Pyrolysis Thesis

8/13/2019 Pyrolysis Thesis

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Soot Formation Modeling during

Hydrocarbon Pyrolysis and Oxidation

behind Shock Waves

DISSERTATION

submitted to the

Combined Faculties for the Natural Sciences and for Mathematics

of Rupertus Carola University of Heidelberg, Germany

for the degree ofDoctor of Natural Sciences

presented by

M.Sc. Iliyana Ivanova Naydenova

born in Sofia, Bulgaria

Ruprecht-Karls-Universitat Heidelberg

Interdisziplinares Zentrum fur Wissenschaftliches Rechnen

2007


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behind Shock Waves

DISSERTATION

submitted to the

Combined Faculties for the Natural Sciences and for Mathematics

of Rupertus Carola University of Heidelberg, Germany

for the degree ofDoctor of Natural Sciences

presented by

M.Sc. Iliyana Ivanova Naydenova

born in Sofia, Bulgaria

Heidelberg, 11.June.2007

Ruprecht-Karls-Universitat Heidelberg

Interdisziplinares Zentrum fur Wissenschaftliches Rechnen

2007


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behind Shock Waves

Supervisor: Prof. Dr. Dr. h. c. Jurgen Warnatz

Reviewer: Priv. Doz. Dr. Hans-Robert Volpp


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Acknowledgement

It is my great pleasure to acknowledge all the people who helped me directly orindirectly to accomplish this dissertation. First and foremost, I express my deep felt

gratitude towards my supervisor Prof. Dr. Dr. h. c. Jurgen Warnatz for his advice,

encouragement, easy accessibility and freedom of work that leads to the completion

of the thesis.

I am also thankful to Dr. Pavel Vlasov (Institute of Chemical Physics, Russian

Academy of Science) and my colleague Jens Marquetand for the tireless discussions,

useful comments and great support in the development of my work. My special

thanks to Dr. Markus Kraft and Matthew Celnik (Department of Chemical En-

gineering, University of Cambridge) for their fruitful discussions on the problems

of soot formation modelling. Many thanks to Volkmar Reinhardt for his friendly

helping hand in preparating the results for our colleagues from the SFB-568 Project

(Technical University, Darmstadt).

I acknowledge Deutsche Forschungsgemeinschaft for their financial support.

My sincere thanks to Priv. Doz. Dr. Uwe Riedel for his advices and assistance in

solving the administrative obstacles. Thanks to Volker Karbach for the discussions

on reaction kinetics. I also thank to Ingrid Hellwig for her help in organising the

administrative details and to Jurgen Moldenhauer, Joachim Simon and Jan Pitann

for solving computer related problems. It would have not been possible to complete

the work without the help of my coworkers and friends. Space here would not be

enough to mention them all personally.

Finally, I would like to thank to my entire family for their help, boundless love and

faith in me. I owe a heartfelt gratitude to my husband Alexander for his endlesslove, invaluable encouragement, support and assistance in all kind, cheerful sense of

humour and care which has been always important part of my success. Thank you

my friend!


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Abstract

In the present work, soot formation was modeled in conditions typical of shock tubeexperiments. Two different detailed kinetic models (Model-1 and Model-2) were

developed. The models were validated by means of a suitable numerical technique

(discrete Galerkin method). The gas-phase chemistry of soot precursor and particle

formation was described in terms of different pathways. Accordingly, the formation

and evolution of soot particles differs with respect to the type of the species leading

to soot particle inception.

Based on previously described hypotheses [1, 2], two types of soot precursors were

considered in Model-1, polyyne and PAH. Latest experimental investigations of soot

formation in flames and shock tube [3, 4, 5] confirmed that young soot particles are

built primarily of polycyclic aromatic hydrocarbons, and the reaction of aliphatic

species with pre-existing soot surface can be an important factor for the particle

mass growth. Following these conclusions, another detailed kinetic model was de-

veloped (Model-2), where PAH were considered as soot precursors, and aliphatic

species take place only in reactions of surface growth. Both models were validated

against the experimentally obtained concentration profiles of the main gas-phase

species, measured in shock tube experiments. They were further applied for sootformation simulation during pyrolysis and oxidation of various hydrocarbons and

their mixtures behind shock wave, for a wide range of reaction conditions (temper-

ature, pressure and mixture composition). The calculation results were compared

with the usually measured characteristics of soot formation, e. g., induction delay

time, observable rate of soot particle growth, soot particle concentration, diameter,

and soot yield.

For the application in a multi-dimensional computational fluid dynamics (CFD)

code for turbine combustion simulation, merely simple empirical models with fewvariables must be used. Therefore, a two-equation model was developed and im-

plemented in a software package [6] for simulation of time-dependent homogeneous

reaction systems. The model was calibrated by the reaction kinetics of the detailed

chemical mechanisms (Model-1 and Model-2). The complex phenomenon of soot

formation is described in terms of several global steps: inception, growth, coagu-

lation and oxidation, where two differential equations are solved for the temporal

change of soot concentration and soot volume fraction. The simulation results were

compared with the experimentally measured soot characteristics during shock tube

oxidation of various hydrocarbons.


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Kurzfassung

Die vorliegende Arbeit beschreibt die Modellierung der Rubildung unter Bedin-gungen, die typisch fur Stowellenexperimente sind. Zwei unterschiedliche, detail-

lierte kinetische Modelle (Modell-1 und Modell-2) wurden entwickelt und mittels

eines geeigneten numerischen Verfahrens (diskrete Galerkin-Methode) uberpruft.

Die Entstehung des Ruvorlaufers und die Russteilchenbildung wurde jeweils durch

unterschiedliche Reaktionspfade beschrieben.

Auf der Grundlage bereits beschriebener Hypothesen [1, 2] wurden in Modell-1

zwei verschiedene Arten von Ruvorlaufern berucksichtigt: Polyine und polyzyk-

lische aromatische Kohlenwasserstoffe (PAK). Die neuesten experimentellen Unter-

suchungen der Rubildung in Flammen und Stowellenrohren [3, 4, 5] bestatigen,

dass die Ruteilchen hauptsachlich aus PAK gebildet werden. Die Experimente

deuten daraufhin, dass die Reaktion aliphatischer Spezies mit Ruoberflache ein

wichtiger Faktor fur das Massenwachstum der Teilchen ist. Aufgrund dieser Ergeb-

nisse wurde ein detailliertes kinetisches Modell-2 entwickelt, in welchem die PAK

als Ruvorlaufer betrachtet werden. Die aliphatische Spezies sind hier nur an

Oberflachenwachstumsreaktionen beteiligt. Beide Modelle wurden der wichtigsten

Gasphasenspezies Konzentrationsprofile validiert, die in Stowellenexperimentengemessen worden sind. Ferner wurden die Modelle fur Simulation der Rubil-

dung wahrend der Pyrolyse und der Oxidation verschiedener Kohlenwasserstoffe und

ihrer Mischungen hinter der Stowellen fur ein breites Spektrum von Reaktionsbe-

dingungen angewandt. Die Ergebnisse der Berechnungen wurden mit Messwerten

(Zundverzugszeit, Bildungsgeschwindigkeit des Ruteilchen, Ruteilchenkonzentra-

tion und Durchmesser) verglichen, die ublich bei Experimenten gemessen werden.

Fur die mehrdimensionale Simulation der Verbrennung in Gasturbinen kann lediglich

ein einfaches empirisches Russbildungs Modell mit wenigen Variablen verwendetwerden. Hierfur wurde ein Zwei-Gleichungsmodell entwickelt und in ein Soft-

warepaket [6] fur die Simulation zeitabhangiger, raumlich homogener Reaktionssys-

teme implementiert. Das Modell wurde anhand der Reaktionskinetik der detail-

lierten chemischen Mechanismen (Modell-1 und Modell-2) kalibriert. Der komplexe

Prozess der Rubildung wurde mittels einiger globaler Schritte beschrieben: Keim-

bildung, Wachstum, Koagulation und Oxidation, wobei zwei Differentialgleichun-

gen fur die zeitliche Anderung der Rukonzentration und des Ruvolumenbruchs

gelost werden. Die Simulationsergebnisse wurden mit der experimentell gemesse-

nen Rucharakteristika der Oxidation unterschiedlicher Kohlenwasserstoffe in einem

Stowellenrohr verglichen.


