The Development and Validation of a Simplified Soot Model ......Figure 4.1 – Schematic of the...

The Development and Validation of a Simplified Soot Model for use in Soot Emissions Prediction in Natural Gas Fuelled Engine Simulations

by

Justin Jeekee Shum

A thesis submitted in conformity with the requirements for the degree of Masters of Applied Science

Graduate Department of Mechanical & Industrial Engineering University of Toronto

© Copyright 2012 by Justin Jeekee Shum

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Abstract

The Development and Validation of a Simplified Soot Model for use in Soot Emissions Prediction in Natural Gas Fuelled Engine Simulations

Justin Jeekee Shum

Masters of Applied Science Graduate Department of Mechanical & Industrial Engineering

University of Toronto 2012

This study employs a novel approach in order to satisfy the need in industry for a computationally

inexpensive means to modelling soot formation in engines fuelled by natural gas. The complex geometries

found in practical combustion devices along with the requirement to solve turbulent, chemically reacting,

and multi-phase flows necessitates this goal. A two-equation model, which tracks soot mass and soot

number density, is employed. The goal is to apply this model in engine simulations at Westport Innovations,

an industry partner.

Experimental data is used to validate the model in various operating conditions. Numerical data

obtained from a detailed sectional soot model is also used to augment available validation data, especially

with respect to soot formation/oxidation mechanisms. The developed model shows good agreement

compared to experimental data and the detailed sectional soot model among all cases considered and will

be further tested and applied in Westport’s natural gas engine simulations.

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Acknowledgements First and foremost, the author would like to acknowledge his supervisor Professor M.J. Thomson for his

valued guidance, direction, and support as without it, this project would not have been possible. The author

would also like to acknowledge the assistance provided by Professor S.B. Dworkin as his expertise in soot

modelling was most helpful in many stages of this work. Special thanks are also given to Dr. Q. Zhang who

provided his prior work on the two equation soot model. The author also acknowledges the Natural

Sciences and Engineering Research Council of Canada, Westport Innovations, and Dr. B. Wasmund for

financial support. The author would also like to thank Dr. N. Slavinskaya and Professor U. Riedel of the

German Aerospace Centre (DLR) and Dr. J. Huang of Westport Innovations for providing the chemical

mechanisms, thermodynamic data, and transport data for methane/air combustion. The author would also

like to further recognize Dr. J. Huang's contributions to the development and validation of the simplified

soot model. Further acknowledgements are given to Dr. G. McTaggart-Cowan, Professor S. Rogak, and the

rest of the APC team for their feedback and advice. The SciNet HPC Consortium is also acknowledged for

providing the computational resources necessary to complete this project. SciNet is funded by: the Canada

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

Research Fund - Research Excellence; and the University of Toronto. Many thanks are also warranted to the

author's peers, colleagues, and friends at the Combustion Research Laboratory who all found the time to

provide support in too many ways to list. Finally, the author would like to acknowledge his family, friends,

and Varsity fencing teammates/coaches for their endless encouragement, helpfulness, and much needed

distractions.

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Contents

1. Introduction ................................................................................................................................................1

1.1. Motivation ...........................................................................................................................................2

1.2. Objectives............................................................................................................................................3

2. Background and Literature Review .............................................................................................................5

2.1. An Introduction to Soot ......................................................................................................................5

2.1.1. Soot Characteristics and overview ..............................................................................................5

2.1.2. Soot nucleation/inception ..........................................................................................................7

2.1.3. Soot particle surface growth .......................................................................................................9

2.1.4. Soot particle oxidation ............................................................................................................. 10

2.1.5. Soot particle agglomeration..................................................................................................... 11

2.2. Current Approaches to Soot Modelling ........................................................................................... 11

2.2.1. Empirical Models ...................................................................................................................... 12

2.2.2. Detailed Models ....................................................................................................................... 14

2.2.3. Semi-empirical models ............................................................................................................. 16

2.2.4. Particle size distribution and soot aerosol dynamics ............................................................... 19

2.3. Laminar Coflow Diffusion Flame ...................................................................................................... 21

2.4. Turbulent Combustion Modelling .................................................................................................... 23

3. Model Development Methodology ......................................................................................................... 28

3.1. Experimental Datasets for Model Validation ................................................................................... 28

3.2. Comparison to Sectional Detailed Soot Chemistry Model ............................................................... 29

3.3. Experimental Cases Considered ....................................................................................................... 31

3.4. Considerations for use in Westport Simulations ............................................................................. 32

3.4.1. Coupling of Soot Model to Gas Phase Species Consumption .................................................. 33

3.4.2. Coupling to Radiation Heat Transfer ........................................................................................ 33

4. Mathematical Formulation ...................................................................................................................... 34

4.1. Computational Domain .................................................................................................................... 34

4.2. Governing Equations ........................................................................................................................ 34

4.2.1. Conservation of mass ............................................................................................................... 35

4.2.2. Conservation of momentum .................................................................................................... 35

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4.2.3. Conservation of species ........................................................................................................... 35

4.2.4. Conservation of energy ............................................................................................................ 36

4.3. Diffusivity of Gaseous Species ......................................................................................................... 36

4.4. Radiation Heat Transfer ................................................................................................................... 37

4.5. Simplified Soot Model ...................................................................................................................... 38

4.5.1. Soot transport equations ......................................................................................................... 42

4.6. Numerical Method ........................................................................................................................... 42

4.6.1. Mesh and boundary conditions ............................................................................................... 43

4.6.2. Parallel Computation ............................................................................................................... 45

4.7. Detailed Sectional Soot Model......................................................................................................... 46

5. Development and Validation of Soot Model ........................................................................................... 48

5.1. Chapter Outline ................................................................................................................................ 48

5.2. Sensitivity Analysis of Parameter Terms in Simplified Model.......................................................... 48

5.3. Model Development at 1 atmosphere using the Smooke et al. [61] data set ................................. 56

5.3.1. Comparisons to numerical data from detailed sectional soot model ..................................... 58

5.4. Model Development at 1 atmosphere using the Schittkowski et al. [76] dataset .......................... 64


5.5. Results at elevated pressures .......................................................................................................... 70


5.6. Preliminary Results in Engine Simulations ....................................................................................... 74

5.7. Model Improvements ...................................................................................................................... 74

5.7.1. Oxidation Mechanism of Soot Model ...................................................................................... 75

5.7.2. Updated Model Parameters and Improved Results ................................................................ 78

5.8. Effect of Coupling – Gas Phase Species Consumption and Radiation .............................................. 87

5.9. Computational Cost Comparison ..................................................................................................... 89

6. Concluding Remarks ................................................................................................................................. 91

6.1. Conclusion/Summary ....................................................................................................................... 91

6.2. Future Work ..................................................................................................................................... 92

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List of Figures

Figure 2.1 – Example of soot aggregate structure in diesel exhaust. Taken from [6]. .......................................6

Figure 2.2 – Representation of soot formation in premixed flames. Adapted from [19]. ..................................8

Figure 2.4 – Five major components of soot modelling. ................................................................................. 12

Figure 2.5 – Typical setup of a laminar coflow diffusion flame adapted from [61] ......................................... 22

Figure 2.6 – Soot formation zones in a coflow diffusion flame along with soot aggregate structure evolution.

