Log-normal Model Particle Diameter (nm) Figure 2.5: A bimodal, lognormal/power law, distribution with the following parameters: α = 3.35, k = 5.14, η = 2.85, σ = or normal, distribution but derived over a log scale rather than a linear scale. It is defined as: f (d p ) = N 0 d p σ 2π exp [ (ln(d p) η) 2 2σ 2 ]. (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(d p ) ), σ is the standard deviation of the same, and N 0 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

49 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 powerlaw distribution. In this case, larger particles are mostly represented by the lognormal distribution while smaller particles are captured by a power-law. ( f (d p ) = N 0 αd k 1 p + [ d p σ 2π exp (ln(d p) η) 2 ] ) 2σ 2 (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 evaluation 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 representation 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 M r = i=0 m r i N i. (2.4) Here, N i represents the number of particles with size m i. 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

50 Table 2.1: Resolved statistical moments of the experimental distribution of Figure 2.1. Moment Value Units M E11 # M 1 M 2 M 3 M 4 M E E E E E-120 m kg 3 m 3 kg 2 #m 3 kg 3 # 2 m 3 kg 4 # 3 m 3 kg 5 # 4 m 3 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 dm r dt ( ) 6 ks = π πρ s r 1 l=0 ( ) r r l M l l+2/3. (2.5) Details and derivation of this equation will be provided later in this work. Suffice it to say that dm r dt = g(m l+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

51 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 M l+2/3 = i=0 m l+2/3 r max /2 i N i i=1 m l+2/3 i w i. (2.6) This is an approximation and the higher the value of r max, the number of resolved moments, the more accurate the approximation. We directly calculate the values of the weights (w i ) and abscissas 1 (m i ) of the quadrature with the resolved whole moments M r = r max /2 w i m r i. (2.7) i=1 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, r max =6, and resolve the fractional moment M 2/3. This leads to a series of equations M 0 = w 1 m w 2m w 3m 0 3, M 1 = w 1 m w 2 m w 3 m 1 3,... (2.8) M 5 = w 1 m w 2m w 3m 5 3. 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]. 1 Abscissa is a general term used in all quadrature method of moments. In the case of a particle size distribution abscissa is a size quantity. 38

52 Table 2.2: Resolved weights and abcissas of the 6 resolved moments in Table 2.1. Variable Value w E10 w E11 w E8 m E-27 m E E-26 m 3 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 M 2/3 = m 2/3 1 w 1 + m 2/3 2 w 2 + m 2/3 3 w 3 = 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. Direct 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 simultaneously 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 particle 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

53 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 logm p = L p (logm 0,logM 1,...,logM rmax ), (2.10) L p (logm 0,logM 1,...,logM rmax ) = r max i=0 r max p j logm i j=0 i j. (2.11) j i Note that the Lagrangian interpolation in Equation 2.10, is interpolating between logrithmic values 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 M 2/3 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

54 probability, Bayesian inference is rooted in Bayes Law, which is derived from an axiom of conditional 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), (2.13) P(B) 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). (2.14) f (y I) 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 naturalgas 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

55 Table 2.3: Experimental data for example gas-reactor. Experiment Fuel Input (kg) Energy Output (J) E E E E E E E E E E E E E E E9 Table 2.4: Ranges over which a & b parameters were analyzed for the example gas-reactor. Parameter Low Range High Range a -3.0E7 5.0E7 b -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 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

56 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 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 1/100 2 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

57 Figure 2.6: This is a model-informed prior of the ab joint probability space as informed by the basic linear model used in gas-reactor example. 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, r 2 = [y i f (x i )] 2, (2.17) i where r 2 is that residual error, y i are data, and f (x i ) are model outputs. In forming a modelinformed 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, r 2, 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 r 2 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 y i 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 (x i ) values we then vary ab joint and solve for F again with the 44

58 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 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 parameter 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 2σ 2 π exp (y γ)2 2σ 2. (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 experimental 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

59 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 analysis 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) 1 σ, (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 (y 0 x) f (y 1 x)... f (y n x). (2.21) In the use of the Gaussian distribution, Equation 2.18 becomes: f (y x,i) = ( n z p(y z γ,σ,i) = σ n z 1 exp z=1 2π 2σ 2 n z z=1 (y z u z ) 2 ), (2.22) where n z represents the number data points to be analyzed. In this case, a separate γ z 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

60 b (J) 2E9 1E9 0-1E9-2E7 0 2E7 a (J/kg) 4E7 Figure 2.7: 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. The likelihood for the ab joint probability space as computed using the Gaussian distribution 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 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

61 In the case of the gas-burner, we can evaluate the marginal likelihood across the evaluated domain of a and b, f (E I) = a high b high a low b low f (E a,b,i) f (a I) f (b I)db da. (2.24) 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 (x 1,x 2 ) = f (x 1,x 2,x 3 )dx 3, (2.25) would result in a two dimensional PDF. Should we want to have only a one dimensional PDF for the x 1, we would simply integrate over the x 2 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. However, 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 parameters, 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

62 b (J) 2E9 1E9 0-1E9-2E7 0 2E7 a (J/kg) 4E7 Figure 2.8: 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. 4E-8 3E-8 2E-8 1E-8 a Marginal Probability 0-5E7-2.5E E7 5E7 a (J/kg) 6E-10 4E-10 2E-10 b Marginal Probability 0-2E9 0 2E9 4E9 b (J) 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

63 Table 2.5: Calibrated parameters from the Bayesian inference for the simple gas-reactor example. Parameter Unit Calibrated Value 95% Credible Interval a J/kg 1.18E7-1.19E7 < a < 3.44E7 b 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, multimodal 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 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

64 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

65 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. However, 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 oxidation 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

66 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 models 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] applied a soot model in the CFD software FLASHCHAIN, designed to predict the primary and secondary 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 oxidation. 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 (Y S ), mass fraction of tar (Y T ), and the number of soot particles per kilogram of gas (N S ). These three variables are used to describe the soot and tar PSDs as mono-dispersed distributions. 53

67 The model applies equations for conservation of mass for soot and tar and conservation of particle number for soot, (ρg uy S ) = ( µ Sc ) Y S + S YS, (3.1) (ρg uy T ) = ( µ Sc ) Y T + S YT, (3.2) (ρg un S ) = ( µ Sc ) N S + S NS. (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, S YS = ṙ FS ṙ OS, (3.4) S YT = ṙ FT ṙ FS ṙ GT ṙ OT, (3.5) S NS = (N a /m C C min )ṙ FS ṙ AN. (3.6) These processes include tar formation (ṙ FT ), tar oxidation (ṙ OT ), tar gasification (ṙ GT ), soot formation (ṙ FS ), soot oxidation (ṙ OS ), and soot aggregation (ṙ AN ). Above, C min is the number of carbon atoms in the incipient soot particle and m C 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, ṙ FT = SP tar, (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

68 Table 3.1: Transport equation source terms in the Brown Model. term A E (kj/mole) source ṙ FT N/A N/A Source term for tar ṙ OT (m 3 /kg s) 52.3 Shaw et al. [171] ṙ GT (1/s) Ma [120] ṙ FS (1/s) Ma [120] ṙ OS (kg K 1/2 /m 2 atm s) Lee et al. [109] ṙ AN 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 SP tar. 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, ṙ OT = (ρ g Y T ) (ρ g Y O2 ) A OT e E OT /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, ṙ GT = (ρ g Y T ) A GT e E GT /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

69 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, ṙ FS = (ρ g Y T ) A FS e EFS/RT, (3.10) also used in the Brown model. The constants from Equations 3.9 and 3.10 were tuned to fit experimental data. Rates of soot oxidation, ṙ OS = SA v,s p O2 T 1/2 A OSe E OS/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 O 2 concentrations and particle surface area across a laminar diffusion flame. Measured rates were used to calibrate Equation 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, ( ) 2/3 SA v,s = (N S ρ g )πd 2 6YS p = (N S ρ g )π, (3.12) πn S ρ s assuming all particles to be spherical and the density of soot, ρ s to be 1900 kg/m 3. Rates of soot aggregation, ( ) 1/6 ( 6mC 6kB T ṙ AN = 2C a πρ s ρ s ) 1/2 ( ) ρg Y 1/6 S (ρ g N S ) 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 m C 56