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I

Contents

1 INTRODUCTION 1

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

1.2 Objectives and structure of the thesis . . . . . . . . . . . . . . . . . . 6

2 FUNDAMENTALS OF PHYSICAL CHEMISTRY 9

2.1 Homogeneous reaction system . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.2 Chemical kinetics . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.3 Analysis of reaction mechanisms . . . . . . . . . . . . . . . . . 16

3 SOOT FORMATION 19

3.1 Gas phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.1 First aromatic ring formation . . . . . . . . . . . . . . . . . . 21

3.1.2 Growth of aromatics by HACA . . . . . . . . . . . . . . . . . 24

3.1.3 Growth of aromatics by other species . . . . . . . . . . . . . . 25

3.1.4 Oxidation of aromatics . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Particulate phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28


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II

3.2.1 Soot particle inception . . . . . . . . . . . . . . . . . . . . . . 28

3.2.2 Soot particle growth . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.3 Soot particle coagulation . . . . . . . . . . . . . . . . . . . . . 31

3.2.4 Soot particle oxidation . . . . . . . . . . . . . . . . . . . . . . 32

3.2.5 Soot agglomeration . . . . . . . . . . . . . . . . . . . . . . . . 33

4 DISCRETE GALERKIN METHOD 35

4.1 Theory of the discrete Galerkin method . . . . . . . . . . . . . . . . . 36

4.2 Program package MACRON . . . . . . . . . . . . . . . . . . . . . . 39

5 DETAILED KINETIC MODELS OF SOOT FORMATION 42

5.1 Description of Model-1 . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.1.1 Gas-phase reaction mechanism . . . . . . . . . . . . . . . . . . 42

5.1.2 Soot precursors and particle inception, surface growth, coag-

ulation and oxidation . . . . . . . . . . . . . . . . . . . . . . . 44

5.2 Results Model-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.2.1 Validation of the model . . . . . . . . . . . . . . . . . . . . . . 50

5.2.2 Hydrocarbon pyrolysis behind shock waves . . . . . . . . . . . 59

5.2.3 Hydrocarbon oxidation behind shock waves . . . . . . . . . . . 74

5.3 Description of Model-2 . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.3.1 Gas-phase reaction mechanism . . . . . . . . . . . . . . . . . . 77

5.3.2 Soot precursor and particle inception, growth, coagulation and

oxidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.4 Results Model-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83


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III

5.4.1 Validation of the model . . . . . . . . . . . . . . . . . . . . . . 83

5.4.2 Hydrocarbon pyrolysis behind shock waves . . . . . . . . . . . 99

5.4.3 Hydrocarbon oxidation behind shock waves . . . . . . . . . . . 107

6 SIMPLIFIED MODEL OF SOOT FORMATION 114

6.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.1.1 The temporal change of soot concentration . . . . . . . . . . . 116

6.1.2 The temporal change of the soot volume fraction . . . . . . . 117

6.1.3 Rate laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.1.4 Soot quantities . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7 Conclusion and future prospects 130

Appendix 133

Appendix A. List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Appendix B. List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . 144


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1

Chapter 1

INTRODUCTION

1.1 Motivation

Soot formation has been of interest to combustion scientists and engineers at least

since the 19th century. Initially soot was valued for its heat- and light-producing

properties and for its relation to carbon-black manufacturing [7]. The carbon black

is used in the production of automotive tires, as a reinforcing agent for rubbers, to

colour printing ink, painting, paper and plastics. The smoke produced by sooting

flames was only an annoyance until the early 1970s. At that time the dangerous

health effects associated with soot and the polycyclic aromatic hydrocarbons (PAH)

that usually accompany soot formation, came to be known, and soot became an

unwanted by-product of combustion. Longwell [8] pointed out that the interest in

controlling soot emissions was due to the understanding that soot particles can ad-

sorb harmful PAH onto their surfaces. Small soot particles can be breathed deeply

into the lungs, where they can do substantial damage. Combustion related particu-late matter is associated with a host of severe impact such as heart attacks, stroke,

cardiovascular death [9] and lung cancer [10] in adults. In children, fine particles

are associated with upper and lower respiratory impact, as well as retardation of

lung growth and crib death [11]. Soot particles from Diesel engines adsorbed onto

their surface metals and toxic substances such as cancer-causing aldehydes and PAH,

while many PAH are known to be carcinogenic or mutagenic. Traffic studies suggest

increased rates of respiratory and cardiovascular disease and risk of premature death

near busy urban streets or highways. Therefore, a great attention was drawn to the

chemistry of soot, PAH and hydrocarbons like 1,3-butadiene, benzene, and tolueneby the scientists allover the world. Thus, the chemistry of rich flames, particularly


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1. INTRODUCTION 2

that involved with hydrocarbon growth into PAH and soot, became one of the most

active research areas in combustion chemistry.

In soot formation modeling, several principle proposals are known, which describe

the nature of soot particle inception. According to them, different types of species

are ranged as potential precursors, leading to soot particle inception: polyacetylenes

or polyynes [12, 13, 14, 15, 16, 17, 2, 18], ions [19, 20], and polycyclic aromatic

hydrocarbons [21, 1, 22, 23, 24].

The investigation of the role of acetylene in soot formation dates back to about

hundreds years ago. The reason why many experimentalists suggested the poly-

acetylenes as contingent soot precursors is that several experimental investigations,

performed in the 1960s and 1970s, showed the existence of hydrocarbons having

molecular mass in excess of 250 g/mol. They appear at the end of the reaction

zone, in the region right before the appearance of the first particles [12] and [25].

Unlike the PAH, these species disappear rapidly during the soot growth, and are

no longer detected at the end of the reaction zone. Some authors suggested that

such species could be polyacetylenes [26, 27, 28]. The development of this idea can

be summarised as follows: Berthelot et al. [29] and Lewes et al. [30] emphasised

the importance of C2H2 in thermal decomposition reactions. Porter [3] proposed

the hypothesis of carbon formation from acetylene through its simultaneous poly-merisation and dehydrogenation. Haynes and Wagner [25] pointed out that the

investigations of the absorption profiles for pre-soot species in pyrolysis and oxi-

dation of different fuels and indicate the presence of species capable of absorbing in

the visible and ultraviolet before soot becomes observable. Cundall et al. [26, 27, 28]

analysed the shape of some spectra and suggested that the absorbers are predom-

inantly polyacetylenes, most probably C10H2 and C12H2. These species have been

measured mass-spectrometrically by Kistiakowsky et al. [31, 32] as products of C2H2

pyrolysis. They and other authors [33] concluded that the reaction proceeds as:

C2H2=C4H3=C4H2=C6H2 =C8H2 ... (1.1)

Bohne and Wagner [34] experimentally observed that in premixed flat flames of

C2H2, C2H4, C3H8, C6H6, and C2H5OH in fuel-rich mixtures higher polyacytylenes

are formed, where such molecules up to C12H2 have been detected experimentally.

Homann and Wagner [12] investigated the hydrocarbons occurring in the region of

carbon nucleation in acetylene and benzene/oxygen flames and discussed the role

of polyacetylenes and polycyclic aromatics in the process of particle inception. The

authors suggested that the soot precursors can be derived by the following scheme:

C6H2+ C2H =C8H3=C8H2+ H =branching =cyclization =... (1.2)


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1. INTRODUCTION 3

Kern et al. [35, 36] measured the product profiles during pyrolysis of acetylene, bu-

tadiene, benzene and toluene. The authors found that the the main products are the

polyynes C4H2, C6H2, C8H2. Nevertheless, the polyacetylene hypothesis, describing

the soot inception by means of the formation of long and stable polyacetylene chains,

has not been elaborated further until the work of Krestinin et al. [15]. The authors

developed a detailed kinetic model of soot formation called polyyne model, regard-

ing the high reactivity of these species in polymerisation reactions. The polyyne

model is applied for soot formation simulation during pyrolysis of C2H2 [16]. A

modified and extended version is further applied for soot formation modeling dur-

ing pyrolysis of different hydrocarbons in reactive flow experiments [17, 2, 18]. The

model treats soot formation as a process of chemical condensation (polymerisation)

of supersaturated polyyne vapour (C2nH2) and describes the formation of youngsoot particlesandmature soot particles, and the transformation between them. The

authors stated that compared to the rather slow multistage increase in the number

of aromatic rings in the PAH, the polyynes grow in a simple and fast way typically

in reactions like

C2nH2+ C2H = C2n+2H2+ H. (1.3)

Calcote [19] argued that the polyacetylenes did not grow sufficiently rapidly to ac-

count for the almost instantaneous formation of soot. He claimed further that the

reactions of neutral species were not fast enough and suggested an ionic mechanism.In the model chemi-ions are assumed to be the precursors of soot on which free rad-

icals, polyacetylenes, and PAH repeatedly add through fast ion-molecule reactions.

Calcote claimed that H3O+ was the dominant ion in near-stoichiometric and lean

flames, and C3H+3 in rich flames, and proposed a kinetic scheme with the elementary

reactions which produce the primary ions in the system.

Simultaneously with the above described hypotheses, many authors accepted that

the PAH are the only possible soot precursors. Thomas [37] stated that the process

of transformation of precursors to soot particles must involve species that must be

stable enough thermodynamically to survive extreme conditions like high tempera-

ture and high pressure combustion environment. In addition to this, they must be

sufficiently reactive to promote the growth of larger molecules on fairly short time

scales (e. g., a few milliseconds in shock tube experiments). Miller [38] pointed

out that the highly reactive criterion can be accommodated by supposing that free

radicals can be formed from stable molecules by abstracting hydrogen atoms. There-

fore, a molecular structure is needed, stable enough to grow in flame environments.

