Soot Formation Modeling during Hydrocarbon Pyrolysis and - IWR

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 of

Doctor of Natural Sciences

presented by

M.Sc. Iliyana Ivanova Naydenova

born in Sofia, Bulgaria

Ruprecht-Karls-Universitat Heidelberg

Interdisziplinares Zentrum fur Wissenschaftliches Rechnen

2007



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 of

Doctor 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



behind Shock Waves

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

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

Acknowledgement

It is my great pleasure to acknowledge all the people who helped me directly or

indirectly 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 endless

love, 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!

Abstract

In the present work, soot formation was modeled in conditions typical of shock tube

experiments. 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 soot

formation 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 few

variables 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.

Kurzfassung

Die vorliegende Arbeit beschreibt die Modellierung der Rußbildung unter Bedin-

gungen, die typisch fur Stoßwellenexperimente 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 Rußvorlaufers 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 Rußvorlaufern berucksichtigt: Polyine und polyzyk-

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

suchungen der Rußbildung in Flammen und Stoßwellenrohren [3, 4, 5] bestatigen,

dass die Rußteilchen hauptsachlich aus PAK gebildet werden. Die Experimente

deuten daraufhin, dass die Reaktion aliphatischer Spezies mit Rußoberflache 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 Rußvorlaufer betrachtet werden. Die aliphatische Spezies sind hier nur an

Oberflachenwachstumsreaktionen beteiligt. Beide Modelle wurden der wichtigsten

Gasphasenspezies Konzentrationsprofile validiert, die in Stoßwellenexperimenten

gemessen worden sind. Ferner wurden die Modelle fur Simulation der Rußbil-

dung wahrend der Pyrolyse und der Oxidation verschiedener Kohlenwasserstoffe und

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

dingungen angewandt. Die Ergebnisse der Berechnungen wurden mit Messwerten

(Zundverzugszeit, Bildungsgeschwindigkeit des Rußteilchen, Rußteilchenkonzentra-

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 verwendet

werden. 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 Rußbildung wurde mittels einiger globaler Schritte beschrieben: Keim-

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

gen fur die zeitliche Anderung der Rußkonzentration und des Rußvolumenbruchs

gelost werden. Die Simulationsergebnisse wurden mit der experimentell gemesse-

nen Rußcharakteristika der Oxidation unterschiedlicher Kohlenwasserstoffe in einem

Stoßwellenrohr verglichen.

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

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

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

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 toluene

by the scientists allover the world. Thus, the chemistry of rich flames, particularly

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)

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 young

soot particles and mature 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

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. D’Alessio 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 of H-abstraction which activates the aromatic molecule, and acetylene ad-

dition which 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 of

molecular 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

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 with

respect 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 kinetic Monte-Carlo and 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-

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.

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 macromolecular

reaction 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 the

formation, 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.

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).

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 diluting

the 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 reflected

shock waves [28, 32, 35, 36, 14, 78, 73, 79, 80, 81, 82, 83, 5]. The shock tube as

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 included

in 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 in T . 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 enthalpy

4H0, the enthropy 4S0 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 + a2 · T + a3 · T 2 + a4 · T 3 + a5 · T 4. (2.1)

Here a1, ..., a5 are constants, and R is the gas constant. In addition, two integration

constants are needed to compute enthalpies and entropies, where a∗6 ·R = H0

298 and

a∗7 ·R = S0

298,

H0

T = a∗6 ·R +

∫ T

T ′=298K

C0

pdT′ and S

0

T = a∗7 ·R +

∫ T

T ′=298K

C0

p

T ′dT ′. (2.2)


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

Eq. (2.1), with an a6 different from a∗6,

H0

T (T )

R= a6 + a1 · T +

a2

2· T 2 +

a3

3· T 3 +

a4

4· T 4 +

a5

5· T 5. (2.3)

The coefficient a6 can be defined setting T = 298 K and demanding that H0(298 K)

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 a∗7,

S0

T (T )

R= a7 + a1 · lnT + a2 · T +

a3

2· T 2 +

a4

3· T 3 +

a5

4· T 4. (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

T at 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.


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

S∑s=1

ν(e)rs As

kr−→S∑

s=1

ν(p)rs As, (2.5)

then the rate law for the formation of species i in reaction r is given by the expres-

sion,

(∂ci∂t

)

chem,r

= kr

(ν

(p)ri − ν

(e)ri

) S∏s=1

cν(e)rs

s . (2.6)

Here ν(e)rs and ν

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

ucts, and cs, the concentration of the S different 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 and

mixture 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)


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=kf

kr

. (2.10)

The expression on the left hand side corresponds to the equilibrium constant(Kc = kf

kr

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

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

Kp =pc

C · pdD

paA · pb

B

. (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(−4R G

0/RT

), (2.12)

where the standard Gibs free energy 4RG0

is calculated by the reaction enthalpy

4H0and entropy 4S0

,

4G0= 4H0 − T · 4S0

. (2.13)

The equilibrium constants Kc and Kp are related by

Kc =kp

(RT )4v, (2.14)

where 4v is the difference between the stoichiometric coefficients of the reaction

(4v = (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 experimentally

for many reactions that a plot of lnk against 1/T gives a straight line with a slope

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


and the interception at 1/T = 0 is equal to natural logarithm of the collision co-

efficient, lnA. Here, A is defined as the pre-exponential factor or the frequency

factor, the parameter Ea is the activation energy. These two parameters are called

Arrhenius parameters and are independent on the temperature. According to this

investigations the rate coefficient is usually expressed by a simple formula called

Arrhenius law,

lnk = lnA− Ea

RT, (2.15)

where after applying antilogarithm the following expression is derived:

k = A · exp

(− Ea

RT

). (2.16)

More recently, accurate measurements showed a temperature dependence of the

pre-exponential factor A, which is usually small in comparison to the exponential

dependence,

k = AT b · exp

(− Ea

RT

). (2.17)

The activation energy Ea corresponds to the energy barrier to be overcome during

the reaction. Its maximum value corresponds to the bond energies in the molecule

(e.g. for dissociation reaction, the Ea is approximately equal to the bond energy of

the bond, which is split), but it can be much smaller or even zero, if new bonds are

formed simultaneously while the old bonds are breaking. If b in Eq. (2.15) is known,

the Ea can be determined from the slope of the plot of ln(k/T b) versus 1/T .

Pressure dependence of rate coefficients

The rate coefficients of dissociation (unimolecular) and recombination (trimolecu-

lar) reactions are found to be pressure-dependent. This is an indication that these

reactions are not elementary; in fact they are a sequence of reactions. In the sim-

plest case, the pressure dependence can be introduced using the Lindemann model

[88, 67]. According to this model, a unimolecular decomposition is only possible,

if the energy in the model is sufficient to brake the bond. Therefore, it is nec-

essary that before the decomposition energy is added to the molecule by collision

with other molecules (for excitation of the molecule), noticed usually as M. After

that, the excited molecule may decompose into products, or it can deactivated by a

collision,

A + Mka−→ A∗ + M (activation)


A∗ + Mk−a−−→ A + M (deactivation) (2.18)

A∗ ku−→ P (roducts) (unimolecular reaction)

The rate equations for this case are given by

d [P]

dt= ku [A∗] and

d [A∗]dt

= ka [A] [M]− k−a [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] + ku

andd [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 the low-pressure range, the concentration of the collision partners M is very small

and k−a[M] << ku. The rate law of the reaction appears to be of second order,

d [P]

dt= ka · [A] [M] = k0 · [A] [M] (2.21)

with a low pressure rate coefficient usually named k0. Then, the reaction rate is

proportional to the concentrations of species A and the collision partners M, because

the activation of the A species is slow, it is the rate-limiting process at low pressures.

In the high-pressure range, the collision partner M has a higher concentration and,

therefore k−a[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 Theory

of Unimolecular Reactions (Robinson and Holbrook [89], Atkins [87], Golden [90],

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


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 yields fall-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 coefficient

as 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 T ∗∗which

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

n− d · (logPr + c)

]2}−1

with

c = −0.4− 0.67logFcent, n = 0.75− 1.27 logFcent, d = 0.14, Pr =k0 · [M]

k∞.

This can then be used to compute the desired result

k = k∞ ·(

Pr

1 + 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


• Reaction flow analysis - determines the characteristic reaction paths.

The information obtained by these methods can be used to reduce the reaction

mechanism by eliminating the unimportant reactions.

Sensitivity analysis

The rate laws for a reaction mechanism consisting of R reactions between S species

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 is

changed, 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 =∂ci∂kr

, (2.25)

and the relative sensitivity as,

Ereli,r =

kr

ci

∂ci∂kr

=∂lnci∂lnkr

. (2.26)


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].

19

Chapter 3

SOOT FORMATION

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

bons including PAH, and soot are 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 particles and 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

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 internal

microstructure 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.


3.1 Gas phase

Following the PAH hypothesis, a mechanism of soot formation should consist of

several stages. Starting, e. g., with an aliphatic fuel, the fuel molecules are first

broken down into smaller hydrocarbon molecules and free radicals either by pyrolysis

or oxidation reactions. Then the key step occurs, the formation of the first aromatic

ring in the system, usually benzene or phenyl. This first ring appears as the nucleus

for the formation and growth of PAH, following different mechanisms [95, 13, 43, 99].

If the reaction mechanism is composed of all possible elementary reactions in the

system, this mechanism should be valid for all conditions like temperatures, pressures

and mixture compositions. The construction of such mechanism is a very difficult

task and complete mechanisms are rarely available related to specific problems [67].

3.1.1 First aromatic ring formation

Flame experiments [3, 4, 20, 22, 98, 101] show that the formation and growth of

aromatics bridges the main combustion zone gas-phase molecular chemistry and

soot particle formation. The high temperature chemistry of aromatics received great

attention in the last decades. The primary focus is on the formation of the first

aromatic ring from small aliphatics, because this step is suggested to be the rate-

limiting step to higher PAH [13, 43, 102, 1, 103, 104, 46]. Some of the most famous

are the even-carbon-atom pathways as Frenklach showed in his review paper [105],

which involve the addition of acetylene to n-C4H3 and n-C4H5 radicals. Frenklach

referred to the reaction

n-C4H5 + C2H2 → C6H6 + H (3.1)

suggested by Bittner and Howard [106], Weissman and Benson [107], and Cole et

al. [108], as an important cyclisation step, where C6H6 is benzene. The authors

supposed that such reaction should have one or more intermediate steps, whereas,

at least under flame conditions, the reaction predominantly occurs as written. At

about the same time, but independently of the investigators cited above, Callear and

Smith [19, 109] experimentally showed the viability of Reaction (3.1) as a cyclisation

step. They studied the reaction of H with acetylene at low temperatures and found

large quantities of benzene in the products. The authors suggested that the Reaction

(3.1) is part of a mechanism, which includes Reactions (3.2 and 3.3),

H + C2H2 → C2H3 (3.2)

C2H3 + C2H2 → n-C4H5. (3.3)


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 + C2H2 → C6H5 (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 much

lower.

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)


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 from radical + molecule

reactants. 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 energy

barrier 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 C2H2 addition

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 + CH3 → C6H6 (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)


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 C4HX species [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, and

c-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 by

Frenklach 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


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 Ai is 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 a

hydrogen 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 particularly

stable aromatics, called islands of stability [13] or stabilomers [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


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−+C2H2 → A3 + 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 of

two-ring aromatics, specifically naphthalene [130, 128, 104, 144, 145, 38],

c-C5H5 + c-C5H5 → C5H5C5H4 + H (3.23)

C5H5C5H4 → naphthalene + H. (3.24)

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

in the system,

C6H5CH2 + C3H3 → naphthalene + 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 + O2 → C6H5O + O (3.26)

C6H5O → c-C5H5 + 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-C5H5 → A3 + 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.


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, while

five-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 oxidising

agent 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 rich

in 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 small

quantities in high-temperature pyrolytic environment, molecular oxygen promotes


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 like soot particle inception, surface growth and oxidation, coagulation, and

agglomeration which are briefly described in the following sections.

3.2.1 Soot particle inception

The soot particle inception is 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, oxidation and 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 homogeneous

particle 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]. The chemical (reac-

tive) condensation considers the process of continuous reactions of macro-molecular

precursors as the driving mechanism of homogeneous soot particle inception.


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 D’Anna 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 and

with small radicals. D’Anna 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-


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],

dfV

dt= ksg(f

∞V − fV ), (3.30)

where ksg is a fitted surface growth rate coefficient and f∞V is a fitted parameter

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

effect for both parameters (ksg and f∞V ) 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],

dfV

dt= 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 soot

growth 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 Faeth’s 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 soot

concentration profiles for the C/O = 0.79 flame was three times higher than their

experimental results, which was in excellent agreement with interpolated values


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-PAH

and acetylene-soot.

3.2.3 Soot particle coagulation

The coagulation is usually expressed as a process of sticking of 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.

dnk

dt=

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

mass mk (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 = β(mi,mj, ...)ninj. (3.33)


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’ is

1

2

∑

i+j=k

Nij =1

2

∑

i+j=k

β(mi,mj)ninj. (3.34)

The molecule ’k’ can lose its identity due to collision with other molecules at the

rate∞∑i=1

Nik = nk

∞∑i=1

β(mi,mk)ni. (3.35)

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

coagulation is given as,

β(mi,mj) =

√6kBT

µi,j

(ri + rj)2 (3.36)

= 2.2

(3

4πρ

)1/6√

6kBT

ρ

√1

mi

+1

mj

(m

1/3i +m

1/3j

)2

,

where µi,j = mimj/(mi +mj) is the reduced mass, ri is the radius of the molecules

in the classes i and ρ 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

4π

)1/6 (6kBT

ρsoot

)1/2

·G ·α. (3.37)

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

densed particle density, α is a factor related to the polydisperse nature of the system,

and G 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 α = 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.


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= γi · Zi · as; i = O,OH,O2, (3.38)

where γi = reaction probability when molecule i hits the soot surface, Zi = collision

number of molecule i per unit time and area, and as = soot surface per 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

number N of primary particles and the maximum length L of the aggregates can be

derived as

N = kf · (L/3dp)Df , (3.39)


where kf is 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.

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 macromolecular species can be calculated by solving an associated

set of differential equations called a countable system of ordinary differential equa-

tions (CODEs), where particle size has to be treated separately. The problem is

that such systems have very high or even infinite dimensions, and an efficient nu-

merical solution by standard ODE software is not possible. Standard computational

approaches for solving such systems include several techniques: large scale stiff inte-

gration, lumping techniques, statistical moment treatment, and continuous modeling.

Deuflhard and Wulkow [180] suggested the so-called discrete Galerkin method, which

considers a flexible and efficient solution of the kinetics of polymerisation reactions.

In general, two types of polymerisation reactions are considered in the literature,

condensation (step-reaction polymerisation) and addition or chain-reaction poly-

merisation (free-radical polymerisation). Condensation takes place between two

molecules to form one larger molecule, with a possible elimination of a small species,

e.g., water. The free-radical polymerisation involves chain reactions in which the

chain-carrying species may be a radical or an ion. In a short time, many monomers

are added to the growing chain, and the reaction stops when, e.g., two radicals react

4. DISCRETE GALERKIN METHOD 36

to end each other’s activity.

It has been shown that soot formation can be numerically treated by analogy to the

polymerisation reactions [68, 69, 70, 60, 71] with the use of the discrete Galerkin

method. The addition of a monomer M to the polymer P[S] can be described by

P[S] + Mkr−→ P[S + 1], S = 1, ..., N, (4.1)

where N is the maximum chain length (polymer index, degree) considered in the

model. The size of the truncation index N is initially unknown; usually practical

considerations can limit it up to a given value.

In the current model, the soot formation was modeled by analogy to the process

of free-radical polymerisation, where a variety of macromolecular reactions as ini-

tiation, growth, termination, degradation, reverse polymerisation, coagulation and

transformation of the type of the polymer had to be modeled. All macromolecular

processes were treated with the discrete Galerkin method. The method is based on

an error-controlled expansion of the size distribution function of a macromolecule

into orthogonal polynomials of a discrete variable, in particular the polymer degree

or the number of monomers added to the growing macromolecule. It was thoroughly

studied by Sojka and described in [71]. In the present work the previously developed

numerical approach was applied for soot formation simulation during hydrocarbon

pyrolysis and oxidation in homogeneous conditions.

4.1 Theory of the discrete Galerkin method

The Galerkin method is a well known method for converting a differential equation

into a linear algebra problem, or a high-dimensional linear system of equations may

be projected to a lower dimensional system. These small systems are easier to solve,

but their solution is only an approximation to the original problem.

Let us(t) denote the concentration of macromolecules of chain length S at time t.

The sequence u1(t), u2(t), ... can be written as distribution u(t) = (us(t), S =

1, 2, ...). As mentioned above, the kinetics of a macromolecular reaction process can

be represented by a countable system of ordinary differential equations (abbreviated

as CODEs) of the form

u′s(t) = (Au(t))S (4.2)

with a given initial distribution us(0). The key point for the construction of the

discrete Galerkin method [180] is the treatment of the polymer degree (chain length)


S as a discrete variable. Formally, it can be written as

(f, g) :=∞∑

S=1

f (S, ρ) g (S, ρ)ψ (S, ρ) , S = 1, 2, ... (4.3)

where f and g are functions of the discrete variable S = 1, 2, ... and ψ(S, ρ) is a

given (positive) weight function with a time dependent parameter ρ. This product

induces an associated norm,

‖f‖ := (f, f)1/2. (4.4)

and an associated orthogonal basis {lj(S, ρ)}j = 0, 1, 2, ... of polynomials of the

discrete variable S with the following characteristics:

(lj, lk) =∞∑

S=1

lj (S, ρ) lk (S, ρ)ψ (S, ρ) = γkδjk, γk > 0, j, k = 0, 1, 2, ... (4.5)

Here γk depends on the choice of the orthogonal basis, and δk is the Kronecker

symbol.

An unique representation of an unknown distribution P (S, t) of the polymerisation

index S of the polymer P [S] with respect to time is described by Deuflhard and

Wulkow [180, 71]:

P (S, t) := ψ (S, ρ) ·∞∑

k=0

ak (t, ρ) lk (S, ρ) , S = 1, 2, ... (4.6)

Here, ψ(S, ρ) is the weight function, ak(t, ρ) are the time-dependent coefficients with

a free parameter ρ. Because of the polynomial orthogonality for given P (S, t), the

coefficients ak(t, ρ) can be obtained by

aj (t, ρ) =1

γj

〈lj (S, ρ) , P (S, t)〉 , j = 0, 1, ... (4.7)

Truncation of the expansion Eq. (4.6) after n terms leads to the so called Galerkin

approximation

P (n) (S, t, ρ) = ψ (S, ρ) ·n∑

k=0

ak (t, ρ) lk (S, ρ) , S = 1, 2, ... (4.8)

in dependence of the truncation index n. This approximation depends on the pa-

rameter ρ as well as on the choice of the truncation index. It is important to notice

that n must be initially given for every type of polymer. This approximation has the

structure of the method of lines [181], an approach used for the treatment of par-

tial differential equations (PDEs). The so-called space discretisation is performed,


which leads to a system of ODEs of fixed dimension. These differential equations are

also stiff and need to be solved by an efficient stiff-stable integrator. The benefit of

the Galerkin method is that the newly arising stiff system has a drastically smaller

dimension than the original one. To achieve this goal, the Galerkin method involves

two important characteristics, which are crucial for the success of the method.

1. A sophisticated choice of the weight function ψ and the parameter ρ provides a

good approximation and decreases n to the needed number. Thus, it is important to

keep n as small as possible. In the case of free-radical polymerisation a reasonable

probability density function is chosen [182], which in the chemical literature is known

as the Schultz-Flory distribution(ψ (S, ρ) = (1− ρ) · ρS−1, 0 < ρ < 1, S = 1, 2, ...

).

The orthogonal polynomials associated with the weight function are the discrete

Laguerre polynomials [71]. There are two natural requirements to normalise a weight

function ψ(S, ρ) such that its zeroth and first order statistical moments v0 and v1

coincide with the corresponding moments µ0, µ1 of an unknown distribution P (S, ρ)

at a fixed time t:

a) υ0 (ρ) :=∞∑

S=1

ψ (S, ρ) = 1 (4.9)

b) υ1 (ρ) :=∞∑

S=1

Sψ (S, ρ) =µ1 (t)

µ0 (t).

The normalisation Eq. (4.9 a) ensures that ψ(S, ρ) has a probability distribution,

and condition Eq. (4.9 b) gives an implicit definition of ρ = ρ(t). Thus, the ith

moment µi can be calculated by

µi (t) :=⟨Si, P (S, t)

⟩=

∞∑S=1

SiP (S, t) , i = 0, 1, ... (4.10)

Further development of the Si in the orthogonal function lk(S, ρ)

Si =i∑

k=0

biklk (S, ρ) , k = 0, ..., i (4.11)

leads to a relation between µi and ak(t)

µi (t) =i∑

k=0

bikak (t) γk. (4.12)

As a consequence, the weight function ψ is then time-dependent just as the wanted

distribution P (S, ρ). The conditions Eq. (4.9) imply

a) a0 (t) ≡ µ0 (t) (4.13)


b) a1 (t) ≡ 0.

2. The quality of the Galerkin approximation is controlled through an error estima-

tion, which is calculated at the end of each simulation. The relative error εn (t) of

the approximation (4.8) can be calculated by solving

εn (t)2 :=

∑∞S=1

(P (n) (S, t, ρ)− P (S, t)

)2/ψ (S, ρ)2

∑∞S=1 P (S, t)2 /ψ (S, ρ)2 =

a2n+1 (t) γn+1∑n+1k=0 ak (t)2 γk

. (4.14)

As the statistical moments, two of the main characteristics of a polymer, are cal-

culated, the mean chain length Sn and the mean mass Sm of the size distribution

P (S, t) of a polymer P are

Sn (t) =µ1 (t)

µ0 (t)and Sm (t) =

µ2 (t)

µ1 (t). (4.15)

The time variation of the size distribution function P (S, t), which depends on the

chemical kinetics of the macromolecular species, results in a complete system of

ODEs for each coefficient aj(t, ρ). In general, they take the form [180, 71]

daj

dt=

1

ρj·⟨lj (S, ρ) ,

dP (S, t)

dt

⟩=

1

ρj·∞∑

S=1

lj (S, ρ) · dP (S, t)

dt. (4.16)

4.2 Program package MACRON

The discrete Galerkin technique suggested by Deuflhard and Wulkow [183] for the

solution of ODEs, describing the kinetics of polymerisation reactions, has been suc-

cessfully applied for the case of free-radical polymerisation [69]. The first software,

that includes the analytical and numerical prerequisites of this method in combi-

nation with the solution of the set of ODEs generated by the gas-phase chemical

kinetics, was the so-called program package MACRON (MACROmolecular reaction

kiNetics), proposed by [68]. This package combines the discrete Galerkin techniques

for the simulation of macromolecular reactions with the software environment of

LARKIN [184, 182, 185, 186] for the numerical treatment of large systems of ODEs

arising in chemical reaction kinetics. The gas-phase reactions as well as the macro-

molecular reaction steps are entered by the user in familiar form, and the preparation

of the Galerkin method (analytical preprocessing) is performed. For this purpose, a

list of typical macromolecular reaction steps (e.g., nucleation, chain addition, trans-

fer reactions, termination process, coagulation) have been implemented [68]. The

extended and revised version of this software [71] is used to model the particle dy-

namics during thermal decomposition of Fe(CO)5 , pure carbon clusters formation


during pyrolysis of C3O2 [60, 72], and soot formation in n-heptane rich oxidation

[71]. The size of the chemical system is restricted by the available computer mem-

ory. In its current version, it can handle up to 5000 gas-phase elementary reactions

between 500 species and 200 macromolecular reactions. For the interpretation of the

gas-phase elementary reactions, the kinetic parameter field consists of one to three

numbers, which denote the Arrhenius parameters: A, Ea, and n arising from the

Arrhenius equation (k = A · T n · exp(−Ea/RT ), see Chapter 3). The macromolecu-

lar reactions are characterised by species, whose empirical formula contains square

brackets (e.g., P[]). A typical example of such a reaction is

A + B ⇒ P[1] + C (A, Ea, n, p).