Adapted from [10]. .......................................................................................................................................... 23

Figure 3.1 – Workflow diagram of project ....................................................................................................... 28

Figure 3.2 – Adapted workflow diagram of project ......................................................................................... 30

Figure 3.3 – Schematic of high pressure combustion rig taken from [78]. ...................................................... 32

Figure 3.4 – (a) Table summarizing burner geometry and operating conditions of experiments (b) Diagram of

coflow burner defining and .................................................................................................................... 32

Figure 4.1 – Schematic of the computational domain (greyed out area) super-imposed on a diagram of a

typical laminar coflow diffusion flame. Also illustrated is the orientation of the coordinate system used.

Note that the illustration is not to scale. Taken from [75]. ............................................................................. 34

Figure 4.2 – Diagram of typical non-uniform mesh employed in the simulations presented. Adapted from

[75]. .................................................................................................................................................................. 45

Figure 5.1 – Diagram of the “wing” and “centreline” regions of a typical flame. ........................................... 50

Figure 5.2 – Sensitivity of sooting behaviour to the pre-exponential value of soot inception, . Solid

lines show peak values at the wing of the flame and dashed lines show values in the centreline of the flame.

A logarithmic scale is applied to the Y-axis. ..................................................................................................... 50

Figure 5.3 – Sensitivity of sooting behaviour to the pre-exponential value of soot surface growth, .

Solid lines show peak values at the wing of the flame and dashed lines show values in the centreline of the

flame. A logarithmic scale is applied to the Y-axis. Legend is in Figure 5.2. .................................................... 52

Figure 5.4 – Sensitivity of sooting behaviour to the pre-exponential value of soot oxidation via O2, .



Figure 5.5 – Sensitivity of sooting behaviour to the pre-exponential value of soot oxidation via OH, .



Figure 5.6 – Sensitivity of sooting behaviour to the selected incipient particle diameter, with the default

value at 12 nm. Solid lines show peak values at the wing of the flame and dashed lines show values in the

centreline of the flame. A logarithmic scale is applied to the Y-axis. Legend is in Figure 5.2. ........................ 55

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Figure 5.7 – Sensitivity of sooting behaviour to the selected agglomeration rate, . Solid lines show peak

values at the wing of the flame and dashed lines show values in the centreline of the flame. A logarithmic

scale is applied to the Y-axis. Legend is in Figure 5.2. ..................................................................................... 56

Figure 5.8 – Soot volume fraction profiles at different axial heights above the burner. Z1, Z2, Z3, and Z4

correspond to heights of 2.0, 2.25, 2.5, 2.75 cm for experimental measurements and computations from the

simplified model for the Smooke et al. [61] flame. For computations from the sectional model, 0.6 cm was

added to each axial height to account for the delay in PAH formation. ......................................................... 58

Figure 5.9 – Contours of soot volume fraction (ppm) generated using the simplified model (left) and

detailed sectional model (right) for the Smooke et al. [61] flame. ................................................................. 59

Figure 5.10 – Contours of soot number density of aggregates (#/cc) generated using the simplified model

(left) and detailed sectional model (right) for the Smooke et al. [61] flame. .................................................. 60

Figure 5.11 – Contours of soot aggregate mass averaged diameters (nm) generated using the simplified

model (left) and detailed sectional model (right) for the Smooke et al. [61] flame. ...................................... 61

Figure 5.12 – Example of a pathline of maximum soot. .................................................................................. 62

Figure 5.13 – Diagram of methodology used to compare inception mechanisms between the simplified code

and the detailed code. ..................................................................................................................................... 63

Figure 5.14 – Calculated plots of soot inception rate, surface growth rate, and oxidation along the pathline

of maximum soot for the Smooke et al. [61] flame. The Y-axis is plotted on a logarithmic scale. ................. 63

Figure 5.15 – Contours of soot volume fraction side by side with experimental measurements in the

Schittwkowski et al. [76] flame. Experimental measurements are on the left side of the flame and

computations are on the right side of the flame. Results of the simplified model are shown in (a) and results

of the detailed model are shown in (b). .......................................................................................................... 65

Figure 5.16 – Contours of soot particle diameter in the Schittkowski et al. [76] flame. (a) shows the

experimental measurements of primary particle diameter on the left and the calculated contour of primary

particle diameter on the right from the detailed model. (b) shows the calculated contour of mass averaged

aggregate particle diameter from the detailed model on the left and the simplified model on the right. .... 67

Figure 5.17 – Contours of soot particle number density in the Schittkowski et al. [76] flame. (a) Shows the

experimental measurements of primary particle number density on the left and the calculated primary

particle number density on the right from the detailed model. Different scales are used for each half of the

flame. (b) Shows the calculated contour of aggregate particle number density from the detailed model on

the left and the simplified model on the right. ................................................................................................ 68

Figure 5.18 – Calculated plots of soot inception rate, surface growth rate, and oxidation along the pathline

of maximum soot for the Schittkowski et al. [76] flame. The Y-axis is plotted on a logarithmic scale. .......... 70

viii

Figure 5.19 – Contours of soot volume fraction (ppm) with experimental measurements made by Joo and

Gülder [79]. Experimental measurements are on the left and calculated contours from the simplified model

are on the right. Operating pressures are at 10 atm (a), 20 atm (b), and 40 atm (c). ..................................... 71

Figure 5.20 – Contours of soot volume fraction (ppm) with experimental measurements made by Joo and

Gülder [79]. Experimental measurements are on the leftt and calculated contours from the detailed model

are on the right. Operating pressures are at 10 atm (a), 20 atm (b), and 40 atm (c). ..................................... 72

Figure 5.21 – (a) Graph illustrating the correction factors employed. (b) The effect of the correction factors

on the rates predicted by the utilized oxidation models. ................................................................................ 76

Figure 5.22 – Integrated soot volume fraction (ppm · cm2) profiles of the Santoro et al. [74] flames of the F2

Non-smoking flame in (a) and the F4 smoking flame in (b). ............................................................................ 77

Figure 5.23 – Peak values of soot volume fraction predicted by the simplified model compared to

experimental results in the centreline (a) and the wing (b) of the Smooke et al. [61] and Schittkowski et al.

[76] flames. ...................................................................................................................................................... 80

Figure 5.24 – Peak values of mass averaged aggregate particle diameters predicted by the simplified model

compared to the detailed model in the centreline (a) and the wing (b) of the Smooke et al. [61] and

Schittkowski et al. [76] flames. ........................................................................................................................ 81

Figure 5.25 – Peak values of aggregate particle number density predicted by the simplified model compared

to the detailed model in the centreline (a) and the wing (b) of the Smooke et al. [61] and Schittkowski et al.

[76] flames. ...................................................................................................................................................... 83

Figure 5.26 – Peak values of soot volume fraction predicted by the simplified models and detailed model

compared to experimental results from Joo and Gülder [79] in the centreline (a) and wing (b) of the flame.