Rummel and Veh [39] proposed that the major role of the PAH is due to their ther-modynamic stability, whereas Thomas [37] suggested that the essential soot precur-

sors are conjugated polyene radicals that grow into polybenzenoid radicals and soot


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1. INTRODUCTION 4

by adding other unsaturated species. Glassman [115] had a similar point of view

and proposed a special role for 1,3-butadiene in the PAH growth. DAlessio et al.

and Minutolo et al. [40, 41] investigated high molecular mass structured formed

in the main oxidation zone of rich premixed flames and rich flames below the soot

threshold limit. The authors [41] detected the existence of high molecular struc-

tures transparent to the visible radiation in both the pre-inception zone of sooting

flames and in flames below the soot formation limit. They stated that the onset of

ultra-violet fluorescence within the main oxidation zone implies that the formation

of these species is a very fast process and can be considered as a polymerisation of

small aromatic groups activated by the presence of oxidising agents.

The modeling of PAH and soot formation and growth in combustion was consider-ably influenced by the work of Frenklach et al. [13, 21, 1]. The authors suggested

a detailed kinetic mechanism of PAH formation and growth called H-abstraction-

C-addition (HACA). According to this model, the aromatics grow by a two-step

process ofH-abstractionwhich activates the aromatic molecule, and acetylene ad-

ditionwhich propagates molecular growth and cyclisation (see Chapter 3.1.2). The

formation and evolution of soot particles is mathematically described using the

method of moments [1]. As Miller said in his review paper [38], these authors con-

verted the qualitative ideas into a quantitative chemical kinetic model. Fernklach

et al. [13, 42] first considered the soot formation modeling in shock tube pyrolysis

of acetylene. The authors developed a detailed kinetic model of PAH formation and

growth including branching reactions of aliphatics, similar to those showed in scheme

1.2, leading to cyclisation (ring closing reactions), where aromatic compounds are

formed. Furthermore, Frenklach et al. and Wang et al. considerably modified the

existing kinetic scheme and extended the modeling to the pyrolysis and oxidation

of different fuels in flames [43, 21, 1, 44, 45, 46].

In the last decade, the idea supporting the PAH as the principle soot precursors

gains more evidences due to the recent development of the experimental techniques

[4, 47, 48, 49] and the numerical models [50, 51, 52, 24, 53, 54, 55]. This provides the

possibility for an extensive research, providing more details of the different stages

of soot formation.

Kronholm [56] studied the molecular weight growth pathways of fuel-rich combus-

tion and suggested that the distinction between the largest PAH molecules and the

smallest (young) soot particles is arbitrary. In his study, Kronholm assumed the

concept that PAH and soot can be treated analogously in a general formulation ofmolecular weight growth. He develop a model of PAH growth and soot nucleation

that treated large PAH similarly to soot aerosols. These aerosols are further lumped


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1. INTRODUCTION 5

into average property particles, called BINs, with a molecular weight between 100

amu and 1600 amu. To distinguish large PAH and soot particles, a specific molar

mass of 1600 amu is assumed as upper limit.

This approach was further developed and applied for soot formation simulation

of various combustion systems [57, 24, 53]. Richter et al. [24] proposed a de-

tailed kinetic model of PAH and soot particle formation in a laminar premixed ben-

zene/oxygen/argon low/pressure flame. The authors used the sectional technique to

model the particulate-phase chemistry. They defined large PAH and carbonaceous

particles with diameter up to 70 nm as classes called BINs, covering given mass

ranges. The number of carbon and hydrogen atoms corresponding to their average

mass are assigned to each class. The change of the C/H ratio is calculated withrespect to the particle size. Soot particle inception takes place in reactions of PAH

radicals with PAH molecules and among PAH radicals. The authors stated that at

about 75 % of the final particle mass is due to the process of surface growth, where

the reaction of acetylene with particle radicals is the major growth route. The model

provides information about the size, mass and content (C/H ratios) of the particles,

but cannot predict soot particle structure.

Violi [52] proposed an atomistic model for particle inception, which is a combination

of kineticMonte-Carloand molecular dynamic methods. The model is applied to in-vestigate the growth of aromatic compounds up to the nano-size range in chemically

specific way. This approach preserves the atomic scale structures like bonds, bond

angles and dihedral angles as the soot precursors evolve into three-dimensional struc-

tures. This technique was applied to aliphatic (C2H2) and aromatic (C6H6) flame

environments. The calculations give information about the similarities and differ-

ences of soot precursor structure, morphology and the H/C ratios in aromatic and

aliphatic flames.

Morgan et al. [55] developed a detailed particle model, which simulates the size

distribution of soot particles in laminar flames with the use of stochastic numeri-

cal methods. The model is applied to simulate flames with bimodal and unimodal

particle size distribution and provides useful information about the change in mor-

phology between the particles from these two types of flames. These results provide

evidence on the importance of interplay among the processes like nucleation, coag-

ulation and surface growth, which is previously studied by [50, 51, 54]. The authors

stated that the transition of spherical growth in fractial-like objects can be related

to the nucleation, as it provides the appearance of very small, primary soot particles.

Several authors suggested models which combine two different pathways of soot for-


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1. INTRODUCTION 6

mation, HACA and polyyne. The model suggested by Vlasov and Warnatz [58]

combines the HACA mechanism of PAH growth [21, 59] with the polyyne model of

soot formation [2], and the model of pure carbon clusters formation [60]. This ap-

proach is applied for soot formation simulation in pyrolysis of various hydrocarbons

and their mixtures in conditions typical for shock tube experiments. An extended

version was used for soot formation modeling during shock tube oxidation of dif-

ferent hydrocarbons [61, 62, 63, 64]. This model is in detail described in Chapter

6. Similarly, Wen et al. [65] developed a detailed kinetic model which is again a

combination of both PAH and polyyne pathways of soot formation and simulated

the nano-particle inception and growth in pyrolysis of C6H6 behind shock waves,

using the sectional technique.

Numerous theoretical models simulate particle formation and evolution in different

types of flames but soot formation modeling in terms of short time scales, e.g.,

the shock tube experiments, takes place in a few milliseconds. This restriction

made it difficult to model soot formation with the use of the traditional HACA

model. It required the investigation and the development of various chemical rection

routes of soot precursor formation and growth together with an adequate kinetic

representation of these processes.

1.2 Objectives and structure of the thesis

Unburned hydrocarbons and soot are typical pollutants formed during combustion,

although these species do not exist in the initial fuel. It is known that the main rea-

son for the appearance of such products is the inappropriate combustion conditions:

time, temperature, and turbulence [66, 67]. Variation of the mixture compositions

and the reaction conditions improve the results for some of the components but

increases the amount of the others. To solve the problem it is necessary to answer

the question, how these compounds are generated and why they were not consumed

during the combustion process.

The goal of the current work is the investigation of the processes and mechanisms

leading to the formation of gas-phase precursors and soot particles. Accordingly,

a detailed kinetic model had to be developed for soot formation simulation during

pyrolysis and oxidation of hydrocarbons and their mixtures at spatially homogeneous

conditions. The model had to be validated against experimentally obtained data,available in the literature. Furthermore, a simplified model of soot formation had

to be developed and calibrated by the detailed scheme.


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1. INTRODUCTION 7

A short historical overview of various hypotheses proposing different types of gas-

phase species as the potential soot precursors is presented in Chapter 1, together

with a brief description of several basic kinetic mechanisms of soot formation.

The development of a kinetic mechanism is based on the concept of elementary re-

action. Therefore, knowledge of the physical and chemical fundamentals are needed

for the adequate description of the reaction systems, in accordance to its thermo-

dynamics, chemical kinetics, and the special features of the combustion facility (see

Chapter 2). In the current work, the soot formation was studied at homogeneous

conditions, in particular shock tube experiments.

The hypothesis regarding the PAH as the most probable soot precursors gains more

evidences in the last decade. Various reaction pathways and mechanisms leading

to PAH formation and growth are presented in Chapter 3, together with a short

description of the physical and chemical processes describing the soot formation:

soot-particle inception, growth and oxidation, coagulation, and agglomeration.

The mathematical representation of the soot formation was performed by a suitable

numerical technique, discrete Galerkin method (see Chapter 4). The method is

previously implemented in a program package, MACRON [68], for the treatment

of large systems of ordinary differential equations, arising from the macromolecularreaction kinetics. It is initially proposed for simulation of polymerisation reactions,

but is also successfully applied for soot formation modeling in shock tube [69] and

flame experiments [70]. An important feature of this approach is the so called

lumping technique, which describes soot formation analogously to the process of free-

radical polymerisation [60, 71, 72]. This technique is based on an approximation of

the distribution function for the degree of polymerisation and a repeating reaction

cycle for the particle growth.