Here, the fourth parameter p denotes the number of carbon atoms added to the

polymer species. To define the rate coefficients of the reversible reactions, two

possible descriptions are used:

• A reversible reaction is followed by two kinetic parameter fields, where the

first assigns the forward reaction and the second the reverse reaction.

• If a reaction is written in both directions, the reversible rate coefficient is cal-

culated via the equilibrium constant. This is only possible if thermodynamical

data (included in the THERMO file) is available for all species appearing in

the equation.

The characteristic parameters of the polymer species as the soot volume fraction fV ,

mean particle diameter D and the soot yield Y can be computed from the values of

particle number density N, mean chain length Sn, and the mean mass of the polymer

size distribution Sm.

The total particle number density N is

N = NA ·NP∑i=1

[P[]i]. (4.17)

Here NA is Avogadro’s number, [P[]i] is the molar concentration of the polymers

P[]i, and NP is the chain length.

The soot volume fraction fV is calculated by

fV =NA

ρ·

NP∑i=1

[P[]i] · (m0,i + (Sn,i − 1) ·mS,i, (4.18)


where ρ is the density of graphite, m0,i is the mass of a monomer from the polymer

P[]i, mS,i is the mass of the smallest momoner added to the polymer P[]i, and Sn,i

is the the mean chain length of the P[]i.

The soot particle diameter D is

D =3

√6 fV

π N. (4.19)

The soot yield Y can be computed as

Y =

∑NP

i=1[P[]i] · (NC,0,i + (Sn,i − 1) ·NC,S,i)∑NHC

j=1 [HC]t0,j·NC,HC,j

, (4.20)

where NC,0,i denotes the number of C atoms included in a monomer, NC,S,i is the

number of C atoms in a momoner added to the polymer P[]i, NHC is the maximum

number of the fuel molecules, [HC]t0,jis the initial concentration of the fuel, and

NC,HC,j is the number of C atoms in a fuel molecule. Information about the differ-

ent parameters included in the input blocks as well as the specific subroutines are

described in detail by Sojka in [71].

The advantage of this approach is that it combines the well-known methods to

simulate the reactions of microheterogeneous particles in such a way that the discrete

character of each elementary transformation is preserved. These transformations

are represented as elementary chemical reactions for the particles of all sizes. The

particles react also with the gas-phase species, and, thus a connection between the

gas-phase chemistry and the soot particles is provided during the whole calculation.

In the current work, two different kinetic schemes were introduced in MACRON to

model the formation and evolution of soot particles during pyrolysis and oxidation

of various hydrocarbons and their mixtures behind shock waves at wide range of

reaction conditions [187, 61, 62, 63, 64, 188].

42

Chapter 5

DETAILED KINETIC MODELS

OF SOOT FORMATION

Last decade, the HACA mechanism of polyaromatic hydrocarbons formation and

growth [21, 1] was accepted as the standard model, and many soot formation models

are based on it. The implementation of a specific model in different codes usually

causes modifications of some important characteristics or steps, which may ruin the

basic idea of the original model. Therefore, it is necessary that every mechanism has

to be described in detail, in particular its differences and similarities to the source

mechanism. In this chapter, two different detailed kinetic models of soot formation

are described, as well as their validation and application in homogeneous conditions.

For simplicity, the models are named Model-1 and Model-2, and with these names

they are referred to further in the text.

5.1 Description of Model-1

5.1.1 Gas-phase reaction mechanism

A detailed kinetic model of soot formation during pyrolysis of various hydrocarbons

in shock-tube experiments is developed [58]. The model was further extended with

a part of the mechanism of n-heptane oxidation [189] and applied for soot formation

simulation in CH4, C3H8, and n-C7H16 rich oxidation behind reflected shock waves

[187, 62]. A set of reactions of C1 to C2 species were also added from the mechanism

of [190]. As a result, the complete detailed kinetic scheme consists of approximately

5. DETAILED KINETIC MODELS OF SOOT FORMATION 43

1850 gas-phase reactions between 186 species and 100 heterogeneous reactions in-

cluding four ensembles of micro-heterogeneous particles. The gas-phase reaction

mechanism includes a complete set of reactions of polyaromatic hydrocarbon (PAH)

formation, described in [46] for laminar premixed acetylene and ethylene flames with

all modifications presented in [23]. In addition to this, the reaction mechanisms of

acetylene pyrolysis [191, 79], the gas-phase mechanism of polyyne species forma-

tion [16, 17, 2], and a set of the gas-phase reactions for small pure carbon clusters

formation up to C30 [60, 192, 72] were included.

The gas-phase kinetic mechanism of polyaromatic hydrocarbon formation describes

the pyrolysis and oxidation of C1 to C2 species, the formation of higher linear hydro-

carbons up to C6 species, the formation of benzene and higher PAH up to pyrene and

the oxidation pathways of the aromatic species. Benzene and phenyl molecules are

formed by interaction between C4HX species with acetylene, by cyclisation of C6HX

species, and the combination of C3H3 propargyl radicals (see Chapter 3.1.1). This

reaction was treated as an overall single irreversible step, with the rate coefficient

fitted to the experimental species profiles against which the model was validated

[23]. As mentioned in [23, 105], the kinetic simulations revealed high reversibility of

chemical reactions leading to the formation and growth of the aromatic rings. Only

for particularly stable compounds, like acenaphthalene or pyrene, the formation re-

actions can be assumed as irreversible. In the kinetic model considered, several even-

and odd-carbon-atom paths for polyaromatics formation were implemented together

with a consistent set of kinetic and thermodynamic data. The formation pathway

of PAH starts with benzene and follows the HACA mechanism. The polycyclic aro-

matic compounds growth up to pyrene by acetylene addition and the ring-forming

addition of vinylacetylene (C4H4) to aromatic radicals. Whereas the vinylacetylene

addition channel contributes mostly to the production of naphthalene (A2), phenan-

threne (A3) is formed by ring-ring condensation reactions in which biphenyl (P2)

is formed by the addition of benzene (A1) to phenyl (A1-) (see Chapter 3.1.2 and

3.1.3).

The gas-phase mechanism described by Krestinin et al. in [16, 17, 2], as the so

called polyyne pathway, was also implemented into the model. The most important

pathway of polyyne growth is the subsequent increase of the carbon chain by a sub-

stitution reaction C2nH + C2mH2 = C2n+2mH2 + H, n,m = 1, 2, ...; n + m < 6,

whose rate coefficients are close to the gas-kinetic collisions. Acetylene and its radi-

cal C2H play the key role in the polyyne growth process. The appropriate chemical

environment, that consists of H, C2H, C2H3 and C4H3 radicals responsible for the

polyyne growth, depends on the type and the structure of the fuel molecules. The


polyyne molecules grow up to C12H2 and the average size of the polyyne radicals

C2nH increases. Large polyyne radicals react with each other or with higher homo-

logues, forming a variety of aggregates, which are no longer straight and planar and

may contain ring closures. This ring closure results in the appearance of new free

valencies, which enable these aggregates to add still more polyynes. Most of the

small polyyne molecules and radicals react to form higher homologues. Finally, the

radicals necessary to form larger polyynes are consumed. The radicals formed by

molecular dissociation are trapped by the carbon particles. As a result, the radical

concentration reduces, which ends the formation of higher polyynes and diminishes

the rate of soot particle inception. In general, the direct decomposition of acetylene

and small polyynes on the surface of the carbon particles is much slower in compar-

ison to their condensation. These specific steps describe the polyyne submodel of

the gas-phase soot precursor formation and growth of Model-1.

5.1.2 Soot precursors and particle inception, surface

growth, coagulation and oxidation

The formation and evolution of soot precursors and soot particles is described

within the framework of the discrete Galerkin technique suggested by Deuflhard

and Wulkow [180, 68], which is briefly described in Chapter 4.

A key aspect of the soot formation process is the deposition of soot mass through

reactions of gaseous species with the soot particle surface. Frenklach and Wang

[21], suggested a detailed kinetic mechanism for soot particles surface growth. Their

mechanism is based on the postulate of the chemical similarity between analogous

surface and gas-phase reactions of carbonaceous species. For soot particles this

means that chemical reactions at the soot particle surface are similar to those of

large polycyclic aromatic hydrocarbons (PAH). The essence of this mechanism is

the HACA growth on the armchair edge of PAH [193]. In Model-1, only the basic

mechanism of surface growth presented in [23] was considered. Following [23], the

first precursors are incepted in condensation reactions of pyrene molecules. As an

extention to this process, the reactions among pyrene, phenanthrene and biphenyl

molecules and their radicals [58] were included. Two different types of particles were

considered for the precursors formed from the PAH - particles with dehydrogenated

C* sites (active) and saturated particles with C-H sites (inactive). The formation

and consumption of active sites on the soot particle surface occur in the reactions

with H2/H and H2O/OH/O2 species. Surface growth is provided by the reactions of

soot particles with active sites with C2H2 [23], accompanied by H abstraction, and by


the reactions of pyrene, phenanthrene, and naphthalene condensation on the active

soot particles [58]. The soot particles with active sites participate in coagulation

reactions, where only a free-molecular regime of coagulation was considered.

In the polyyne pathway, the soot precursors are formed in polymerisation reactions

of higher polyynes (C12H2, C10H2). In the current model, only the high tempera-

ture path of initiation reactions between neutral polyyne molecules and reactions

of particle growth with participation of polyyne molecules and their radicals were

adopted from the original model proposed by Krestinin et al. [16, 17, 2]. Re-

vised thermodynamic data for the polyyne molecules included in the gas-phase part

of the mechanism were taken from the literature [194] or calculated using group

additivity techniques. A complete list of the reactions with participation of micro-

heterogeneous soot precursors and soot particles is presented in Table 5.1. The

surface growth in the polyyne pathway occurs via reactions of soot precursors and

particles with the most reactive gas-phase species: C2H2, C2H, C2, C4H2, C4H, C4,

C6H2 , C6H, C6, C8H2, C8H, C8 , C10H2, C10H, C10, C12H2, C12H, and C12. Ac-

cording to the main assumption of the polyyne model, a growing particle generates

active sites in each act of interaction with the gas-phase species. Therefore only

one type of active soot particles was considered in the model. Soot particles formed

through the polyyne pathway react also with PAH and coagulate with each other

(see Table 5.1).


Table 5.1: Mechanism of formation, growth, coagulation andtransformation of soot precursors and soot particles (Model-1)

Reaction A(a) n(a) EA p(b) Ref.

HACA pathway of soot formation

Soot precursors formation

A4 + A4 → PR[1] 1.558E+13 0.5 0.0 32 (c,d)

A4 + A4- → P[1] 1.558E+13 0.5 0.0 32 (c,d)

A4- + A4- → PR[1] 1.558E+13 0.5 0.0 32 (c,d)

A4 + A3 → PR[1] 1.558E+13 0.5 0.0 30 (c,d)

A4- + A3 → P[1] 1.558E+13 0.5 0.0 30 (c,d)

A3 + A3 → PR[1] 1.558E+13 0.5 0.0 28 (c,d)

P2 + P2 → PR[1] 1.558E+10 0.0 0.0 24 (c,e)

Growth of soot precursors with active sites

P[N ] + C2H2 → P[N+1] + H 8.0E+7 1.56 15.88 2 (f,g)

P[N ] + A4 → P[N+1] 4.5E+12 0.5 0.0 16 (f,g)

P[N ] + A3 → P[N+1] 4.5E+12 0.5 0.0 14 (f,g)

P[N ] + A2 → P[N+1] 4.5E+12 0.5 0.0 10 (f,g)

Activation - Deactivation of soot precursors

PR[N ] + H → P[N ] + H2 4.17E+13 0.0 54.34 (h,i)

P[N ] + H2 → PR[N ] + H 3.90E+12 0.0 45.98 (h,i)

PR[N ] + OH → P[N ] + H2O 1.00E+10 0.73 5.98 (h,i)

P[N ] + H2O → PR[N ] + OH 3.68E+8 1.14 71.48 (h,i)

P[N ] + H → PR[N ] 2.00E+13 0.0 0.0 (h,i)

P[N ] + O2 → PR[N ] + H2O +H2O 2.20E+12 0.0 31.35 (h,i)

Transformation of soot precursors to soot particles

P[N ] → S[N ] 1.0E+6 0.0 0.0 (j,k)

Coagulation of soot precursors with active sites

P[N ] + P[M ] → P[N+M ] 4.50E+12 0,5 0.0 (l)

Polyyne pathway of soot formation


C8H2 + C8H2 → C[1] 4.0E+13 0.0 163 16 (m,n)

C10H2 + C10H2 → C[1] 4.0E+13 0.0 84 20 (m,n)

C12H2 + C12H2 → C[1] 4.0E+13 0.0 17 24 (m,n)

C12H2 + C12H2 → C[1] 4.0E+13 0.0 17 22 (m,n)

Growth of soot precursors

C[N ] + C2H2 → C[N+1] + H 8.0E+7 1.56 15.88 2 (o,p)

C[N ] + C2H2 → C[N+1] + H2 4.0E+13 0.0 133.1 2 (o,p)



C[N ] + C2H → C[N+1] + H 4.0E+13 0.0 0.0 2 (o,p)

C[N ] + C2H → C[N+1] 4.0E+13 0.0 0.0 2 (o,p)

C[N ] + C2 → C[N+1] 4.0E+13 0.0 0.0 2 (o,p)

C[N ] + C4H2 → C[N+1] + H2 4.0E+13 0.0 50.2 4 (o,p)

C[N ] + C4H2 → C[N+1] + H 8.0E+7 1.56 15.88 4 (o,p)

C[N ] + C4H → C[N+1] + H 4.0E+13 0.0 0.0 4 (o,p)

C[N ] + C4H → C[N+1] 4.0E+13 0.0 0.0 4 (o,p)

C[N ] + C4 → C[N+1] 4.0E+13 0.0 0.0 4 (o,p)

C[N ] + C6H2 → C[N+1] + H2 4.0E+13 0.0 33.5 6 (o,p)

C[N ] + C6H2 → C[N+1] + H 8.0E+7 1.56 15.88 6 (o,p)

C[N ] + C6H → C[N+1] + H 4.0E+3 0.0 0.0 6 (o,p)

C[N ] + C6H → C[N+1] 4.0E+13 0.0 0.0 6 (o,p)

C[N ] + C6 → C[N+1] 4.0E+13 0.0 0.0 6 (o,p)

C[N ] + C8H2 → C[N+1] + H2 4.0E+13 0.0 12.6 8 (o,p)

C[N ] + C8H2 → C[N+1] + H 8.0E+7 1.56 15.88 8 (o,p)

C[N ] + C8H → C[N+1] + H 4.0E+13 0.0 0.0 8 (o,p)

C[N ] + C8H → C[N+1] 4.0E+13 0.0 0.0 8 (o,p)

C[N ] + C8 → C[N+1] 4.0E+13 0.0 0.0 8 (o,p)

C[N ] + C10H2 → C[N+1] + H2 4.0E+13 0.0 0.0 10 (o,p)

C[N ] + C10H2 → C[N+1] + H 8.0E+7 1.56 15.88 10 (o,p)

C[N ] + C10H → C[N+1] + H 4.0E+13 0.0 0.0 10 (o,p)

C[N ] + C10H → C[N+1] 4.0E+13 0.0 0.0 10 (o,p)

C[N ] + C10 → C[N+1] 4.0E+13 0.0 0.0 10 (o,p)

C[N ] + C12H2 → C[N+1] + H2 4.0E+13 0.0 0.0 12 (o,p)

C[N ] + C12H2 → C[N+1] + H 8.0E+7 1.56 15.88 12 (o,p)

C[N ] + C12H → C[N+1] + H 4.0E+13 0.0 0.0 12 (o,p)

C[N ] + C12H → C[N+1] 4.0E+13 0.0 0.0 12 (o,p)

C[N ] + C12 → C[N+1] 4.0E+13 0.0 0.0 12 (o,p)

C[N ] + C4H4 → C[N+1] + H2 + H2 2.0E+12 0.0 115.9 4 (o,p)

C[N ] + A4 → C[N+1] 4.5 E+12 0.5 0.0 16 (q)

C[N ] + A3 → C[N+1] 4.5E+12 0.5 0.0 14 (q)

C[N ] + A2 → C[N+1] 4.5E+12 0.5 0.0 10 (q)


C[N ] → S[N ] 1.0E+6 0.0 0.0 (k)

Coagulation of soot precursors

C[N ] + C[M ] → C[N+M ] 4.50E+12 0.5 0.0 (l)

Growth of soot particles

S[N ] + C2H2 → S[N+1] + H 8.0E+7 1.56 15.88 2 (p)



S[N ] + C2H2 → S[N+1] + H2 4.0E+13 0.0 133.1 2 (p)

S[N ] + C2H → S[N+1] + H 4.0E+13 0.0 0.0 2 (p)

S[N ] + C2H → S[N+1] 4.0E+13 0.0 0.0 2 (p)

S[N ] + C2 → S[N+1] 4.0E+13 0.0 0.0 2 (p)

S[N ] + C4H2 → S[N+1] + H2 4.0E+13 0.0 50.2 4 (p)

S[N ] + C4H2 → S[N+1] + H 8.0E+7 1.56 15.88 4 (p)

S[N ] + C4H → S[N+1] + H 4.0E+13 0.0 0.0 4 (p)

S[N ] + C4H → S[N+1] 4.0E+13 0.0 0.0 4 (p)

S[N ] + C4 → S[N+1] 4.0E+13 0.0 0.0 4 (p)

S[N ] + C6H2 → S[N+1] + H2 4.0E+13 0.0 33.5 6 (p)

S[N ] + C6H2 → S[N+1] + H 8.0E+7 1.56 15.88 6 (p)

S[N ] + C6H → S[N+1] + H 4.0E+13 0.0 0.0 6 (p)

S[N ] + C6H → S[N+1] 4.0E+13 0.0 0.0 6 (p)

S[N ] + C6 → S[N+1] 4.0E+13 0.0 0.0 6 (p)

S[N ] + C8H2 → S[N+1] + H2 4.0E+13 0.0 12.6 8 (p)

S[N ] + C8H2 → S[N+1] + H 8.0E+7 1.56 15.88 8 (p)

S[N ] + C8H → S[N+1] + H 4.0E+13 0.0 0.0 8 (p)

S[N ] + C8H → S[N+1] 4.0E+13 0.0 0.0 8 (p)

S[N ] + C8 → S[N+1] 4.0E+13 0.0 0.0 8 (p)

S[N ] + C10H2 → S[N+1] + H2 4.0E+13 0.0 0.0 10 (p)

S[N ] + C10H2 → S[N+1] + H 8.0E+7 1.56 15.88 10 (p)

S[N ] + C10H → S[N+1] + H 4.0E+13 0.0 0.0 10 (p)

S[N ] + C10H → S[N+1] 4.0E+13 0.0 0.0 10 (p)

S[N ] + C10 → S[N+1] 4.0E+13 0.0 0.0 10 (p)

S[N ] + C12H2 → S[N+1] + H2 4.0E+13 0.0 0.0 12 (p)

S[N ] + C12H2 → S[N+1] + H 8.0E+7 1.56 15.88 12 (p)

S[N ] + C12H → S[N+1] + H 4.0E+13 0.0 0.0 12 (p)

S[N ] + C12H → S[N+1] 4.0E+13 0.0 0.0 12 (p)

S[N ] + C12 → S[N+1] 4.0E+13 0.0 0.0 12 (p)

S[N ] + C4H4 → S[N+1] + H2 + H2 2.0E+12 0.0 115.9 4 (p)

S[N ] + A4 → S[N+1] 4.5E+12 0.5 0.0 16 (q)

S[N ] + A3 → S[N+1] 4.5E+12 0.5 0.0 14 (q)

S[N ] + A2 → S[N+1] 4.5E+12 0.5 0.0 10 (q)

Coagulation of soot particles

S[N ] + S[M ] → S[N+M ] 4.50E+12 0.5 0.0 (l)


(a) The rate coefficients are presented by the Arrhenius equation:

k = AT nexp (−EA/RT ), where A (cm3,mol, s), T (K), and

EA (KJ/mol).

(b) Index p denotes the number of carbon atoms added to a particle in

each act of interaction with carbon-containing gas-phase species.

(c) Notation PR[1] denotes the concentration of the soot precursors

formed through the HACA pathway containing p carbon atoms

(p = 28− 32) without active sites on their surface. P[1] denotes

the concentration of the PAH precursors with active sites on their

surface (p = 24− 32).

(d) Rate coefficient is similar to the rate coefficient of soot particle

inception proposed in [23].

(e) This reaction was added to improve the coincidence of the calcu-

lated results and experimentally measured values.

(f) Notation P[N ] denotes the concentration of the active soot pre-

cursors after N acts of interaction with various carbon-containing

gas-phase species.

(g) The chosen rate coefficient is similar to the rate coefficient of soot

particle growth proposed in [23].

(h) Notation PR[N ] denotes the concentration of the inactive soot pre-

cursors, where index N corresponds to the number of interactions of

active soot particles P[N ] with carbon-containing gas-phase species.

(i) The chosen rate coefficient is similar to the rate coefficient of trans-

formation of soot particles proposed in [23].

(j) Notation S[N ] denotes the concentration of active soot particles

formed through the polyyne pathway of soot formation and by

transformation reactions of the P[N ] and C[N ] into S[N ] particles.

(k) The chosen rate coefficient is similar to the rate coefficient of inter-

nal transformation of pure carbon clusters into soot-like particles

proposed in [72].