......................................................................................................................................................................... 84

Figure 5.27 – Peak values of mass averaged aggregate diameter as predicted by the simplified models and

detailed model in the centreline (a) and wing (b) of the Joo and Gülder [79] flame. An uncertainty of

is assumed for the detailed code calculations. ................................................................................................ 86

Figure 5.28 – The effect of coupling on the predicted peak soot volume fractions in the wing and centreline

(CL) of the Smooke et al. [61] flame and the Schittkowski et al. [76] flame. .................................................. 87

Figure 5.29 – Graph illustrating the factor of increase in calculated peak values of soot volume fraction in

the wings and centreline (CL) of the high pressure Joo and Gülder [79] flames. From left to right, the 10, 20,

and 40 atm cases are shown plotted with respect to the maximum soot volume fractions measured in the

experiment. A logarithmic scale is applied to the Y-axis. ................................................................................ 88

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List of Tables

Table 2.1 – List of reactions in the HACA surface growth sequence. Adapted from [13]. .............................. 10

Table 4.1 – Summary of reaction rate constants in the Arrhenius form , where units

are in g, cm, mol, s, K. ...................................................................................................................................... 40

Table 4.2 – Summary of geometric properties of meshes used in laminar coflow flame simulations. ........... 44

Table 4.3 – A summary of the major differences between the employed simplified soot model and a

previously developed detailed sectional model. ............................................................................................. 46

Table 5.1 – List of major parameters in simplified two equation soot model. ................................................ 49

The Smooke et al. [61] data set was used as a baseline case upon which to investigate the sensitivity of soot

volume fraction, soot aggregate averaged diameters, soot number density, soot inception rate, soot surface

growth rate, and soot oxidation rate (henceforth collectively referred to as “sooting behaviour”) to the

parameters listed in Table 5.1. Simulations were run where a single parameter was modified while all other

parameters were held constant. Then, the effect on the aforementioned soot details relative to the baseline

case with initial parameters listed in Table 4.1 and Section 4.5 was recorded. This process was repeated for

each parameter listed in Table 5.1. Peak values of sooting behaviour were recorded in both the wing and

the centreline of the flame (illustrated in Figure 5.1)...................................................................................... 49

Table 5.2 – Calculated peak mass averaged aggregate particle diameters (nm) in the Joo and Gülder [79]

flames. .............................................................................................................................................................. 73

Table 5.3 – Calculated peak aggregate particle number densities (#/cc) in the Joo and Gülder [79] flames. 73

Table 5.4 – Summary of parameters in simplified model. See Equation (5.3) for definition of correction

factor (identical for both O2 and OH). Changes are highlighted in red and underlined. .............................. 79

Table 5.5 – Representative comparison of computational costs of running the 2-D laminar flame code with

the detailed section soot model and the simplified two equation soot model. ............................................. 90

x

Nomenclature

Oxidation correction factor parameter

Agglomeration constant

Minimum number of carbon atoms found in a soot particle

Constant pressure specific heat of the mixture

Constant pressure specific heat of the species

Constant pressure specific heat of soot

Primary particle diameter

Mass averaged aggregate diameter of soot

Activation energy

Oxidation correction factor term

Soot volume fraction

Integrated soot volume fraction

Gravitational acceleration

Specific enthalpy of the species

Specific enthalpy of soot

Total number of gaseous species present in the chemical mechanism

Mass flow rate

Molar mass of carbon

Mass of soot

Soot number density

Aggregate number density

Primary particle number density

Avogadro’s number

Pressure

Total radiation heat transfer

Radial direction

Inner radius of fuel tube

Outer radius of fuel tube

Temperature

Oxidation correction factor parameter

Axial velocity

xi

Radial Velocity

Diffusive velocities in the Radial direction of the species

Diffusive velocities in the axial direction of the species

Radial thermophoretic velocities of soot particles

Axial thermophoretic velocities of soot particles

Molecular weight of the species

Molecular weight of soot

Mass fraction of the species

Soot mass fraction

Axial direction

Mixture fraction

Greek Symbols

Portion of available surface sites on soot particle for chemical reaction

Scalar dissipation

Thermal conductivity of the mixture

Boltzmann’s constant

Dynamic viscosity

Molar production rate of the species per unit volume

Molar production rate of soot per unit volume

Collisional efficiency of OH molecules for soot oxidation in simplified model

Mixture density

Density of soot

xii

Acronyms

CL Centreline (of flame)

CRL Combustion Research Laboratory (at the University of Toronto)

DOM Discrete Ordinate Method (Radiation Model)

HACA Hydrogen-Abstraction-Carbon-Addition

ILDM Instrinsic Low-Dimensional Manifold

NOx Nitric Oxide

PAH Polycyclic Aromatic Hydrocarbon

SLFM Steady/Stationary Laminar Flamelet Model

ULFM Unsteady Laminar Flamelet Model

1

1. Introduction

The work described in this thesis represents one component of a collaborative research project

between the Combustion Research Laboratory (CRL) supervised by Professor Thomson at the

University of Toronto, Professor Rogak and his research group at the University of British Columbia, and

Westport Innovations with the aim to aid in the research and development of natural gas fuelled heavy

duty compression ignition engines.

The development of combustion systems is a complex and time consuming endeavour – while the

basic operating principles of the technology used today have not changed significantly over the last few

decades, there is the ongoing need to address the growing demands of consumers while satisfying

stringent environmental policies. Despite recent advances in research and development made to move

away from combustion-based devices in an effort to mitigate the effects of global warming and limit

harmful pollutant emissions, alternative technologies are unlikely to become viable for widespread use

for many years to come [1]. This is especially true in long-distance transportation applications where

the use of conventional fossil fuels is necessitated by their high energy density relative to alternative

sources such as battery storage. As a result, combustion systems must be able to meet the ever

increasing legislated limits on harmful pollutant emissions such as nitric oxides (NOx), carbon

monoxide, volatile organic compounds (VOCs), and particulate matter (PM).

The design of a combustion system is often an iterative process from the early stage of conception,

its eventual design and manufacture, and finally, testing and validation. As such, projects can quickly

become both time consuming and expensive as individual prototypes and test rigs are either

manufactured or purchased along with the necessary data extraction and analysis tools. To mitigate

this, numerical models and predictive tools are often used to assist in the development of combustion

systems. Another advantage of numerical modelling is that it provides insight to combustion

2

phenomena, fluid flows, and other behaviour that is normally not directly observable by experimental

procedures. This type of insight can help to design more efficient and less polluting systems.

1.1. Motivation

The focus of this work lies in the development and validation of a numerical model that can

accurately predict the emissions of the combustion-generated particulate matter known as soot. The

aim is then to apply this model in compression-ignition natural gas fuelled engine simulations in order

to gain a better understanding of how and why soot is emitted from these engines. This knowledge will

facilitate the ability for rapid and iterative engine design in the difficult task of balancing performance

and pollutant emissions.

The formation of soot particulate matter in combustion and its subsequent release into the

atmosphere has received attention in research and industry due its adverse environmental and health

effects. Individual soot particles can often be formed on the scale of 100 nanometers or less. This poses

a significant health risk as inhalation of soot particles to the lungs can cause inflammation and cancer.

These fine particles are then able to migrate through the bloodstream and damage other vital organs

such as the heart or brain [2]. Soot is also known to play a major role in global warming phenomena

and is thought to be a major contributor to global warming effects behind CO2 [3]. From a design

standpoint, the deposition of soot particles on equipment can also pose a problem for issues such as

maintenance or thermal loading (due to the high thermal absorptivity of soot particles). Also important

to note is that the formation of soot in itself represents a degree of inefficiency in converting the

energy contained in a fuel to useful work. Thus, it is easy to see why accurately predicting soot

formation and emission in combustion devices is an important endeavour.

The mechanisms of soot production in combustion are very complex and not yet fully understood.