Soot formation modeling usually needs a detailed kinetic scheme, describing theformation, growth and oxidation of the gas-phase soot precursors, and a soot-particle

model. In Chapter 5, two detailed kinetic mechanisms are presented, which contain

different approaches of the gas-phase precursors formation and the formation and

evolution of the particulate-phase. Both mechanisms are described together with

the literature sources for the relevant kinetic and thermodynamic properties of the

gas-phase species and the reaction kinetics of the macromolecular reactions. The

mechanisms were validated against experimental data available in the literature for

the concentration profiles of various gas-phase species, the induction delay time, soot

growth rate, particle concentration, diameter, and soot yield, measured in shock-

tube experiments.


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1. INTRODUCTION 8

The detailed reaction mechanisms usually consist of thousands of elementary reac-

tions between hundreds of species. Such reaction scheme cannot be directly used

for CFD simulations of three-dimensional systems, because it exceeds available com-

puter capacities. Therefore, reduced reaction mechanisms are needed, which describe

the chemical reaction system using small number of variables. For the application in

a multi-dimensional CFD code for gas-turbine combustion simulation, an empirical

model was developed which describes the complex process of soot particle formation

and evolution in terms of two differential equations (see Chapter 6).


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9

Chapter 2

FUNDAMENTALS OF

PHYSICAL CHEMISTRY

2.1 Homogeneous reaction system

According to the macroscopic properties of a system (temperature, pressure, con-

centrations, viscosity, electro-conductivity etc) it can be characterised as spatially

homogeneous or heterogeneous. If the properties of the system are the same or

change gradually in every point (part), it is defined as homogeneous. Shock wave

reactors are an example of homogeneous reaction system. The shock tubes are used

by many kineticist as a high temperature reactor to obtain rate coefficient data un-

der homogeneous conditions. The shock tube experiment has the advantages that

it provides a nearly one-dimensional flow with practically instantaneous heating of

reactions, high dilution of the reactants by an inert gas and high sensitivity of the

diagnostic techniques employed to monitor species. The main advantage of dilutingthe reactants with an inert gas is that the exothermicity or endothermicity of the

reactants involved will not greatly alter the constant temperature conditions during

the investigation. On the other hand, by using very low initial reactant concentra-

tions, the influence of subsequent reactions can be avoided or reduced. This allows

the study of only one or two elementary reactions with high accuracy, without being

strongly disturbed by fast secondary reactions [73, 74].

Soot formation has been studied widely in laminar flames [75, 19, 25, 40, 12, 22, 76,

3, 77, 4, 48, 49], but numerous experiments have been performed behind reflectedshock waves [28, 32, 35, 36, 14, 78, 73, 79, 80, 81, 82, 83, 5]. The shock tube as


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2. FUNDAMENTALS OF PHYSICAL CHEMISTRY 10

a wave reactor provides an excellent environment for the study of particle nucle-

ation and growth from the vapour phase at high temperatures. It is a convenient

technique to investigate the effect of controlling and varying the initial conditions

like temperature, pressure, and mixture composition on the size and yield of the

produced particles. The observation time is relatively short, usually limited to a

few milliseconds.

A detailed kinetic model of soot formation usually consists of two general parts, a

gas-phase kinetic scheme describing the fuel destruction and soot precursor forma-

tion, and a soot model describing the particulate-phase chemistry. Such detailed

kinetic mechanism of hydrocarbon combustion consists of several hundreds or even

thousands of chemical elementary reactions. For each reaction and species includedin the mechanism a set of kinetic and thermodynamic data is needed. Nowadays,

experimentally observed or calculated thermodynamic data of a large number of

species as well as reaction rate coefficients are available in the literature.

2.1.1 Thermodynamics

Thermodynamics studies the different forms of energy transformation, which makes

it possible to analyse quantitatively these phenomenon and gives useful predictions

of the system behaviour. For the needs of numerical simulations the thermodynamic

properties are often stored as polynomials inT. If it is possible these values are based

on experimental data, but most of the data is derived from theoretical calculations

based on number of semi-empirical schemes relating thermodynamic properties to

molecular structure [84].

Heat capacities expressed as NASA polynomials (Stull and Prophet [85], Kee et

al. [84], Burkat [86], Warnatz et al [67] etc.) are used to calculate the enthalpyH0, the enthropy S0 and the equilibrium constant KC. Usually the molar heat

capacities C0

p(C0

p= C0

V + R) are expressed as polynomials of fourth order in T,

C0

p

R =a1+ a2T+ a3T

2 + a4T3 + a5 T

4. (2.1)

Herea1, ...,a5are constants, andR is the gas constant. In addition, two integration

constants are needed to compute enthalpies and entropies, where a6 R= H0

298 and

a7R= S0

298,

H0

T =a6R+

TT=298K

C0

pdT and S

0

T =a7R+

TT=298K

C0

p

TdT. (2.2)


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The enthalpy at any temperature T follows from integration of the heat capacity,

Eq. (2.1), with an a6 different from a6,

H0T(T)R

=a6+ a1T+a2

2 T2 + a

3

3 T3 +a

4

4 T4 + a

5

5 T5. (2.3)

The coefficienta6 can be defined settingT= 298 K and demanding that H0(298K)

is equal to the enthalpy of formation at 298 K.

The entropy at any temperature T follows from integration of the heat capacity

divided by temperature T, with an a7 different from a7,

S0

T(T)

R =a7+ a1ln T+ a2T+a3

2 T2

+a4

3 T3

+a5

4 T4

. (2.4)

The coefficient a7 can be defined by setting T = 298 K and determining that

S0

T(298 K) is equal to the entropy at 298 K. Thus seven coefficients define C0

p,

H0

T , and S0

Tat any temperature T.

2.1.2 Chemical kinetics

Chemical kinetics deals with rates of chemical reactions. It explains how rapidly

reactants are consumed and products formed, how the rate responds to changes

in the conditions or the presence of catalyst, and the step by which a reaction

takes place. The reason for studying the reaction rates is of practical importance

in order to predict how quickly a reaction mixture reaches equilibrium. The study

of reaction rates helps to understand the mechanism of a single reaction as well as

complex reactions. The rate depends on variables under our control such as pressure,

temperature, and presence of catalyst. Therefore, we might be able to control it by

the appropriate choice of conditions.

Rate law and elementary reactions

Detailed reaction mechanisms are based on the concepts of the elementary reaction,

which has the advantages that the reaction order is always constant (in particular,

independent of time and of experimental conditions) and can be easily determined.

It is only necessary to look at the reaction molecularity that denotes the number

of species, which form the reaction complex (the transition state between reactants

and products). In general the molecularity equals the order of elementary reactions.


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A rate law describes an empirical formulation of the reaction rate in particular, the

rate of formation or consumption of a species in a chemical reaction. The reaction

rate of an elementary reaction could be experimentally obtained, but only for a given

temperature range. The rate coefficients beyond that temperature can be calculated

using the Arrhenius equation (Eq. 2.16).

If the equation of an elementary reaction r is given by

Ss=1

(e)rs Askr

Ss=1

(p)rs As, (2.5)

then the rate law for the formation of species iin reaction ris given by the expres-

sion,cit

chem,r

=kr

(p)ri

(e)ri

Ss=1

c(e)rs

s . (2.6)

Here (e)rs and (p)rs denote stoichiometric coefficients of reactants (educts) and prod-

ucts, and cs, the concentration of the Sdifferent species s.

The rate law can always be specified for an elementary reaction mechanism. If the

reaction mechanism is composed of all possible elementary reactions in the system,

then it is a complete mechanism and is valid for all conditions (temperatures andmixture compositions), but such mechanisms are rarely available.

Relation of forward and reverse reactions

Chemical reactions move towards a dynamic equilibrium in which both educts and

products are present in significant concentrations, but no net change occurs. In such

cases, thermodynamics can be used to predict the equilibrium composition under

any reaction conditions.

For example, the chemical reaction (2.7) runs in both directions,

A + B C + D, (2.7)

where A and B denote the educts of the reaction, C and D the reaction products,

and kf and kr are the rate coefficients of the forward and reverse respectively. The

reaction rate with respect to the production of the species A is expressed by the

equation

d[A]

dt =kr[C]

c[D]d. (2.8)


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At the chemical equilibrium, the forward and reverse reaction have the same rates,

which can be expressed by the equation

kf[A]a[B]b =kr[C]c[D]d, (2.9)

or,

[C]c[D]d,

[A]a[B]b =

kfkr

. (2.10)

The expression on the left hand side corresponds to the equilibrium constantKc=

kfkr

of the reaction. For a gas-phase reaction, the equilibrium rate constant

can be expressed by the species partial pressure [87] as

Kp=pcCp

dD

paApbB

. (2.11)

The equilibrium composition correspond to a minimum in the Gibs energy plotted

against the extent of reaction [67]. If the location of this minimum is established,

the relation between the equilibrium constant and the standard Gibs energy of the

reaction can be derived. In this way, the equilibrium constant Kp can be also

calculated through the thermodynamic data by

Kp= exp

RG0

/RT

, (2.12)

where the standard Gibs free energy RG0

is calculated by the reaction enthalpy

H0and entropyS

0,

G0

=H0

T S0. (2.13)

The equilibrium constantsKc and Kp are related by

Kc= kp

(RT)v

, (2.14)

where v is the difference between the stoichiometric coefficients of the reaction

(v= (c+ d)(a + b)) , and R is the ideal gas constant.