(l) The rate coefficient of coagulation was adopted from [72].

(m) Notation C[1] denotes the concentration of the precursors

formed through the polyyne pathway containing p carbon atoms

(p = 16− 22) with active sites on their surface.

(n) The rate coefficient was adopted from [16, 17].

(o) Notation C[N ] denotes the concentration of the precursors formed

through the polyyne pathway with active sites on their surface after

N acts of interaction with carbon-containing gas-phase species.

(p) The rate coefficients were adopted from [16, 17, 23].


(q) The chosen rate coefficient is equal to the rate coefficient of the

coagulation reactions.

5.2 Results Model-1

5.2.1 Validation of the model

In the context of the detailed kinetic model (Model-1), the process of soot formation

during acetylene and benzene pyrolysis demonstrates the most pronounced differ-

ences with respect to the fuel structure and the reaction pathways included in the

scheme. Kern et al. [35, 36] investigated the pyrolysis of acetylene and benzene and

observed that, although both types of fuel have different chemical structure, the

main gas-phase products are acetylene and several small polyynes. In the current

model the higher polyynes were considered as soot precursors. Thus, to validate

the detailed kinetic model, the calculated results of the concentration profiles of the

species measured in [35, 36] were compared with the experimental measurements

behind reflected shock waves. In Figure 5.1 the concentration profiles of C2H2 decay

and the main products C2H2, C4H2 , and C6H2 formed during pyrolysis of a 3.2%

C2H2 diluted in a (99%Ne/1%Ar) mixture at temperature 2030 K and pressure 0.39

bar are presented. For the case of acetylene pyrolysis, the results of simulations are

in very good agreement with the experiment [35].

Kern et al. [36] studied also the kinetic of benzene thermal decomposition with the

use of three different techniques, time-of-flight mass spectrometry (TOF), atomic

resonant absorption spectroscopy (ARAS), and laser-schlieren density gradient mea-

surements (LS) in shock-tube experiments. Benzene decay profiles measured and

calculated at three different temperatures are plotted in Figure 5.2. The authors

reported that the main species observed in the experiments were C6H6, C2H2, and

C4H2. They described the process of benzene dissociation by means of the reac-

tions C6H6 → C6H5 + H and C6H5 → C2H2 + C4H3, whose rate coefficients were

calculated together with an overall rate of C6H6 decomposition, occurring by the

direct ring rupture process C6H6 → C2H2 + C4H4. For the results of calculation

in the case of C6H6 pyrolysis, a particular difference in the C2H2 (Figure 5.3) and

C4H2 (Figures 5.4) formation profiles was observed at high temperatures. Several

additional calculations showed that the agreement improves if the reactions of soot

particle inception and surface growth with participation of polyyne molecules and

radicals are excluded from the kinetic scheme.


Figure 5.1: Concentration profiles of the main gas-phase species measured (closed sym-bols) [35] and simulated (open symbols and lines) during pyrolysis of 3.2 % C2H2/Ne/Armixture, at T = 2030 K, and p = 0.39 bar behind reflected shock waves: (squares) C2H2,(triangles) C4H2 · 2, (inverse triangles) C6H2 · 10.

Figure 5.2: Concentration profiles of C6H6 decay measured (closed symbols) [36] andsimulated (open symbols and lines) for a mixture of 2.1 % C6H6, diluted in argon at p =0.52 bar for three different temperatures: (circles) 1704 K, (triangles) 1942 K, (squares)2192 K.


Figure 5.3: Time history of C2H2 concentration measured (closed symbols) [36] andsimulated (open symbols and lines) at conditions as in Figure 5.2.

Figure 5.4: Time history of C4H2 concentration measured (closed symbols) [36] andsimulated (open symbols and lines) at conditions as in Fig.5.2.

Integral Reaction Flow Analysis (IRFA) and global sensitivity analysis were per-

formed during pyrolysis for both types of reaction systems, C2H2 and C6H6. The

IRFA diagrams show only the main routes of soot precursor formation, by means of

the different pathways implemented in the kinetic mechanism (HACA and polyyne).


In the case of acetylene pyrolysis, the reaction routes of soot precursor formation

Figure 5.5: Integral reaction flow analysis of the PAH formation pathways during pyrolysisof a 4.62% C2H2/Ar mixture at T = 2000K, p = 6.0 bar, and reaction time 0.003 s.

Figure 5.6: Integral reaction flow analysis of the polyyne formation pathways duringpyrolysis of a 4.62% C2H2/Ar mixture at T = 2000 K, p = 6.0 bar, and reaction time0.003 s.

(PRsoot[1] and Csoot[1] ) are shown in Figures 5.5 and 5.6. The main routes of soot

precursor formation through PAH (HACA pathway) are indicated in Figure 5.5.

Benzene, as the first aromatic ring in the system, is formed at about 50% through


Figure 5.7: Sensitivity analysis with respect to benzene during pyrolysis of acetylene atconditions as in Figure 5.5.

Figure 5.8: Sensitivity analysis with respect to phenanthrene during pyrolysis of acetyleneat conditions as in Figure 5.5.

the propargyl radical (C3H3) recombination. The rate-determining steps with re-

spect to benzene formation (Figure 5.7) are the vinylacetylene (C4H4) decomposition

and the C3H3 recombination,

H + C4H4 = CH3 + C3H2, (5.1)

C3H3 + C3H3 = A1. (5.2)

The rate coefficients for both reactions k = 5.0 · 1012 in cm3mol-1s-1 were adopted

from [23]. In Model-1, Eq. (5.2) was written as a one-step irreversible reaction.

Phenanthrene is formed preferably through the C2H2 addition to biphenyl radical,

which is an example for the significant contribution of the ring-ring condensation

reactions in the PAH mass growth. Phenanthrene and pyrene are the main species

engaged in the process of soot precursors formation. Both molecules are mostly


Figure 5.9: Sensitivity analysis with respect to pyrene during pyrolysis of acetylene atconditions as in Figure 5.5.

Figure 5.10: Sensitivity analysis with respect to C12H2 during pyrolysis of acetylene atconditions as in Figure 5.6.

produced following the HACA model. Nevertheless, in the case of acetylene pyroly-

sis, the PAH route based only on the HACA [46] cannot compete with the polyyne

pathway. The rate-limiting step with respect to phenanthrene (A3) and pyrene (A4)

formation is again Reaction (5.1). Vinylacetylene is one of the key species for the

PAH formation and growth. Important for the PAH growth are also the reactions of

1- and 4-phenanthrene radical formation, which after C2H2 addition forms pyrene,

and the propargyl radical recombination (see Figures 5.8 and 5.9).

In the case of acetylene pyrolysis, the polyyne pathway of soot formation dominates

(Figure 5.6). Starting from C2H2 as a parent molecule, the polyyne molecular mass

growth follows the sequence of reactions of substitution described in Section 5.1.1 of

the Chapter. The soot particle precursors are formed in reactions of higher polyyne

polymerisation, like 1,3,5,7,9-decapentayne (C10H2) and 1,3,5,7,9,11-dodecahexayne


(C12H2), which later interact with C2H2, C4H2 and provide the major contribution

in the soot particle surface growth. The sensitivity analysis showed that the rate-

determining reactions with respect to the polyyne molecules participating in the soot

model (C10H2 and C12H2 ) are related to the formation of ethynyl radical (C2H) and

C8H2, as these two species play the key role for the the formation of higher polyynes

(Figure 5.10). The reaction

C2H2 + H = C2H + H2 (5.3)

was adopted from the C1-C4 mechanism of hydrocarbon combustion [190], with

the rate coefficient k = 2.0 · 109T 1.64exp(−126.788/RT ) in cm3mol-1s-1, with EA in

KJmol-1.

Figure 5.11: Integral reaction flow analysis of the PAH formation pathways during py-rolysis of a 1.54% C6H6/Ar mixture at T = 2000 K, p = 6.0 bar, and reaction time 0.003s.

Soot formation starts much earlier in benzene pyrolysis where the PAH route of

soot formation dominates (Figures 5.11 - 5.12). The main reason is the absence of

the rate-limiting cyclisation step with respect to the PAH formation and growth in

comparison to the C2H2 pyrolysis. During benzene pyrolysis, most of the benzene

molecules are destroyed by analogy to the mechanism described by Kern et al. [36],

but some of them survive the termal decomposition. These are used as a kernel for

PAH growth by HACA.

Critical for the polyyne formation are the benzene dehydrogenation and the phenyl


Figure 5.12: Integral reaction flow analysis of the polyyne formation pathways duringpyrolysis of a 1.54% C6H6/Ar mixture at T = 2000 K, p = 6.0 bar, and reaction time0.003 s.

Figure 5.13: Sensitivity analysis with respect to biphenyl during pyrolysis of benzene atconditions as in Figure 5.11.

radical decomposition (Figure 5.12). They are thoroughly studied in [1]. Subse-

quently, O-benzyne is formed, which decomposes to 1-buten-3-yn-1-yl (n-C4H3) and

C2H2. Frenklach and Wang [1] suggested that in rich flames the n-C4H3 radical is

rapidly consumed to form its resonantly stabilised i-C4H3 radical.

Paralelly, the polyacetylene mass growth follows the well-known sequence of sub-

stitution reactions up to (C10H2) and (C12H2) formation. The global sensitivity

analysis confirmed that the rate-limiting reaction with respect to the C10H2 and


Figure 5.14: Sensitivity analysis with respect to phenanthrene during pyrolysis of benzeneat conditions as in Figure 5.11.

Figure 5.15: Sensitivity analysis with respect to C12H2 during pyrolysis of benzene atconditions as in Figure 5.12.

C12H2 formation is the phenyl radical decomposition, leading to O-benzyne,

A1− = H + c-C6H4, (5.4)

whose ring structure is opened in a propagation reaction with H atom, forming the

linear species 3-hexen-1,5-diyne (l-C6H4),

H + c-C6H4 = H + l-C6H4. (5.5)

Both steps are described by Wang and Frenklach in [45] and in the model of Appel

et al. [23]. In the current mechanism, the following values for the rate coefficient

(k = 2.4 · 1060T -13.66exp(−123.428/RT ) in cm3mol-1s-1 , with EA in KJmol-1[23] and

k = 1.4 ·1054T -11.7exp(−144.348/RT ) in cm3mol-1s-1 , with EA in KJmol-1 [23]) were

used for the Reactions (5.4) and (5.5) respectively.


Figure 5.16: Sensitivity analysis with respect to C10H2 during pyrolysis of benzene atconditions as in Figure 5.12.

5.2.2 Hydrocarbon pyrolysis behind shock waves

Usually, several parameters of soot formation such as induction time (τ), soot yield

(SY), and soot growth rate coefficient (kf) are measured from the experimentalists.

They are known to be very sensitive to the chemical structure of the pyrolysed

hydrocarbon. The induction time τ [s] is defined as the intersection point of the

tangent at the inflection point of the soot yield curve with the time axis. The

soot yield is defined as a ratio of the converted to soot carbon to the total carbon

content in the parent mixture. The soot yield profiles, after the inflection point,

can be approximated by an empirically obtained first order rate law, where the

parameter kf [s−1] is the soot growth rate coefficient, which can be interpreted as

an effective measure of the active lifetime of soot particles, assuming that they are

losing their reactivity.

A direct comparison of the experimental results with the calculated results for the

induction time, the soot yield and the observable rate of soot growth, obtained by

extinction techniques during pyrolysis of various hydrocarbons and their mixtures,

was performed. Various mixtures were investigated during pyrolysis of methane,

ethylene, acetylene, benzene, benzene/acetylene, and acetylene/hydrogen, diluted

in argon or neon mixtures behind shock waves [195, 81, 82, 78].


Pyrolysis of acetylene and ethylene

The temperature dependences of the soot yield and the Arrhenius-type plots for the

induction delay time during the pyrolysis of C2H2/Ar mixtures at a pressure 50.7

bar, and C2H4/Ar at a pressure 50.0 bar for different carbon atom concentrations

Figure 5.17: Temperature dependence of the soot yield measured (closed symbols) [195]and simulated (open symbols and lines) during pyrolysis of C2H2/Ar mixtures at p =57.0 bar for three different C atom concentrations: (inverse triangles) [C] = 3.8 [mol/m3],(circles) [C] = 1.7 [mol/m3], (squares) [C] = 0.9 [mol/m3].

Figure 5.18: Induction delay time in measured (closed symbols) [195]and simulated (opensymbols and lines) during pyrolysis of C2H2/Ar mixtures at conditions as in Fig. 5.17.


Figure 5.19: Temperature dependence of the soot yield measured (closed symbols) [195]and simulated (open symbols and lines) during pyrolysis of C2H4/Ar mixtures at p =50.0 bar for three different C-atom concentrations: (inverse triangles) [C] = 7.4 [mol/m3],(circles) [C] = 4.7 [mol/m3], (squares) [C] = 4.0 [mol/m3]. Open diamonds denote thecalculated results for C2H6/Ar mixture: p = 50.0 bar, [C] = 4.0 [mol/m3].

Figure 5.20: Induction delay time measured (closed symbols) [195] and simulated (opensymbols and lines) during pyrolysis of C2H4/Ar mixtures at p = 50.0 bar for three differ-ent C-atom concentrations: (triangles) [C] = 7.4 [mol/m3], (circles) [C] = 4.7 [mol/m3],(squares) [C] = 4.0 [mol/m3].

are presented in Figures 5.17 - 5.20. The model predictions are in agreement with

the experimentally measured data. In the case of C2H2 and C2H4 pyrolysis, the


polyyne pathway was found to play a dominant role in the soot formation process.

These conclusions were observed by simply excluding one or another submodel from

the soot formation scheme during the calculations. According to these tests, the

kinetic scheme cannot predict the experimentally measured values of the soot yield

if, the polyyne pathway is excluded from the model and only the HACA pathway is

active.

Pyrolysis of methane and benzene

Methane is a fuel with many practical applications, known as one of the major

components of natural gas. Combustion products of methane have the highest

H2O/CO2 ratio due to its highest H/C ratio. It is also considered to be among

the least-polluting fuels available. However, previous studies showed that even in

non-sooting methane flames the concentration of PAH molecules may be significant

[196].

Tanke [195] studied the soot formation in CH4/Ar mixtures at different pressures

with the cw-laser extinction technique (Figures 5.21 and 5.22). The model predic-

tions coincide well with the experimentally measured values of the induction time at

low temperatures, whereas at higher temperature both the induction delay time and

the soot yield are overpredicted. Additional investigations showed that in the case

of CH4 pyrolysis the HACA pathway dominates in the soot formation, especially at

low temperatures. With the temperature increase the contribution of the polyyne

pathway in the soot formation becomes equal to the HACA. A similar effect was

observed also in the case of C6H6 pyrolysis. Two different sets of experiments were

simulated for the case of benzene pyrolysis [195, 81, 82].

In the work of Tanke [195], the conversion of benzene to soot is measured by the

attenuation of the light beam from a HeNe laser (λ = 632.8 nm). The light ex-

tinction profiles are converted into soot yield profiles with the help of Beer’s law.

The refractive index and density of soot particles are given as m = 1.57 - 0.56i and

1.86 g/cm3. The experiments are carried out under elevated pressure, near 50.0 bar,

for various carbon atom concentrations in the initial mixture (4.0, 1.0, 0.8, and 0.4

mol/m3) at a reaction time of 1.5 ms. In these experiments, the temperature and

concentration dependences of the soot yield and the induction delay time are quanti-

tatively determined. For these particular cases, the experimentally measured curve

for the maximum soot yield appears at about 1800 K, whereas the kinetic model

predicts this maximum at about 2000 K (Figure 5.23). The calculated induction


Figure 5.21: Temperature dependence of the soot yield measured (closed symbols) [195]and simulated (open symbols and lines) during pyrolysis of CH4/Ar mixtures for severaldifferent carbon atom concentrations: (circles) p = 55.0 bar, [C] = 6.4 [mol/m3]; (squares)p = 55.0 bar, [C] = 3.4 [mol/m3]; (inverse triangles) p = 120.0 bar, [C] = 4.0 [mol/m3];(triangles) p = 25.0 bar, [C] = 3.0 [mol/m3]; (diamonds) p = 55.0 bar , [C] = 1.7 [mol/m3].

Figure 5.22: Induction delay time measured (closed symbols) [195] and simulated (opensymbols and lines) during pyrolysis of CH4/Ar mixtures at conditions as in Figure 5.21.

times coincide very well with the experimental measurements (Figure 5.24).


Figure 5.23: Temperature dependence of the soot yield measured (closed symbols)[195] and simulated (open symbols and lines) during pyrolysis of C6H6/Ar mixtures atp = 50.0 bar, and tr = 1.5ms, for four different C atom concentrations: (circles) [C] =4.0 [mol/m3], (squares) [C] = 1.0 [mol/m3]; (triangles) [C] = 0.8, (diamonds) [C] = 0.4[mol/m3].

Figure 5.24: Induction delay time measured (closed symbols) [195] and simulated (opensymbols) during pyrolysis of C6H6/Ar mixtures at conditions as in Figure 5.23.

Strake et al. [81, 82] measured the formation of soot particles during benzene py-

rolysis for four different C6H6/Ar mixtures (0.25%, 0.5%, 1%, and 2% C6H6) at


Figure 5.25: Temperature dependence of the soot yield measured (closed symbols) [81, 82]and simulated (open symbols and lines) during pyrolysis of C6H6/Ar mixtures at p =1.2 bar, tr = 1.3ms, and for three different reactive mixtures: (squares) 2 %, (circles) 1 %,(triangles) 0.5 %.

Figure 5.26: Induction delay time, measured (closed symbols) [81, 82] and simulated(open symbols) during pyrolysis of C6H6/Ar mixtures at conditions as in Figure 5.25.

pressures 1.0 bar - 1.3 bar behind a reflected shock wave.


For the cw-laser extinction measurements, a conventional 20 mW HeNe laser (λ

= 632.8 nm) is used. The refractive index and density of soot particles are given

as m = 1.90 - 1.0i and 1.86 g/cm3. The model predictions (Model-1) are in very

good agreement with the experimentally measured soot yield and induction delay

time for all mixtures presented in Figures 5.25 and 5.26. Possible reasons for the

different behaviour of the model with respect to the different sets of experimental

data [81, 195] could be:

• the choice of the refractive indexes ,

• different experimental conditions (pressure),

• insufficient choice of the kinetic data with respect to the pressure dependence

of the rate coefficients included in the model.

It is important to note that the extinction method requires knowledge about the

soot particle refractive index (m), which is subject to large uncertainty. There is

a considerable difference between the refractive indexes used in both experiments

[81, 195]. Smyth and Shaddix [197] studied the influence of the choise of m on the

final results for the amount of soot measured with an extinction technique. They

pointed out that the soot volume fraction may differ by almost a factor of 2, using

different values for m.

The experimentally measured (by the time-resolved LII method), and calculated

temperature dependences of the mean soot particle radius during pyrolysis of ben-

zene/argon mixtures are shown in Figure 5.27 for three different benzene concen-

trations and tr = 1000 µs. The results of the calculations show the same trend as

the experiment. The particle radius increases with increasing benzene concentration

and follows a bell-shaped curve with respect to the temperature but the maximum

temperature is overpredicted by the model. The time-resolved mean soot particle ra-

dius was experimentally studied with the use of the laser induced incandescence (LII)

technique during pyrolysis of various benzene/argon mixtures. A direct comparison

between the experimentally measured data and the simulated values is presented in

Figure 5.28 at p = 1.2 bar and T = 2000 K. A tendency of increasing the mean

particle diameter with increasing the hydrocarbon concentration was observed in

both experiment and calculations. A particular difference between the experimen-

tally measured and calculated values of the mean soot particle radius was observed

at low and high temperatures. At high temperatures, the detailed kinetic model

(Model-1) usually overestimates the soot yield and the mean particle radius. Due to

one of the main assumptions of the model, after a particular time all soot particles


Figure 5.27: Mean particle radius measured (closed symbols) [81, 82] and calculated(open symbols and lines) during pyrolysis of three different C6H6/Ar mixtures: (squares)2 %, (circles) 1 %, (triangles) 0.5 %, for tr = 1.0 ms at p = 1.2 bar.

Figure 5.28: Time history of the mean particle radius measured (closed symbols) [81,82] and calculated (open symbols and lines) during pyrolysis of four different C6H6/Armixtures: (squares) 2 %, (circles) 1 %, (triangles) 0.5 %, (inverse triangles) 0.25 % atp = 1.2 bar.


are considered as active which allows them to react with the gas-phase molecules

up to the end of the observation time. In practice, soot particles become inactive

with time and temperature, and their growth is interrupted. To solve this problem

systematically, it is necessary to consider the kinetics of the non-stationary distri-

bution of active sites on the surface of soot particles, which at the current status of

the Galerkin technique was not possible. Therefore, in Model-1 only two extremes

were considered: active and inactive soot particles.

The difference observed at low temperatures cannot be explained only by the insuffi-

cient surface growth rate of soot particles in the kinetic model, because the increase

of this rate will result in reduced induction delay times and in increased soot yield.

A rapid heating of soot particles may produce chemical and physical changes of the

soot particle structure: the formation of unusual shell structures, the formation of

pores and porous material in the inner core. These changes in the soot morphology

can alter the heat transfer characteristics of the heated soot, because the pores in-

side the spheres will reduce their volume and allow the particles to cool faster than

the equivalent solid particles. Implementation of these phenomena in the present

model would improve considerably the predictions, but there is still a great lack of

detailed understanding and interpretation of the soot formation process.

Pyrolysis of benzene/acetylene and acetylene/hydrogen mixtures

Knorre et al. [78] studied the influence of C2H2 additives on the soot formation

during pyrolysis of C6H6 at elevated pressure (6.0 bar-60.0 bar) with the cw-laser

extinction technique. In Figure 5.29 the simulated results of the soot formation

time history during pyrolysis of a benzene/acetylene = 3:1 mixture, at pressure

60 bar for four different temperatures, are plotted against the experimentally ob-

served data. The model overestimates the soot yield at high temperatures, but

it clearly demonstrates the tendency of decreasing the induction time with the

temperature increase. This effect was also confirmed in Figure 5.30, where the

Arrhenius-type plots of the experimentally measured and calculated results of the

induction time (τ) are presented for various benzene/acetylene mixtures and com-

pared to the pure C2H2/Ar and C6H6/Ar mixtures. A good agreement between

the experimentally measured and calculated values of the induction time for acety-

lene/argon mixture was observed, and a particular discrepancy for benzene/argon

and benzene/acetylene/argon mixtures. The experimentally determined [78] induc-

tion period for the different benzene/acetylene/argon mixtures lies between those

for the pure hydrocarbons. In contrast to the experimentally measured results, the


Figure 5.29: Soot formation time history measured (closed symbols) [78] and simulated(open symbols and lines) during pyrolysis of a diluted in argon C6H6/C2H2 = 3/1 mixture([C] = 1.2·10−6[mol/cm3]) at p = 60 bar for various temperatures: (circles) T = 1676 K,(squares) T = 1789 K, (triangles) T = 1806 K, (inverse triangles) T = 1880 K.

model predictions show that τ for all benzene/acetylene mixtures is shorter than

that obtained for pure benzene/argon mixtures. Interestingly, the acetylene is the

key species in both HACA and polyyne pathway of soot precursor formation and

growth in Model-1.