Modelling soot formation in various combustion applications is an ongoing area of research that has

received much attention of late [4],[5],[6],[7],[8],[9]. Due to the highly complex nature of soot

3

formation, these models are often computationally expensive as they attempt to recreate our current

understanding of fundamental soot formation/oxidation mechanisms. This study employs a simplified

approach in order to satisfy the need in industry for a computationally inexpensive approach to

modelling soot formation in engines fuelled by natural gas, such as those at Westport. As an engine

designer for natural gas fuelled compression ignition engines, Westport is interested in low

computational cost soot modelling techniques that can allow them to quickly evaluate new engine

concepts. The complex geometries found in practical combustion devices along with the requirement

to solve turbulent, chemically reacting, and multi-phase flows drives the goal of reducing the

computational cost of soot modelling.

The use of natural gas, which is mostly composed of methane, is rapidly becoming an important

alternative fuel in transportation applications. There is a growing interest in using natural gas as a fuel

due to its wider availability (and subsequently, affordability) and lower carbon footprint per unit of

energy. Potential greenhouse gas emissions reductions have been estimated to be over 20% per vehicle

in large scale industrial applications simply by switching from conventional fuels to natural gas [10].

Natural gas has also been demonstrated to be a cleaner fuel with less harmful pollutant emissions

compared to conventional fuels such as diesel or gasoline [11]. In addition, natural gas can be derived

sustainably from biomass and can also be collected from landfills, which is sometimes referred to as

synthetic natural gas or renewable natural gas. Thus, a switch to natural gas technology could also

allow for the use of sustainable fuels.

1.2. Objectives

The objectives of the present work are as follows:

1) Develop a simplified yet robust soot model that can be applied (i.e. is computationally

tractable) in more complex engine simulations, such as those being carried out at Westport

4

Innovations. Specifically, the aim is to apply this model in natural gas fueled compression

ignition engines.

2) Validate the developed model in a variety of operating conditions for key soot characteristics

such as soot volume fraction, soot number density, and soot particle diameters. If necessary,

improve model performance and document this process.

3) Investigate any key issues for soot model behaviour – especially when applied to engine

simulations at Westport. For example, can the model predict the observed experimental soot

emissions? If not, can qualitative trends in soot emissions with changing engine parameters

such as load condition be reproduced? Resolve these issues if possible; otherwise, identify

possible alternatives or methods for soot model application in engine simulations.

5

2. Background and Literature Review

This chapter explores some of the background information necessary to fully understand the

present work and its associated goals. Current understanding of soot and its formation/oxidation is

explored as well as the different approaches that are generally employed to model and predict soot

emissions.

2.1. An Introduction to Soot

Soot is best described as a type of airborne pollutant that is commonly created as a by-product of

hydrocarbon combustion. Unlike other common pollutants such as NOx or CO, which are emitted in the

gaseous phase, soot is unique in that it is emitted as a solid – usually consisting of a complex aggregate

structure composed of many smaller soot particles. These individual soot particles, which are

commonly referred to as “primary particles”, are roughly spherical, mainly composed of carbon, [12]

and are graphitic in nature [13]. Soot emissions from combustion applications such as diesel engines

are often recognized as dark-black exhaust plumes. The presence of soot within a flame is usually

characterized by the yellow-orange glow given off by the flame due to the luminosity of soot particles

within the flame.

2.1.1. Soot Characteristics and overview

Many experimental observations have shown that soot primary particles range in size from 10 – 50

nanometers (nm) [13],[14],[15]. Soot aggregate structure is fractal in nature and can consist of

anything from a few primary particles to hundreds of primary particles. An example of the soot

aggregate structure is shown in Figure 2.1. The macroscopic density of a soot particle is usually

considered to be between 1.8 [16] to 2.0 [12] grams/cm3. Finally, the overall characteristics of soot

particles do not seem to be strongly dependent on the type of flame, fuel, application, or other types of

operating conditions involved [17].

6

Soot in combustion can be characterized in a variety of ways, the most common being soot volume

fraction ( ), soot number density , and diameter of primary particles ( . Soot volume fraction,

which is a unit-less term, is defined as the volume of soot divided by the volume of gas. Soot number

density describes the number of soot particles per unit of volume (ex.

) or sometimes as

the number of soot particles per unit of mass (ex.

). In addition, soot number density can

refer to the total number of aggregates, denoted as or it can refer to the total number of primary

particles, denoted as (per volume/mass of the mixture). As seen in Figure 2.1, the number of

primary particles per aggregate can often vary by a large amount, even within the same type of

flame/application. This is one of the reasons accurately modelling soot can be very difficult, as one

needs to keep track of a population of soot particles that can vary in both diameter and aggregate

structure.

Figure 2.1 – Example of soot aggregate structure in diesel exhaust. Taken from [12].

7

Soot formation and oxidation processes have been reviewed by many authors; some of which

include the reviews by Glassman [17], Richter and Howard [13], Haynes and Wagner [18], Stanmore et

al. [12], and Appel et al. [19]. While the exact details of soot formation and oxidation is still an area of

debate, the overall process of soot formation and oxidation is generally agreed to start with precursor

formation, particle nucleation (or inception), followed by the parallel processes of soot particle surface

growth, particle agglomeration, and soot particle oxidation.

2.1.2. Soot nucleation/inception

Soot nucleation is one of the least understood mechanisms of soot formation, but there is a

general consensus that nucleation occurs due to the combination of polycyclic aromatic hydrocarbons

(PAHs) that transition to the solid phase [18],[19],[20]. This important step is actually preceded by the

pyrolysis of the fuel itself to give rise to various so-called “precursor species” that provide the input for

PAH species. Several precursor species have been identified, with acetylene (C2H2) typically receiving

the most attention [13]. The smallest aromatic species, benzene (C6H6), has also received much

attention as a precursor species due to the fact that PAHs can be considered to be grown from benzene

[13] via the addition of acetylene. It is thought that one of the bottlenecks to soot inception is the

formation of the first aromatic ring (such as benzene or phenyl) and not surprisingly, early work by

Glassman [17] showed that aromatic based fuels had a higher tendency to soot when compared to

alkanes, alkenes, and alkynes. Glassman [17] identified acetylene as an important precursor species to

soot formation and a later review by Richter and Howard [13] also reiterated the same point. Much

attention in recent research has been directed at determining exactly how PAHs form and grow from

their parent fuels [19],[20],[21]. In Figure 2.2, which illustrates a representation of soot formation in

premixed flames, one can observe an example of PAH growth in the “molecular zone” of the diagram.

There is still some debate over which chemical reaction pathways are the most important in forming

the first aromatic ring – some of which is outlined in [4],[21],[22],[23].

8

Regardless of the PAH formation route considered, once they are formed, they will continue to

grow due to subsequent chemical reactions with gaseous species and also collisions with other PAH

molecules to form PAH dimers, trimers, etc. At a certain size, the PAH molecules condense and

transition to a solid state – this results in a nascent soot particle, or in other words, soot particle

nucleation. The details of this process are poorly understood as experimental observations of this

phenomenon are difficult since these large PAH molecules still have relatively small diameters on the

order of 1 nm [13]. Soot inception/nucleation will ultimately add to both the number of soot particles

formed and the total mass of soot formed.

Figure 2.2 – Representation of soot formation in premixed flames. Adapted from [24].

9

2.1.3. Soot particle surface growth

Once the PAH molecules have transitioned into solid soot particles, they can continue to grow in

size due to heterogeneous chemical reactions with gaseous species on the surface of the soot particle.