Temperature dependence of rate coefficients

It is a characteristic of the chemical reactions that their rate coefficients depend

strongly and in a nonlinear way on the temperature [67]. It is found experimentallyfor many reactions that a plot of lnk against 1/Tgives a straight line with a slope

that is characteristic of the reaction [87]. The slope is equal to the relation Ea/RT


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A+ M kaA + M (deactivation) (2.18)

A kuP (roducts) (unimolecular reaction)

The rate equations for this case are given by

d[P]

dt =ku[A

] and d [A]

dt =ka[A] [M] ka[A

][M] ku[A] . (2.19)

Assuming quasistationary concentrations for the highly unstable species, A* is in a

quasi-steady state (d[A] /dt 0). Then, the concentration of the activated species

[A*] and the rate of the product P are given by

[A] = ka[A] [M]

k

a[M] +kuand

d[P]

dt =

kuka[A] [M]

k

a[M] +ku. (2.20)

Two extremes can be distinguished, reaction at very low and very high pressures.

In thelow-pressure range, the concentration of the collision partners M is very small

and ka[M] ku the apparent first order rate law can be obtained,

d[P]

dt =ka[A] [M] =k0[A] [M] (2.22)

with a high pressure rate coefficient k. Here, the reaction rate does not depend on

the concentrations of the collision partners, because at high pressures collisions occur

more often then at low pressures, while at high pressures collisions occur very often

and, thus the decomposition of the activated molecule A* is rate-limiting instead of

the activation.

The Lindemann mechanism illustrates the fact that the reaction order of the com-

plex (non-elementary) reactions depends on the chosen conditions. Nevertheless,

between these two extremes exists a wide transition area, which depends also on the

nature of the species. For smaller molecules this area is observed at higher pressures

and is wider than for the bigger species with higher molecular weight. This area

cannot be described by the simple Lindemann theory. More accurately, the pres-

sure dependence of the unimolecular reactions can be obtained using the Theoryof Unimolecular Reactions (Robinson and Holbrook [89], Atkins [87], Golden [90],

Warnatz [67]). This theory takes into account that not only one activated species


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can be defined, but a large number of activated molecules with different levels of

activation (e.g. vibration or rotation).

If the rate law of a unimolecular reaction is written as d[P]/dt = k[A], then the

rate coefficient k depends on the pressure, and the temperature. The theory of

unimolecular reactions yieldsfall-off curves, which describe the pressure dependence

of k for different temperatures. Usually the logarithm of the rate coefficient is

plotted versus the logarithm of the pressure. The appropriate treatment of pressure-

dependent reactions is important because many experiments on reaction kinetics are

at atmospheric or lower pressure while many combustion processes run at elevated

pressure. An often used formalism is the F-Center treatment of Troe (Gilbert et al.

[91], Warnatz [67]), where ten parameters are used to determine a rate coefficientas specified temperature and pressure. One set of coefficients give the high-pressure

modified Arrhenius parameters, another set the low-pressure modified Arrhenius

parameters, and a third set containing four parameters a , T, T, and Twhich

are used to determine the F-center value (describing the center of the fall-off range),

Fcent= a exp

T

T

+ exp

T

T

+ (1 a)exp

T

T

. (2.23)

The value F is calculated via

logF= logFcent

1 +

logPr+ c

nd (logPr+ c)

21

with

c= 0.40.67logFcent, n= 0.751.27 logFcent, d= 0.14, Pr=k0[M]

k.

This can then be used to compute the desired result

k= k

Pr1 +Pr

F.

2.1.3 Analysis of reaction mechanisms

Detailed reaction mechanisms for different hydrocarbons may consist of several thou-

sand elementary reactions. Depending on the system of interest, many of these reac-

tions can be neglected. Thus, analysis methods, which eliminate negligible reactions,

are of particular interest:

Sensitivity analysis - identifies the rate-limiting steps


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Reaction flow analysis- determines the characteristic reaction paths.

The information obtained by these methods can be used to reduce the reactionmechanism by eliminating the unimportant reactions.

Sensitivity analysis

The rate laws for a reaction mechanism consisting ofR reactions between Sspecies

can be written as a system of first order ordinary differential equations [67],

dcidt

=Fi(c1,...,cS; k1, ..., kR) i= 1, 2,...,S (2.24)

ci(t= t0) =c0i .

The time t is the independent variable, the concentrations ci of species i are the

dependent variables, and kr are the parameters of the system; c0i denote the initial

conditions at t0.

The solution of the differential equation system (2.24) depends on the initial condi-

tions as well as on the parameters of the system. If one of the initial parameters ischanged, i.e., one of the rate coefficients of the elementary reactions, then the solu-

tion, i.e., the values of the concentrations at time t, will be influenced. For many of

the elementary reactions, a change in its rate coefficients has nearly no effect on the

time-dependent solution (this shows that quasi-steady state or partial equilibrium

are active). If this reaction has to be included in the mechanism, there is no need

of a highly accurate rate coefficient. On the other hand, for a few of the elemen-

tary reactions, changes in its rate coefficients have large effects on the outcome of

the system. Accordingly, accurately obtained rate coefficients are necessary. These

several important reaction steps are called rate-determining or rate-limiting steps.

The dependence of the solution ci on the parameters kr is called sensitivity. The

absolute sensitivity is defined as,

Ei,r = cikr

, (2.25)

and the relative sensitivity as,

Ereli,r =kr

ci

ci

kr

= lnci

lnkr

. (2.26)


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Reaction flow analysis

The reaction flow analysis (RFA) shows the percentage of the contribution of re-action r (r= 1,...,R) to the formation (or consumption) of the chemical species s

(s= 1,...,S). Thus, a reaction flow diagram can be built, showing the main reaction

paths for the formation (or consumption) of the species of interest. There are two

different types of analysis, the integral and the local reaction flow analysis. The

integral reaction flow analysis considers the overall formation or consumption dur-

ing the combustion process. The results for homogeneous time-dependent systems

are, e.g., integrated over the whole reaction time. The local reaction flow analy-

sis considers the formation and consumption of species locally. For a homogeneous

time-dependent system the result is calculated with respect to the specific times

[67].


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19

Chapter 3

SOOT FORMATION

Due to the incomplete combustion various undesired products like NOX, hydrocar-

bons includingPAH, andsootare formed. The reason for that are the unfavourable

combustion conditions of time, temperature, and turbulence. Because, the present

work is concentrated on the modeling of soot precursors and soot particle formation,

the important stages of these processes will be discussed in the following chapter.

To give an answer to the question, how the gas-phase precursors and soot parti-cles are formed in combustion, every stage starting from the very beginning of the

combustion processes has to be studied until the entire mechanism is completed.

A characteristic time scale of the soot particle formation is tens of milliseconds in

flames and several milliseconds behind shock waves. Examined under an electron

microscope, soot appears as necklace-like agglomerates composed of a selection of

small, basic particles with nearly spherical structure [92, 93]. Individual Diesel soot

particulates vary in shape from clusters of spherules to chains of spherules, where a

soot cluster may contain as many as 4000 spherules. The size of spherules varies in

diameter from 10 to 80 nm, but mostly lies between 15 and 30 nm. The spherules

are called primary soot particlesand the cluster- or chain-like soot aggregates are

defined as secondary particles, composed of tends to hundreds of primary spherical

particles [25]. The transmission electron microscopy studies show that the primary

soot particles have a layered structure and consist of numerous concentric crys-

tallites [94]. The X-ray diffraction analysis indicates that the carbon atoms of a

primary soot particle are packed into hexagonal face-centered arrays commonly de-

scribed as platelets. These platelets are arranged in layers to form crystallites, and

there are typically 2-5 platelets per crystallite. The mean layer spacing is 3.55 nm,only slightly larger than that of graphite [95]. The thickness of crystallites is about


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3. SOOT FORMATION 20

1.2 nm [95], and there are the order of 103 crystallites per primary soot particle.

The crystallites are arranged in a layered structure, parallel to the particle surface.