A direct comparison between the experimentally measured and the calculated re-

sults of the normalised soot growth rate coefficient (kf/[C]) for benzene, acetylene,

benzene/acetylene (B/A), and acetylene/hydrogen mixtures at pressure 60 bar is

presented in Figure 5.31. The model properly describes the difference of the kf

values for benzene and acetylene at the low temperatures, but cannot describe

the sharp decrease of the kf values for the pure acetylene/argon and some ben-

zene/acetylene/argon mixtures at high temperatures. As discussed above, these

discrepancies can be described by the fact that an additional approach, describing

the dynamics of the distribution of the active sites on the soot particles surface,

is needed. The experimentally measured [78] and calculated temperature de-

pendencies of the soot yield in pyrolysis of benzene/argon, acetylene/argon and

benzene/acetylene/argon mixtures are presented in Figures 5.32 to 5.34. The model

overestimates the soot yield for the case of benzene and benzene/acetylene mix-

tures at the maximum soot yield and higher temperatures, but describes well the

bell-shaped temperature dependence of the soot yield. The influence of acetylene


Figure 5.30: Arrhenius-type plot of the experimentally measured (closed symbols)[78] and calculated (open symbols) induction time τ for mixtures with various ben-zene/acetylene ratios (B/A) at pressure 60 bar: (circles) benzene, [C] = 4.0·10−6 mol/cm3;(squares) acetylene, [C]= 4.0 · 10−6 [mol/cm3]; (inverse triangles) B/A = 10/1, [C]=9.0 · 10−6 [mol/cm3]; (diamonds) B/A = 1/1, [C]= 5.0 · 10−6 [mol/cm3]; (triangles) B/A= 2.5/1, [C]= 12.0 ·10−6 [mol/cm3]; (hexagons) benzene; only the HACA pathway of sootformation was active, [C]= 4.0 · 10−6 [mol/cm3].

additives on the soot formation in benzene/argon mixtures demonstrates a complex,

nonlinear character. The C2H2 added to benzene results in a decrease of the induc-

tion time for all investigated mixtures. The concentration of soot particles and soot

precursors decreased steadily from the maximum value for the pure benzene/argon

mixture to the minimum value for the pure acetylene/argon mixture. The model

does not demonstrate significant pressure dependence of the soot yield, but shows

an essential dependence of the induction delay time, the soot growth rate constant,

and the soot yield on the carbon atom concentration. It predicts properly the in-

duction time for pure benzene/argon mixtures, but considerably overestimates the

soot yield. The maximal soot yield appears at higher temperatures in the case of

pure benzene and benzene/acetylene mixtures. On the other hand, the calculated

values of τ (Figure 5.30) in the case of acetylene pyrolysis differ from the experi-

mentally measured ones, whereas the simulated soot yield is in good agreement with

the experiments.

The temperature dependence of the soot yield was calculated for a pure ben-

zene/argon mixture, when only the HACA route was active, and compared with


Figure 5.31: Temperature dependence of the normalised observable rate of soot parti-cle growth (kf/[C]) measured (closed symbols) [78] and calculated (open symbols) duringpyrolysis of benzene, acetylene, benzene/acetylene (B/A), and acetylene/hydrogen mix-tures at pressure 60 bar: (circles) benzene, [C] = 4.0 · 10−6 [mol/cm3]; (squares) [C] =4.0·10−6 [mol/cm3]; (inverse triangles) B/A = 10/1, [C]= 9.0·10−6 [mol/cm3]; (diamonds)B/A = 1/1, [C]= 5.0·10−6 [mol/cm3]; (triangles) B/A = 2.5/1, [C]= 12.0·10−6 [mol/cm3];(hexagons) C2H2/H2 = 1/1, [C]= 2.0 · 10−6 [mol/cm3].

Figure 5.32: Temperature dependence of the soot yield measured (closed symbols)[78] and calculated (open symbols and lines) during pyrolysis of C6H6/Ar, [C] = 4.0·10−6[mol/cm3], and C2H2/Ar, [C] = 4.0 ·10−6[mol/cm3] mixtures at p = 6.0 bar.


Figure 5.33: Temperature dependence of the soot yield measured (closed symbols) [78]and calculated (open symbols and lines) during pyrolysis of C6H6/C2H2/Ar mixtures,(triangles) B/A = 2.5/1 [C] = 9.0 ·10−6 [mol/cm3], (inverse triangles) B/A = 1/1, [C] =5.0 ·10−6[mol/cm3], (diamonds) B/A =1/2.5 [C] = 9.0 ·10−6[mol/cm3] at p = 6.0 bar/cm3.

Figure 5.34: Temperature dependence of the soot yield obtained during pyrolysis ofbenzene ([C] = 4.0 · 10−6 mol/cm3) at p = 6 bar: (closed circles) the experimental mea-surements [78], (open circles and line) the calculated results performed with Model-1,(open hexagons and line) the results of calculation performed with Model-1, when onlythe HACA pathway of soot formation (Table 5.1) was active.


Figure 5.35: Temperature dependence of the soot yield measured (closed symbols)[78] and calculated (open symbols and lines) during pyrolysis of C2H2, C2H4, andC2H2/H2diluted in argon mixtures: (squares) C2H2, [C] = 4.0·10−6[mol/cm3]; (cir-cles) C2H4, [C] = 4.0·10−6[mol/cm3]; (inverse triangles) C2H2/H2 = 1/1, [C] = 4.0·10−6

[mol/cm3]; (triangles) C2H2/H2 = 1/1[mol/cm3], [C] = 2.0·10−6[mol/cm3] mixtures atp = 6.0 bar.

the experimentally measured and simulated results of the soot yield for the same

mixture composition and reaction conditions (Figure 5.34). Within the low tem-

perature range, the pure HACA route considerably underestimates the soot yield,

but describes it quite satisfactory at higher temperatures, whereas Model-1 overes-

timates the SY. Additional tests showed that the best agreement, between Model-1

and the experimentally measured soot yield can be achieved if the polyyne pathway

is excluded from the surface growth reaction mechanism.

In Figure 5.35 the experimentally measured [78] temperature dependences of the soot

yield in pyrolysis of acetylene/argon, ethylene/argon, and acetylene/hydrogen/argon

mixtures was compared with the calculated results. The model predictions confirmed

the experimentally obtained soot suppression effect on the soot yield of the hydrogen

additives to the acetylene/argon mixtures. As a result, the induction time becomes

longer (Figure 5.30) and the soot yield lower (Figure 5.35) in comparison to the pure

acetylene/argon mixture.


5.2.3 Hydrocarbon oxidation behind shock waves

Oxidation of methane, n-propane and n-heptane

Soot formation is experimentally studied [198] during rich oxidation of CH4, C3H8,

and n-C7H16/Ar mixtures behind shock waves. A combination of extinction-

scattering techniques (λ = 488 and 632.8 nm) is used from the authors [198] for

the time-resolved measurements of soot particle diameter and number density, and

a traditional extinction technique is applied to determine the soot yield. In Figure

5.36, the experimentally measured and calculated values for the soot yield of the rich

oxidation of CH4, C3H8, and n-C7H16/O2/Ar mixtures are presented. The mixtures

are selected in such way that the difference in the soot yield for the three hydrocar-

bons must be insignificant. Nevertheless, the model prediction in the case of propane

oxidation shows different behaviour of the soot yield in comparison to the experimen-

tal data [198]. In the case of hydrocarbon combustion, most of the fuel molecules

Figure 5.36: Temperature dependence of the soot yield measured (closed symbols) [198]and calculated (open symbols and lines) during rich oxidation of CH4, [C] = 7.6 [mol/m3];C3H8 [C] = 6.0 [mol/m3]; n-C7H16 [C] = 5.9 [mol/m3] at φ= 5 and p = 40 bar.

are destroyed by oxidation or thermal decomposition, and numerous intermediate

species are formed. In this way, a competition between the molecular growth and

oxidative reactions occurs. Although most of the intermediates are oxidised, many

of the carbon-containing species may participate in the molecular growth process.

Oxidative reactions lead to the formation of various oxygen-containing intermedi-


Figure 5.37: Temperature dependence of the experimentally measured (closed sym-bols) [198] and calculated (open symbols and lines) soot yield during rich oxidation ofn-C7H16/Ar mixture (99 % Ar, 0.3125 % C7H16, and 0.6875 % O2) at constant argonconcentration: (circles) p = 30 bar, (squares) p = 40 bar, (diamonds) p = 50 bar.

ates and products like CO, CO2, and H2O. As a result, the soot yield decreases

compared to the case of hydrocarbon pyrolysis.

The influence of the carbon atom concentration on the soot yield was studied exper-

imentally [198] and numerically in n-heptane rich oxidation behind reflected shock

waves. The effect of increased C-atom concentration is presented in Figure 5.37,

where the soot yield is plotted against the temperature for a mixture of 99 % Ar,

0.3125 % C7H16, and 0.6875 % O2, and the pressure is varied from 30 bar, 40 bar,

to 50 bar. The simulations showed the weakest rise of the soot yield at the highest

pressures (p = 50 bar).

5.3 Description of Model-2

Recent progress in the development and understanding of the chemistry of poly-

cyclic aromatic hydrocarbons [52, 24, 199, 200, 160] provided foundation for further

improvement of the kinetic modeling of soot formation.

Violi [52] suggested a theoretical model for PAH growth and soot particle inception


in aliphatic and aromatic flames, using kinetic Monte-Carlo and molecular dynamics

techniques. The authors proposed a model, describing the growth of polycyclic aro-

matic hydrocarbons, in which the structural parameters like bonds, bond angles, and

dihedral angles are preserved as the soot precursors evolve into three-dimensional

structures. The model is applied in acetylene and benzene premixed flames simula-

tion and describes the differences in the soot precursors born in these systems with

respect to their H/C ratio, particle sphericity, and depolarisation ratio. In this work

[52], two main reaction sequences were considered as possible pathways that con-

tribute to the aromatic growth and soot particle inception. The PAH grow through

the traditional HACA mechanism [46] and a radical-molecule sequence of reactions,

involving five-membered ring PAH, present at large concentrations in the reacting

mixture. In the context of the model [52], soot precursors with different structure are

formed in the case of benzene and acetylene flames. The results of their calculations

showed that the structures formed in aromatic flames are mostly species with 3D

characteristics, whereas the species formed in aliphatic flames have mainly planar

structures, and the contribution of C2H2 for this case is significant. These results are

in agreement with the findings reported by Homann et al. [20, 3]. The authors [20]

determined experimentally the C/H ratios in both acetylene and benzene flames. In

the case of acetylene flames, the results confirmed that soot nucleation starts with

the formation of PAH structures, while further growth results in the formation of

PAH with unique structures of totally condensed hexagons. Their investigation of

benzene flames [3] shows a variety of possible PAH structures possessing different

C/H ratios. An important conclusion is that either in benzene or acetylene flames

the observed C/H ratios of the hydrocarbon species formed considerably differ from

those that would be observed in the case of fast polymerisation of polyynes.

Frenklach et al. [21, 105] introduced the hypothesis of chemical similarity, confirming

that the surface of soot particles is assumed to look like the edge of a large PAH

molecule covered with C/H bonds. Abstraction of H atom from the surface releases

active sites by forming surface radicals. Richter et al. [57, 24] also described the

formation of active sites on PAH molecules as a result of H atom abstraction. These

sites provide a chemical basis for reactive coagulation of PAH compounds with each

other and with small radicals.

Recent experimental work of Oktem et al. [4] confirmed that the freshly nucleated

soot is built primarily of PAH. The authors observed that the soot nucleation and

the mass growth at the early stage are dominated by the aromatics, and that the

reaction of aliphatic species with pre-existing soot surface must be an important

factor at the later stages of soot mass growth.


The goal of the proposed work was the development of a detailed kinetic model of

soot formation based on the comprehensive models of PAH formation and growth

[57, 24, 23, 42, 201]. The concepts of soot particle nucleation was described through

a combination of different pathways of PAH formation and growth, soot particle

inception published in [52, 24, 199, 200, 160], and the traditional HACA route

of PAH and soot particle growth [21, 1, 23]. The model was initially developed

and applied for soot formation simulation in shock-tube pyrolysis and oxidation of

toluene, n-heptane, and toluene/alcohol mixtures [188]. It was further extended

and applied for soot formation simulation of methane, ethylene, benzene and n-

heptane pyrolysis and oxidation of methane and propane in conditions typical of

shock tube experiments. The main idea taken from the models listed above is that

the particle mass increases by their reactions with the gaseous species simultaneously

with particle size growth by collision reactions among the PAH species.

5.3.1 Gas-phase reaction mechanism

The developed kinetic model consists of a gas phase reaction mechanism, which

describes the pyrolysis and oxidation of the parent hydrocarbons, and the formation

and growth of PAH through different reaction pathways up to coronene. The gas-

phase reaction mechanism is a combination of the recently evaluated mechanism of

C1-C4 hydrocarbon oxidation [190], the C2 mechanism of Appel et al. [23], and a

set of reactions of C3-, C5-,C6- and C7-hydrocarbons presented in [57, 201, 189].

The formation and growth of polycyclic aromatic hydrocarbons is based on the

HACA model evaluated for laminar premixed acetylene and ethylene flames [1, 46]

with all modifications presented in [23]. The PAH nomenclature from [1, 46, 23]

was adopted in the current mechanism. The abbreviation Ai was use to describe

a PAH, where the index i denotes the number of benzene rings included in the

PAH molecule. Additional reaction paths of PAH formation and growth for PAH

between benzene and pyrene (A1-A4) published in [24], were included in the reaction

scheme. The large polycyclic aromatics up to coronene (A4-A7) were introduced by

several reaction paths, adopted from [24]. As a result, the following pathways of

PAH formation and growth were incorporated in the reaction mechanism:

• the alternating H-abstraction/C2H2-addition (HACA) route, resulting in a suc-

cessive growth of PAH,

• the combination reactions of phenyl with C6H6,

• the cyclopentadienyl recombination,


• the ring-closure reactions of aliphatic hydrocarbons,

• the recombination reactions between AiR and aliphatic radicals (R is a small

aliphatic radical).

The reaction mechanism contains approximately 2350 direct and reverse elemen-

tary reactions between 230 different species. The corresponding thermodynamic

data for all gas-phase species was taken from the available literature sources of the

mechanisms cited above and the thermodata database of Burcat and Ruscic [202].

The main differences between the gas-phase mechanism of Model-2 and the gas-

phase mechanism of Model-1 (see Sections 5.1.1 and 5.1.2 in this Chapter) are as

follows:

• A set of reactions of polyyne hydrogenation and decomposition [42, 43] was in-

cluded in Model-2. This strategy provided different behaviour of the polyynes,

whose concentration decreases approximately by one order of magnitude with

increasing number of carbon atoms in the polyyne molecule. The higher

polyynes (C10H2 and C12H2) were excluded from the scheme because of lack

of thermodynamic data.

• Numerous reactions with participation of aliphatic hydrocarbons (C3, C5, C6,

C7, and C8) [57, 189, 201, 42, 43] leading to the formation of benzene, phenyl

and higher PAH were included in the model (the reactions in Chapter 3.1.1 -

3.1.3).

• The variety of reaction paths of PAH formation and growth included in Model-

2 made it possible to describe the process of soot formation in the case of

pyrolysis and oxidation of hydrocarbons with both aliphatic and aromatic

structures, without necessity to include the polyynes in the particle inception.

5.3.2 Soot precursor and particle inception, growth, coagu-

lation and oxidation

The formation, growth, oxidation and coagulation of soot precursors and soot par-

ticles are described with the use of the discrete Galerkin method [180, 68, 71],

described in Chapter 4. The large PAH were considered as soot precursors. Their

mean diameter has value between 1 nm and 2 nm. These values are in agree-

ment with the experiential measurements of young soot particle size reported in


[81, 82, 24, 203, 204]. The C/H ratios of the soot precursors formed during these

reactions are close to the upper boundary of the C-H diagram reported in [3]. In

the present model the precursors are formed mainly by radical-molecule reactions

of different PAH including species with five- and six-membered rings [57, 24, 52].

The radical-radical reactions play a minor role; therefore they were reduced to a

minimum. These reactions result in the formation of polyaromatic molecules (soot

precursors) containing from 24 to 46 carbon atoms, which are stabilised by the for-

mation of new chemical bonds. All reactions with participation of soot precursors

and particles are specified in Table 5.2. The precursors grow by the HACA mecha-

nism, where the initiation of PAH radicals occurs in H-abstraction reactions with H

and OH radicals described in details in [24]. Termination reactions with H atoms,

H2, and H2O molecules were also included in the model, whose rate coefficients were

adopted from the literature [24]. The reaction flow analysis showed that more than

70 % of the surface growth is due to C2H2 addition by the reaction described in

the model of Appel et al. [23], with the same rate coefficient. Species like C2H2,

C4H2, C6H2, and different PAHs are measured in the pyrolysis and oxidation of

aliphatic and aromatic hydrocarbons [35, 36, 5], and are found in rather high con-

centrations. Therefore, growth reactions with the participation of those species,

and various PAH molecules and their radicals, were also included in the model. The

particle mass increases by their reactions with the gaseous species, simultaneously

with particle-size growth by collision reactions among the PAH species. Soot pre-

cursors are oxidised by O and OH radicals and transformed into soot particles in a

first-order reaction of internal conversion, with the formation of new chemical bonds

[58, 65]. In this transformation reaction, the number of active sites in the reacting

system is preserved. The active soot particles grow by interactions with C2H2, C4H2,

and C6H2, and with PAH molecules and radicals. All types of particles participate

in coagulation reactions for the case of a free-molecular regime. The soot-particle

oxidation takes place in reactions with O and OH radicals [24]. All reactions with

microheterogeneous particles and the respective rate coefficients are listed in Table

2. The rate coefficients were taken from the literature [23, 24, 61]. They were chosen

to coincide with the reaction classes and the size of the PAH involved in a particular

reaction. No investigation of the thermodynamic properties of the aerosol structures

in this model has been done. Therefore, forward and reverse reactions were written

separately.


Table 5.3: Mechanism of formation, surface growth, coag-

ulation, oxidation and transformation of soot precursors

and soot particles (Model-2)



Molecule-radical reactions

A2C2HA + A2C2HA* → C-H[1] + H 2.0E+12 0.5 0.0 24 (c)

A2C2HB + A2C2HB-3 → C-H[1] + H 2.0E+12 0.5 0.0 24 (c)

A2R5 + A2R5- → C-H[1] + H 2.0E+12 0.5 0.0 24 (c)

P2 + A2R5- → C-H[1] + H 2.0E+12 0.5 0.0 24 (c)

P2 + A2C2HA* → C-H[1] + H 2.0E+12 0.5 0.0 24 (c)

P2 + HA2R5 → C-H[1] + H 2.0E+12 0.5 0.0 24 (c)

BIPHEN + BIPHENH → C-H[1] + H 2.0E+12 0.0 0.0 24 (c)

A3R5 + INDENYL → C-H[1] + H 2.0E+12 0.0 0.0 25 (c)

A2R5HA + A2R5YNE3-4 → C-H[1] + H 2.0E+12 0.0 0.0 26 (c)

P2 + A2R5YNE3-4 → C-H[1] + H 2.0E+12 0.0 0.0 26 (c)

P2 + A3-1 → C-H[1] + H 2.0E+12 0.0 0.0 26 (c)

P2 + A3-4 → C-H[1] + H 2.0E+12 0.0 0.0 26 (c)

A3 + P2- → C-H[1] + H 2.0E+12 0.0 0.0 26 (c)

P2 + A3R5-7 → C-H[1] + H 2.0E+12 0.0 0.0 28 (c)

A2R5YNE1 + A2R5YN3-4 → C-H[1] + H 2.0E+12 0.0 0.0 28 (c)

A2R5YNE3 + A2R5YN3-4 → C-H[1] + H 2.0E+12 0.0 0.0 28 (c)

A2R5YNE1 + A2R5YN4-3 → C-H[1] + H 2.0E+12 0.0 0.0 28 (c)

A2R5YNE4 + A2R5YN3-4 → C-H[1] + H 2.0E+12 0.0 0.0 28 (c)

A2R5YNE5 + A2R5YN3-4 → C-H[1] + H 2.0E+12 0.0 0.0 28 (c)

A3R5 + A2R5- → C-H[1] + H 2.0E+12 0.0 0.0 28 (c)

A3C2H + A2R5- → C-H[1] + H 2.0E+12 0.0 0.0 28 (c)

A3 + A3L-1 → C-H[1] + H 2.0E+12 0.0 0.0 28 (c)

A3 + A4- → C-H[1] + H 2.0E+12 0.0 0.0 30 (c)

P2 + A4- → C-H[1] + H 2.0E+12 0.0 0.0 30 (c)

A4 + P2- → C-H[1] + H 2.0E+12 0.0 0.0 30 (c)

A3 + A3C2H2 → C-H[1] + H 2.0E+12 0.0 0.0 30 (c)

A3R5 + A3R5-7 → C-H[1] + H 2.0E+12 0.0 0.0 32 (c)

A3LR5 + A3LR5-S → C-H[1] + H 2.0E+12 0.0 0.0 32 (c)

A4 + A4- → C-H[1] + H 2.0E+12 0.0 0.0 32 (c)

A4 + A4- → C-H[1] + H 2.0E+12 0.0 0.0 32 (c)

A5 + A3R5-10 → C-H[1] + H 2.0E+12 0.0 0.0 32 (c)

A3LR5 + A5- → C-H[1] + H 2.0E+12 0.0 0.0 32 (c)



A4 + A5- → C-H[1] + H 2.0E+12 0.0 0.0 36 (c)

A4 + A6- → C-H[1] + H 2.0E+12 0.0 0.0 36 (c)

A5 + A4- → C-H[1] + H 2.0E+12 0.0 0.0 36 (c)

A6 + A3R5-10 → C-H[1] + H 2.0E+12 0.0 0.0 38 (c)

A6 + A3R5-10 → C-H[1] + H 2.0E+12 0.0 0.0 38 (c)

A6 + A4- → C-H[1] + H 2.0E+12 0.0 0.0 38 (c)

A5 + A5- → C-H[1] + H 2.0E+12 0.0 0.0 40 (c)

A7 + A3R5-10 → C-H[1] + H 2.0E+12 0.0 0.0 40 (c)

A7 + A4- → C-H[1] + H 2.0E+12 0.0 0.0 40 (c)