It is generally agreed upon that acetylene plays a major role in contributing to soot surface growth, as

demonstrated in [19],[21],[25]. The amount of available soot surface area for reactions to occur also

plays a vital role in determining the amount of surface growth that occurs. Surface growth via

acetylene has been described by Frenklach et al. in [19],[21],[26] as the Hydrogen-Abstraction-Carbon-

Addition (HACA) reaction sequence, where the C-H bonds on the surface of soot particles can react

with gaseous species. The reactions contained in HACA are based off of analogous PAH gas phase

reactions and are shown in Table 2.1. represents a reaction site on the surface of the soot

particle where a carbon atom is bonded with a hydrogen atom; i.e. it is hydrogenated and non-reactive.

These hydrogenated sites can later become dehydrogenated through hydrogen abstraction via H atoms

or OH molecules as seen in S1 and S2 (in Table 2.1). represents a de-hydrogenated

(reactive/active) site on the surface of the soot particle that can accept acetylene molecules and

subsequently result in the growth of the soot particle (reaction S4). Alternatively, these de-

hydrogenated sites can be re-hydrogenated (S3) or even oxidized (S5 or S6), in which case the soot

particle becomes smaller. Technically speaking, reactions S5 and S6 are examples of a soot oxidation

mechanism, which will be explored further in Section 2.1.4. A phenomenon known as “ageing” has also

been observed in various experiments – in these experiments, the tendency of a soot particle to

undergo surface growth was observed to decline with increasing particle growth [21]. This can be

explained by the HACA mechanism as a reduction in the availability of active sites on a soot particle, a

decrease in H atom concentration, and/or an arrival at an equilibrium state for H atoms in the mixture.

10

Table 2.1 – List of reactions in the HACA surface growth sequence. Adapted from [19].

PAHs have also been proposed to contribute to surface growth in soot particles, in a mechanism

commonly referred to as PAH condensation [19],[27]. Similar to how PAH molecules can collide with

one another to form nascent soot particles; PAH molecules could also collide with existing soot

particles and condense on the surface of them. Macadam et al. [27] showed that in acetylene-lean

conditions, surface growth via PAH condensation was especially important. However, in acetylene-rich

conditions, surface growth via acetylene was dominant. Regardless of the avenue of soot growth, it is

generally agreed that soot surface growth is the dominant mechanism in forming additional soot mass

in a flame.

2.1.4. Soot particle oxidation

Soot particle oxidation is the mechanism by which soot particles can be oxidized and converted

back into gaseous species. As with soot surface growth, the amount of soot surface area available for

oxidizing agents to attack the soot particles plays a role in determining the rate of oxidation.

Competition between soot inception and surface growth mechanisms against soot oxidation

mechanisms ultimately determines whether or not a flame emits soot particles; i.e. if a flame is

smoking or non-smoking. Of the various species that can contribute to soot particle oxidation, O2 and

OH are generally regarded to be the most important [28],[29],[30],[31],[32]. O2 is generally a major

contributor under fuel-lean conditions, while OH is the dominant specie in fuel-rich conditions [33].

Oxidation by other species, such as the oxygen radical O has been investigated [28] and gasification of

soot via other species such as H2O and NO2 has also been shown to be possible [12]. Some studies have

11

also demonstrated that soot oxidation can lead to changes in the aggregate structure as well, such as

fragmentation of the aggregate into smaller structures [33],[34].

2.1.5. Soot particle agglomeration

Soot particle agglomeration is a unique mechanism of soot growth in that soot nucleation, surface

growth, and oxidation are mainly chemical processes where as agglomeration is mainly a physical

process. An exception to this is that PAH condensation is also a physical process. Agglomeration is best

described as the increase in soot particle size due to the collision of two or more soot particles. For

small (and newly formed particles), collisions may actually lead to a phenomenon referred to as

coalescence, where the two particles collide and merge into a larger spherical particle due to their

liquid-like behaviour [35]. Larger particles that collide can stick to one another and form complex

fractal-like aggregate structures as seen in Figure 2.1. Depending on the circumstances of the collision

and the particles involved, some intermediary result can occur, where the particles partially merge and

form a “bridge” or “neck” [16]. It is also worth noting that not all collisions will necessarily result in

merging or sticking of the particles involved as observed by Kellerer et al. [36]. D’Alessio et al. [37]

noted that under certain flame temperature conditions, particles might not stick due to a thermal

rebound effect – these observations contrast from an earlier belief that all collisions had a 100%

sticking efficiency. Ultimately, soot particle agglomeration will generally only affect the total number of

soot particles formed with negligible effect on the total mass of soot formed.

2.2. Current Approaches to Soot Modelling

Approaches to soot modelling can generally be categorized into three different types of

approaches: (i) empirical approaches, (ii) semi-empirical approaches, and (iii) detailed approaches. Soot

modelling was first extensively reviewed by Kennedy [38] and the continued development of new soot

modelling techniques remains an active area of research today [5],[6],[8],[20],[39]. Figure 2.3 illustrates

five major components of a soot model developed for a combustion system. A flow solver is needed to

calculate the solutions to the basic conservation equations to give the correct fluid flow field, while

12

combustion chemistry is needed to account for ongoing chemical reactions between species present in

the simulation. As soot formation is highly coupled to temperature, radiative effects from the flame

and soot particles often need to be considered. Finally, the components of the soot model itself must

also be included – this can be divided into two major parts: (1) Mechanisms which account for soot

formation/oxidation and (2) Mechanisms that consider the interactions between different soot

particles and also the population distribution of soot particle sizes. While all five components are

required for a full detailed approach, many empirical and semi-empirical approaches neglect one of

more of these components in the interest of reducing complexity and computational costs.

Figure 2.3 – Five major components of soot modelling.

2.2.1. Empirical Models

Empirical models are usually based solely on direct correlations between operating conditions and

the amount of soot that is emitted – i.e. all five components shown in Figure 2.3 could be neglected. In

the case where a flow solver is neglected, the soot model could be solely a function of combustion

input parameters, such as engine load or fuel input in the case of an empirical model for an engine.

They usually have very little computational cost and are quick to implement and run. As a result, they

are typically used in applications where it is not computationally feasible to include a more detailed

13

model, such as in a diesel engine or gas turbine. These types of combustion applications already use

significant computational resources to compute solutions for chemically reacting turbulent flows, and

the addition of a detailed soot model would make the calculations intractable. Due to the nature of

correlation, empirical models cannot be applied to applications or operating conditions that are

significantly different from the baseline from which the model was developed. They also fail to give any

insight on the specifics of soot formation; for example, one would not be able to determine precisely

where/when/why soot is formed in a diesel engine. For these reasons, empirical models may not be

practical for predictive purposes where engine geometries and operating parameters may change

radically from case to case. However, that is not to say that empirical models serve no useful purpose.

One application where empirical models are useful are in diagnostics systems where a user can be fed

real time information on how heavily their engine is sooting based on parameters such as combustion

temperatures. Since the engine operating conditions are not expected to change radically compared to

prescribed conditions for which the model is calibrated for, soot emissions can be accurately predicted

without the need for a complex measurement setup.