Dislocation of five- and seven-member rings produce surface wrinkling. The lay-

ered structure of soot particles is also characteristic of pyrolytic graphite, which is

though to be responsible for its unusually high resistance to oxidation. Analysed

under high-resolution transmission electron microscopy, two distinct parts of a pri-

mary Diesel soot particle can be identified, an outer shell and a inner core [96]. The

platelet model mentioned above applies to the outer shell. However, the inner core

contains fine particles with a spherical nucleus surrounded by carbon networks with

a bending structure. This indicates that the outer shell, composed of graphitic crys-

tallites, is of a rigid structure, while the inner core is chemically less stable due to

the thermodynamic instability of its structure. Heat treatment can alter the internalmicrostructure of the particles [25]. Particles produced in situ are quite different

from those formed in exhaust gases [97]. Soot contains at least 10 % by mole or

atomic fraction of hydrogen. The considerable hydrogen content corresponds to an

empirical composition formula of C8H for soot [92]. The H/C ratio is around 1 for

the young soot particles.

Soot formation is a complex process, which involves many chemical and physical

steps. A detailed kinetic model of soot formation usually contains two general

parts, gas-phase chemistry, initiating the soot precursors, and particulate-phase

model, which is the less explored area in soot formation theory. Several differ-

ent types of species have been defined as the key gaseous precursors to soot, poly-

acetylens or polyynes, ionic species and polycyclic aromatic hydrocarbons (see Chap-

ter 1.1). Recent studies stated that the PAH are the most probable soot precursors

[44, 22, 98, 99, 20, 3, 52, 100, 24, 4, 5]. Several authors suggested that particle

inception occurs through formation of aromatic-aliphatic-linked hydrocarbons [101]

or PAH with five membered rings [5], which later graphitise forming more compact

structures. The homogeneous inception of large molecular precursors is a still incom-

pletely studied area. The soot particle size increases in reactions of surface growth

by the active sizes on the particle surface. Coagulation forms larger particles, where

during agglomeration the primary particles stick to each other, forming chain-like

aggregates.


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The same interpretation was given by Miller et al. [38, 110], and stands till the

present day.

Frenklach et al. [13, 42, 43, 111, 112, 113, 114, 115] stated that cyclisation occurs

primarily through the reaction

n-C4H3+ C2H2C6H5 (3.4)

where C6H5 is phenyl. Reaction (3.4) was suggested as a key step in the forma-

tion of the first aromatic ring in a detailed kinetic scheme used for simulation of

pyrolysis of acetylene behind shock waves [13, 42, 43, 111]. The authors confirmed

also the importance of Reaction (3.1) at lower temperatures. Miller and Melius

[116, 102, 117, 113, 38], stated that the Reactions (3.1) and (3.4) occur less prob-ably, because these species should be rapidly transformed to their corresponding

resonantly stabilised isomers, iso-C4H3 and iso-C4H5. Instead, they emphasised on

the importance of resonantly stabilised free radicals (RSFRs), such as propargyl

(C3H3), in forming aromatics and PAH in flames. They proposed an odd-carbon-

atom pathway via the recombination reaction of two propargyl radicals,

C3H3+ C3H3 C6H6 (or C6H5+ H). (3.5)

The propargyl radical is an exceptionally stable radical and for a long time was

assumed to be the species with the main role in aromatics formation [118, 67].

Miller et al. [119] showed through quantum chemical calculations that the chemical

activation of the educt might be sufficient to overcome the enormous potential energy

barriers to its cyclisation. They explained the stability of the RSFRs as reduced

reactivity, especially with respect to O2. The RSFRs generally form weaker bonds

than do ordinary free radicals, particularly with stable molecules (O2) [120, 38]. The

second factor that makes RSFRs less reactive with O2 is that there is a potential

energy barrier in the entrance channel for the addition of O2 to a RSFR, whereas

the corresponding potentials for the O2 adding to ordinary free radicals are muchlower.

Miller [38] pointed out on the work of Moriarty et al. [121] and Moskaleva et al.

[122] which proposed the reaction

C3H3+ C2H2 c-C5H5 (3.6)

as an important cyclisation step. Once formed, the cyclopentadienyl radical

(c-C5H5) reacts rapidly to form benzene [123, 104, 124]. Melius et al. [104] suggested

a mechanism of benzene formation through fulvene (C5H4CH2),

c-C5H5+ CH3 ...C5H4CH2 (3.7)


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C5H4CH2+ H C6H6+ H. (3.8)

However, Miller [38] stated that Reaction (3.6) is typical of the class of reactions in

which a collisionally stabilised radical is formed as a product fromradical + moleculereactants. It is observed that such reactions shift their equilibria in the 1400 K-1700

K temperature range to favour the reactants, particularly if the radical reactant is

resonantly stabilised.

Flame calculations [57] showed that Reaction (3.6) is actually a source of propargyl,

rather than a source of cyclopentadienyl. The c-C5H5 is mainly formed from the

oxidative mechanism, discussed in Section 3.1.3 of the same Chapter (Reactions 3.26

and 3.27), and Reaction (3.6) goes in the reverse direction for temperatures above

1500 K. Miller [38] concluded that, at lower temperatures, the potential energybarrier existing in the inlet channel of such radical-molecule reactions makes them

too slow to be effective. Such equilibrium shifts could have important consequences

for certain steps involved in the growth of PAH, mostly the process of C2H2addition

in the periodic sequence of HACA.

Other efficient odd-carbon-atom cyclisation reactions have been suggested in [104,

123, 125, 124, 105]:

c-C5H5+ CH3 C6H6+ H + H (3.9)

c-C5H5+ c-C5H5 naphthalene + H + H. (3.10)

Pope and Miller [126] described the reactions

i-C5H3+ CH3C6H6 (3.11)

C5H4CH2 (3.12)

C6H5+ H (3.13)

which could be at least partially responsible for benzene formation.

Marinov et al. [127, 128, 129], investigated different flames and suggested that

reactions involving 2 RSFRs as reactants are primarily responsible for the formation

of the first aromatics containing one or two rings. Particularly prominent is the

Reaction (3.5) and reactions involving radical-substituted propargyls (RCCCH2).

The R can be both a small aliphatic or aromatic radical [111, 130].

The reaction between allyl and propargyl discussed in [128, 126],

C3H3+ C3H5 C5H4CH2+ H + H, (3.14)


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leads to C5H4CH2 formation, which is found to play an important role for the PAH

formation in many flames. Reaction (3.14) is actually a short version of a two-step

process, the second of which is a dissociation producing fulvene and a hydrogen

atom. The fulvene produced by Reaction (3.14) can be converted to benzene by

H-atom-assisted isomerisation [104] as described in the Reactions (3.7 and 3.8).

Kazakov et al. [131] showed that the formation of the first aromatic ring via reac-

tions of C6HX species as well as the ring-ring reactions play a significant role with

increasing the pressure. Such reactions were considered in many kinetic mechanisms

[42, 43, 46, 23, 24].

The formation of single-aromatic-ring compounds is a very common area of investi-

gations, but it may not be the rate-limiting step [132, 105]. Frenklach [105] suggested

that the growth of higher PAH can be initiated by the direct formation of multi-

ring PAH, bypassing the formation of the benzene ring, like, e.g., Reaction (3.10).

Such alternative proposal includes also formation of aromatics from condensation of

polyacetylenes C2nH2 [25], combination of C4HXspecies [19], as well as combination

of larger radicals [13, 42].

At present, the most important reactions in forming the first and second

rings in flames of aliphatic fuels appear to be C3H3+ C3H3, C3H3+ C3H5, andc-C5H5+ c-C5H5 [133, 128, 134]. However, except for propargyl recombination, not

enough theoretical or experimental work has been done on these reactions. The

kinetics of reactions involving RSFRs, cyclic species, and unsaturated, conjugated

molecules in general is under investigation [135, 136, 137, 138, 139, 140].

3.1.2 Growth of aromatics by HACA

Stein [141] and Stein and Fahr [142] calculated equilibrium as a function of atomic

structure, temperature, and partial pressures of H2 and C2H2. They found that at

high temperature, the most stable species thermodynamically, as the carbon number

is increased, lie in a sequence of peri-fused polybenzenoid molecules with occasional

five-membered rings around the edges. From these results the authors suggested

that such molecules and their radicals are the primary intermediates in the soot

formation process.