70 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 implemented 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 package 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 hundreds/thousands/millions of cells. Navier-Stokes transport equations and equations of species conservation 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 equations. 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 Fröhlich 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 particulate 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

71 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 tabulated by mixture fraction and enthalpy before simulation with an assumed equilibrium gas composition. 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 fullydeveloped turbulent flow. In addition, oxy-coal experiments were conducted previously measuring soot volume fraction in this unit [185] providing an opportunity for model validation 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 O 2, a O 2 /CO 2 mixture, or air, while the primary carrier gas is usually CO 2 or N 2. 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

72 Figure 3.1: Diagram of the downward burner and draft portion of the oxy-fuel combustor at the University 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 CO 2 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 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

73 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 Skyline High-Vol Bit Table 3.3: Flow rates for the two simulated experiments. SUFCO Skyline Bituminous Coal Bituminous Coal Primary Inlet Coal: (kg/hr) Coal: 6.80 (kg/hr) CO 2 : 5.40 (kg/hr) CO 2 : 10.8 (kg/hr) O 2 : 1.04 (kg/hr) O 2 : 2.08 (kg/hr) T: 300 (K) T: 300 (K) Secondary O 2 : 7.48 (kg/hr) O (kg/hr) T: 489 (K) T: 489 (K) Purge CO 2 : 3.08 (kg/hr) CO 2 : 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

74 (a) (b) (c) (d) max 75% 50% 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 steadystate with regard to soot formation. 61

75 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 temperatures 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 CO 2. 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

76 (a) (b) (c) max 75% 50% 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

77 (a) (b) (c) max 75% 50% 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, N S, from the Brown model. Image (c) shows the soot volume fraction, f v,s computed from the modeled Y S, f v,s = (ρ g /ρ s )Y S. (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

78 (a) (b) max 75% 50% 25% min Figure 3.5: Results of the SUFCO coal simulations, showing (a) the CO 2 mole fraction (max = 1, min = 0) and (b) O 2 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. Abián 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, CO 2 and H 2 O, in an oxy-fuel environment promoted the gasification of soot particles. Figure 3.5 shows the mole fractions of CO 2 and O 2 ; note the high concentration of CO 2 and low concentration of O 2 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 O 2 layer, they stopped being consumed in the simulation, where in reality these particles were continually consumed through interactions with CO 2. To rectify this problem, a gasification sub-model was added to the Brown model to reflect the consumption of soot via CO 2 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 CO 2 in a thermogravimetric analysis set-up. 65

79 (a) (b) max 75% 50% 25% min Figure 3.6: Results of the SUFCO coal simulations with soot gasification, showing (a) the soot particle number (max = 1E5, min = 5E16) and (b) soot volume fraction (max = 6 ppmv, min = 0 ppmv). 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, H 2 O concentrations would also increase dramatically and a gasification term as a result of soot/h 2 O 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

80 (a) (b) (c) max 75% 50% 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, logarithmic scaling), and (c) large particle number density (max = 6.0E8, min = 1.0E1, logarithmic scaling). (a) (b) (c) max 75% 50% 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 = 0.24 ppmv, min = 0 ppmv). 67

81 Soot Volume Fraction (ppmv) Simulation Validation Flame Root Flame Middle Flame Tip Experiments Distance Along Line of Sight (m) Figure 3.9: Line of sight measurements of the soot volume fraction across the flame. Solid lines represent optical measurements while dotted line represent simulation results. Blue is at the root of 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 lineof-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

82 Table 3.4: Comparisons the average soot volume fraction across the flame from optical measurements and simulations. Flame Location Average Soot Volume Fraction (ppmv) Experimental Simulation Relative Error Root % Middle % Tip % 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 mechanism. A more detailed soot consumption model needs to explored. This consumption model needs to include soot gasification from interactions with CO 2 at a minimum and preferably with H 2 O 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

83 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 gasification 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 O 2 molecule with the particle surface [109], and oxidation models were developed based on the O 2 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 O 2 and/or OH, it is difficult to experimentally differentiate between oxidation via O versus oxidation by O 2 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 O 2 or OH. In recent years there has been an increased interest in oxygen-enriched combustion (oxyfuel combustion) as a means of enabling carbon capture [187, 22]. Oxy-fuel combustion often involves higher temperatures and higher H 2 O and CO 2 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 oxyfuel combustion, where the higher H 2 O and CO 2 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 particle surface reactivity. Researchers have developed mechanisms reflecting the many elementary 70

84 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 nonpremixed 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 consumption 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 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 + O 2 CO 2, (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 caution when considered for simulations outside of flames and may not be appropriate for detailed 71

85 simulations with fully resolved physics. A full mechanism for soot oxidation may contain hundreds 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 complete 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 O 2 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 oxidation 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: r ox = 1 ( ] ) EO2 T 0.5 A O2 P O2 exp[ + A OH P OH. (4.3) RT Here, r ox (kg/m 2 s) 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 O 2 and OH. Similar in form to previously developed models [74], it contains three fitted parameters: A O2, E O2, and A OH. Equation 4.3 assumes the following: 72

86 1. Oxidizing Agents Oxidation is assumed to occur by O 2 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 O 2 and O [96]. In turbulent flames, higher mixing rates may allow for greater interaction between O 2 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 O 2 rates in TGA experiments, and so O oxidation is not explicitly considered here. Concentrations of O 2 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 activation energy. This effective activation energy is what is used for the O 2 reaction, while the effective activation energy for the OH reaction is considered to be negligibly small because OH is such an effective oxidizer [74] Oxidation Data Experiments measuring soot oxidation have been carried out in many forms, and the literature 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

87 Table 4.1: Studies from which oxidation data were extracted for model development. Study Fenimore and Jones, 1967 [47] Neoh et al., 1981 [141] Ghiassi et al., 2016a [65] Kim et al., 2004 [100] Kim et al., 2008 [101] Garo et al., 1990 [61] Puri et al., 1994 [151] Xu et al., 2003 [200] Lee et al., 1962 [109] Chan et al., 1987 [27] Higgins et al., 2002 [84] Kalogirou and Samaras, 2010 [96] Sharma et al., 2012 [170] Data Points Oxidizing Agent Experiment 3 O 2 & OH Pre-mixed Ethylene Flame 14 O 2 & OH Laminar Methane Diffusion Flame 54 O 2 & OH Premixed Varied-Fuel Flame 2 O 2 & OH Laminar Ethylene Diffusion Flame 3 O 2 & OH Laminar Ethylene Diffusion Flame 6 O 2 & OH Laminar Methane Diffusion Flame 15 O 2 & OH Laminar Methane Diffusion Flame 15 O 2 & OH Laminar Mixed Hydrocarbon Diffusion Flames 29 O 2 & OH Laminar Mixed Hydrocarbon Diffusion Flame Temperature (K) O 2 TGA O 2 Tandem Differential Mobility Analyzer O 2 TGA O 2 TGA

88 the different experiments used for this study including the experimental method and the number of data points. ways. Each of these experiments was performed differently and results were presented in different 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: r ox = η im cr,i C i v, (4.4) 4 where η i is the collision efficiency of species i, m cr,i is the mass of carbon removed due to the oxidation by species i per mole of species i, C i 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

89 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 fuellean 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 O 2 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 laserinduced fluorescence in a methane-air laminar diffusion flame. Temperatures, species partial pressures, and oxidation rates were all reported. Reported values of O 2 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 O 2 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 O 2 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 monodispersed particles, collected from an ethylene diffusion flame, were subjected to an elevated temperature 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: r ox = ρ s(d 1 d 2 ), (4.5) 2t where the density of the soot particles (ρ s ) 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

90 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, O 2 concentrations, and calculated rate constant (k) values of a single-step Arrhenius equation: r ox,rep = kx n O 2. (4.6) Kalogirou assumed a n=0.75 order dependence of O 2, while Sharma assumed a 1.0 order dependence and used the partial pressure of O 2 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: r ox = r ox,repρ s d 1, (4.7) 6 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 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 + CO 2 2CO, (4.8) C + H 2 O CO + H 2, (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 gasification 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