A5 + A6- → C-H[1] + H 2.0E+12 0.0 0.0 42 (c)

A6 + A5- → C-H[1] + H 2.0E+12 0.0 0.0 42 (c)

A6 + A6- → C-H[1] + H 2.0E+12 0.0 0.0 44 (c)

A7 + A5- → C-H[1] + H 2.0E+12 0.0 0.0 44 (c)

A7 + A6- → C-H[1] + H 2.0E+12 0.0 0.0 46 (c)

Radical-radical reactions

A3R5-7 + A2R5- → C-H[1] + H 2.0E+12 0.0 0.0 ?? (c)

A3R5-7 + A3R5-10 → C-H[1] + H 2.0E+12 0.0 0.0 32 (c)

A4- + A4- → C-H[1] + H 2.0E+12 0.0 0.0 32 (c)

A5- + A5- → C-H[1] + H 2.0E+12 0.0 0.0 40 (c)

A5- + A6- → C-H[1] + H 2.0E+12 0.0 0.0 42 (c)

A6- + A6- → C-H[1] + H 2.0E+12 0.0 0.0 44 (c)

Soot precursors activation-deactivation

C[N] + H → C-H[1] 0.2E+14 0.0 0.0 (d)

CH[N ] + H → C[N ] + H2 0.417E+14 0.0 54.34 (d)

CH[N ] + 0H → C[N ] + H2O 0.18E+14 0.0 19.14 (c)

C[N ] + H2 → C-H[N ] + H 3.9E+11 0.0 45.98 (e)*

C[N ] + H2O → C-H[N ] + OH 0.199E+11 0.0 43.98 (c)*

Growth of soot precursors

C[N ] + C2H2 → C[N+1] + H 8.0E+07 1.56 15.9 2 (d)

C[N ] + C4H2 → C[N+1] + H2 4.0E+13 0.0 50.2 4 (e)

C[N ] + C6H2 → C[N+1] + H2 4.0E+13 0.0 33.5 6 (e)

C[N ] + A3 → C[N+1] 0.12E+13 0.0 18.03 14 (c)

C[N ] + A4 → C[N+1] 0.12E+13 0.0 18.03 16 (c)

C[N ] + A5 → C[N+1] 0.12E+13 0.0 18.03 20 (c)

C[N ] + A6 → C[N+1] 0.12E+13 0.0 18.03 22 (c)

C[N ] + A7 → C[N+1] 0.12E+13 0.0 18.03 24 (c)

C[N ] + A3- → C[N+1] 0.18E+14 0.0 0.46 14 (c)

C[N ] + A4- → C[N+1] 0.18E+14 0.0 0.46 16 (c)



C[N ] + A5- → C[N+1] 0.18E+14 0.0 0.46 20 (q)

C[N ] + A6- → C[N+1] 0.18E+14 0.0 0.46 22 (c)

Oxidation of soot precursors

C[N+1] + OH → C-H[N ] + CO + H 0.659E+13 0.5 0.0 1 (c)

C[N+1] + O → C-H[N ] + CO 0.672E+13 0.5 0.0 1 (c)

C[N+1] + OH → C[N ] + CO + H 0.659E+13 0.5 0.0 1 (c)

C[N+1] + O → C[N ] + CO 0.672E+13 0.5 0.0 1 (c)

Coagulation of soot precursors

C[N ] + C[M ] → C[N+M ] 4.50E+12 0.5 0.0 (e)

C-H[N ] + C-H[M ] → C-H[N+M ] 4.50E+12 0.5 0.0 (e)

C[N ] + C-H[M ] → C[N+M ] 4.50E+12 0.5 0.0 (e)


C[N ] → S[N ] 5.0E+04 0.0 0.0 (e)*

C-H[N ] → S-H[N ] 5.0E+04 0.0 0.0 (e)*

Reactions of soot particles

Soot particles activation-deactivation

S[N ] + H → S-H[N ] 0.2E+14 0.0 0.0 (d)

S-H[N ] + H → S[N ] + H2 0.2E+15 0.0 54.34 (c,d)

S-H[N ] + OH → S[N ] + H2O 0.18E+15 0.0 19.14 (c)

S-H[N ] + H2 → -H[N ] + H 3.9E+11 0.0 45.98 (e)*

S[N ] + H2O → S-H[N ] + OH 0.199E+12 0.0 43.98 (c)*

Growth of soot particles

S[N ] + C2H2 → S[N+1] + H 1.2E+09 1.56 15.88 2 (c,d)*

S[N ] + C4H2 → S[N+1] + H2 4.0E+13 0.0 50.2 4 (e)

S[N ] + C6H2 → S[N+1] + H2 4.0E+13 0.0 33.5 6 (e)

S[N ] + A7 → S[N+1] + H 0.12E+14 0.5 0.0 24 (c)

S[N ] + A6 → S[N+1] + H 0.12E+14 0.5 0.0 22 (c)

S[N ] + A5 → S[N+1] + H 0.12E+14 0.5 0.0 20 (c)

S[N ] + A4 → S[N+1] + H 0.12E+14 0.5 0.0 16 (c)

S[N ] + A3 → S[N+1] + H 0.12E+14 0.5 0.0 14 (c)

S[N ] + A6- → S[N+1] + H 0.18E+14 0.5 0.0 22 (c)

S[N ] + A5- → S[N+1] + H 0.18E+14 0.5 0.0 20 (c)

S[N ] + A4- → S[N+1] + H 0.18E+14 0.5 0.0 16 (c)

S[N ] + A3- → S[N+1] + H 0.18E+14 0.5 0.0 14 (c)

Oxidation of soot particles

SH[N+1] + OH → SH[N ] + CO + H 0.659E+14 0.5 0.0 1 (c)

SH[N+1] + O → SH[N ] + CO 0.672E+14 0.5 0.0 1 (c)

S[N+1] + OH → S[N ] + CO + H 0.659E+14 0.5 0.0 1 (c)



S[N+1] + O → S[N ] + CO 0.672E+14 0.5 0.0 1 (c)

Coagulation of soot particles

S[N ] + S[M ] → S[N+M ] 4.50E+12 0.5 0.0 (e)

SH[N ] + SH[M ] → SH[N+M ] 4.50E+12 0.5 0.0 (e)

S[N ] + SH[M ] → S[N+M ] 4.50E+12 0.5 0.0 (e)

(a) Rate coefficients are expressed by the Arrhenius equation (k = A ·T nexp(−EA/RT )) in cm3mol−1s−1, where A (cm3,mol, s), T (K), and EA (KJ/mol).

(b) Index N denotes the number of carbon atoms, which are incorporated into a

particle in each act of interaction with carbon-containing gas-phase species.

(c) Rate coefficients were adopted from the mechanism of Richter et al. [24].

(d) Rate coefficients were adopted from the mechanism of Appel et al. [23].

(e) Rate coefficients were adopted from Vlasov and Warnatz [58].

* The rate coefficient was modified to provide better representation of the experi-

mental results.

Particles with active sites on the surface were expressed as C[N ] (soot precursors)

and S[N ] (soot particles), CH[N ] and SH[N ] denote the particles without active

sites.

The notation S[N ] shows the concentration of soot particles with active sites after

N acts of interaction with various carbon-containing gas-phase species.

5.4 Results Model-2

5.4.1 Validation of the model

To validate the reaction mechanism (Model-2), the experimentally measured con-

centration profiles of various gas-phase species formed in pyrolysis and oxidation of

different systems were simulated. An integral reaction flow analysis and a global

sensitivity analysis were performed for the case of toluene and n-heptane oxidation.


The experimentally observed and simulated concentration profiles of H atoms, mea-

sured in benzene [205] and phenol [206] thermal decomposition behind reflected

shock waves are presented in Figure 5.38. The kinetic model adequately describes

the concentration profiles and predicts the longer induction delay times at the lower

temperatures.

Figure 5.38: Experimentally measured (closed symbols) and calculated results (opensymbols and lines) of the time-resolved concentration profiles of H atoms measured duringshock-tube pyrolysis of C6H5OH [206] and C6H6 [205].

Vaudevan et al. [207] measured the OH radical concentration during toluene oxi-

dation behind reflected shock wave and extracted the induction delay time values

from the OH profiles. Several concentration profiles obtained in toluene and n-

heptane oxidation were simulated with Model-2 and presented in Figures 5.39, 5.40

and 5.41. The model clearly predicts the shorter ignition delay times with the tem-

perature increase. Following the observations of Vaudevan et al. [207], three regions

can be distinguished in the OH curves. The first region shows a slight increase in

the OH concentration, followed by the appearance of an intermediate plateau at

low temperatures due to the slower toluene decomposition. In the second region,

the OH concentration rises rapidly due to the chain branching and propagation.

In the third region, the rate of formation of OH comes closer to zero. The reac-

tion flow analysis shows that during toluene oxidation the OH radicals are formed

mainly in the reactions H + O2 = O + OH (51%), HO2 + H = OH + OH (21%) and

O + H2 = H + OH (11%), and are consumed in the reactions CO + OH = CO2 + H

(44%) and OH + H2 = H + H2O (19%). The first reaction (H + O2 = O + OH) was

investigated in details by several authors [208, 43, 209, 210]. This reaction is reported


as the basic chain-branching step in the high-temperature combustion mechanism of

C1-C4 hydrocarbon combustion [190], which is also part of the Model-2. The OH

profiles formed in toluene and n-heptane oxidation are shown in Figure 5.41. The

coincidence between the experimentally measured and calculated OH mole fraction

profiles is notable for the toluene oxidation (open squares). In contrast, the model

prediction underestimates the OH concentration in the case of n-heptane oxidation

(open triangles). In the experimental data, a quasistationary level of the OH con-

centration occurs at approximately 300 µs and 800 µs in n-heptane and toluene

oxidation. The effect is reasonably reproduced by the model and probably caused

by the different times of attaining the maximal concentration of CO which consumes

the OH radicals: 232.8 µs for n-heptane and 678.5 µs for toluene [207].

Kern et al. [35] experimentally measured the product concentration profiles during

pyrolysis of toluene, benzene and acetylene with a time-of-flight mass spectrome-

ter. The main products detected from the reflected shock zone were acetylene and

several polyynes (Figures 5.42, 5.43, and 5.44). The pyrolysis of 3.2 % C2H2 was

investigated at a temperature 2030 K and a pressure 0.39 bar. The comparison

between the experimentally measured and calculated profiles showed a very good

agreement for the three measured species (Figure 5.42). The authors stated that

the most important observation in this set of experiments are the concentration

Figure 5.39: Concentration profiles of OH radicals measured (closed symbols) [207] andcalculated (open symbols and lines) during toluene oxidation: φ = 1, 0.1 % C6H5CH3, 0.9% O2, (circles) T = 1689 K, and (triangles) T = 1586 K and p = 1.9 bar.


Figure 5.40: Concentration profiles of OH radicals measured (closed symbols) [207] andcalculated (open symbols and lines) during toluene oxidation: φ = 1, 0.025 % C6H5CH3

+ 0.225 % O2, (triangles) T = 1783 K, p = 1.84 bar; (inverse triangles) T = 1700 K,p = 1.89 bar; (squares) T = 1648 K, p = 2.03 bar; (diamonds) T = 1607 K p = 2.03 bar.

Figure 5.41: Concentration profiles of OH radicals measured (closed symbols) [207] andcalculated (open symbols and lines) during n-C7H16 and C6H5CH3 oxidation: (triangles)130 ppm n-C7H16, T = 1640 K, p = 2.0 bar; (squares) 1250 ppm C6H5CH3, T = 1648 K,p = 2.0 bar.


Figure 5.42: Concentration profiles of the main gas-phase species measured [35] andsimulated in pyrolysis of 3.2 % C2H2/Ne/Ar mixture at T = 2030 K and p = 0.39 barbehind reflected shock waves: (squares) C2H2, (triangles) C4H2 · 2, (inverse triangles)C6H2 · 10.

plateaus observed for the investigated species during the 800 µs of observation time

at temperature exceeding 2000 K.

Interestingly, the same species were also detected as the main gas-phase products

in the case of high temperature benzene pyrolysis. In Figure 5.43, the calculated

benzene decay profile is plotted versus time, together with the experimental data

[36]. The model overestimates the consumption of C6H6 during the first 200 µs. The

simulated concentration profiles of C2H2, C4H2, and C6H2 (Figure 5.44) follow the

same tendency with the respective experiments. Their concentration decreases with

increasing the number of C atoms in the polyyne molecule, but there is still some

difference between the calculated results and the experimentally obtained values.

The total carbon concentration and the main gas-phase species, measured [35] and

simulated during pyrolysis of a toluene/neon mixture (1.8% C6H5CH3, T = 1900 K,

and p = 0.4) behind reflected shock waves are presented in Figures 5.45 and 5.46.

The simulated total balance of carbon atoms present in the reaction mixture and

the toluene decay profile are in good agreement with the experimentally measured

values. However, the concentration profiles of the gas-phase species C2H2, C4H2,

and C6H2 are underpredicted at longer reaction times.


Figure 5.43: Concentration decay profiles of the fuel molecule measured (closed symbols)[36] and simulated (open symbols and lines) during pyrolysis of 2.1 % C6H6 diluted in(99% Ne-1% Ar) mixture at T = 2190 K, p = 0.52 bar.

Figure 5.44: Concentration profiles of the main gas-phase species measured (closed sym-bols) [36] and simulated (open symbols and lines) during pyrolysis of 2.1 % C6H6 atconditions as in Figure 5.43.

Another study of the thermal decomposition of toluene was presented by Colket

and Seery in [211]. The authors investigated experimentally and theoretically the

rich gas-phase chemistry occurring during toluene pyrolysis. The experiments are

performed in a single-puls shock tube for temperatures 1200 K to 1850 K, pressure


Figure 5.45: Fuel-decay concentration profiles measured (closed symbols) [35] and simu-lated (open symbols and lines) during pyrolysis of 1.8 % C6H5CH3 at T = 1900 K, p = 0.4bar.

Figure 5.46: Concentration profiles of the main gas-phase species measured (closed sym-bols) [35] and simulated (open symbols and lines) during pyrolysis of 1.8 % C6H5CH3 atconditions as in Figure 5.45.

10.013 atm and residence time 600 µs. The pyrolytic products are analysed with a

gas-chromatography technique. In the following Figures (5.47 - 5.52), the toluene

decomposition is shown together with the concentration profiles of hydrogen and


Figure 5.47: Experimentally measured [211] concentration of several aliphatic hydrocar-bons and the fuel decay profile detected during pyrolysis of 1 % toluene at pressure 10.013bar and reaction time 600 µs.

Figure 5.48: Concentration profiles of several aliphatic hydrocarbons and the fuel decaycalculated during pyrolysis of toluene at conditions as in Figure 5.47.

hydrocarbons from methane up to pyrene. For most of the investigated species, the

calculated results are in good agreement with the experimentally measured values.

Nevertheless, there are some discrepancies, especially at temperatures above 1600 K.

The consumption of toluene (A1CH3) is overpredicted (Figure 5.50) at T > 1600 K,

as well as the consumption of ethylbenzene (A1C2H5) in Figure 5.48 and indene in


Figure 5.49: Experimentally measured [211] concentration profiles of aromatic hydrocar-bons detected during pyrolysis of 1 % toluene at pressure 10.013 bar and reaction time600 µs.

Figure 5.50: Concentration profiles of aromatic hydrocarbons calculated during toluenepyrolysis at conditions as in Figure 5.49.

Figure 5.52. In the last figure, the phenanthrene (A3) concentration is overestimated,

although the concentration of its structural isomer anthracene (A3L) is close to

the measured values. This effect is caused by the slightly higher concentration

of biphenyl (Figure 5.52) which participates in the main chanel of phenanthrene

formation (see RFA diagram, Figure 5.53).


Figure 5.51: Experimentally measured [211] concentration profiles of various polycyclicaromatic hydrocarbons detected during pyrolysis of 1 % toluene at pressure 10.013 barand reaction time 600 µs.

Figure 5.52: Concentration profiles of various polycyclic aromatic hydrocarbons calcu-lated in toluene pyrolysis at conditions as in Figure 5.51.

Integral reaction flow analysis and sensitivity analysis were carried out for the case

of toluene and n-heptane oxidation at the maximum soot yield temperature. The

reaction flow diagrams (Figures 5.53 and 5.58) show the major reaction routes for

the formation of microheterogenoeus particle.

For the case of toluene oxidation (Figure 5.53), an argon-diluted mixture of 1.5 %

toluene and 1.5 % oxygen was studied at conditions typical of shock tube experiments

(T = 1900 K, p = 2.0 bar, and tr = 2.0 ms). According to these results, toluene is


Figure 5.53: Integral reaction flow analysis of the main pathways of soot precursor in-ception during C6H5CH3/O2/Ar oxidation at T = 1900 K, p = 2.0 bar, and reaction time2.0 ms.

Figure 5.54: Sensitivity analysis with respect to benzene during toluene oxidation atconditions as in Figure 5.53.

consumed in the reactions

A1CH2 + H = A1CH3(43%) (5.6)

A1CH3 + H = A1CH2 + H2(41%), (5.7)

A1CH3 + H = A1 + CH3(10%), (5.8)

where mostly benzyl and a small amount of benzene is formed. The same results

are also observed by Colket and Seery in [211]. Furthermore, benzyl is destructed

to phenyl, which gives benzene, in a hydrogenation reaction

A1CH2 + H = A1- + CH3. (5.9)


Figure 5.55: Sensitivity analysis with respect to ethynylnaphthalene radical (A2C2HB)during toluene oxidation at conditions as in Figure 5.53.

Figure 5.56: Sensitivity analysis with respect to acenaphthylene (A2R5) during tolueneoxidation at conditions as in Figure 5.53.

The phenyl radical is a key species for the PAH formation and growth in the case

of toluene oxidation. Once formed, it delivers benzene and benzyne to the system,

which start different reaction routes of PAH growth (see Figure 5.53). Phenyl hy-

drogenation is the major chanel of benzene formation (51 %), through a third body

reaction suggested by Appel et al. [23]. Another possible but less likely pathway in

this particular case is the propargyl radical recombination [23]. Benzyne (c-C6H4)

forms biphenylene, which further recombines to acenaphthylene (A2R5), a reaction

sequence suggested by Porter and Steifeld [212], Mebel et al. [213], and Richter

et al. [24]. Simultaneously, biphenyl is formed by the ring-ring condensation be-

tween benzene and phenyl, and starts an effective chanel of phenanthrene [23] and

acephenanthrylene (A3R5) production [24].

The sensitivity analysis (Figure 5.54), made with respect to benzene formation,


Figure 5.57: Sensitivity analysis with respect to acephenanthrylene (A3R5) during tolueneoxidation at conditions as in Figure 5.53.

showed that the rate-limiting steps are

A1CH3 + H = A1 + CH3, (5.10)

and

C2H2 + c-C5H5 = A1CH2. (5.11)

Both of them are described by Emdee et al. [214], Richter et al. [215] and Rasmussen

et al. [201], with the rate coefficients k = 1.2 · 1013exp(−21.55/RT ) in cm3mol-1s-1 ,

with EA in KJmol-1 and k = 1.73 · 1017T -1.89exp(−42.84/RT ) in cm3mol-1s-1 , with

EA in KJmol-1 respectively. The rate coefficient of the Reaction (5.11) has nega-

tive temperature dependence. Therefore, with increasing temperature the reverse

reaction will dominate, delivering C2H2 to the system.

The rate-determining reaction with respect to naphthalene and the ethynylnaphtha-

lene radical (5.55) is the formation of 1-phenyl-1,3-butadienyl (A1C4H4),

C3H3 + A1CH2 = A1C4H4 + H, (5.12)

suggested by Marinov et al. [216] and Rasmussen et al. [201], with k = 2.0 ·1012 cm3mol-1s-1. The phenylbutadienyl radical then forms naphthalene by

n-A1C4H4 = A2 + H, (5.13)

with a rate coefficient k = 1.0 · 1010 cm3mol-1s-1 [42]. Marinov et al. [216] and

Rasussen et al. [201] used a temperature dependent coefficient for that reaction

(k = 5.0 · 1037T -7.4exp(−322.08/RT ) in cm3mol-1s-1, with EA in KJmol-1). In the

present work, no additional investigation was carried out to determine the role of


the different rate coefficients of Reaction (5.13). Another way to form naphthalene

is the recombination reaction of benzyl and the propargyl radical

C3H3 + A1CH2 = A2 + 2H. (5.14)

This is an effective channel, which in the case of toluene pyrolysis is the fastest step

leading to naphthalene. The reaction is described also by Colket and Seery [211] in

their kinetic study of toluene pyrolysis in shock-tube experiments as the dominant

reaction for naphthalene production with k = 6.3 · 1011 cm3mol-1s-1. Rasmussen

et al. [201] used the same values, whereas Richter et al. [24] reported k = 3.0 ·1012 cm3mol-1s-1, suggested by D’Anna and Violi [217]. In the present mechanism

the value published by Colket and Seery [211] and Rasmussen et al. [201] was

implemented.

Although acenaphthylene is formed mainly by biphenylene recombination, the reac-

tion with highest sensitivity (Figure 5.56) is

C2H2 + A2-1 = A2R5 + H, (5.15)

with k = 1.9 ·1031T -5.26exp(−90.374/RT ) cm3mol-1s-1, with EA in KJmol-1 [23]. The

rate-determining step with respect to acephenanthrylene (Figure 5.57) formation is

C2H2 + A3-1 = A3R5 + H, (5.16)

with k = 1.83 · 1013T 0.295exp(−62.51/RT ) cm3mol-1s-1, with EA in KJmol-1 [24].

An integral reaction flow analysis and a global sensitivity analysis were performed

during n-heptane oxidation for a n-C7H16/O2/Ar mixture with [C] = 7.89 [mol/m3],

and φ = 5 behind shock wave (T = 1750 K, p = 25.0 bar and tr = 2.0 ms). The

reaction flow diagram (Figure 5.58) shows the major routes of benzene formation,

the PAH growth, and the first particle inception. According to these results, benzene

is generally formed in the reaction of two propargyl radicals, adopted from [23] with

k = 5.0 · 1012 cm3mol-1s-1. Several authors studied the kinetics of this reaction and

the possible products. Scherer [205], e.g., suggested that, depending on the product,

k may vary in a wide range (4.5 ·1012 ≤ k(2C3H3=C6H6) ≤ 9.0 ·1012cm3mol-1s-1). In the

present model, this reaction is written in both directions, whereas in the previously

described model (Model-1) it caused a grate reduction of the soot yield for the case

of acetylene pyrolysis. In the current model another important source of A1 is the

fulvene recombination [201]

C5H4CH2 = A1. (5.17)


Figure 5.58: Integral reaction flow analysis of the main pathways of soot precursor incep-tion during n-C7H16/O2/Ar rich oxidation, [C] = 7.89 [mol/m3], φ = 5, at T = 1750 K,p = 25.0 bar, and reaction time 2.5 ms.