An example of an empirical model is the one developed by Khan et al. [40] for diesel engines. In

this model, Khan and co-workers assumed that the diameters of soot particles did not vary with respect

to operating speeds or loads. They also assumed that the overall formation rate of soot was only

dependent on inception, neglecting soot growth and oxidation, giving the equation:

(2.1)

where is the soot mass density [kg/m3], and are model parameters, is the activation energy of

soot formation set to 1.7 x 105 [kJ/kmol], is the volume of the soot formation zone [m3], is the

volume of the cylinder contents at normal temperature and pressure [m3], is the partial pressure of

unburned fuel [Pa], is the local unburned equivalence ratio, and is the local temperature. As

model parameters were adjusted until results fit the available experimental data, the model performed

14

reasonably well for the given conditions. However, one would expect that any significant departure

from the base set of calibrated data would result in poor performance, due in part to the neglect of

many fundamental soot formation/oxidation mechanisms.

Another example of an empirical model is the approach developed by Hiroyasu et al. [41]. Hiroyasu

and co-workers assumed that soot mass emissions were solely based on pressure, temperature, fuel

concentration, and O2 concentration – neglecting intermediary soot formation/oxidation mechanisms

and also ignoring the calculation of the number of soot particles. They defined the formation rate of

soot mass as:

(2.2)

where is the formation coefficient, is the oxidation coefficient, and

are the local mass

fractions for fuel and oxygen, and is the local mass fraction of soot. and are subsequently

defined as:

(2.3)

(2.4)

where and are model parameters, and is the pressure and operating pressure, and

are activation energies set to 6313 and 7070 [K-1] respectively. As with any empirical model, the

model performed relatively well as long as the conditions did not stray far from the conditions used to

calibrate the model parameters.

2.2.2. Detailed Models

On the opposite end of the spectrum, detailed soot models attempt to replicate the current

fundamental understanding of soot formation/growth/oxidation as described in Section 2.1. Detailed

soot models are typically based on fundamental combustion chemistry and make use of aerosol

15

dynamics theory. An ideal detailed soot model would work for any fuel, combustion application, and

operating condition; but in practice, all detailed soot models are still limited to a range of possible

scenarios for which the model was developed – albeit a much broader range than that of empirical

models. A well developed soot model can be applied to various fuels and applications and has better

general applicability. An example of this is the detailed model demonstrated in [42] which included

complex combustion chemistry to model the formation of PAHs and has since been applied with

success to both ethylene/air and methane/air flames under different operating conditions. Detailed

models are also capable of giving insight into the soot formation process and are also able to provide

information on the population size distribution of soot particles. The disadvantage of using detailed

soot models is that they tend to be very computationally expensive – most detailed soot models are

limited to simulations with simple geometry (1-D/2-D) and laminar flow conditions.

A commonly cited example of a detailed model is the one developed by Frenklach and Wang

[21],[26],[43] of which the chemical kinetic mechanism that describes everything from the pyrolysis of

fuel to the formation of PAHs is an integral component. Further details such as inception via PAH

molecules, growth by the HACA mechanism, oxidation, agglomeration, and aggregate structures were

also considered in this model. It is important to note that chemical kinetics play a major role in the

formation of soot at nearly every phase of soot production (inception, surface growth, and oxidation)

[38] and as such, detailed models almost ubiquitously employ some form of a PAH chemical kinetic

mechanism. Recent efforts such as those by Dworkin et al. [20] and Chernov et al. [44] have been made

in the application of improved PAH chemical mechanisms in detailed soot models.

It should be noted that the divide between a “detailed model” and a “semi-empirical” model is not

clearly defined and many approaches saddle a grey area between the two. An example is the work by

Lindstedt [45] which employs a detailed chemical mechanism and simplified soot chemistry to model

soot formation in ethylene and propane counterflow diffusion flames. Soot nucleation was based on

16

the precursor species of acetylene and benzene, with some focus in the work spent on developing the

chemical kinetic mechanism to accurately predict benzene. Oxidation was modelled considering only

O2 as an oxidative species, using rates developed by Lee et al. [31]. Surface growth via acetylene was

considered – however, the dependence on the surface area of soot was modelled using four different

assumptions. One assumption was to assume that surface growth was linearly dependent on the

surface area of soot. A second assumption was that the surface growth was dependent on both the

surface area and the number of available reaction sites per unit area on the soot particle. The third

assumption was that surface growth was only dependent on the number of particles and the final

assumption was that surface growth was dependent only on the concentration of acetylene and

temperature. Results from Lindstedt’s work [45] showed that the third assumption actually produced

the best results, although the author conceded that it was in part due to the difficultly in modelling the

HACA sequence such that there was confidence in the number of available reaction sites and may have

also been a result of the other model parameters that were selected. Reasonable predictions for both

the ethylene and propane flame were obtained in terms of soot volume fractions and particle

diameters.

2.2.3. Semi-empirical models

Semi-empirical models represent a middle ground between empirical models and detailed models

and provide a compromise between computational costs and the ability to model fundamental soot

formation/oxidation behaviour. Semi-empirical models tend to incorporate many soot

formation/oxidation mechanisms but reduce computational costs by simplifying the chemistry

involved. Where detailed models typically require large chemical kinetic mechanisms detailing

hundreds of reactions necessary to account for the formation of PAHs, semi-empirical models typically

employ simplified chemistry to minimize computational costs.

Fairweather et al. [46] developed a model where nucleation of soot particles was solely based on

the precursor species acetylene, allowing for a reduced chemical mechanism without the need to

17

model PAH formation. This implementation differs from the Lindstedt [45] approach as a simplified

chemical mechanism was used. This model was applied to a turbulent diffusion natural gas/air flame

where chemistry was solved by using a flamelet library (i.e. species concentrations and temperatures

were linked to mixture fraction instead of being explicitly solved). Surface growth was considered only

to occur via C2H2 surface reactions and oxidation was considered to occur only via O2. Further

simplifications were made by neglecting soot aggregate structure and assuming all soot particles were

solid spheres without the fractal aggregate structure seen in Figure 2.1. Finally, it was assumed that

surface growth and oxidation rates were linearly related to the surface area of soot particles. Despite

these simplifications, the model performed satisfactorily and unlike a fully empirical model, it could still

provide some insight to soot formation/oxidation rates and also provide more detailed soot data such

as soot number density and diameters. The model was later updated by Woolley et al. [47] to include

inception via benzene molecules as well and also included additional oxidation via OH. It was applied to

a turbulent methane/air flame as well as a propane/air flame demonstrating good agreement with

experimental results and thereby showing fuel flexibility.

The model developed by Fairweather and coworkers represents a popular two-equation approach

to soot modelling – where one equation is used to track soot volume fraction and a second equation is

used to track soot number density. These two equations typically resemble the following form:

(2.5)

(2.6)

where represents the mass of soot, , and represent the mass of soot formed/destroyed

due to inception, growth, and oxidation, respectively, represents soot number density, and and

represent soot number density from inception and agglomeration, respectively.

represents the model specific constants that are usually calibrated based on the exact mechanisms

18

used to represent the aforementioned soot mechanisms and the application for which the model is

used.

A similar two-equation approach was used by Moss et al. [48] where the major difference was in

the rate equations used to represent inception, surface growth, oxidation, and agglomeration. Like

Fairweather et al. [46], Moss and co-workers [48] assumed that surface growth and oxidation were

linearly dependent on soot surface area. A flamelet library was again used to solve for combustion

chemistry. Unlike the Fairweather et al. [46] model, only OH oxidation was considered. The model was

able to predict reasonable soot volume fractions and soot number densities along the centre-line of an

ethylene laminar diffusion flame, but only after some model parameters were adjusted to match

experimental data. The model was later extended by Brookes and Moss [49] to a turbulent

methane/air jet diffusion flame and compared favourably to experimental results. However, it is

important to note that the parameters of the model were again adjusted to fit the experimental data. A

two equation approach utilizing a form of the Brookes and Moss [49] model was also recently applied

to predicting soot in an automotive diesel engine simulation in a study by Pang et al. [8]. Pang et al.

found that the values for constants in the Brookes and Moss model typically found in literature could

not reproduce satisfactory soot behaviour in the engine and henceforth needed to carefully calibrate

the constants such that the model reproduced experimental results.