91 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 oxidation. Products of oxidation are usually limited to: CO, CO 2, and H 2 O. Gasification reactions, on the other hand, will often include these species along with H 2, small hydrocarbons, alcohols, carbonyls, and other species as products [130]. In oxy-fuel systems, the increased concentrations of CO 2 and H 2 O are of interest. CO 2 is the most commonly considered gasification agent. H 2 O is often considered to be an oxidizer; however, data in the literature has shown that the products of soot/h 2 O reactions are more indicative of gasification than oxidation [130]. Other species are able to gasify as well, such as NO 2, 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 CO 2 and H 2 O 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 biomassderived soots. These experiments, along with a few others found in the literature, are used to form the proposed model of this study. 78

92 This model consists of two additive rate terms for gasification by CO 2 and H 2 O: r gs = r CO2 + r H2 O, (4.10) ( ) r CO2 = A CO2 PCO T 2 ECO2 exp, (4.11) RT r H2 O = A H 2 OPH n ( ) 2 O EH2 O T 1/2 exp. (4.12) RT Rates in these equations are defined in units of (kg/m 2 s). Equation 4.11 represents gasification due to attack by CO 2 with a modified Arrhenius equation dependent on temperature and the partial pressure of CO 2. The CO 2 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 parameters. 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 H 2 O. 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 H 2 O order of reaction. These two equations are analyzed separately because researchers have studied gasification by CO 2 and H 2 O independently 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 differently 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

93 Table 4.2: Studies from which gasification data were extracted for model development. Study # of Data Points Gasifying Agent Temperatures (K) Abián et al., 2012 [1] 14 CO Kajitani et al., 2010 [95] 6 CO Qin et al., 2013 [153] 3 CO Otto et al., 1980 [144] 2 H 2 O & CO Arnal et al., 2012 [8] 6 H 2 O 1273 Chhiti et al., 2013 [31] 28 H 2 O Neoh et al., 1981 [141] 14 H 2 O Xu et al., 2003 [200] 15 H 2 O The following is a brief description of each experiment along with some aspects of the instrumental model used. Abián et al. [1] produced soot particles in an ethylene diffusion flame. These particles were collected and placed in a TGA under a N 2 /CO 2 environment. The partial pressure of CO 2 was 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 CO 2 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

94 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 H 2 O as the gasifying agent and then repeated with CO 2. Rates (µg/m 2 s), partial pressures of the gasifying agents, and temperatures were reported. Otto et al. noted that data collected for CO 2 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, CO 2, and H 2 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 CO 2 concentrations increased. Once again an initial particle diameter of 50 nm was assumed. Chhiti et al. [31] explored soot gasification by H 2 O 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 H 2 O. 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 H 2 O 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 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 H 2 O portion of the gasification model found in Equation As noted in Chapter 2, the effective dimensionality of the system needed for evaluation is n p + 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 n p + 1 dimensional grid stored as an n p + 1 dimensional array. 81

95 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 H 2 O, Equation 4.12, contains n p =3 adjustable parameters: A H2 O, n, and E H2 O. The final ranges over which these and all other parameters were tested are shown in Table 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 E CO2, E H2 O, and n, which had linear spacing. 3. A prior needs to be established. In this study, a uniform prior was used for the model parameters, meaning that all combinations of model parameters had uniform probability. Jeffrey s prior was used for the σ z 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 n p + 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 H 2 O gasification, the modeled rates are computed using using Equation 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 σ z, 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

96 7. The previous three steps are repeated for each point in the n p + 1 dimensional grid. The result for H 2 O gasification is a four dimensional array holding the (unnormalized) posterior PDF for the given experiment. 8. This posterior is then marginalized to remove the σ z 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 σ z direction. The resultant threedimensional 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 posterior, 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 n p 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 H 2 O gasification, with E H2 O as the PoI, we have the numerical equivalent of f (E H2 O y) = f (x y)da H2 Odn. (4.13) 83

97 Table 4.3: Range over which model parameters were tested. Equation Parameter Range A O2 1E-2 to 1E2 E O2 1E5 to 2.51E5 A OH 3.16E-4 to 1E-2 A CO2 1E-18 to 1E-15 E CO2 0 to 3E4 A H2 O 1E2 to 3.16E7 E H2 O 1E5 to 5E5 n 0 to 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 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 A O2 is much more broad than A OH : the marginal PDF of A O2 spans over 2 full orders of magnitude, while that for A OH 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 O 2. As a result, the flame experiments defined A OH the OH Arrhenius constant more distinctly than A O2. E O2 has a sharp peak compared to either A O2 or A OH. This peak is due to the TGA experiments, which were dominated by O 2 oxidation. Slight variations in E O2 had a stronger impact on overall rate than A O2 the O 2 Arrhenius constant variations and 84

98 Marginal Posterior e-2 1e-1 1e0 1e2 1e-2 A O2 E O2 2.0e5 1.7e5 1.5e5 1e-2 1e-1 1e0 1e2 1e-2 A O2 Marginal Posterior e5 1.7e5 2.0e5 E O2 A OH 5e-3 2e-3 1e-3 1e-2 1e-1 1e0 1e2 1e-2 A O2 A OH 5e-3 2e-3 1e-3 1.5e5 1.7e5 2.0e5 E O2 Marginal Posterior e-3 2e-3 5e-3 A OH Figure 4.1: PDFs of each of the oxidation parameters in Equation 4.3. Contours indicate joint PDFs. 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 A OH =1.89E-3 kgk1/2 Pam 2 corresponds to a collision efficiency of 0.15, which is consistent with s 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 between the three different parameters. The top of these three plots shows a heavy correlation between A O2 and E O2. A correlation is to be expected because these two parameters are used in combination to describe the oxidation reaction as occurs by the O 2 molecule. There is a positive correlation between E O2 and A O2, which is consistent with an increase in A O2 being offset by an increase in E O2 for a given rate. The shape of the correlation is consistent with the model form. In contrast to the E O2 /A O2 PDF, the A OH /A O2 and A OH /E O2 PDFs show little correlation between 85

99 Table 4.4: Calibrated parameters for soot oxidation, Equation % Credible Interval Variable Value Units Lower Bound Upper Bound kgk A O2 7.98E E E0 1/2 Pam 2 s E O2 1.77E5 1.57E5 1.94E5 J mol kgk A OH 1.89E E E-3 1/2 Pam 2 s their respective parameter pairs. The correlation that is present is slightly negative so that increases in A O2 and E O2 result in decreases in A OH. These low correlations are due to the nature of the experiments from which data was derived. Oxidation in TGA experiments were due entirely to the O 2 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 O 2 and OH. This figure displays a parity plot of model calculated rates and literature reported rates. The solid line indicates perfect agreement 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 R 2 statistic (coefficient of determination), using log 10 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, R 2 = The NSC model represents the oxidation of graphite by O 2. 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 O 2 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, R 2 = While this combined model does better than the NSC model alone at predicting soot oxidation, the calibrated model is slightly more accurate (R 2 value of 0.75 vs 0.71). 86

100 Calculated Rates (kg/m 2 *s) Fenimore Neoh Ghiassi Kim Garo Puri Xu Lee Chan Higgins Kalogirou Sharma Measured Rates (kg/m 2 *s) Figure 4.2: Comparison of predicted rates of soot oxidation by calibrated, with parameters in Table 4.4, model and those rates collected from the literature. Those experiments that are measured only oxidation by O 2, such as TGA, are filled symbols (R 2 = 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 calibration allows for the quantification of parameter uncertainty as shown in Figure 4.1, such a jointparameter 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 established models in the literature Gasification Model H 2 O Gasification Results for the parameter calibration of H 2 O 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