Figure 5.59: Sensitivity analysis with respect to benzene during n-heptane oxidation atconditions as in Figure 5.58.

The rate-limiting reaction with respect to benzene is the ethynyl radical decompo-

sition

O2 + C2H = CH + CO2, (5.18)

as C2H together with the acetylene are the key species for the PAH formation and

growth [218, 219, 13, 220, 21, 193]. Benzene is oxidised to phenol or phenoxy radical.

The second one decomposes to cyclopentadiene and CO, where the cyclopentadiene

in reactions with O, H and OH is dehydrogenated to the cyclopentadienyl radical


Figure 5.60: Sensitivity analysis with respect to acenaphthylene (A2R5) during n-heptaneoxidation at conditions as in Figure 5.58.

Figure 5.61: Sensitivity analysis with respect to acephenanthrylene (A3R5) during n-heptane oxidation at conditions as in Figure 5.58.

c-C5H5. The recombination of two c-C5H5 radicals is an efficient channel of naphtha-

lene formation, with k = 5.0 · 1012exp(−33.49/RT ) cm3mol-1s-1 and EA in KJmol-1,

suggested by Richter et al. [24]. In the present reaction mechanism, the growth of

PAH follows several basic pathways. Naphthalene is produced except by the c-C5H5

recombination, Reactions 5.13 and 5.14, and also in the reaction

A1 −+C4H4 = A2 + H, (5.19)

which was adopted from the model of Appel et al. [23] with k = 3.3 ·1033T -5.7exp(−106.692/RT ) in cm3mol-1s-1 and EA in KJmol-1 .

The rate-limiting step with respect to acenaphthalene is

A2-1 + C2H2 = A2R5 + H, (5.20)

with k = 1.9 · 1031T -5.26exp(−90.37/RT ) cm3mol-1s-1 and EA in KJmol-1, Appel et

al. [23].


Benzyl and C2H2 react to form indene (see Figure 5.58) which starts the reaction

route to acephenanthrylene (A3R5). A significant amount of phenanthrene (A3) is

produced in from indene and c-C5H5. The rate-limiting steps for the acephenan-

thrylene formation are

A3-1 + C2H2 = A3R5 + H, (5.21)

c-C5H5 + INDENYLE = A3 + 2H, (5.22)

(see [24] and [201]) with the rate coefficients k = 1.83 · 1013T 0.295exp(−62.51/RT )

in cm3mol-1s-1, and k = 1.0 ·1013exp(−33.49/RT ) in cm3mol-1s-1 respectively, where

EA in both terms is in KJmol-1 . The same data was implemented in the current

scheme. The analysis showed a competition between A3R5 and its linear isomer

A3R5L, which influences the A3R5 concentration.

5.4.2 Hydrocarbon pyrolysis behind shock waves

Pyrolysis of ethylene

The temperature dependences of the soot yield and the induction delay time ob-

tained in the pyrolysis of C2H4/Ar at pressure 50.0 bar for several different mix-

tures [195] are presented in Figures 5.62 and 5.63. The calculated results are in

good agreement with the experimentally measured parameters. The model repro-

duces the typical bell-shape type of the soot-yield curve and the broadening effect

on the soot-yield decay with increasing the C-atom concentration in the mixture.

The simulated values of τ are in good agreement with the experimentally measured

ones for the low temperature range, whereas an increase in τ is observed for the

temperatures above the maximal soot yield. This effect is due to the negative tem-

perature dependence of the rate coefficients of many of the reactions leading to PAH

formation, which decreases the speed of formation of the corresponding species at

the higher temperatures and subsequently increases the τ of the soot particle incep-

tion. The same tendency was observed for the τ simulations of all systems studied

with the kinetic scheme Model-2.

Pyrolysis of methane and benzene

The calculated results for the soot yield temperature dependence were compared

with the experimentally measured data (Figure 5.64) in the case of CH4/Ar py-

rolysis at different pressure and carbon atom concentration. The experimentally


Figure 5.62: Temperature dependence of the experimentally measured (closed symbols)[195] and simulated (open symbols and lines) soot yield during pyrolysis of C2H4/Armixtures at p = 50.0 bar for three different C-atom concentrations: (circles) [C] = 7.4[mol/m3], (squares) [C] = 4.7 [mol/m3], (inverse triangles) [C] = 4.0 [mol/m3].

Figure 5.63: Experimentally measured (closed symbols) [195] and simulated (open sym-bols) induction delay time during pyrolysis of C2H4/Ar mixtures at p = 50.0 bar forthree different C-atom concentrations: (triangles) [C] = 7.4 [mol/m3], (circles) [C] = 4.7[mol/m3], (squares) [C] = 4.0 [mol/m3].

measured results were obtained for reaction time 1.5 ms, whereas the calculated

results were based on tr= 2.5 ms. A strong dependence of the soot yield on the

carbon atom concentration was observed. However, in the lean mixtures, the model


underestimates the soot yield and the induction delay time.

Figure 5.64: Temperature dependence of the soot yield measured (closed symbols) [195]and simulated (open symbols and lines) during pyrolysis of CH4/Ar mixtures for severaldifferent carbon-atom concentrations: (circles) p = 55.0 bar, [C] = 6.4 [mol/m3]; (squares)p = 55.0 bar, [C] = 3.4 [mol/m3]; (inverse triangles) p = 120.0 bar, [C] = 4.0 [mol/m3];(triangles) p = 25.0 bar, [C] = 3.0 [mol/m3]; (diamonds) p = 55.0 bar , [C] = 1.7 [mol/m3].

The experimental data [195, 81, 82] described in detail in Chapter 5.2.2 were chosen

to model the soot formation in the pyrolysis of benzene. In Figures 5.65 and 5.66,

the calculated values of the soot yield and the induction delay time during pyrol-

ysis of various C6H6 mixtures at elevated pressure 50 bar were compared with the

experimentally obtained data. The model fairly describes the effect of pressure and

carbon atom concentration on the soot yield although the maximum soot yield is

slightly shifted to the higher temperatures (approximately by 100 K). The shock-

tube measurements of Starke et al. [81, 82] are performed at a pressure of 1.2 bar

and a wide temperature range (1600 K - 2800K). The best agreement between the

calculated and the experimentally measured soot yield was observed for the lean

mixture (0.5 % C6H6). The model predictions reproduced the strong dependence of

the soot yield on the carbon atom concentration, but underestimated the soot yield

for the rich mixtures. In Figure 5.69, the mean particle radius (MPR) is plotted

versus temperature for three different C-atom concentrations which were measured

and calculated at a pressure 1.2 bar and a reaction time 1 ms. The time-resolved

MPR was also studied with Model-2. The simulated and measured results (T = 2000

K and p = 1.2 bar) are presented in Figure 5.70. The maximum MPR simulated


Figure 5.65: Temperature dependence of the experimentally measured (closed symbols)[195] and simulated (open symbols and lines) soot yield during pyrolysis of C6H6/Armixtures at p = 50.0 bar for four different C-atom concentrations: (circles) [C] = 4.0[mol/m3], (squares) [C] = 1.0 [mol/m3]; (triangles) [C] = 0.8 [mol/m3], (diamonds) [C] =0.4 [mol/m3].

Figure 5.66: Experimentally measured (closed symbols) [195] and simulated (open sym-bols) induction delay time during pyrolysis of C6H6/Ar mixtures at conditions as in Figure5.65.


Figure 5.67: Temperature dependence of the soot yield measured (closed symbols) [81]and simulated (open symbols and lines) during pyrolysis of C6H6/Ar mixtures at p =1.2 bar, tr = 1.3ms for three different reactive mixtures: (squares) 2 %, (circles) 1 %,(triangles) 0.5 %.

Figure 5.68: Induction delay time, measured (closed symbols) [81] and simulated (opensymbols) during pyrolysis of C6H6/Ar mixtures at conditions as in Figure 5.67.

with Model-2 is shifted by approximately 200 K (Fig. 5.69) and shows values sim-

ilar to the calculations performed with Model-1 (see Chapter 5.2.2). Both detailed

kinetic schemes (Model-1 and Model-2) underestimate the experimentally measured


Figure 5.69: Mean particle radius measured (closed symbols) [81] and calculated (opensymbols and lines) during pyrolysis of three different C6H6/Ar mixtures: (squares) 2 %,(circles) 1 %, (triangles) 0.5 %, for a fixed reaction time tr = 1.0 ms, at p = 1.2 bar.

Figure 5.70: Time history of the mean particle radius measured (closed symbols) [81] andcalculated (open symbols and lines) during pyrolysis of four different C6H6/Ar mixtures:(squares) 2 %, (circles) 1 %, (triangles) 0.5 %, (inverse triangles) 0.25 % at p = 1.2 bar.

values of the MPR. Nevertheless, the tendencies of increasing the particle size with

the increase of the carbon-atom concentration were satisfactory reproduced.


Pyrolysis of toluene

The soot formation during pyrolysis of various toluene/argon mixtures was exper-

imentally [221] and numerically studied in a shock tube (Figure 5.71) over a wide

temperature range (1600 K - 2400 K) and at a pressure 3.5 bar. The model de-

scribes the temperature dependences of the soot yield in a good way. A strong

influence of the carbon atom concentration on the soot yield was observed in both,

experimentally measured and calculated results.

Figure 5.71: Temperature dependence of the experimentally measured (closed sym-bols) [221] and calculated (open symbols and lines) soot yield during pyrolysis of threeC6H5CH3/Ar mixtures at tr = 2 ms: (triangles) 1.5 % C6H5CH3, p = 3.5 bar; (squares)1.0 % C6H5CH3, p = 3.3 bar; (diamonds) 0.5 % C6H5CH3, p = 2.5 bar.

Pyrolysis of n-heptane

The soot formation during pyrolysis of n-heptane has been recently studied in shock-

tube experiments, using a single-wavelength laser extinction method [222]. The

calculated results were compared with the experimentally measured data of the soot


Figure 5.72: Temperature dependence of the soot yield, measured (closed symbols) [222]and calculated (open symbols and line) during pyrolysis of 0.1 % n-C7H16 diluted in argonmixture at p = 20 bar.

Figure 5.73: Induction delay time, measured (closed symbols) [222] and calculated (opensymbols), during n-C7H16 pyrolysis at conditions as in Figure 5.72.

yield and the induction delay (Figures 5.72 and 5.73). The model fairly represents

the typical bell-shaped curve, the appearance of both the soot-yield maximum and

the induction delay time, but underestimates the values of the maximum soot yield.


5.4.3 Hydrocarbon oxidation behind shock waves

Oxidation of methane, propane, and n-heptane

The detailed kinetic scheme (Model-2) was applied for soot formation simulation

during CH4, C3H8, and n-C7H16 rich oxidation behind a reflected shock wave [198]. A

direct comparison of the simulations with the experimentally measured values of the

soot yield is presented in Figure 5.74. The simulated methane profile underestimates

the soot yield at the given conditions. The same problem was observed and discussed

in the case of methane oxidation (see Figure 5.64).

Figure 5.74: Temperature dependence of the soot yield measured (closed symbols) [198]and calculated (open symbols and lines) during rich oxidation of CH4, [C] = 7.6 [mol/m3];C3H8 [C] = 6.0 [mol/m3]; and n-C7H16 [C] = 5.9 [mol/m3] , at φ= 5 and p = 40 bar.

The experiments were performed at elevated pressures between 15 and 100 bar and

in a wide temperature range (1500 K - 2200 K). The time-resolved soot yield, the

mean soot particles diameter and the logarithm of soot particle number density

measured and simulated during n-heptane rich oxidation behind shock wave are

plotted in Figures 5.75, 5.76 and 5.77. The coincidence between the simulations and

the experiment is notable. Both experimental and theoretical investigations showed

that an increase of the pressure leads to a subsequent increase of the particle number

density and thus the amount of soot.

The temperature dependences of the experimentally measured [198] and calculated


Figure 5.75: Time-resolved soot yield measured (closed symbols) [198] and calculated(open symbols and line) during n-C7H16 rich oxidation, φ= 5, [C] = 7.89 [mol/m3], T =1750 K, p = 25 bar.

Figure 5.76: Time-resolved mean particle diameter measured (closed symbols) [198] andcalculated (open symbols and line) during n-C7H16 rich oxidation at conditions as in Figure5.75.

soot yield during rich oxidation of argon diluted n-heptane/oxygen mixtures are

presented in Figure 5.78. The model underestimates the soot yield at the highest

pressure, but adequately reproduces the dependence of the soot yield on the ini-


Figure 5.77: Time-resolved soot particle number density measured (closed symbols) [198]and calculated (open symbols and line) during n-C7H16 rich oxidation at conditions as inFigure 5.75.

Figure 5.78: Temperature dependence of the soot yield measured (closed symbols)[198] and calculated (open symbols and lines) during shock tube rich oxidation of ann-C7H16/O2/Ar at constant Ar concentration (0.3125 % C7H16, % O2, and 99 % Ar):(circles) p = 30 bar, (squares) p = 40 bar, (diamonds) p = 50 bar.

tial hydrocarbon concentration. The experimentalists observed that an increase of

pressure at constant carbon density leads to smaller particle sizes, even though soot


Figure 5.79: The pressure influence of the soot yield measured (closed symbols) [198] andcalculated (open symbols and line) during shock tube rich oxidation of an n-C7H16/Ar/O2

mixture, [C] = 5.8 [mol/m3], φ= 5 for three different pressures 20 bar, 40 bar and 80 bar.

yield is positively enhanced.

A weak pressure dependence of the soot yield was observed experimentally which is

even less in the calculated results (Figure 5.79), although, the pressure was varied

in a wide range.

Oxidation of toluene, toluene/methanol and toluene/ethanol mixtures

The influence of oxygen and oxygen-containing mixtures on soot formation is ex-

perimentally studied in [221]. The authors investigated several toluene/oxygen,

toluene/methanol and toluene/ethanol mixtures in a reflected shock. The simu-

lated results followed the same tendencies as in the experimental observations. The

addition of oxygen to toluene not only suppresses soot formation, but also shifted

the soot yield to lower temperature (Figure 5.80). However, the model overesti-

mates the soot yield, because the contribution of the surface oxidation reactions

with the chosen rate coefficients was very small in comparison to the oxidation in

the gas-phase.

The influence on soot by methanol and ethanol additives to toluene is shown in Fig-

ures 5.81 and 5.82, respectively. Both methanol and ethanol suppress soot formation,


and this effect becomes significant if methanol exceeds 50% and ethanol exceeds 70%

of the total fuel mixture to be pyrolysed. At high temperatures methanol suppresses

the soot formation more efficiently than ethanol.

Figure 5.80: Temperature dependence of the soot yield experimentally measured (closedsymbols) [221] and calculated (open symbols and lines) during oxidation of three differentC6H5CH3/O2/Ar mixtures at tr = 2 ms: (triangles) 1.5 % toluene, p = 3, 5 bar; (squares)1.5 % toluene + 1.5 % O2, p = 2.0 bar; (diamonds) 1.5 % toluene + 2.5 % O2, p = 2.0bar.


Figure 5.81: Temperature dependence of the soot yield experimentally measured (closedsymbols) [221] and calculated (open symbols and lines) during thermal decomposition ofC6H5CH3/Ar and C6H5CH3/CH3OH/Ar mixtures at tr = 2 ms: (triangles) 1.0 % toluene,p = 3, 3 bar; (squares) 1.0 % toluene + 1.0 % methanol, p = 2.7 bar; (diamonds) 1.0 %toluene + 2.0 % methanol, p = 2.7 bar.


Figure 5.82: Temperature dependence of the experimentally measured (closed symbols)[221] and calculated (open symbols and lines) soot yield during pyrolysis of C6H5CH3/Arand C6H5CH3/C2H5OH/Ar mixtures at tr = 2 ms: (triangles) 1.0 % toluene, p = 3, 3 bar;(inverse triangles) 1.0 % toluene + 1.0 % ethanol, p = 3.0 bar; (squares) 1.0 % toluene +2.0 % ethanol, p = 3.0 bar; (diamonds) 1.0 % toluene + 3.0 % ethanol, p = 3.0 bar.

114

Chapter 6

SIMPLIFIED MODEL OF SOOT

FORMATION

A direct CD simulation of a real three-dimensional technical system, using detailed

chemical mechanisms, is impossible, because it exceeds the available computer ca-

pacities. Therefore, a reduced reaction mechanism is needed, which as accurate as

possible describes the chemical reaction system using a smaller number of variables.

A variety of soot models, including simple empirical correlations, relating the amount

of the particles in the exhaust, to the engine operating parameters, the detailed de-

scriptions of the pre-particle chemistry, and the soot particle dynamics, have been

proposed for engine simulations. Tesner et al. [223] suggested a two-step mechanism

of soot formation, considering the formation of radical nuclei and the soot particles.

The model became significantly popular when several slightly modified versions were

applied to simulate different combustion problems [224, 225]. Hiroyasu et al. [226]

proposed a model in which the net rate of change in soot mass is calculated as the

difference between the rates of soot formation and oxidation. Due to its simplicity,

the model has been widely used in Diesel engine simulations [227, 228, 229]. Koll-

mann et al. [230] suggested a thermochemical model of soot-formation modeling in

a turbulent ethylene diffusion flame. The authors described the thermochemistry of

the flame in terms of a constrained equilibrium model. Soot formation is described

by means of three processes, nucleation, surface growth and oxidation. The author

introduced a temperature dependence of the surface growth by assuming an Arrhe-

nius expression for the growth rate with a given activation energy. Lindstedt [231]

proposed a four-step mechanism of soot formation, in which soot particle nucleation

occurs in reactions of gas-phase species, like benzene and acetylene. Particle growth

6. SIMPLIFIED MODEL OF SOOT FORMATION 115

is described by the process of acetylene addition to the soot particle mass, where

a surface-area function, corresponding to the rate coefficient, is implemented. Ox-

idation is introduced by a single reaction of soot particles with molecular oxygen,

and the soot particles growth is expressed by their coagulation. The formation and

oxidation are described by global reaction rates, and the coagulation rate is derived

from the collision theory. Belardini et al. [227], Kazakov and Foster [232] and [233]

used a modified version of the Lindstedt’s model, in which acetylene is assumed to be

both soot precursor and growth species, and applied it for Diesel-engine simulation.

Such simple models are usually valid only for certain geometries, and sometimes

their implementation in other codes is computationally expensive or not warranted

due to uncertainties in the other models involved into the global one.

In the present work, a semi-empirical model of soot formation [234, 235, 71, 236] was

modified for soot formation modeling of shock tube experiments, and implemented

into a program package for spatially homogeneous reaction system simulation HOM-

REA [6]). The complex process of soot formation is described in terms of several

global steps − nucleation, soot particle growth and coagulation, and particle oxida-

tion. For that purpose two differential equations are solved for the temporal change

of soot particle number density and the soot volume fraction [237].

The goal of the present work is to develop a model that connects the complex

(detailed) hydrocarbon combustion chemistry to such a simple two-equation soot

model and effectively describes the soot formation phenomenon. Such models are

suitable for implementation in the KIVA code of Diesel engine simulation [236] or

CFD codes of turbine combustion simulation. The modeling approach is based on

the implementation of the existing detailed gas-phase chemistry of C1-C7 hydro-

carbons combustion including the variety of reactions pathways of PAH formation,

growth, and oxidation described in details in Model-1 and Model-2 (see Chapter 5).

The simplified soot model has been calibrated by experimental shock-tube data for

methane, propane, n-heptane and toluene oxidation, which were previously studied

with the detailed kinetic schemes (Model-1 and Model-2). In the following Chapter,

the simplified model is described in detail, and the simulations are compared with

the experimentally measured results of the soot characteristics during hydrocarbon

oxidation.


6.1 Model description

The simplified model of soot formation is described in terms of two variables, in

particular the soot volume fraction (fV, the volume of soot/total volume), and the

soot concentration (CS in mol/cm3). The respective source terms initially presented

by Moss et. al [235] take the following form:

Temporal change of soot concentration

dCS

dt= α− β (6.1)

Temporal change of soot volume fraction

dfV

dt= δ + γ − ε (6.2)

6.1.1 The temporal change of soot concentration

The source equation for the temporal change of soot concentration is expressed by

the nucleation (α) and coagulation processes (β). A widely accepted theory is that

PAH molecules and their radicals are the soot particle precursors [13, 42, 43, 21, 1,

3, 38, 4, 135, 52]. Therefore, the process of soot particle inception is described by

means of the key step in the PAH formation − the formation of the first aromatic

ring in the system. Coagulation was introduced as a process of sticking of two

particles by which larger particles are formed.

Nucleation

Nucleation increases the number of soot particles, and thus, the soot concentration.

The key step for the soot particle inception was considered to be the propargyl

radical recombination (C3H3 + C3H3 = C6H6), which delivers the first aromatic ring

(benzene of phenyl) in the system. The reaction flow analysis confirmed that in

many aliphatic fuels this is the reaction producing the highest amount of benzene.

Therefore, the nucleation term has the form

α = αconst · C2C3H3

, (6.3)


where αconst is the rate coefficient of the gas-phase reaction according to the detailed

kinetic models, Model-1 and Model-2,

αconst = 5.0 · 106 m3

mol · s . (6.4)

Coagulation

Coagulation decreases the soot number density and diminishes the concentration.

In the current model, coagulation is described by a collision number β [25, 236] and

the soot particle concentration. This term is derived from the chemical equation −Csoot(N) + Csoot(M) = Csoot(N +M) and can be expressed as

β = βconst · C2S (6.5)

with a rate coefficient,

βconst = 1.0 · 109

(T

K

)1/2m3

mol · s . (6.6)

It was assumed in the model that 10 benzene molecules are needed to build one soot

particle. Therefore, the nucleation term α (Eq. 6.1) was reduced by a factor of 10

(a simplified approach for the complex rate calculation). Finally, Eq. (6.1) takes

the form,

dCS

dt=

α

10− β. (6.7)

It was assumed that, in an act of oxidation, a single soot particle is not completely

destroyed and the number of soot particles remains unchanged. Surface growth influ-

ences the particle size, but does not change the number of soot particles. Therefore,

the last two processes were not included in the term for the temporal change of soot

concentration.

6.1.2 The temporal change of the soot volume fraction

The temporal change of the soot volume fraction (dfV/dt) consists of the following

terms: nucleation, surface growth and oxidation. Nucleation and surface growth

increase the volume fraction whereas oxidation reduces it. Coagulation has no effect

on the volume fraction therefore; it was not included in the equation.


Nucleation

The nucleation process was described by means of the same gas-phase reaction of

propargyl radical recombination (see Eq. 6.3), and the nucleation term is

δ = δconst · C2C3H3

, (6.8)

with a rate coefficient,

δconst = αconst · VS(m3)

2

(mol)2 · s . (6.9)

Here, VS is the volume occupied by a soot particle, formed in the nucleation process,

VS =Msoot

ρsoot

m3

mol, (6.10)

where Msoot is the molar mass (kg/mol) of a soot particle (taken to be that of

C60H60) and ρsoot is the density of soot (1800 kg/m3).