The most popular mechanism of PAH growth is the HACA pathway developed byFrenklach and Wang [21, 1]. The model proposes a repetitive reaction sequence of

two principal steps, 1. Abstraction of an H atom from the reacting hydrocarbon by


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a gaseous H atom,

Ai+ H Ai-+ H2 (3.15)

2. Addition of a gaseous C2H2 molecule to the radical formed,

Ai-+ C2H2 products. (3.16)

The nomenclature of the aromatics is published in [42, 118], where Aiis an aromatic

molecule with i peri-condensed rings, and Ai- is its radical. The key feature of the

first step of HACA is its reversibility. The reverse steps can be the reverse direction

of the H abstraction itself,

Ai-+ H2 Ai+ H (3.17)

or the reaction of combination with a gaseous H,

Ai-+ H Ai. (3.18)

Frenklach [105] stated that the contribution of Reaction (3.18) as compared to the

Reverse (3.17) increases with pressure and molecular size (e.g., the rate coefficient

of Reaction (3.18) approaches its high-pressure limit). Moreover, the reversibility

of the acetylene addition step (Reaction 3.16) determines whether this step will

contribute to molecular growth. For a simple addition, due to the entropy loss,

the reaction is highly reversible, and often runs in the reverse direction. Forming ahydrogen atom as a product [105],

Ai-+ C2H2 products + H, (3.19)

recovers some of the entropy but, in many cases, the reaction is still highly reversible,

e.g.,

Ai-+ C2H2 AiC2H + H. (3.20)

Only when, in addition to the recovery in entropy, the decrease in energy is high

enough, the reaction becomes more irreversible, and in the formation of particularlystable aromatics, calledislands of stability[13] orstabilomers[142], the reaction be-

comes practically irreversible. This coupling between the thermodynamic resistance

of the reaction reversibility and the kinetic driving force is the defining feature of

the HACA model explained in detail in [112].

3.1.3 Growth of aromatics by other species

Glassman [143] suggested that hydrocarbons with conjugated structures and their

derivatives are critical intermediates to soot nucleation. Frencklach [111] showed


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that in the pyrolysis of benzene the growth of the aromatics is initiated by the

formation of biphenyl,

phenyl + benzene biphenyl + H, (3.21)

but the following growth proceeds via acetylene addition,

biphenyl+C2H2A3+ H. (3.22)

The same mechanism appears to play an important role for the PAH growth at

different conditions and fuels [111, 118, 131, 105].

The reactions between resonantly stabilised free radicals, e.g., the recombination of

cyclopentadienyl radicals, became one of the most prominent for the formation oftwo-ring aromatics, specifically naphthalene [130, 128, 104, 144, 145, 38],

c-C5H5+ c-C5H5 C5H5C5H4+ H (3.23)

C5H5C5H4naphthalene + H. (3.24)

The reaction of benzyl with propargyl leads directly to the formation of two rings

in the system,

C6H5CH2+ C3H3naphthalene + H + H. (3.25)

Reactions (3.23-3.25), as well as others mentioned above, are not likely to occur as

elementary steps. This problem is discussed in more detail in [146]. Nevertheless,

the cyclopentadienyl needed for the formation of naphthalene through Reactions

(3.23 and 3.24) is found as a by-product of oxidation in most flames. It is generally

formed by

C6H5+ O2C6H5O + O (3.26)

C6H

5O c-C

5H

5+ CO, (3.27)

where C6H5O is phenoxy. This is an example that the oxygen may also promote

the formation of higher PAH [128, 105]. Marinov et al. [128] described a similar

sequence of steps for modeling the formation of phenanthrene through the reaction

of indenyl with cyclopentadienyl,

naphtoxy = indenyl + CO (3.28)

indenyl + c-C5H5A3+ H + H. (3.29)

The advantage of such mechanisms of PAH growth is that they deflect the thermo-dynamic barriers that exist in forming two- and three-ring aromatics through the

HACA mechanism.


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Frenklach et al. [147] suggested reaction pathways for aromatic ring growth through

the so called migration reactions. The authors studied theoretically different possi-

ble channels, such as enhanced formation of five-member aromatic rings, enhanced

formation of six-member aromatic rings, interconversion of five- and six-member

rings, and migration of the cyclopenta ring along zigzag aromatic edges [105]. All

of these pathways have one critical feature in common: The reaction pathway is in-

duced or assisted by hydrogen atom migration. Moriarty et al.[148] investigated the

kinetics and thermodynamics of several migration reactions by quantum-chemical

calculations. They observed that the derived reaction rates are sufficiently fast for

these reactions to play a role in high-temperature aromatic chemistry. In [149], the

authors studied the five-member ring migration along a graphene edge. They con-

cluded that an important implication of the migration phenomenon is that, whilefive-member rings are constantly being formed on the growing edge, they do not

accumulate, but are rather converted to six-member rings.

3.1.4 Oxidation of aromatics

A process parallel to the aromatics growth is their oxidation. Haynes and Wagner

[25] and Neoh et al. [150] considered the hydroxyl radical as the primary oxidisingagent of soot particles.

Frenklach [105] declared that the primary mechanism is the oxidation of aromatic

radicals by O2, and the oxidation by OH is rather unimportant, at least in laminar

premixed flames [105]. The author further stated that the largest effect in the oxida-

tion of aromatics occurs at the very beginning of their growth, at the phenyl stage.

This is due to the rapidly decreasing concentration of O2 in fuel-rich environments

sustaining aromatics growth. Experimental observations showed that soot inception

appears in the time or space of the main combustion zone, in an environment richin H atoms and poor in O2 molecules.

However, the mechanism of PAH and soot oxidation is still not completely under-

stood. Oxidation of aromatics removes carbon mass from further growth, but more

important is the removal of mass at earlier stages, those preceding the PAH for-

mation. Numerical simulations [42, 43] identify oxidation of C2H3 as the key point

of branching between carbon growth and carbon oxidation. The authors concluded

that the effect of oxidation at this small-molecule level is two-sided. It diverts the

carbon mass from further growth. On the other hand, added in relatively smallquantities in high-temperature pyrolytic environment, molecular oxygen promotes


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formation of soot by building various radicals, and specifically H atoms.This phe-

nomenon is observed in different experimental studies in shock tubes [13], compu-

tational analysis [42], and in diffusion flames [151].

3.2 Particulate phase

In spite of the great effort in understanding the mechanism of hydrocarbons and soot

formation, there are still numerous uncertainties which need to be studied experi-

mentally and theoretically. The formation and evolution of soot particles includes

processes likesoot particle inception,surface growthandoxidation, coagulation, andagglomerationwhich are briefly described in the following sections.

3.2.1 Soot particle inception

Thesoot particle inceptionis a homogeneous process occurring in the gas-phase en-

vironment. According to different investigations, it takes place at molecular masses

between 500 a.m.u. [152], 300-700 a.m.u. [153], 1600 a.m.u. [154] and 2000 a.m.u.

[101], discussed in [67]. Above this values the PAH can be interpreted as solid

particles rather than molecules. These first soot particles are roughly spherical in

shape and have a C/H ratio of about 2. Upon aging, they can coalesce into larger

spherical particles, undergo surface reactions, dehydrogenation,oxidationand coag-

ulation. The soot that is emitted from combustion devices typically has a C/H ratio

of approximately 10 and consists of some sort of agglomerates of spherical particles

that have an underlying graphitic-like structure [13].

Two general mechanisms have been proposed in the literature in which homogeneousparticle inception is considered to be a process of physical condensation or a pro-

cess of chemical (reactive) condensation. The physical condensation suggests that

when the supersaturation of macro-molecular precursors generated by gas-phase re-

actions become sufficiently high, the partial pressure of the precursors forces the

macromolecules to condense physically into liquid-phase soot [155, 156]. The ho-

mogeneous condensation can be approximated by classical nucleation theory, which

gives the number of critical nuclei per unit volume [157, 156]. Thechemical (reac-

tive) condensationconsiders the process of continuous reactions of macro-molecular

precursors as the driving mechanism of homogeneous soot particle inception.


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Frenklach and Wang [158] studied the reactive coagulation of stable PAH. They

treated the coagulation process, starting form pyrene, in the free molecular regime

and considered the coagulation reactions as irreversible. A size-independent en-

hancement factor of 2.2 was used in their calculation of collision frequencies. Once

the PAH monomers have reached a certain size they begin to stick to each other dur-

ing collisions and thus form PAH dimers. These dimers collide with PAH molecules

forming trimers, or with other dimers forming tetramers, and so on. Consequently,

these PAH clusters evolve into solid particles as they increase in size.

Howard [159] and DAnna et al. [160] emphasised on the role of PAH activation

by hydrogen abstraction. The active sites formed on the PAH provide a chemical

basis for reactive coagulation of polycyclic aromatic compounds with each other andwith small radicals. DAnna et. al [160] proposed a model in which the chemical

specificity of the reactive coagulation process was studied. They considered the

radical-molecule reactions between the gas-phase PAH having conjugated double

bonds. In these reactions resonantly stabilised radical intermediates are formed

that continue the addition sequence, forming higher mass species.

The polyyne hypothesis assumes that every radical capable of forming polyyne com-

plexes becomes a center of polymerisation. Following a polyyne molecule and radical

or two polyyne molecules react to form the polyyne complex [2].

Experimentally, the particle inception is characterised by the induction period. In

shock-tube experiments, the soot volume fraction, calculated from the extinction of

light, can be plotted versus measurement time. After the clearly visible passage of

incident and reflected shock, soot growth is delayed by a characteristic induction

time, which is specific for the different hydrocarbons. During that period hydrocar-

bons are transformed into soot particles.