101 Calculated Rates (kg/m 2 *s) Fenimore Neoh Ghiassi Kim Garo Puri Xu Lee Chan Higgins Kalogirou Sharma Measured Rates (kg/m 2 *s) Figure 4.3: Comparison of oxidation rates as predicted by the NSC oxidation model [140] and those rates collected from the literature (R 2 = 0.65) Calculated Rates (kg/m 2 *s) Fenimore Neoh Ghiassi Kim 10-6 Garo Puri Xu Lee 10-8 Chan Higgins Kalogirou Sharma Measured Rates (kg/m 2 *s) Figure 4.4: Comparison of oxidation rates as predicted by the NSC oxidation model combined with Neoh et al.[141] calculated collision efficiency for OH and those rates collected from the literature (R 2 = 0.71). 88

102 Table 4.5: Calibrated parameters for H 2 O gasification of soot, Equation % Credible Interval Variable Value Units Lower Bound Upper Bound kgk A H2 O 6.27E4 8.31E3 2.47E7 1/2 Pa n m 2 s E H2 O 2.95E5 2.66E5 3.26E5 J mol n 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 (normal 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 E H2 O and the log of A H2 O, indicating a close linking between these two parameters, as was noted for the oxidation reaction above. However, there is a much different correlation between the reaction order n and either E H2 O or A H2 O, with nearly round contours until the reaction order n drops to low levels. This shape of contour implies that the H 2 O 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 E H2 O or A H2 O. This indicates that the rates are mostly governed by A H2 O and E H2 O, unless the reaction order is sufficiently low (on the order of 0.5 or less), where the other parameters must be adjusted to compensate. 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

103 Marginal Posterior e2 1e4 1e5 1e7 A H2 O E H2 O 3.5e5 3e5 2.5e5 1e2 1e4 1e5 1e7 A H2 O Marginal Posterior e5 3e5 3.5e5 E H2 O n e2 1e4 1e5 1e7 A H2 O n e5 4e5 7e5 E H2 O Marginal Posterior n Figure 4.5: PDFs of each of the H 2 O gasification parameters in Equation CO 2 Gasification Results for the parameter calibration of CO 2 gasification are shown in Figure 4.7. The two diagonal plots are the marginal PDFs for the two adjustable parameters in Equation 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 E CO2 value of 0 implies that a straight A CO2 with no exponential activation energy term, r CO2 = A CO2 P 0.5 CO 2 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

104 10 0 Calculated Rates (kg/m 2 *s) Measured Rates (kg/m 2 *s) Arnal Chhiti Neoh Otto Xu Figure 4.6: Comparison of predicted rates of soot gasification via H 2 O by calibrated model, parameters in Table 4.5, and those rates collected from the literature (R 2 = 0.87 minus Neoh Data). Table 4.6: Calibrated parameters for CO 2 gasification of soot, Equation % Credible Interval Variable Value Units Lower Bound Upper Bound kg A CO2 3.06E E E-16 Pa 1/2 K 2 m 2 s E CO2 5.56E3 6.04E2 1.95E4 J mol 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, E CO2 and the log of A CO2 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 CO 2 gasification rates. A large amount of scatter is seen in this plot and the model is much less accurate than for the oxidation and H 2 O gasification rates. This discrepancy is due to the combined effects of inconsistencies 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

105 Marginal Posterior e-18 3e-17 1e-16 A CO2 E CO2 2e4 1e4 0 8e-18 3e-17 1e-16 A CO2 Marginal Posterior e4 2e4 E CO2 Figure 4.7: PDFs of each of the CO 2 gasification parameters in Equation Calculated Rates (kg/m 2 *s) Abian Kajitani Otto Qin Measured Rates (kg/m 2 *s) Figure 4.8: Comparison of predicted rates of soot gasification via CO 2 by calibrated model, parameters from Table 4.6, and those rates collected from the literature (R 2 = 0.62). 92

106 Calculated Rates (kg/m 2 *s) Calculated Rates (kg/m 2 *s) Abian Measured Rates (kg/m 2 *s) Otto Measured Rates (kg/m 2 *s) Calculated Rates (kg/m 2 *s) Calculated Rates (kg/m 2 *s) Kajitani Measured Rates (kg/m 2 *s) Qin Measured Rates (kg/m 2 *s) Figure 4.9: Comparison of predicted rates of soot gasification via CO 2 by individually calibrated models 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

107 Marginal Prior 2.5e15 2.0e15 1.5e15 1.0e15 0.5e15 0 1e-18 1e-17 1e-16 1e-15 A CO2 ECO2 3.0e4 2.0e4 1.0e4 Marginal Prior 6.5e-5 5.0e-5 2.5e-5 0 1e-18 1e-17 1e-16 1e-15 A H2O 1.0e e4 2.0e4 3.0e4 E CO2 Figure 4.10: Model-informed priors for the CO 2 gasification model. Derived with mode values at A CO2 =3.06E-17 and E CO2 =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 Rate-Informed Priors The previous section s work was repeated but instead of using uniform priors for each analysis, model-informed priors were used as described in Section 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 parameters 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 CO 2 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

108 Marginal Posterior # e-18 3e-17 1e-16 A CO2 E CO2 2e4 1e4 0 8e-18 3e-17 1e-16 A CO2 Marginal Posterior # e4 2e4 E CO2 Figure 4.11: PDFs of each of the oxidation parameters in Equation 4.3 derived using the modelinformed priors of Figure Contours indicate joint PDFs. Figure 4.10 is the model-informed prior for Equation 4.11, the CO 2 portion of the gasification model. The A CO2 and E CO2 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 A CO2 and E CO2 found in the model where A CO2 is outside the exponential function while E CO2 is within it. Although these ellipses radiate out from the PDF mode, they are not centered on the mode. This is because as A CO2 gets smaller and E CO2 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 A CO2 gets larger and E CO2 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; therefore, smaller values of A CO2 are more probable than larger values, as reflected in the top plot, a marginalized PDF of the A CO2 prior. On the same thread, large values of E CO2 are more probable than small values as seen in the bottom-right plot, a marginalized PDF of the E CO2 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 This figure is very similar to Figure 4.7 indicating the method we used to assign 95

109 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 modelinformed prior. Note in the bottom-left plot, a small kink in the ellipse contours toward the higher values of A CO2. This deformity is consistent with the model-informed prior, which clearly favored higher values of A CO2, 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 CO 2 gasification data. Figure 4.12 shows the model-informed prior for Equation 4.3 for soot oxidation. Unfortunately, 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 CO 2 prior, the A O2 /E O2 correlation is linear when plotted on a semi-xlog plot, consistent with the expectation discussed before. The relations between A OH and the other parameters is more telling, showing that as A OH 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 E OH to equal zero, therefore A OH quickly becomes the most important parameter in determining overall rates. The overall probability of A OH favors smaller values for the same reason stated for the A CO2 parameter in the above paragraphs. Figure 4.13 is also very similar to Figure 4.1, for the same reason stated before for the CO 2 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 modelinformed 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

110 Marginal Prior e-2 1e-1 1e0 1e1 1e2 A O2 EO2 2.5e5 2.0e5 1.5e5 Marginal Prior 4e-5 3e-5 2e-5 1e-5 1.0e5 1e-2 1e-1 1e0 1e1 1e2 A O2 1e-2 1e e5 1.5e5 2.0e5 2.5e5 E O2 140 AOH 3e-3 1e-3 AOH 3e-3 1e-3 Marginal Prior e-4 1e-2 1e-1 1e0 A O2 1e1 1e2 4e-4 1.0e5 1.5e5 2.0e5 E O2 2.5e A OH Figure 4.12: Model-informed priors for the oxidation model. A O2 =7.98E-1, E O2 =1.77E5, and A OH =1.89E-3. Derived with mode values at As model form for the H 2 O gasification model were similar enough to the two models above that no additional insight was expected to be gained from this additional analysis Rate Prediction The results of Sections and 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