Surface growth

Most of the soot in flames is produced as a result of the surface growth (Chapter 5).

The rate of this process depends on the concentration of the available carbon-bearing

species in the particle neighbourhood, on the temperature and on the reactivity of

the particle surface. Therefore, an empirical correlation between the specific surface

growth rates and the concentration of the growth species is needed. The specific

surface growth rate is the volume of the soot added to a particle per unit aerosol

surface area [230]. Previous investigations [161, 1, 24] and the reaction flow analysis,

made for different reaction systems with the use of the detailed chemistry of Model-1

and Model-2, confirmed that the process of acetylene addition to the soot particle

surface dominates the particle surface growth. According to that, the surface growth

term takes into account the attachment of C2H2 molecules onto the soot particle

surface (e.g., the reaction: Csoot + C2H2 = Csoot-H + H [1]. The source term is

derived from the gas kinetic theory and gives a first order growth law suggested in

the literature [87, 25, 236, 71],

γ = σSG · VG · ϑe · (fV,∞ − fV) /fV,∞1

s(6.11)

where

σSG = 1.4 · 10−3 = sticking coefficient of the growth species [71], in particular C2H2,


VG = mG

ρsoot= volume of an adsorbed growth species [m3],

mG = mass of the growth species [kg],

ρsoot = 1800 [kg/m3] = soot density [238, 239, 195, 80],

ϑe = PG√2πmGkBT

·Asoot =√

kBT2πmG

·NA·CG·Asoot = effusion velocity [(1/s m2) · (m2/m3)],

Asoot = π ·D2S ·NS = π ·

(6fv

πNS

)2/3

·NS = (36πCSNA)1/3 · f 2/3V = soot surface density

[m2/m3],

CG = concentration of an adsorbed growth species [mol/m3],

PG = n · kB · T = partial pressure of a growth species [Pa],

kB = R/NA = Boltzman’s constant [m2kg/s2K],

NA = 6.023 · 1023 = Avogadro’s constant [1/mol],

DS = 3

√6fv

πNS= soot diameter [m],

NS = soot particle number density [1/m3],

CS = soot particle concentration [mol/m3],

(fV,∞ − fV) /fV,∞ = empirically obtained term to preserve unrealistically high values

of fV [-], [71].

After the analytical preprocessing, Eq. (6.11) receives the form

γ = γconst · f 2/3V · C1/3

S ·N1/3A · (fV,∞ − fV) /fV,∞, (6.12)

where γconst is the rate coefficient

γconst =σSG

ρsoot

√NARmG

2π· (36π)1/3 · T 1/2 · CC2H2 . (6.13)

Finally, the surface growth term can be expressed by

γ = 58.79 · T 1/2 · CC2H2 · C1/3S · f 2/3

V · (fV,∞ − fV) /fV,∞. (6.14)

The maximum volume fraction fV,∞ can be estimated with an empirical approach.

A curve fit of the fV values is calculated from the experimental results of the soot

concentration measured in flames for numerous hydrocarbons [240, 71] for wide


range of operating conditions (pressure, temperature, and mixture composition). It

gives the maximum volume fraction as a function of the excess C-atom concentration

Csurplus(1018/cm3) in the reacting mixture,

fV,∞ = 1.45 · 10−6 · C1.7surplus. (6.15)

The excess C-atom concentration ( Csurplus) can be obtained by computing the excess

C atoms available over a critical C to O ratio[(

CO

)crit

]. This ratio is a specific

characteristic for every fuel and varies with the pressure [25, 67]. Accordingly, the

Csurplus can be calculated by [71],

Csurplus = NA ·(NC,fuel · [Cfuel]t0 −

(C

O

)

crit

·NO,oxidizer · [Ooxidizer]t0

), (6.16)

where NC,fuel represents the number of the C atoms in a fuel molecule, NO,oxidiser is

the number of O atoms in an oxidiser molecule, and [Cfuel]t0 and [Ooxidiser]to are the

initial concentrations of the fuel and the oxidiser, respectively. The disadvantage of

this approach in this particular case is that the Csurplus is measured in flames and no

detailed information about the(

CO

)crit

ratio for the investigated hydrocarbons was

found for shock-tube experiments.

Sojka [71] suggested an empirically obtained value for the maximum volume fraction,

fV,∞ = 1.02 · 10−5, (6.17)

obtained in n-heptane rich oxidation behind shock wave. For the calculations pre-

sented in this chapter, the same value was used. This enabled better coincidence

between the experimentally measured and the calculated results of the temporal

change of the soot yield, the particle diameter and the number density in the case

of n-heptane rich oxidation.

Oxidation

The oxidation term is calculated similarly to the surface growth, but in this

case the sticking of OH on the soot particle surface was taken into account in-

stead of C2H2. Only one type of oxidation reaction is considered in the model

(Csoot-C + OH → Csoot + CHO) [236]. The oxidation term then is given by

ε = εconst · f 2/3V · C1/3

S ·N1/3A . (6.18)

The related rate coefficient is calculated by

εconst =σOX

ρsoot

√NARmOH

2π· (36 · π)1/3 · T 1/2 · COH. (6.19)


In this expression mOH is the mass and COH is the molar concentration of the OH

radicals, respectively. The sticking coefficient of OH is σOX = 0.1 [236, 67]. For the

oxidation term can be obtained by

ε = 3.4 · 103 · T 1/2 · COH · f 2/3V · C1/3

S

1

s. (6.20)

The nucleation term for the temporal change of the soot volume fraction was reduced

by a factor of 10, analogous to Eq. (6.7), and finally Eq. (6.2) can be written as

dfV

dt=

δ

10+ γ − ε. (6.21)

6.1.3 Rate laws

The rate laws for the gas-phase species, included in the different terms, are influenced

by the soot formation. Therefore, additional terms for the temporal change of the

concentration of C3H3, C2H2, and OH were introduced in the model (Eq. 6.22 -

6.24),

dCC3H3

dt=

dCC3H3

dt− 2 · α, (6.22)

dCC2H2

dt=

dCC2H2

dt− γ

ρsoot

mC2H2NA

, (6.23)

dCOH

dt=

dCOH

dt− ε

ρsoot

mOHNA

. (6.24)

In this way, an interaction between the gas-phase and the particulate phase chem-

istry is taken into account during the entire simulation.

6.1.4 Soot quantities

The soot volume fraction (fV), the soot particle concentration (CS) and the particle

diameter (DS) are the three characteristics of soot formation usually measured by

the experimentalists. These three parameters are mutually dependent, and each of

them can be calculated if the values of the other two are known [67];

fV

CS

= NA · π6·D3

S . (6.25)

The soot particle diameter can be calculated by [67]

DS = 3

√6 · fV

π · CS ·NA

, (6.26)


In the current model, the particle shape was assumed to be spherical, and the particle

number density was calculated by [67]

NS =fV · ρsoot ·NA

MC

. (6.27)

Here, MC is the molar mass of a C atom and ρsoot is the soot particle density, taken

to be equal to 1.8 in g/cm3, the density of graphite [238, 239, 195, 80].

The soot yield expresses the fraction of carbon appearing as soot, which usually is

calculated by

SY =[C]soot

[C]total

, (6.28)

where [C]soot is the soot concentration

[C]soot = NS, (6.29)

and [C]total is the total carbon atom concentration in the system

[C]total = [Cfuel]initial ·NC,fuel ·NA. (6.30)

Here, [Cfuel]initial is the initial fuel concentration and NC,fuel is the number of C atoms

in a fuel molecule.

6.2 Results

Rich oxidation of methane, propane and n-heptane

The simplified soot model was applied to predict soot formation in methane,

propane, n-heptane and toluene oxidation for a wide range of operating conditions

(T =1600 K - 2300 K, and p = 2 bar - 40 bar) in shock-tube experiments [198]. The

gas-phase chemistry is modeled with the use of the two detailed kinetic schemes

described in Chapter 5 (Model-1 and Model-2). These mechanisms were validated

against the experimentally measured values of the ignition delay time, the soot yield,

the mean particle diameter, and the concentration profiles of different species dur-

ing pyrolysis and oxidation of various hydrocarbons in shock-tube experiments for

a wide range of operating conditions (see Chapter 5). The mechanism of C1-C4 hy-

drocarbon combustion, introduced in the model, is validated against experimental

measurements of the ignition delay time and flame velocity, measured with different

techniques (shock tube, reactive flow experiments and laminar flat flames) [190].


The simulations of the gas-phase chemistry and the simplified soot model are per-

formed simultaneously during the whole reaction time. The results obtained with

the use of the gas-phase chemistry of the detailed kinetic model (Model-1) are de-

scribed as SSM-1 and those performed with the detailed kinetic model (Model-2) as

SSM-2.

A direct comparison between the model predictions and the experimentally measured

soot characteristics soot yield, mean particle diameter and mean particle number

dencity in the case of n-heptane rich oxidation [198] is presented in Figures 6.1 -

6.3. Both models describe fairly well the time history of the soot yield (Figure 6.1).

The soot yield simulated with SSM-1 is in better agreement with the experiment

within the longer reaction time. This effect is caused by the higher acetylene con-

centration (see Figure 6.5). The SSM-2 slightly underestimates the experimentally

obtained soot yield at the longer reaction time (tr = 1 ms - 2.5 ms) because of the

slower surface growth, but better matches the initial data at the early stages of the

soot particle inception (tr = 0.0 ms - 0.5 ms). Similar results were observed for

the simulated particle mean diameter. Although both models (SSM-1 and SSM-

2) underestimate the experimentally measured particle diameter, the calculations

performed with SSM-2 show better agreement over the whole reaction time.

Figure 6.1: Time-resolved soot yield, measured (closed symbols) [198] and calculated(open symbols and lines) in n-C7H16 rich oxidation, [C] = 7.89 [mol/m3], φ= 5, T = 1750K, p = 25 bar and tr = 2.5 ms.

The time-dependent concentration profiles of the nucleation (C3H3), the surface


Figure 6.2: Time-resolved mean particle diameter, measured (closed symbols) [198] andcalculated (open symbols and lines) in n-C7H16 rich oxidation at conditions as in Figure6.1.

Figure 6.3: Time-resolved soot particle number density, measured (closed symbols) [198]and calculated (open symbols and lines) in n-C7H16 rich oxidation at conditions as inFigure 6.1.

growth species (C2H2) and the oxidation agent (OH), involved in the models, are

presented in Figures 6.4 - 6.6. According to them, the faster rise of the particle

size, observed with SSM-1 at tr = 0.0 ms - 0.5 ms, can be explained with the

faster nucleation rate, influenced by the higher C3H3 concentration. Although the

C2H2 concentration is considerably higher in the case of SSM-1, there is no big


Figure 6.4: Time-dependent profiles of the C3H3 concentration, calculated at the condi-tions as in Figure 6.1: (circles) SSM-1 and (diamonds) SSM-2.

Figure 6.5: Time-dependent profiles of the C2H2 concentration, calculated at conditionsas in Figure 6.1: (circles) SSM-1 and (diamonds) SSM-2.

influence on the soot particle diameter calculated with the SSM-1, which is even

slightly smaller in comparison to the results of SSM-2. The soot particle oxidation

usually takes place at longer reaction time; therefore in the current calculations the

effect of oxidation is minor. This process will be important in the case of flames or

Diesel-engine simulations.

The soot particle number density is similarly described by both models. It starts


Figure 6.6: Time-dependent profiles of the OH concentration, calculated at the conditionsas in Figure 6.1: SSM-1 (circles) and SSM-2 (diamonds).

Figure 6.7: Temperature dependence of the soot yield, measured (closed symbols) [198]and calculated (open symbols and lines) in rich oxidation of n-C7H16 [C] = 5.9 [mol/m3],φ= 5, p = 40 bar and tr = 2.5ms: (circles) experiment, (triangles) Model-1 and (diamonds)Model-2.

with a rapid increase of the particle size due to the fast nucleation at the beginning

of the reaction time. Then the particle concentration shows a smooth decrease

and reaches a plateau after tr = 1 ms due to the drop in the concentration of the

nucleation species (C3H3). The behaviour of the soot particle concentration curves


Figure 6.8: Temperature dependence of the soot yield, measured (closed symbols) [198]and calculated (open symbols and lines) in rich oxidation of C3H8 [C] = 6.0 [mol/m3], φ=5, p = 40 bar and tr = 2.5 ms: (circles) experiment, (triangles) Model-1 and (diamonds)Model-2.

Figure 6.9: Temperature dependence of the soot yield, measured (closed symbols) [198]and calculated (open symbols and lines) in rich oxidation of CH4 [C] = 7.6 [mol/m3], φ=5, p = 40 bar and tr = 2.5 ms: (circles) experiment, (triangles) Model-1 and (diamonds)Model-2.

follows the shape of the propargyl radical profiles (see Figure 6.4).


The temperature dependence of the soot yield was also simulated with the use of

the models described above (SSM-1 and SSM-2). In Figure 6.7, the experimentally

measured and calculated soot yield are compared for the case of n-heptane rich

oxidation ([C] = 5.9 mol/m3) at a pressure of 40 bar and in a temperature range

1600 K - 2300 K. In this case, both models shift the maximum soot yield towards

high temperatures (by about 200 K). Nevertheless, SSM-2 matches the maximum

soot yield very well, whereas the SSM-1 overestimates it by about 30 %. This effect

is caused by the higher rates of both nucleation and surface growth with respect

to the higher amount of the reactive species in the gas-phase surrounding of SSM-

1. Similar calculations were performed in the oxidation of argon-diluted methane

and propane mixtures. The results show the similar tendencies as in the case of

n-heptane oxidation (Figures 6.8 and 6.9).

Toluene oxidation

In the case of toluene oxidation (Figure 6.10), the temperature dependence of the

soot yield was simulated during oxidation of a low-pressure (toluene/O2 = 1/1)

mixture only with the SSM-2. Due to the fact that the detailed kinetic scheme

(Model-1) was not further extended to describe the chemistry of toluene, the Model-

1 was not considered in this simulations. Accordingly, the experimentally measured

[221] results were plotted against the numerical simulations performed with the

detailed kinetic model (Model-2) with the program package MACRON (see Chapter

5), and the SSM-2 with the modified version of the program HOMREA [237]. Both

models (Model-2 and SSM-2) overestimate the maximum soot yield. In the case of

Model-2, it is caused by the small rate of the soot particle oxidation assumed in the

soot submodel, as it was described in Chapter 5. The soot yield simulated with the

SSM-2 is highly overestimated, and the maximum soot yield is shifted by about 200

K in the high-temperature range. The same effect of shifting the maximum soot

yield towards the high temperatures was found also in the case of methane, propane

and n-heptane oxidation (Figures 6.7, 6.8 and 6.9) in the calculations performed with

both SSM-1 and SSM-2. It is caused by the rate of soot particle surface growth. A

reasonable way to avoid this problem can be the implementation of reverse surface

growth and the effect of the temperature dependence on the process.


Figure 6.10: Temperature dependence of the soot yield, measured (closed symbols) [221]and calculated (open symbols and lines) in oxidation of an C6H5CH3/O2/Ar mixture (1.5% toluene + 1.5 % O2), p = 2.0 bar and tr = 2 ms: (circles) experiment, (diamonds)SSM-1 (triangles) Model-2.

130

Chapter 7

Conclusion and future prospects

Two different rection mechanisms (Model-1 and Model-2) were developed and ap-

plied for soot formation simulation at homogeneous conditions. Both models differ

with respect to the concepts describing the gas-phase and the particulate-phase

chemistry.

Model-1 involves two different pathways of soot formation (polyyne and HACA),

which required the parallel development of two diverse reaction paths and the corre-

sponding gas-phase environment in the reacting mixture. Soot formation simulation

in a time-dependent system, e.g., a shock tube, needs the kinetic representation of

very fast processes that lead to soot particle nucleation at reaction times of several

milliseconds. The combination between the HACA pathway of PAH and soot parti-

cles formation and growth and the process of fast polymerisation of polyyne species

offered an efficient solution to the problem. Nevertheles, there are no thermody-

namic data and precise reaction kinetics of the higher polyynes (above C8H2).

Latest experimental studies [99, 3, 100, 4, 5] proposed the PAH as the preferable soot

precursors. Several authors [4, 5] suggested that the aliphatic compounds, including

some polyynes, may play an important role in the earlier stages of soot formation

at the PAH growth as well as in the soot particle mass growth. The role of the

polyynes in the soot formation is not yet clearly understood. They are believed to

be destroyed to smaller species at high temperatures. However, all these conclusions

need further experimental and theoretical investigations.

Following these observations, a new detailed kinetic model of soot formation (Model-

2) was developed. The polyyne molecules were included in several reaction channels

of aromatic molecules formation and in the soot particle surface growth. Particle

7. Conclusion and future prospects 131

inception occurs in radical-radical and radical-molecule reaction of PAH, includ-

ing polyaromatics containing five-membered rings, formed through various reaction

pathways. Model-2 was successfully applied for the soot formation simulation in

pyrolysis and oxidation of different hydrocarbons and their mixtures.

Due to the different nature of the soot precursors in Model-1 and Model-2, the soot

yield and the induction delay time simulated with Model-1 were slightly overesti-

mated in some of the cases, whereas Model-2 underestimated them at high tem-

peratures. Additional investigations and improvements are needed, which requires

the implementation of newly obtained kinetic and thermodynamic data of the exist-

ing reaction paths, additional pathways of PAH formation and growth, and deeper

investigation and understanding of the complexity of the soot formation process.

It is important to note that all numerical results were obtained while keeping the

reaction mechanisms and the entire sets of rate coefficients unchanged for all calcu-

lations performed with Model-1 and Model-2. The calculations were performed for

hydrocarbons with different chemical structures and hydrocarbon mixtures within a

wide range of conditions (with respect to temperature, pressure, mixture composi-

tion and reaction time). The comparison between the calculated and experimentally

measured results demonstrated qualitatively and in most cases quantitatively good

agreement for the usually measured parameters of soot formation. The temperature

dependence of the soot yield is described by a bell-shaped curve, usually situated

between 1500 k and 2500 K. This interval varies according to the type of the fuel

and the reaction conditions. The gas-phase soot precursors are formed at tempera-

tures above 1000 K. They go through a maximum at about 1600 K - 1700 K that

is followed by a subsequent decrease in their concentrations at higher temperatures

due to their thermal destruction and oxidation.

An empirical (two-equation) model was applied for soot formation simulation dur-

ing shock-tube oxidation of various hydrocarbons, with the use of the gas-phase

chemistry of two different detailed kinetic schemes (Model-1, and Model-2). The

gas-phase chemistry and the soot model were simultaneously calculated. The nu-

merical results confirmed that the soot formation strongly depends on the kinetic

representation of the gas-phase chemistry. The different kinetic schemes determined

a distinct behaviour of the gas-phase species chosen as soot precursors, growth or

oxidative reagents. The choice of the reaction kinetics and thermodynamic data

plays a crucial role for the quality of the reaction mechanism, and it is of great

importance to keep these data up to date with respect to the actual experimental

and theoretical results.

7. Conclusion and future prospects 132

In spite of the great effort in understanding the soot formation phenomenon and the

complex processes related to it, there are still lots of uncertainties in the gas-phase

chemistry of the soot precursor formation and growth, the transition between the

gas-phase and the particulate chemistry, and the soot particle surface chemistry.

For the optimisation of theoretical models and combustion facilities with respect to

pollutant formation, it is necessary to study all these processes in detail.

The models described in this work can be improved in the following directions:

• Experimental and theoretical investigations of new reaction channels of PAH

formation and growth.

• Implementation of newly evaluated kinetic and thermodynamic data.

• Validation of the model against experimentally measured data for the gas-

phase chemistry and the soot characteristics in flames.

• Implementation of the distribution of the active sites on the soot particle

surface.

• Deeper understanding and implementation of the mechanisms of particle in-

ception and growth, in particular the transition of the gas-phase to the liquid

and then to the particulate-phase.

• The simplified soot model needs the implementation of the effect of tempera-

ture on the soot particle nucleation and surface growth.