3.2.2 Soot particle growth

The greater part of soot (> 95 %) is formed by surface growth rather than soot

inception [67]. It is assumed that particle growth is similar to the formation of

PAH, and acetylene and PAH are accepted to be the two main potential agents

responsible for soot surface growth. The problem is that surface growth is not a gas-

phase reaction of small molecules, but a heterogeneous process, where adsorption

and desorption processes at the surface have to be considered as well.

Because of the lack of precise data, phenomenological approaches are used to sim-


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ulate this process. Mass growth of soot in premixed flames typically rises to an

asymptotic value even though C2H2 is present and temperatures are high in the

region of no mass growth. Wagner described it through a first order differential

equation [161, 67],

dfVdt

=ksg(fV fV), (3.30)

where ksg is a fitted surface growth rate coefficient and fV is a fitted parameter

which represents the ultimate volume fraction of soot formed. The temperature

effect for both parameters (ksg and fV) have been empirically determined [67].

Harris and Weiner [162] studied several premixed acetylene-air flat flames and pre-

mixed ethylene/air flames. The authors observed that only C2H2 satisfies the re-

quirements for a soot growth reactant and proposed a simple model in which soot

mass growth rate is proportional to soot surface area and acetylene concentration

[163, 164, 67],

dfVdt

=kC2H2 pC2H2 S, (3.31)

where S is the soot surface area density (in, e.g., m2/m3) and pC2H2 is the partial

pressure of the gas-phase acetylene. PAHs were not measured because they were

believed to have insufficient concentrations and could not be counted as possible sootgrowth reactants. One of the most important result showed by Harris and Weiner

is that the specific surface growth rate is only weakly dependent on stoichiometry,

compared with the total growth rate. The authors stated that the much higher

total growth rate of soot in richer flames was almost entirely due to the increased

surface area available, while the concentration of growth species was similar in all

of the flames, which is confirmed by Xu and Faeths experimental data obtained at

similar condition [165]. Harris and Weiner extended their conclusion and claimed

that there was no depletion of growth species by surface growth. They considered

acetylene as the dominant growth species because its concentration was high enough

to account for the mass increase provided by surface growth. They found that the

PAH concentration changed sharply with stoichiometry, and the PAH concentrations

were at about 100 times higher in benzene flames than in flames of aliphatic fuels

[12], but the soot growth rates in both flames were similar [166].

Behish et al. [167] did not agree with the above conclusions. They believed that

most (95% or more) of the soot growth occurs by PAH addition. The authors

repeated the particular flames investigated by [162] and found that previous sootconcentration profiles for the C/O = 0.79 flame was three times higher than their

experimental results, which was in excellent agreement with interpolated values


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from optical measurement of Feitelberg [168] in similar rich ethylene flames. They

identified 26 PAHs, which accounted for 49% of the total PAH mass using high

performance liquid chromatography (HPLC). They assumed that PAH growth was

the net effect of acetylene addition to PAH and PAH addition to soot, while soot

growth resulted from addition of acetylene and PAH, ignoring oxidation in view of

the fuel-rich post-flame conditions.

Kazakov and Frenklach [169] numerically analysed the contribution of acetylene and

PAH to soot particle surface growth and concluded that a model with acetylene as

surface growth species is not contradicted by the experiments of Benish et al. [167].

They stated that the difference between the results obtained by [169] and [167] comes

from the different assumptions of the collision efficiencies between acetylene-PAHand acetylene-soot.

3.2.3 Soot particle coagulation

The coagulation is usually expressed as a process ofstickingof two particles, which

are glued together by a common outer shell generated by deposition similarly to

surface growth. Coagulation takes place only for relatively small particles, which

are characterised by high rates of growth (up to a diameter of 10 nm in low pressure

premixed systems [170, 171, 67]. The rate of a sticking process can be calculated by

solving Smoluchowski equation[172], following the assumptions:

1. The soot particles are small in comparison to the gas mean free path,

2. each collision of two particles results in coagulation,

3. all particles are spherical.

dnkdt

=1

2

i+j=k

Nij

i=1

Nik. (3.32)

Here, nk represents the number density of new molecules in the size class k, with

massmk(the molecule of starting class in the molecular size spectrum, e. g., a PAH

monomer), resulting from the collision between two other molecules of different

classes i and j. Nij denotes the rate of collision between molecules of classes i and

j, defined by

Nij =(mi, mj,...)ninj. (3.33)


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The collision of two molecules leads to the formation of a new molecule k, with

the summed mass of the two contributing molecules mk = mi+ mj . The rate of

formation of the new molecules k is1

2

i+j=k

Nij =1

2

i+j=k

(mi, mj)ninj. (3.34)

The molecule kcan lose its identity due to collision with other molecules at the

rate

i=1

Nik =nk

i=1

(mi, mk)ni. (3.35)

(mi, mj) is a size-dependent collision frequency factor, which for free-molecular

coagulation is given as,

(mi, mj ) =

6kBT

i,j(ri+ rj )

2 (3.36)

= 2.2

3

4

1/66kBT

1

mi+

1

mj

m1/3i + m

1/3j

2,

where i,j =mimj /(mi+ mj) is the reduced mass, ri is the radius of the moleculesin the classes i and is the density of these molecules.

Graham [173, 67], studied soot coagulation in shock-heated hydrocarbon/argon mix-

tures and showed a coagulation rate, expressed in terms of the rate of decrease of

the particle number density [n],

dn

dt =

5

6ktheoryf

1/6V [n]

11/6,with ktheory= 5

12

3

4

1/66kBTsoot

1/2 G . (3.37)

Here, fV is the soot volume fraction, kB is the Boltzmann constant, is the con-

densed particle density,is a factor related to the polydisperse nature of the system,andG is a factor accounting for the increase in collision cross-section over the hard-

sphere value due to electronic and dispersion forces. Graham suggested that G= 2

for spherical particles and for self-preserving size distribution = 6.55.

3.2.4 Soot particle oxidation

The process of soot particle oxidation is parallel to the surface growth. In fact,oxidation is also a surface reaction, which in principal should be treated as catalytic

combustion [67]. Potential soot oxidants are O, O2, OH, and CO2.


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Frenklach stated that the major oxidation process occurs at the very beginning of

soot particle growth, which is the soot particle nucleation period, where a rapidly

decreasing concentration of O2 in fuel-rich environments is observed [105].

According to Neoh et al. [174] and Lucht et al. [175], the hydroxyl radical is the

most abundant oxidising species under fuel-rich condition. The authors stated that

OH could suppress soot formation via oxidative destruction of precursors, and OH

concentration might be an important factor in soot precursor kinetics. Lucht et al.

[175] concluded that OH is the limiting oxidative reactant under fuel-rich condition

as the soot decreases with an increase in OH concentration.

Experimental studies performed by Liu et al. [176] showed that CO2

has chemical

dilution, and thermal effects on soot formation reduction. They suggested that the

chemical mechanism of CO2 addition might be to promote the concentrations of

oxygen atom and hydroxyl that in return increase the oxidation of soot precursors

in soot formation regions. Vandooren et al. [177] studied experimentally the CO2

addition to rich but non-sooting CH4/O2/Ar premixed flames and showed that the

reaction CO2+ H = CO + OH is responsible for the promoted hydroxyl concentra-

tion. They also observed that the concentration of acetylene decreases as a result of

CO2 addition.

However, due to the lack of data on the mechanism of soot particle oxidation, a

one-step treatment is often used, assuming the rate law for the CO formed given as

[67]

d[CO]

dt =iZi as; i= O, OH, O2, (3.38)

where i =reaction probabilitywhen molecule i hits the soot surface, Zi =collision

numberof moleculei per unit time and area, and as = soot surfaceper unit volume.

3.2.5 Soot agglomeration

Soot agglomeration takes place in the late phase of soot formation when, due to lack

of surface growth, coagulation is no longer possible [67]. As a result, open structured

aggregates are formed, containing from 10 to 100 primary particles (spherules) and

characterised by a log-normal size distribution [178, 67]. A relationship between the

numberNof primary particles and the maximum length L of the aggregates can be

derived as

N=kf (L/3dp)Df , (3.39)


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where kfis a constant fractal prefactor, dp the primary particle diameter, and Df a

fractal dimension around 1.8 [179, 67]. In the current work, agglomeration was not

considered in the models.


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35

Chapter 4

DISCRETE GALERKIN

METHOD

A detailed chemical mechanism of soot formation has to describe the reaction ki-

netics of both the gas- and the particulate-phase. Usually, the gas-phase chem-

istry model contains large number of elementary reactions between hundreds of

species. The formation and evolution of the macromolecular species (the hetero-

geneous, particulate-phase) needs to be treated simultaneously with the gas-phase

chemistry. The temporal change of the gas-phase species concentration and the dis-

tribution of the macromolecu

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