111 Marginal Posterior e-2 1e-1 1e0 1e2 1e-2 A O2 E O2 2.0e5 1.7e5 1.5e5 1e-2 1e-1 1e0 1e2 1e-2 A O2 Marginal Posterior # e5 1.7e5 2.0e5 E O2 A OH 5e-3 2e-3 1e-3 1e-2 1e-1 1e0 1e2 1e-2 A O2 A OH 5e-3 2e-3 1e-3 1.5e5 1.7e5 2.0e5 E O2 Marginal Posterior e-3 2e-3 5e-3 A OH Figure 4.13: PDFs of each of the oxidation parameters in Equation 4.3 derived using the modelinformed priors of Figure 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 P O2 = 21,300 Pa and P OH = 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 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

112 Probability Rate (kg/m 2 s) Figure 4.14: PDF of the calculated gasification rate in Higgins experiment where the flame data was 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 temperature 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, Figure 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

113 Rate of Soot Oxidation (kg/m 2 s) Model Predicted Measured Rate 90% Credible Interval Temperature of the Flame (K) Figure 4.15: Comparison of the model predicted oxidation rate with confidence bounds versus the measured 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 uncertainty 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 oxidation 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

114 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 requires 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 calibration should yield the same results [91]. Both methods are based on the use of the Gaussian Distribution found in Equation Because σ is a nuisance parameter, to maximize the probability of this distribution the numerator of the exponential should be minimized: max(p(y z,i µ z (x),σ z )) = min ( (y z,i µ z (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 parametermarginal 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-squaredsum 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 consumption model were analyzed including a collision-efficiency model, simple Arrhenius equations, and modified Arrhenius equations, with varied temperature and concentration dependencies. 101

115 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 errors. 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 H 2 O gasification, but only marginal results for CO 2 gasification when considering all data sets. Individual data sets could be fit with much more accuracy. The R 2 values for the oxidation, H 2 O, and CO 2 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 uncertainty in the calibrated model parameters. This is especially true for the performance of the CO 2 gasification. 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 O 2 + Neoh OH combined oxidation model R 2 = 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

116 CHAPTER 5. DETAILED MODELING OF SOOT FROM SOLID FUELS This chapter presents a developed physics-based detailed model for predicting soot formation 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, Ntotal PAH n bins = i=0 N PAH i, (5.1) where N PAH i 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. N PAH i not just PAHs formed from light gases. refers to all precursors within a given section, As the molecular weight range of the precursor PSD remains roughly fixed and sufficiently narrow ( 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 dependent on system configuration, sometimes growing to very broad ranges. Thus using a sectional 103

117 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 statistical moments that describe a PSD, M r = i=1 m r i N i, (5.2) where M r is the resolved r th moment, m i is the molecular weight of particle i, and N i 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 logm p = L p (logm 0,logM 1,...,logM n ), (5.3) L p (logm 0,logM 1,...,logM n ) = n i=0 logm i n j=0 j i p j i 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 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 dn PAH i dt = r f ormi r nucli r depoi r cracki + r growthi r consumei, (5.5) 104

118 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 = R pyrene δ (m pyrene m i ) + R pyrolysisi. (5.6) PAH formation from light gas precursors, R pyrene, 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 determine the production rate of pyrene in the gas-phase. The molecular weight of pyrene is 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, R pyrolysisi 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 r nucli = n bins β PAH j=1 i, j Ni PAH N PAH j. (5.7) 105

119 where β PAH i, j represents the frequency of collision between the two sectional species; in the freemolecular collision regime, β PAH i, j is given by β PAH i, j = 2.2 πk B T 2µ i, j ( di + d j ) 2, (5.8) µ i, j = m im j m i + m j, (5.9) d i = C h m 1/2 i, (5.10) C h = d A 2 3m C, (5.11) where k B is Boltzmann s constant, T is temperature, µ i, j is the reduced mass of species i and j, d i is the collision diameter of species i, d A is the diameter of a single aromatic ring (0.28 nm), m C 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 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 rate, Soot growth via precursor deposition is modeled with the following precursor-soot collision r depoi = j=1 β i, j N soot j Ni PAH, (5.12) where β i, 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) r depoi = 2.2 πkb T 2 ( C 2 h m1/2 i M soot 0 + 2C h C a C s M1/3 soot +C2 scam 2 1/2 i M2/3 soot ) N PAH i. (5.13) 106

120 Here, C s and C a are the spherical soot collision diameter and the particle shape deviation from spherical ( ) 6 1/3 C s =, (5.14) πρ s C a = (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 C d is a proportionality constant determined by a Monte-Carlo fitting to be [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 gaseousfuel 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

121 Precursor Phenol R 1 Napthalene Toluene R 2 R 3 R 4 Light Gases R 5 Benzene Figure 5.1: Basic outline of PAH thermal cracking. Table 5.1: Reactions and reaction rates used in precursor cracking scheme (rates in kmole m 3 s, concentrations in kmole J, and activation energies in m 3 mole K ). Reaction Rates C 6 H 6 O CO + 0.4C 10 H C 6 H 6 R 1 = k 1 [C 6 H 6 O] ( ) + 0.1CH H 2 k 1 = exp RT C 6 H 6 O + 3H 2 O 2CO + CO 2 + 3CH 4 R 2 = k 2 [C 6 H 6 O] ( ) k 2 = exp RT C 10 H 8 + 4H 2 O C 6 H 6 + 4CO + 5H 2 R 3 = k 3 [C 10 H 8 ][H 2 ] ( 0.4 ) k 3 = exp RT C 7 H 8 + H 2 C 6 H 6 + CH 4 R 4 = k 4 [C 7 H 8 ][H 2 ] 0.5 ( ) k 4 = exp RT C 6 H 6 + 5H 2 O 5CO + 6H 2 + CH 4 R 5 = k 5 [C 6 H 6 ] ( ) k 5 = exp RT 108

122 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, ( 31.1 r cracki = (5.16) 94 k 1X phe + k 2 X phe k 3X napth [H 2 ] ) 92 k 4X tol [H 2 ] k 5 X ben Ni PAH, where k n values are given in Table 5.1. Details for this equation s derivation are given in A.0.3. [H 2 ] is the concentration of H 2 measured in kmole. X m 3 phe, X napth, X tol, and X ben are the mole fractions of surrogate precursors. The difficulty in using this sub-model lies in specifying the X phe, X napth, X tol, and X ben 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 H 2 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 X phe, X napth, X tol, and X ben in subsequent soot simulations. Temperature, H 2, and total initial precursor 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 phenol, 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 components, 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 polyphenolicbenzenes. 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

123 Concentration (#/m 3 ) [10 19 ] Time (ms) Type Mole Fraction Time (ms) Phenol Toluene Naphthalene Benzene Figure 5.2: Result of numerical study considering the evolution of precursors from Pittsburgh #8 coal at 1800 K as found in Section Results were 0.004, 0.283, 0.503, and for X phe, X napth, X tol, and X ben 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 Precursor Growth Particles are able to either increase or decrease in mass through interactions with the surrounding 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 radical 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

124 Figure 5.3: Diagram of the complete HACA mechanism illustrating growth of a benzene ring. Table 5.2: Surface growth mechanism where k i = AT n exp ( ) E RT [7]. No. Reaction A ( m 3 kmol s K n ) n E ( J mole ) 1 C H + H C + H ,392 1R C H + H C + H ,024 2 C H + OH C + H 2 O ,932 2R C H + OH C + H 2 O ,093 3 C + H C H C + C 2 H 2 C H + H ,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 radical 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/m 2 s) takes the form R HACA = 2m C k 4 [C 2 H 2 ]αχ C. (5.17) 111

125 χ C 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 implementations 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, where µ 1 = M 1 M 0, and a and b are given as ( ) a α = tanh + b, (5.18) log µ 1 a = T, (5.19) b = T. (5.20) The χ C value is computed using steady-state assumptions of the HACA mechanism in Table 5.2 k 1 [H] + k 2 [OH] χ C = 2χ C H k 1 [H 2 ] + k 2 [H 2 O] + k 3 [H] + k 4 [C 2 H 2 ]. (5.21) χ C H is the number density of sites on the particle surface available for reaction, estimated to be sites/m 2 [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 r growthi = R HACASi PAH The surface area, S PAH i, of a precursor molecule is [186] N PAH i m i. (5.22) S PAH i = N PAH C,i, (5.23) N PAH C,i = m i m C. (5.24) 112