133

List of Figures

5.1 Concentration profiles of the main gas-phase species measured (closed sym-

bols) [35] and simulated (open symbols and lines) during pyrolysis of 3.2

% C2H2/Ne/Ar mixture, at T = 2030 K, and p = 0.39 bar behind re-

flected shock waves: (squares) C2H2, (triangles) C4H2 · 2, (inverse trian-

gles) C6H2 · 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.2 Concentration profiles of C6H6 decay measured (closed symbols) [36] and

simulated (open symbols and lines) for a mixture of 2.1 % C6H6, diluted

in argon at p = 0.52 bar for three different temperatures: (circles) 1704 K,

(triangles) 1942 K, (squares) 2192 K. . . . . . . . . . . . . . . . . . . . 51

5.3 Time history of C2H2 concentration measured (closed symbols) [36] and

simulated (open symbols and lines) at conditions as in Figure 5.2. . . . . . 52

5.4 Time history of C4H2 concentration measured (closed symbols) [36] and

simulated (open symbols and lines) at conditions as in Fig.5.2. . . . . . . 52

5.5 Integral reaction flow analysis of the PAH formation pathways during py-

rolysis of a 4.62 %C2H2/Ar mixture at T = 2000K, p = 6.0 bar, and

reaction time 0.003 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.6 Integral reaction flow analysis of the polyyne formation pathways during

pyrolysis of a 4.62% C2H2/Ar mixture at T = 2000 K, p = 6.0 bar, and

reaction time 0.003 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.7 Sensitivity analysis with respect to benzene during pyrolysis of acetylene

at conditions as in Figure 5.5. . . . . . . . . . . . . . . . . . . . . . . . 54

5.8 Sensitivity analysis with respect to phenanthrene during pyrolysis of acety-

lene at conditions as in Figure 5.5. . . . . . . . . . . . . . . . . . . . . . 54

. LIST OF FIGURES 134

5.9 Sensitivity analysis with respect to pyrene during pyrolysis of acetylene at

conditions as in Figure 5.5. . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.10 Sensitivity analysis with respect to C12H2 during pyrolysis of acetylene at


5.11 Integral reaction flow analysis of the PAH formation pathways during py-

rolysis of a 1.54 %C6H6/Ar mixture at T = 2000K, p = 6.0 bar, and

reaction time 0.003 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.12 Integral reaction flow analysis of the polyyne formation pathways during

pyrolysis of a 1.54% C6H6/Ar mixture at T = 2000 K, p = 6.0 bar, and

reaction time 0.003 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.13 Sensitivity analysis with respect to biphenyl during pyrolysis of benzene


5.14 Sensitivity analysis with respect to phenanthrene during pyrolysis of ben-

zene at conditions as in Figure 5.11. . . . . . . . . . . . . . . . . . . . . 58

5.15 Sensitivity analysis with respect to C12H2 during pyrolysis of benzene at

conditions as in Figure 5.12. . . . . . . . . . . . . . . . . . . . . . . . . 58

5.16 Sensitivity analysis with respect to C10H2 during pyrolysis of benzene at


5.17 Temperature dependence of the soot yield measured (closed symbols) [195]

and simulated (open symbols and lines) during pyrolysis of C2H2/Ar mix-

tures at p = 57.0 bar for three different C atom concentrations: (inverse

triangles) [C] = 3.8 [mol/m3], (circles) [C] = 1.7 [mol/m3], (squares) [C]

= 0.9 [mol/m3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.18 Induction delay time in measured (closed symbols) [195]and simulated

(open symbols and lines) during pyrolysis of C2H2/Ar mixtures at con-

ditions as in Fig. 5.17. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60




tures at p = 50.0 bar for three different C-atom concentrations: (inverse

triangles) [C] = 7.4 [mol/m3], (circles) [C] = 4.7 [mol/m3], (squares) [C] =

4.0 [mol/m3]. Open diamonds denote the calculated results for C2H6/Ar

mixture: p = 50.0 bar, [C] = 4.0 [mol/m3]. . . . . . . . . . . . . . . . . . 61

5.20 Induction delay time measured (closed symbols) [195] and simulated (open

symbols and lines) during pyrolysis of C2H4/Ar mixtures at p = 50.0 bar

for three different C-atom concentrations: (triangles) [C] = 7.4 [mol/m3],

(circles) [C] = 4.7 [mol/m3], (squares) [C] = 4.0 [mol/m3]. . . . . . . . . . 61

5.21 Temperature dependence of the soot yield measured (closed symbols)

[195] and simulated (open symbols and lines) during pyrolysis of CH4/Ar

mixtures for several different carbon atom concentrations: (circles) p =

55.0 bar, [C] = 6.4 [mol/m3]; (squares) p = 55.0 bar, [C] = 3.4 [mol/m3];

(inverse triangles) p = 120.0 bar, [C] = 4.0 [mol/m3]; (triangles) p =

25.0 bar, [C] = 3.0 [mol/m3]; (diamonds) p = 55.0 bar , [C] = 1.7 [mol/m3]. 63


symbols and lines) during pyrolysis of CH4/Ar mixtures at conditions as

in Figure 5.21. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63



tures at p = 50.0 bar, and tr = 1.5ms, for four different C atom con-

centrations: (circles) [C] = 4.0 [mol/m3], (squares) [C] = 1.0 [mol/m3];

(triangles) [C] = 0.8, (diamonds) [C] = 0.4 [mol/m3]. . . . . . . . . . . . 64


symbols) during pyrolysis of C6H6/Ar mixtures at conditions as in Figure

5.23. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64


[81, 82] and simulated (open symbols and lines) during pyrolysis of

C6H6/Ar mixtures at p = 1.2 bar, tr = 1.3ms, and for three different

reactive mixtures: (squares) 2 %, (circles) 1 %, (triangles) 0.5 %. . . . . . 65


5.26 Induction delay time, measured (closed symbols) [81, 82] and simulated

(open symbols) during pyrolysis of C6H6/Ar mixtures at conditions as in

Figure 5.25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.27 Mean particle radius measured (closed symbols) [81, 82] and calculated

(open symbols and lines) during pyrolysis of three different C6H6/Ar mix-

tures: (squares) 2 %, (circles) 1 %, (triangles) 0.5 %, for tr = 1.0 ms at

p = 1.2 bar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.28 Time history of the mean particle radius measured (closed symbols) [81, 82]

and calculated (open symbols and lines) during pyrolysis of four different

C6H6/Ar mixtures: (squares) 2 %, (circles) 1 %, (triangles) 0.5 %, (inverse

triangles) 0.25 % at p = 1.2 bar. . . . . . . . . . . . . . . . . . . . . . . 67

5.29 Soot formation time history measured (closed symbols) [78] and simu-

lated (open symbols and lines) during pyrolysis of a diluted in argon

C6H6/C2H2 = 3/1 mixture ([C] = 1.2·10−6[mol/cm3]) at p = 60 bar

for various temperatures: (circles) T = 1676 K, (squares) T = 1789 K,

(triangles) T = 1806 K, (inverse triangles) T = 1880 K. . . . . . . . . . . 69

5.30 Arrhenius-type plot of the experimentally measured (closed symbols) [78]

and calculated (open symbols) induction time τ for mixtures with vari-

ous benzene/acetylene ratios (B/A) at pressure 60 bar: (circles) benzene,

[C] = 4.0 · 10−6 mol/cm3; (squares) acetylene, [C]= 4.0 · 10−6 [mol/cm3];

(inverse triangles) B/A = 10/1, [C]= 9.0 · 10−6 [mol/cm3]; (diamonds)

B/A = 1/1, [C]= 5.0 · 10−6 [mol/cm3]; (triangles) B/A = 2.5/1, [C]=

12.0 · 10−6 [mol/cm3]; (hexagons) benzene; only the HACA pathway of

soot formation was active, [C]= 4.0 · 10−6 [mol/cm3]. . . . . . . . . . . . 70

5.31 Temperature dependence of the normalised observable rate of soot parti-

cle growth (kf/[C]) measured (closed symbols) [78] and calculated (open

symbols) during pyrolysis of benzene, acetylene, benzene/acetylene (B/A),

and acetylene/hydrogen mixtures at pressure 60 bar: (circles) benzene, [C]

= 4.0 ·10−6 [mol/cm3]; (squares) [C] = 4.0 ·10−6 [mol/cm3]; (inverse trian-

gles) B/A = 10/1, [C]= 9.0 ·10−6 [mol/cm3]; (diamonds) B/A = 1/1, [C]=

5.0 · 10−6 [mol/cm3]; (triangles) B/A = 2.5/1, [C]= 12.0 · 10−6 [mol/cm3];

(hexagons) C2H2/H2 = 1/1, [C]= 2.0 · 10−6 [mol/cm3]. . . . . . . . . . . 71



and calculated (open symbols and lines) during pyrolysis of C6H6/Ar, [C]

= 4.0 ·10−6[mol/cm3], and C2H2/Ar, [C] = 4.0 ·10−6[mol/cm3] mixtures

at p = 6.0 bar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.33 Temperature dependence of the soot yield measured (closed sym-

bols) [78] and calculated (open symbols and lines) during pyrolysis

of C6H6/C2H2/Ar mixtures, (triangles) B/A = 2.5/1 [C] = 9.0 ·10−6

[mol/cm3], (inverse triangles) B/A = 1/1, [C] = 5.0 ·10−6[mol/cm3], (dia-

monds) B/A =1/2.5 [C] = 9.0 ·10−6[mol/cm3] at p = 6.0 bar/cm3. . . . . 72

5.34 Temperature dependence of the soot yield obtained during pyrolysis of

benzene ([C] = 4.0 · 10−6 mol/cm3) at p = 6 bar: (closed circles) the

experimental measurements [78], (open circles and line) the calculated re-

sults performed with Model-1, (open hexagons and line) the results of

calculation performed with Model-1, when only the HACA pathway of

soot formation (Table 5.1) was active. . . . . . . . . . . . . . . . . . . . 72


[78] and calculated (open symbols and lines) during pyrolysis of C2H2,

C2H4, and C2H2/H2diluted in argon mixtures: (squares) C2H2, [C]

= 4.0·10−6[mol/cm3]; (circles) C2H4, [C] = 4.0·10−6[mol/cm3]; (in-

verse triangles) C2H2/H2 = 1/1, [C] = 4.0·10−6 [mol/cm3]; (triangles)

C2H2/H2 = 1/1[mol/cm3], [C] = 2.0·10−6[mol/cm3] mixtures at p = 6.0

bar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73


and calculated (open symbols and lines) during rich oxidation of CH4, [C]

= 7.6 [mol/m3]; C3H8 [C] = 6.0 [mol/m3]; n-C7H16 [C] = 5.9 [mol/m3] at

φ= 5 and p = 40 bar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.37 Temperature dependence of the experimentally measured (closed symbols)

[198] and calculated (open symbols and lines) soot yield during rich oxida-

tion of n-C7H16/Ar mixture (99 % Ar, 0.3125 % C7H16, and 0.6875 % O2)

at constant argon concentration: (circles) p = 30 bar, (squares) p = 40

bar, (diamonds) p = 50 bar. . . . . . . . . . . . . . . . . . . . . . . . . 75

5.38 Experimentally measured (closed symbols) and calculated results (open

symbols and lines) of the time-resolved concentration profiles of H atoms

measured during shock-tube pyrolysis of C6H5OH [206] and C6H6 [205]. . 84


5.39 Concentration profiles of OH radicals measured (closed symbols) [207] and

calculated (open symbols and lines) during toluene oxidation: φ = 1, 0.1

% C6H5CH3, 0.9 % O2, (circles) T = 1689 K, and (triangles) T = 1586 K

and p = 1.9 bar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85


calculated (open symbols and lines) during toluene oxidation: φ = 1, 0.025

% C6H5CH3 + 0.225 % O2, (triangles) T = 1783 K, p = 1.84 bar; (inverse

triangles) T = 1700 K, p = 1.89 bar; (squares) T = 1648 K, p = 2.03 bar;

(diamonds) T = 1607 K p = 2.03 bar. . . . . . . . . . . . . . . . . . . . 86


calculated (open symbols and lines) during n-C7H16 and C6H5CH3 oxida-

tion: (triangles) 130 ppm n-C7H16, T = 1640 K, p = 2.0 bar; (squares)

1250 ppm C6H5CH3, T = 1648 K, p = 2.0 bar. . . . . . . . . . . . . . . . 86

5.42 Concentration profiles of the main gas-phase species measured [35] and

simulated in pyrolysis of 3.2 % C2H2/Ne/Ar mixture at T = 2030K and

p = 0.39 bar behind reflected shock waves: (squares) C2H2, (triangles)

C4H2 · 2, (inverse triangles) C6H2 · 10. . . . . . . . . . . . . . . . . . . . 87

5.43 Concentration decay profiles of the fuel molecule measured (closed sym-


% C6H6 diluted in (99% Ne-1% Ar) mixture at T = 2190 K, p = 0.52 bar. 88



% C6H6 at conditions as in Figure 5.43. . . . . . . . . . . . . . . . . . . 88

5.45 Fuel-decay concentration profiles measured (closed symbols) [35] and sim-

ulated (open symbols and lines) during pyrolysis of 1.8 % C6H5CH3 at

T = 1900 K, p = 0.4 bar. . . . . . . . . . . . . . . . . . . . . . . . . . . 89



% C6H5CH3 at conditions as in Figure 5.45. . . . . . . . . . . . . . . . . 89

5.47 Experimentally measured [211] concentration of several aliphatic hydro-

carbons and the fuel decay profile detected during pyrolysis of 1 % toluene

at pressure 10.013 bar and reaction time 600 µs. . . . . . . . . . . . . . . 90


5.48 Concentration profiles of several aliphatic hydrocarbons and the fuel decay

calculated during pyrolysis of toluene at conditions as in Figure 5.47. . . . 90

5.49 Experimentally measured [211] concentration profiles of aromatic hydro-

carbons detected during pyrolysis of 1 % toluene at pressure 10.013 bar

and reaction time 600 µs. . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.50 Concentration profiles of aromatic hydrocarbons calculated during toluene

pyrolysis at conditions as in Figure 5.49. . . . . . . . . . . . . . . . . . . 91

5.51 Experimentally measured [211] concentration profiles of various polycyclic

aromatic hydrocarbons detected during pyrolysis of 1 % toluene at pressure

10.013 bar and reaction time 600 µs. . . . . . . . . . . . . . . . . . . . . 92

5.52 Concentration profiles of various polycyclic aromatic hydrocarbons calcu-

lated in toluene pyrolysis at conditions as in Figure 5.51. . . . . . . . . . 92

5.53 Integral reaction flow analysis of the main pathways of soot precursor

inception during C6H5CH3/O2/Ar oxidation at T = 1900 K, p = 2.0 bar,

and reaction time 2.0 ms. . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.54 Sensitivity analysis with respect to benzene during toluene oxidation at


5.55 Sensitivity analysis with respect to ethynylnaphthalene radical (A2C2HB)

during toluene oxidation at conditions as in Figure 5.53. . . . . . . . . . . 94

5.56 Sensitivity analysis with respect to acenaphthylene (A2R5) during toluene

oxidation at conditions as in Figure 5.53. . . . . . . . . . . . . . . . . . 94

5.57 Sensitivity analysis with respect to acephenanthrylene (A3R5) during

toluene oxidation at conditions as in Figure 5.53. . . . . . . . . . . . . . 95

5.58 Integral reaction flow analysis of the main pathways of soot precursor

inception during n-C7H16/O2/Ar rich oxidation, [C] = 7.89 [mol/m3], φ =

5, at T = 1750K, p = 25.0 bar, and reaction time 2.5 ms. . . . . . . . . . 97

5.59 Sensitivity analysis with respect to benzene during n-heptane oxidation at



5.60 Sensitivity analysis with respect to acenaphthylene (A2R5) during n-

heptane oxidation at conditions as in Figure 5.58. . . . . . . . . . . . . . 98

5.61 Sensitivity analysis with respect to acephenanthrylene (A3R5) during n-

heptane oxidation at conditions as in Figure 5.58. . . . . . . . . . . . . . 98


[195] and simulated (open symbols and lines) soot yield during pyrolysis

of C2H4/Ar mixtures at p = 50.0 bar for three different C-atom concentra-

tions: (circles) [C] = 7.4 [mol/m3], (squares) [C] = 4.7 [mol/m3], (inverse

triangles) [C] = 4.0 [mol/m3]. . . . . . . . . . . . . . . . . . . . . . . . 100

5.63 Experimentally measured (closed symbols) [195] and simulated (open

symbols) induction delay time during pyrolysis of C2H4/Ar mixtures at

p = 50.0 bar for three different C-atom concentrations: (triangles) [C] =

7.4 [mol/m3], (circles) [C] = 4.7 [mol/m3], (squares) [C] = 4.0 [mol/m3]. . 100


[195] and simulated (open symbols and lines) during pyrolysis of CH4/Ar

mixtures for several different carbon-atom concentrations: (circles) p =

55.0 bar, [C] = 6.4 [mol/m3]; (squares) p = 55.0 bar, [C] = 3.4 [mol/m3];

(inverse triangles) p = 120.0 bar, [C] = 4.0 [mol/m3]; (triangles) p =

25.0 bar, [C] = 3.0 [mol/m3]; (diamonds) p = 55.0 bar , [C] = 1.7 [mol/m3]. 101


[195] and simulated (open symbols and lines) soot yield during pyrolysis of

C6H6/Ar mixtures at p = 50.0 bar for four different C-atom concentrations:

(circles) [C] = 4.0 [mol/m3], (squares) [C] = 1.0 [mol/m3]; (triangles) [C]

= 0.8 [mol/m3], (diamonds) [C] = 0.4 [mol/m3]. . . . . . . . . . . . . . . 102

5.66 Experimentally measured (closed symbols) [195] and simulated (open sym-

bols) induction delay time during pyrolysis of C6H6/Ar mixtures at con-

ditions as in Figure 5.65. . . . . . . . . . . . . . . . . . . . . . . . . . . 102



tures at p = 1.2 bar, tr = 1.3ms for three different reactive mixtures:

(squares) 2 %, (circles) 1 %, (triangles) 0.5 %. . . . . . . . . . . . . . . . 103


5.68 Induction delay time, measured (closed symbols) [81] and simulated (open

symbols) during pyrolysis of C6H6/Ar mixtures at conditions as in Figure

5.67. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.69 Mean particle radius measured (closed symbols) [81] and calculated (open

symbols and lines) during pyrolysis of three different C6H6/Ar mixtures:

(squares) 2 %, (circles) 1 %, (triangles) 0.5 %, for a fixed reaction time

tr = 1.0 ms, at p = 1.2 bar. . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.70 Time history of the mean particle radius measured (closed symbols) [81]

and calculated (open symbols and lines) during pyrolysis of four different

C6H6/Ar mixtures: (squares) 2 %, (circles) 1 %, (triangles) 0.5 %, (inverse

triangles) 0.25 % at p = 1.2 bar. . . . . . . . . . . . . . . . . . . . . . . 104


[221] and calculated (open symbols and lines) soot yield during pyrolysis

of three C6H5CH3/Ar mixtures at tr = 2 ms: (triangles) 1.5 % C6H5CH3,

p = 3.5 bar; (squares) 1.0 % C6H5CH3, p = 3.3 bar; (diamonds) 0.5 %

C6H5CH3, p = 2.5 bar. . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.72 Temperature dependence of the soot yield, measured (closed symbols) [222]

and calculated (open symbols and line) during pyrolysis of 0.1 % n-C7H16

diluted in argon mixture at p = 20 bar. . . . . . . . . . . . . . . . . . . 106

5.73 Induction delay time, measured (closed symbols) [222] and calculated

(open symbols), during n-C7H16 pyrolysis at conditions as in Figure 5.72. . 106


and calculated (open symbols and lines) during rich oxidation of CH4, [C]

= 7.6 [mol/m3]; C3H8 [C] = 6.0 [mol/m3]; and n-C7H16 [C] = 5.9 [mol/m3]

, at φ= 5 and p = 40 bar. . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.75 Time-resolved soot yield measured (closed symbols) [198] and calculated

(open symbols and line) during n-C7H16 rich oxidation, φ= 5, [C] = 7.89

[mol/m3], T = 1750 K, p = 25 bar. . . . . . . . . . . . . . . . . . . . . . 108

5.76 Time-resolved mean particle diameter measured (closed symbols) [198] and

calculated (open symbols and line) during n-C7H16 rich oxidation at con-



5.77 Time-resolved soot particle number density measured (closed symbols)

[198] and calculated (open symbols and line) during n-C7H16 rich oxidation



and calculated (open symbols and lines) during shock tube rich oxidation

of an n-C7H16/O2/Ar at constant Ar concentration (0.3125 % C7H16, %

O2, and 99 % Ar): (circles) p = 30 bar, (squares) p = 40 bar, (diamonds)

p = 50 bar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.79 The pressure influence of the soot yield measured (closed symbols) [198]

and calculated (open symbols and line) during shock tube rich oxidation of

an n-C7H16/Ar/O2 mixture, [C] = 5.8 [mol/m3], φ= 5 for three different

pressures 20 bar, 40 bar and 80 bar. . . . . . . . . . . . . . . . . . . . . 110

5.80 Temperature dependence of the soot yield experimentally measured (closed

symbols) [221] and calculated (open symbols and lines) during oxidation

of three different C6H5CH3/O2/Ar mixtures at tr = 2 ms: (triangles) 1.5

% toluene, p = 3, 5 bar; (squares) 1.5 % toluene + 1.5 % O2, p = 2.0 bar;

(diamonds) 1.5 % toluene + 2.5 % O2, p = 2.0 bar. . . . . . . . . . . . . 111

5.81 Temperature dependence of the soot yield experimentally measured (closed

symbols) [221] and calculated (open symbols and lines) during thermal

decomposition of C6H5CH3/Ar and C6H5CH3/CH3OH/Ar mixtures at

tr = 2 ms: (triangles) 1.0 % toluene, p = 3, 3 bar; (squares) 1.0 % toluene +

1.0 % methanol, p = 2.7 bar; (diamonds) 1.0 % toluene + 2.0 % methanol,

p = 2.7 bar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112


[221] and calculated (open symbols and lines) soot yield during pyrolysis

of C6H5CH3/Ar and C6H5CH3/C2H5OH/Ar mixtures at tr = 2 ms: (tri-

angles) 1.0 % toluene, p = 3, 3 bar; (inverse triangles) 1.0 % toluene + 1.0

% ethanol, p = 3.0 bar; (squares) 1.0 % toluene + 2.0 % ethanol, p = 3.0

bar; (diamonds) 1.0 % toluene + 3.0 % ethanol, p = 3.0 bar. . . . . . . . 113

6.1 Time-resolved soot yield, measured (closed symbols) [198] and calculated

(open symbols and lines) in n-C7H16 rich oxidation, [C] = 7.89 [mol/m3],

φ= 5, T = 1750 K, p = 25 bar and tr = 2.5 ms. . . . . . . . . . . . . . . 123


6.2 Time-resolved mean particle diameter, measured (closed symbols) [198]

and calculated (open symbols and lines) in n-C7H16 rich oxidation at con-


6.3 Time-resolved soot particle number density, measured (closed symbols)

[198] and calculated (open symbols and lines) in n-C7H16 rich oxidation at


6.4 Time-dependent profiles of the C3H3 concentration, calculated at the con-

ditions as in Figure 6.1: (circles) SSM-1 and (diamonds) SSM-2. . . . . . 125

6.5 Time-dependent profiles of the C2H2 concentration, calculated at condi-

tions as in Figure 6.1: (circles) SSM-1 and (diamonds) SSM-2. . . . . . . 125

6.6 Time-dependent profiles of the OH concentration, calculated at the condi-

tions as in Figure 6.1: SSM-1 (circles) and SSM-2 (diamonds). . . . . . . 126


and calculated (open symbols and lines) in rich oxidation of n-C7H16 [C]

= 5.9 [mol/m3], φ= 5, p = 40 bar and tr = 2.5ms: (circles) experiment,

(triangles) Model-1 and (diamonds) Model-2. . . . . . . . . . . . . . . . 126


and calculated (open symbols and lines) in rich oxidation of C3H8 [C] =

6.0 [mol/m3], φ= 5, p = 40 bar and tr = 2.5 ms: (circles) experiment,



and calculated (open symbols and lines) in rich oxidation of CH4 [C] =

7.6 [mol/m3], φ= 5, p = 40 bar and tr = 2.5 ms: (circles) experiment,


6.10 Temperature dependence of the soot yield, measured (closed sym-

bols) [221] and calculated (open symbols and lines) in oxidation of an

C6H5CH3/O2/Ar mixture (1.5 % toluene + 1.5 % O2), p = 2.0 bar and

tr = 2 ms: (circles) experiment, (diamonds) SSM-1 (triangles) Model-2. . . 129

144

List of Tables

5.1 Mechanism of formation, growth, coagulation and transformation of soot

precursors and soot particles (Model-1) . . . . . . . . . . . . . . . . . . 46

5.3 Mechanism of formation, surface growth, coagulation, oxidation and

transformation of soot precursors and soot particles (Model-2) . . . . 80

Eidesstattliche Erklarungen

Ich erklare hiermit, dass ich die vorgelegte Dissertation selbst verfasst und mich

keiner anderen als der von mir ausdrucklich bezeichneten Quellen und Hilfen bedient

habe.

Heidelberg, 29.03.2007

Iliyana Naydenova

146

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