Hong et al. [50] used aspects of the Fairweather et al. [46] model (namely the inception and

oxidation mechanisms) to model soot formation in a diesel engine. A skeletal form of a detailed

mechanism for n-heptane was used to calculate combustion chemistry. However, instead of using a

simplified acetylene-only based approach for surface growth, a series of surface growth reactions

based on available reaction sites was used. In addition, a method of moments was used in order to

allow for the tracking of particle size distributions by assuming a log normal distribution. While

19

quantitative results were under-predicted in the test cases and the diesel engine, the authors were

satisfied with the qualitative trends reproduced.

Besides the above-mentioned applications, a two equation soot model has also been applied in

laminar methane co-flow diffusion flames [51],[52], laminar methane opposed flow diffusion flames

[53], laminar ethylene diffusion co-flow flames [6],[54],[55], laminar ethylene opposed flow diffusion

flames [56], turbulent ethylene diffusion co-flow flames [9], laminar heptane opposed flow diffusion

flames [57], and laminar acetylene co-flow diffusion flames [58]. A smaller number of applications were

also at high pressure [6],[8],[51],[53],[57] as detailed measurements of soot at high pressure are not

readily available in the literature. In terms of handling combustion chemistry, some approaches used a

flamelet library [9],[47],[49],[52] approach while others calculated detailed chemistry

[6],[51],[53],[55],[56]. Other forms of reduced chemical kinetics were also used in [8],[46],[54],[58]. As

evidenced by the wide variety of applications, the two equation model shows promise in terms of

widespread applicability and low computational cost. A direct comparison of computational costs

between an implementation of the two equation model and a detailed soot model is shown in Section

5.9.

2.2.4. Particle size distribution and soot aerosol dynamics

One of the challenges of soot modelling, besides handling the complex soot chemistry, is how to

track the size and aggregate structure of every soot particle that is formed. In processes with

multiphase flow, additional equations are required to describe changes to the population of particles

which can evolve due to interactions with other particles or due to chemical reactions with species in

other phases.

In the case of soot modelling, the population of soot particles can be affected by the parallel

processes of nucleation, surface growth, agglomeration, particle fragmentation, and oxidation. These

processes can lead to a complex population of differing particles that vary in size, shape, and structure,

20

making it computationally expensive to model accurately. The approach to soot modelling can be said

to be split into two parts: the interaction between soot particles and the gas phase species (i.e. soot

kinetics detailed in the above sections) and the interaction between soot particles (i.e. soot aerosol

dynamics) [59].

There are currently a few major methods that are employed to handle soot particle dynamics [39].

One obvious approach would be to directly model each individual particle in the population of

particles, coined as a “continuous” model approach. While the continuous model is accurate, the

computational costs are large [60] and the implementation of such a model is also restricted to simple

zero to one dimensional cases [39]. A second approach is to model the population of soot particles into

discrete sections or “bins”. The discretization of the particle size distribution allows for computational

tractability in more complicated scenarios compared to a continuous approach. Sectional models still

provide good accuracy if an adequate number of sections is used to represent the particle size

distribution, but the drawback is that each additional variable used to describe the population (e.g.

volume, surface area, etc.) increases the number of needed equations exponentially [39]. As such, a

single parameter, such as the particle mass per section bin along with a spherical particle assumption is

used in most applications of the sectional model. A third approach is a stochastic approach, where the

population of particles is determined by using a stochastic algorithm such as the Monte Carlo method.

However, while this approach has shown success in laminar flames, this method is also very

computationally expensive and is not generally considered for turbulent cases [61]. Another approach

of interest is the Method of Moments (MOM) where evolution equations for moments of the

population distribution are solved instead of explicitly solving the population distribution. In the MOM,

a compromise is made between accuracy and computational costs. Instead of calculating the exact

distribution of a particle population, mean quantities (i.e. moments) are computed [39]. A moment can

be thought of as a measure of varying aspects in a distribution depending on the order of the moment.

Thus, knowledge of all moments from 0 to ∞ in essence fully describes the distribution function itself

21

[62]. It has been noted that 3-6 moments are generally sufficient for an accurate soot calculation [59].

A final “approach” is to neglect tracking the particle distribution altogether, as Fairweather et al. [46]

did in his implementation of a two equation soot model. In this model, it was assumed that all soot

particles were spheres of identical diameter within a control volume (i.e. a monodisperse spherical

particle assumption). While this approach can save computational cost, it introduces errors into the

model as the above-mentioned assumption is questionable. This can lead to some inaccuracy in

predicting the available soot surface area for soot kinetics such as surface growth and oxidation. To

handle this error, simplified models that neglect particle aerosol dynamics typically account for these

errors by adjusting model parameters and constants in order to compensate for the calculated soot

surface areas.

2.3. Laminar Coflow Diffusion Flame

In the development of our soot model, it is important to consider the need for a combustion

configuration that is amenable to iterative numerical experimentation, has extensive validation data,

and is representative of typical combustion conditions within compression ignition engines. While it

would be ideal to develop the model in a compression ignition engine configuration, the complicated

geometry, reciprocating motion, and turbulent flow would make numerical experimentation difficult

due to the associated high computational costs. In addition, due to the nature of combustion in

engines, detailed spatial measurements of soot characteristics are nearly non-existent. For these

reasons, a steady axisymmetric laminar coflow diffusion flame was selected. This combustion

configuration was chosen in part due to its simple laminar flow field which lends itself well to

conducting numerous detailed numerical experiments. In addition, the mixing of fuel and air in a

diffusion flame can be comparable to processes that occur in compression ignition engines; although

admittedly, the effects of turbulent mixing would not be present in a laminar flame. A schematic of a

laminar coflow diffusion flame can be seen in Figure 2.4.

22

Figure 2.4 – Typical setup of a laminar coflow diffusion flame adapted from [63]

The laminar coflow flame also provides an additional benefit in that soot formation/oxidation

mechanisms in the flame span a relatively wide region, which allow for multi-dimensional detailed

measurements of soot characteristics. Figure 2.5 illustrates the typical evolution of a soot particle as it

undergoes soot formation/oxidation mechanisms in a diffusion flame. In can be seen that inception

occurs in the early part of the flame, near the edge of the luminous envelope (the edge of the flame is

usually referred to as the “wing” of the flame). This is not surprising as in diffusion flames, soot is

generally formed in high temperature, fuel rich conditions between 1300 – 1600 K [64]. The soot

particles that are formed then continue to grow as they progress upwards in the flame, before being

oxidized as the temperature in the flame increases. In cases where there is insufficient oxidation in the

flame, some soot particles can escape the wings of the flame, causing a “smoking” flame. In cases

where there is enough oxidation to fully oxidize the soot particles formed, the flame is referred to as a

“non-smoking” flame.

23

Figure 2.5 – Soot formation zones in a coflow diffusion flame along with soot aggregate structure evolution. Adapted from [16].