126 Precursor Consumption We model the consumption of precursors via oxidation and gasification. Oxidation of a particle surface is an exothermic reaction between surface carbon/hydrogen atoms and oxidizing gases (O 2 and OH here), leading to products of combustion: CO 2, H 2 O, 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 H 2 O or CO 2, and results in a more diverse array of gaseous products 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/m 2 s) are given in Equations 4.3 and Both rates are mass consumption per unit surface area of the particles (kg/m 2 s). Similar to the growth term in Equation 5.22, the consumption of particle number is accomplished by converting the mass consumed into an equivalent number of particles from a PSD section, r consumei = ( ) Roxidation + R gasi f ication S PAH i Ni PAH. (5.25) m i Soot As mentioned above, the soot PSD is represented using the method of moments. Moment rates are determined by a series of submodels, dm r dt = Nu r + Gr r + Dp r +Cg r, (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 describes the process of this coagulation and its effect on the precur- 113

127 sor PSD. The expression for its effect on the soot PSD is similar [58], Nu r = n bins i=1 n bins j=i β i, j (m i + m j ) r Ni PAH N PAH j, (5.27) where β i, j again represents the frequency of collision between precursor species i and j, it is computed using Equation 5.8. Soot Coagulation particles [53] Coagulation of soot particles is computed based on the collision frequency between soot Cg r = 1 2 r 1 ( k)( r k=1 i=1 j=1 m k i m r k j β i, j N i N j ). (5.28) ( r ) k is the binomial coefficient. Note, that coagulation does not effect the first PSD moment, thus Cg 1 = 0. The β i, 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. β i, j in the continuum flow regime is ( βi, C j = K C m 1/3 i + m 1/3 j + K C [ ]) m 2/3 i + m 2/3 j (m 1/3 i + m 1/3 j ), (5.29) leading to coagulation source terms in the continuum regime for r = 0 and r 2, Cg c 0 = K c[m M 1/3 M 1/3 + K c(m 1/3 M 0 + M 2/3 M 1/3 )], (5.30) Cg c r = 1 2 K c r 1 k=1 ( ) r [2M k M r k + M k k+1/3 M r k 1/3 + M k 1/3 M r k+1/3 + K c(m k 1/3 M r k + M k M r k 1/3 + M k+1/3 M r k 2/3 + M k 2/3 M r k+1/3 )], (5.31) where the K c = 2k B T /3µ and K c = 2.514λ f /(C s C a ), and µ is the gas viscosity. C s and C a are evaluated using Equations 5.14 and Fractional moments are computed using Lagrangian interpolation among logarithms of integer moments using Equation

128 Coagulation in the free molecular regime is more difficult as the β i, j expression is ( ) ( β f i, j = K f m 1/3 i + m 1/3 2 1 j + 1 ) 1/2, (5.32) m i m j and results in a non-expandable form of summations in Equation Therefore, a grid function is established and evaluated using Lagrangian interpolation [53], Cg f 0 = 1 2 K f f (0,0) 1/2, (5.33) Cg f r = 1 2 K f r 1 k=1 ( ) r f (k,r k) k 1/2, (5.34) where the K f = εc 2 ac 2 s πkb T /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 ( ) k ( ) m x i m y j m 1/3 i + m 1/3 2 j Ni N j. (5.35) j=1 m i m j 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 Cg r = Cgc rcgr f Cg c r +Cgr f. (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

129 in Sections and The rate of change of the number density of particle i is given by dn i dt = k s m (N i 1S i 1 N i S i ). (5.37) k s is the rate of a surface reaction (HACA, oxidation, or gasification) and is equal to R HACA, - R oxidation, or -R gasi 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 = 2m C, while for oxidation/gasification m = m C. Applying moments, the net soot growth/consumption moment source term for r 1 is derived to be Gr r = πc 2 s r 1 k s m m2/3 d 0 k=0 ( ) r ( m) r k M k k+ d. (5.38) For r = 0, Gr 0 = 0. Precursor deposition was discussed in Section and the moment source term for r 1 is πkb T Dp r = r 1 k=0 ( ) r ( Ch 2 k MPAH r k+1/2 Msoot k ) +2C h C a C s Mr k PAH Msoot k+1/3 +C2 scam 2 r k 1/2 PAH Msoot k+2/3, (5.39) where the C h, C s, and C a constants were given in Equations 5.11, 5.14, and For r = 0, we have Dp 0 = 0. The precursor PSD moment is calculated across all resolved sections M PAH n bins j = i=1 m j i 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 = S 0 i=1 ( mi m 0 ) d N i = S 0 m d M d, (5.41) 0 116

130 where S 0 and m 0 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 M 0, M 1, and M d, d = log µ d log µ 1, (5.42) where µ d = M d M 0 and µ 1 = M 1 M 0. 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 = m 2/3 0 Nu 0. (5.44) The deposition source term is determined by Lagrangian interpolation of the Dp i terms for the resolved integer moments Dp d = L d (logdp 1,logDp 2,logDp 3 ). (5.45) Surface growth and consumption terms require the use of another grid function g k. The source term is Gr d = πc 2 s k s ( ) m m2/3 d 0 g d M 2 d, (5.46) with details and derivations given in A.0.5. As in Equation 5.37, k s is the rate of a surface reaction (HACA, oxidation, or gasification) and is equal to R HACA, -R oxidation, or -R gasi f ication. Similar to f (x,y) l in Equation 5.35, g k is computed at integer values and used to interpolate to g d. The grid 117

131 function g k needed in Equation 5.46 is g k = k i=0 ( ) k m k i M i i+ d, (5.47) where m represents the mass of carbon change resulting from a single reaction ( m = 2m C for HACA, and m = m C for oxidation and gasification). In using this aggregation model, Balthasar and Frenklach [11] note that constituent particles of the evolving aggregate are assumed to have point contacts with each other and, consequently, 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 M 1 and M d remain stationary, but M 0 decreases. The decreasing number of particles pushes M 0 toward M 1 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 M 0 and M 1. Therefore, as particle coagulation affects one end of that scale, M 0, it must effect M d as well. As the proposed submodel for particle coagulation in Section 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 coagulation scheme, submodels resolve the coagulation rate for both the free-molecular and continuum regimes. The continuum regime moment source term, Cg c d = K c ( 1 2 h d ( 2M 0 M d + M 1/3 M d 1/3 + M 1/3 M d +1/3 + K C [ M0 M d 1/3 + M 2/3 M d +1/3 + M d 2/3 M 1/3 + M 1/3 M d ]) ), (5.48) uses a grid function h k in order to interpolate to h d using Lagrange interpolation as before h k = i=0 ( ) k mi + m j (2 + m 1/3 i m 1/3 j + m 1/3 i m 1/3 j j=0 + K C [ m 1/3 i + m 1/3 i m 2/3 j + m 2/3 i m 1/3 j + m 1/3 j Coagulation in the free-molecular regime, ( ) Cg f 1 d = K f 2 f (0,0) d +1/2 f ( d,0) 1/2, ]) N i N j. (5.49) (5.50) 118

132 uses the grid function given in Equation 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 Equation 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 system 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 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 CH 4, H 2, and dilution N 2. Coal particles were steadily added to the center of the flame with an N 2 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, resulting 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

133 Char-Leg Filter Cyclone/ Char Collector Soot-Leg Filter Virtual Impactor Cooling Water Cooling Water Quench Nitrogen Suction Probe Water Bath Water Traps Quartz Tower Char Stream Soot Cloud To Air Flowmeters F F Flat- Flame Burner Oxidizer Fuel Vacuum Pumps Coal Particles from the Feeder Figure Particle collection and separation system. 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 Pittsburgh #8 High- Volatile A Bituminous Illinois #6 High- Volatile A Bituminous 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

134 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 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

135 Table 5.4: Precursor species fractions as described in Section for the coal experiments. Temp (K) Coal Mole Fraction Phenol Toluene Naphthalene Benzene 1650 Utah Hiawatha Pittsburgh # Illinois # Utah Hiawatha Pittsburgh # Illinois # Utah Hiawatha Pittsburgh # Illinois # PSD are transported in the z-direction by advection via the following the balance equations d(u z N PAH i ) dz = dnpah i, (5.51) dt d(u z M r ) dz = dm r, (5.52) dt assuming negligible axial diffusion relative to advection, and no significant pressure differential. Velocities, u z, 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 C 2 H 2, H, H 2, O 2, OH, CO 2, and H 2 O. Chemical equilibrium at the local experimental temperature was assumed for these gas phase species using the ABF mechanism discussed in Section The production rate of pyrene was computed from this gas state using the rate from the ABF mechanism, and any produced pyrene was added to the precursor PDF as described in Section 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

136 30 Utah Hiawatha Pittsburgh #8 Illinois #6 Yield (%) K exp 1650 K sim z (mm) 1800 K exp 1800 K sim K exp 1900 K sim Figure 5.5: 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). As described in Section for the thermal cracking submodel, precursors were characterized 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 experimental 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 experimental 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

137 prediction models [69]. The curve shapes found in the Figure are indicative of reaction mechanisms 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 nucleate 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 temperature 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. C 2 H 2, 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

138 Particle Diameter (nm) K 1800 K 1900 K z (mm) Figure 5.6: Average particle collision diameter across the flame portion of the Pittsburgh # 8 coal experiments as predicted by the simulation Particle Shape Factor K 1800 K 1900 K z (mm) 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

139 40 Utah Hiawatha Pittsburgh #8 Illinois #6 30 Yield (%) K exp 1650 K sim 0 50 z (mm) K exp 1800 K sim Maximum Sooting Potential K exp 1900 K sim Figure 5.8: Soot mass yield with an additional maximum sooting potential solid line representing the 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 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

140 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 maximum 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

141 30 Utah Hiawatha Pittsburgh #8 Illinois #6 Yield (%) z (mm) z (mm) K exp 1650 K sim 1800 K exp 1800 K sim 1900 K exp 1900 K sim 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 particleparticle 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 N 2 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 experimental 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 collection 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

142 the extended simulation does not capture the additional agglomeration happening within the flame due to the particle concentration distribution discussed previously. This additional flame agglomeration 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 causing 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 system itself. There are complex flow dynamics occurring in the collections system (gas dilution, expansion, 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 informative. 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 Biomass System Trubetskaya et al. [190], collected soot from a fast-pyrolysis drop-tube reactor which gasified 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

143 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) Beechwood (Hardwood) Wheat Straw Collected particles were analyzed in a number of ways: elemental analysis, ash compositional analysis, FTIR spectroscopy, X-ray diffraction, thermogravimetric analysis, N 2 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 reported 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 fractions 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 computed 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 Table 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

144 Table 5.6: Precursor species fractions as described in Section for the biomass experiments.. Temp ( C) Biomass Fraction Phenol Toluene Naphthalene Benzene 1250 Pinewood Beechwood Wheat Straw Pinewood Beechwood Wheat Straw Pinewood Beechwood Wheat Straw 8 Yield (%) C 1400 C 1250 C 1400 C 1250 C 1400 C 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 difference 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 assumed 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

145 Percentage (%) Percentage (%) Percentage (%) Pinewood 1250 C Pinewood 1400 C Beechwood 1250 C Beechwood 1400 C Wheat Straw 1250 C Particle Diameter (nm) Wheat Straw 1400 C Particle Diameter (nm) Figure 5.11: Blue bars represent experimentally measured particle-size distributions and red lines represent 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

146 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 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 lognormal 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 proposed. This model has a number of advantages for predictability in a wide variety of flames. Researchers 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 inorganics, 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

147 or through chemical consumption. Currently, this model does not account for any particle fragmentation. Section 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 undesirable 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 detailed 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 presented 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 particle 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

148 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 computationally 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 sootprecursor molecules (N tar ), the number density of soot particles (N soot ) and the mass density of soot particles (M soot ), 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 precursor 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 resolution 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 monodispersed 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

149 Rates of generation/consumption for each of these terms is defined by many of the same submodels found in the detailed model, dn tar dt = r PI 2r SN r PD r TC N tar r PS, (6.1) dn soot dt = r SN r SC, (6.2) dm soot dt = r SN + m tar r PD + π ( 6msoot πρ s ) 2/3 N sootr SS. (6.3) These equations include terms for precursor inception (r PI ), precursor deposition (r PD ), thermal cracking (r TC ), soot nucleation (r SN ), soot coagulation (r SC ), and surface reactions (r PS and r SS ) 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 negligible 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 adaptation (CPDbio) [112] were used as gold standards to which the sooting potential model was calibrated. 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

150 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 parameters 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, galactogluco-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 pressures 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 y tar = a + bt g + cp + dtg 2 + ep 2 + f T g P + gtg 3 + htg 2 P + it g P 2 + jp 3 k + lt g + mp + ntg 2 + op 2 + pt g P + qtg 3 + rtg 2 P + st g P 2 +tp 3, (6.4) m tar = a + bt g + cp + dtg 2 + ep 2 + f T g P + gtg 3 + htg 2 P + it g P 2 + jp 3 k + lt g + mp + ntg 2 + op 2 + pt g P + qtg 3 + rtg 2 P + st g P 2 +tp 3, (6.5) were fitted. In Equations 6.4 and 6.5, T g 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 y tar = y tar,cell y cell + y tar,hw/hc y hw/hc + y tar,sw/hc y sw/hc + y tar,hw/lig y hw/lig + y tar,sw/lig y sw/lig, (6.6) m tar = m tar,cell y cell + m tar,hw/hc y hw/hc + m tar,sw/hc y sw/hc + m tar,hw/lig y hw/lig + m tar,sw/lig y sw/lig. (6.7) 137

151 Table 6.1: Sooting potential model for biomass with calibrated parameters for Equations 6.4 and 6.5. T g and P are the gas temperature (K) and log-pressure (log(atm)) respectively. Component Cellulose Hardwood Hemicellulose Softwood Hemicellulose Hardwood Lignin Softwood Lignin Model y tar,cell = -1.57E T g 0.022T 2 g +8.00T g P+3.60E-5T 3 g 0.036E-2T E T g +11.2T g P+4.53E-5Tg Tg 2 P m tar,cell = -3.06E T g+1.05e4p 1.84E3P T g P+461.8P T g 0.145T g P 0.021T g P P 3 y tar,hw/hc = -5.21E5+3.12E3T g 0.382T 2 g 1.08E3T g P+0.207T 2 g P 5.75E3T g 2.65E3T g P 1.45E-4Tg Tg 2 P m tar,hw/hc = 236.7T gp E4P T g P P 3 y tar,sw/hc = 7.05E T g 1.29E-5P+0.233Tg E-5Tg E5+91.0T g 3.22E5P+0.725Tg E-4Tg 3 m tar,sw/hc = 6.41E4P T g P+26.0T g P E4P E3P T g P+0.072T g P P 3 y tar,hw/lig = 9.04E4 76.2T g 3.43E4P+6.03E-3T 2 g +36.6T g P+7.69E-6T 3 g 0.011T 2 g P 1.37E T g 3.66E4P+0.012Tg T g P+1.00E-5Tg Tg 2 P m tar,hw/lig = 4.78E6 8.40E3T g+7.36t 2 g +3.39E6P T g P 1.23E-3T 3 g T g P E5P T g +1.47E4P T g P P 3 y tar,sw/lig = 9.15E T g 3.00E5P+0.070T 2 g T g P+1.32E-T 3 g 0.046T E6 1.10E3T g 3.02E5P+0.22Tg T g P 0.047Tg 2 P m tar,sw/lig = 9.15E T g 3.00E5P T 2 g T g P+1.318E-5T 3 g 0.046T E E3T g 3.02E5P+0.219Tg T g P Tg 2 P g P g P g P Note that this sooting potential model neglects the behavior of extractives in biomass in part because 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 potential model follows the trends of CPDbio with good agreement (R 2 =0.811 and for soot mass 138

152 Tar Mass Yield Predictions hardwood_hemicellulose softwood_hemicellulose hardwood_lignin softwood_lignin cellulose Tar Size Predictions (g/mole) Model Predicted Model Predicted CPD Predicted CPD Predicted Figure 6.1: Comparison between results given by CPDbio versus the proposed sooting potential empirical 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 (R 2 =0.811) and the right plot shows the comparison for tar mass size (R 2 =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 component of varying 1 3C NMR parameters. These parameters may be obtained through a correlation 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< O C <0.35), H/C atomic ratio (0.3< H C <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 y tar = P O C 223.9OC H C 107.3HC V V PC H 0.521PV 5.32H C V P E3O C 2.46E3OC H C 266.3HC V V PH C 462.5O C H C O C V 17.8H C V (6.8) 139

153 1.0 Tar Mass Yield Predictions 800 Tar Size Predictions Model Predicted Model Predicted CPD Predicted CPD Predicted Figure 6.2: Comparison between results given by CPD versus the proposed sooting potential empirical model. The left plot shows the comparison for tar mass yield (R 2 =0.794) and the right plot shows the comparison for tar mass size (R 2 =0.854). and m tar = 3.12E T g E5O C 8.48E5H C E5HC V 0.221T g V 6.39E5O C H C E3H C V T g O C 1.77E3H C E3HC E-3T gp 0.024T g H C 5.27E-T g V PV 361.0O C H C 3.83H C V (6.9) In these equations P is the logarithm of the pressure measured in atmospheres. O C and H C are the atomic ratio of oxygen and hydrogen to carbon respectivily. V is the mass percent of volatile matter in the parent coal. T g 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 (R 2 =0.794 and 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

154 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 r v (kg/m 3 s), of which there are many developed models [202, 159, 104, 50]. r T I = y tarṙ v m tar, (6.10) 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 cracking 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, ( 31.1 r cracki = 94 k 1X phe + k 2 X phe k 3X napth [H 2 ] ) 92 k 4X tol [H 2 ] k 5 X ben N PAH. (6.11) Justification for this model was given previously in the preceding chapter. The difficulty in this submodel is designating values for X phe, X napth, X tol, and X ben. 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, H 2 concentration, and initial precursor number density, to determine both parameter sensitivity and to derive a simple empirical 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 precursor number density. The other three parameters, oxygen mass fraction, aromatic/aliphatic carbon ration, and H 2 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<N tar (#/m 3 )<1E25. Observe that at low temperatures 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 comparison to soot nucleation mechanisms. On the other hand, at high temperatures the thermal cracking 141

155 Mole Fraction Temperature Variation Temperature (K) Concentration Variation Concentration (#/m^3) Phenol Toluene Naphthylene Benzene Figure 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, x i = tanh( a + bt + cc + dt 2 + ec 2 + f TC ) + g + ht + ic + jt 2 + kc 2 + ltc, (6.12) m 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 parameters were tuned, negligible ones (those with an influence on the final x i 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 x i parameters, X phe = 1 tanh( T C) C, 6 (6.13) X napth = 1 2 tanh( E-4 T C C E-7 T E-5 TC ) T 1.69E-7 T 2, (6.14) 142

156 Precursor Type Fraction Prediction Phenol Toluene Naphthalene Benzene Model Predicted Numerical Study Figure 6.4: Comparison between empirical model and numerical study for predicting precursortype fractions. The black straight 45 represents a perfect agreement between the two (R 2 =0.919). X tol = 1 3 tanh( T 1.08 C C T C ) T C E-6 TC, X ben = 1 X phe X napth X tol, (6.15) (6.16) 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 Soot Nucleation Soot nucleation occurs through the coalescence of two precursor molecules to form an incipient soot particle r SN = εβ P Ntar. 2 (6.17) Here β PAH 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

157 compute the frequency of collision between two molecules in the free-molecular regime β P = d 2 PAH 8πkB T m tar. (6.18) d PAH, the effective diameter of the precursor, can be computed using a geometric relationship assuming that the precursor is highly condensed [58] d PAH = d A 2mtar 3m C. (6.19) Given these definitions, Equation 6.17 can be expanded and then simplified r SN = 4εd2 A N2 tar 2πkB T m tar. (6.20) 3m C 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 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 r PD = εβ ps N tar N soot, (6.21) using a frequency of collision, β ps, between precursors and particles. ε is the Van der Waals enhancement factor. We compute the frequency of collision assuming a free-molecular flow regime β ps = (d soot + d PAH ) 2 πkb T 2m tar. (6.22) 144

158 d PAH is the effective diameter of the precursors and is computed using Equation 6.19 and d soot is the effective diameter of the soot particles, d soot = ( 6msoot πρ s ) 1/3, (6.23) where m soot is the mass of individual soot particles defined as m soot = M soot N soot. Substituting the collision frequency and effective diameters back into Equation 6.21 yields r PD = ε k B T [ m 1/2 tar ( 2 π ) 1/6 ( ) 2/3 ( ) 3msoot 3 1/2 ( +d A π 1/3 6msoot ρ s m C ρ s ) 2/3 +d 2 A ( 2πmtar 9 (6.24) ) ] 1/2 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. N tar N soot Surface Reactions This model considers three types of surface reactions: surface growth through the established hydrogen-abstraction-carbon-addition mechanism (HACA), consumption through oxidation, and gasification. HACA is a literature-established growth mechanism [7, 56, 129, 128] described previously in Section The overall reaction rate is given in Equation Use of this equation requires the first two moments of a PSD. For soot this is not a problem as N soot and M soot 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, N tar, and the second can be computed using the assumed molecular size, M tar = m tar N tar. (6.25) Oxidation and gasification rates are resolved using the work of Chapter 4, given in Equations 4.3 and The total effect of all three surface reactions are the sum of the individual processes, r SS or r PS = R HACA R oxidation R gasi f ication. (6.26) 145

159 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 Coagulation Particle-particle coagulation only affects the number density term of soot particles as total 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 r CS = β S Nsoot. 2 (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 freemolecular 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 = 2λ soot d soot (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, where d soot is computed from Equation β f S = 8πkB T εd2 soot (6.29) m soot 146

160 In the continuum flow regime, the frequency of particle collision was modeled by Seinfeld and Pandis [167] as where µ is the gas viscosity. β c S = 8k BT 3µ ( Kn), (6.30) Should we be firmly in the free-molecular flow regime, Kn < 0.1, then we use β f S in Equation 6.27 to model the coagulation rate. If we are firmly in the continuum flow regime, Kn > 10, then we use β c S weighted combination in Equation in Equation If we are in the transition regime, 0.1 < Kn < 10, then we use a βs t = β S c 1 + Kn + β f S 1 + 1/Kn (6.31) 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 Coal Flat-Flame Burner Details of this system were given in Section 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 O 2, 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

161 Number Density (#/m 3 ) Particle Number Density z (mm) PPBv Soot Volume Fraction Detailed Model Simplified Model z (mm) Figure 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 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 predict 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 simplified 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

162 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 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 Table 3.2; inlet flow rates were double those shown in Table 3.3. Fuel density was 1300 kg/m 3 and the dry/ash free fuel enthalpy was taken as E6 J/kg. The fuel is represented by a particle distribution resolved using DQMoM with 3 quadrature nodes at 20, 120, and 240 µm in diameter. 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) dv dt = A devol 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 for A devol, E devol, and V respectively. Char oxidation is modeled using a global reaction rate dm char dt = A char P n O 2 exp ( Echar RT ), (6.33) where M char is the consumption rate of carbon per m 2 of available surface area for oxidation. Values for A char, E char, and n were taken from work done by Murphy and Shaddix [138] and were (kg/m 2 atm n ), 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

163 Soot Volume Fraction Soot Particle Number Density Detailed Model Simplified Model Detailed Model Simplified Model max 75% 50% 25% min Figure 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 #/m 3, min = 0 #/m 3 ), and soot particle number density from simplified model (max = 1E21 #/m 3, min = 0 #/m 3 ). 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 ppmv across the entire domain and 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 ppmv across the entire domain and ppmv along the reactor centerline. Figure 6.6 shows the same 2-dimensional cross-section of the reactor 150