2.4. Turbulent Combustion Modelling

While the work presented in this thesis focuses on laminar flame simulations, it is important to

consider the complexities and complications turbulent combustion modelling brings, as the eventual

goal of the work is to apply the soot model in turbulent engine simulations. Turbulent combustion

theory and modelling is a large field of research in its own right and as such, the details of turbulent

combustion modelling will only be briefly discussed. Extensive reviews of turbulent combustion theory

and modelling have been conducted by several researchers, such as the efforts by Bilger et al. [65],

Pitsch [66], and Buckmaster et al. [67].

Turbulent combustion modelling can be very difficult and computationally expensive as one needs

to simultaneously calculate the complex, evolving fluid fields as well as the ongoing chemical reactions.

24

While solving for the fluid field and chemistry simultaneously is computationally tractable for laminar,

steady flames, this is not the case for turbulent flames. Instead, chemistry is often de-coupled and/or

simplified in order to reduce computational costs. Some important methods used in non-premixed

turbulent combustion modelling are discussed below.

The first simplification is the assumption that combustion in the turbulent flame is entirely

dominated by mixing related phenomena. A variable referred to as mixture fraction ( ) is often used to

quantify the local degree of mixing. Mixture fraction can be defined as the local mass of material that

originated in the fuel stream divided by the local total mass of the mixture [68]. Thus in the case of a

coflow diffusion flame, will be equal to 1 in the fuel stream and equal to 0 in the oxidizer stream. The

mixture fraction will vary between 0 and 1 throughout the flame and is said to be a “conserved

scalar”, i.e. the scalar variable is conserved at every point in the flow and there is no

creation/consumption of . The general idea of the mixing dominated assumption is that the

instantaneous temperature and species composition could be related to the mixture fraction. In the

case of Reynolds/Favre-averaged approaches of modelling turbulent flow, the average reaction rate

can be obtained by weighting the instantaneous reaction rates related to the probability density

function (PDF) of mixture fraction. Unfortunately, soot (and many other pollutants) does not correlate

well with mixture fraction [38]. This is due to the fact that soot chemistry is relatively slow and is not in

chemical equilibrium. Furthermore, soot particles do not diffuse in the same manner as gaseous

species [38]. Therefore, while the assumption of assuming mixing-dominated combustion is

computationally feasible and realistic for many applications, it is problematic when attempting to

model soot formation/oxidation accurately. Later studies showed that with the assumption of fast

chemistry (i.e. when chemical reaction rates are fast compared to fluid mixing rates and species quickly

reach their chemical equilibrium levels) one could relate the reaction rates in flames to the rate of

scalar dissipation ( ) of the rate of molecular mixing ( ) [65]. The variable is used to describe the

instantaneous local departure from equilibrium observed in the diffusion flame – a high indicates a

25

high rate of removal of heat and species due to turbulent mixing. At a critical , often denoted the

rate of heat removal becomes too high and the flame is quenched.

A method that was developed and is closely related to the concept of mixing-dominated turbulent

combustion is the laminar flamelet model. The key assumption of this approach is that the flame

reaction zones are thinner than the smallest length scales of turbulence – if this is true, then the

turbulent flame can be said to be made up of a collection of smaller laminar flames (i.e. flamelets) [65].

The implication of this is that the local instantaneous species composition and temperature can be

derived from a laminar flame that has the same mixture fraction and scalar dissipation. In the

Stationary (or Steady) Laminar Flamelet Model (SLFM), the parameters of mixture fraction and scalar

dissipation are used to generate a library of values for temperature, species composition, and even

reaction rates [65]. This library is usually pre-calculated for a range of and in order to save time

during the simulation and is therefore a favourable approach to reducing computational costs. During

the CFD calculation, the mixture fraction and scalar dissipation (along with flow fields) would be solved

independently of temperatures and species compositions, which could then later be looked up in the

flamelet library. Unfortunately, this indirect nature of solving for species compositions and

temperatures presents a problem for soot modelling as it is not possible to include the effects of

species consumption due to soot formation/oxidation kinetics. A larger problem is that the SLFM has

difficulty accurately predicting slow-forming species like NOx despite a relaxed dependence on the

assumption of equilibrium chemistry. This is due to the fact that the flamelet structures cannot

respond instantaneously to changes in the scalar dissipation ( ). As a result, the SLFM is not

appropriate for applications where chemical time scales are comparable (i.e. on the same order of

magnitdue) to the flow time scale [69]. In other words, the SLFM is not appropriate for slow chemistry

applications (like soot) due to its fast chemistry assumptions.

26

The inability of the SLFM to handle transient, non-equilibrium effects has led to the development

of an Instationary (or Unsteady) Laminar Flamelet Model (ILFM/ULFM). In this approach, unsteady

flamelets are not pre-processed, but instead solved in conjunction with the CFD simulation. One

example of an unsteady flamelet model is the one developed by Pitsch et al. [70],[71] which was

successfully applied to both a turbulent ethylene jet diffusion flame and turbulent hydrogen jet

diffusion flame. Pitsch and coworkers accounted for the "history" of a flamelet by calculating a

"flamelet time" which was related to the distance to the fuel nozzle and the axial velocity of the flow

[71]. This calculated flamelet time could then be used to resolve the unsteady terms in the transport

equations for species mass fractions and temperature. By using this method, Pitsch and coworkers

were able to use the ULFM to account for transient affects such as radiative heat transfer [71].

The SLFM and ULFM are widely used in turbulent combustion modelling studies due to the

advantage of reduced computational costs and relatively good ability to predict species composition

and temperatures. However, it is important to remember the inherent assumption behind both

flamelet models – that is, that the reaction zone must be thinner than the smallest length scale of

turbulence. If this is not true, an alternative method must be used. One such alternative method is the

Intrinsic Low-Dimensional Manifold (ILDM), which was first developed by Maas and Pope [72]. Maas

and Pope showed that for a given chemical system with a given number of species and reactions, one

could decouple the chemical reactions with the fastest timescales, greatly reducing the number of

variables needed to describe the system [73]. This is a reasonable assumption as the range of chemical

timescales in a simulation can often extend across many orders of magnitude. If enough time passes,

the slower time scales begin to dominate the overall behaviour of the system and the faster reactions

can be assumed to be at equilibrium – the manifold represents a subspace of the entire reaction

system where this behaviour is observed [72]. Maas and Pope demonstrated that all reaction

trajectories tended to move towards this manifold regardless of the initial conditions specified. Thus,

the number of necessary calculations can be reduced by only solving for the slow chemistry that occurs

27

on this manifold and neglecting the relatively short amount of time it takes for the reaction trajectory

to reach the manifold [72]. As a result, one could feasibly create a low dimension tabulation of the

chemical system as a function of a smaller number of so-called “progress variables” (arbitrary

parameters) [69] instead of trying to account for every coupled species and reaction, which can result

in a table that would need over 30 dimensions for just a simple methane-air flame [73]. The dimension

of the ILDM is related to how much time must pass before the reaction trajectories reach the manifold

– the shorter the time, the higher the dimension required [74]. The ILDM approach has been applied

with some success in applications such as a turbulent methane diffusion jet under high pressure [75]

and also in compression ignition engine simulations at Westport Innovations. Compared to the flamelet

approach, the ILDM requires more computational resources to tabulate and to utilise within the

simulation (due to a higher number of variables in the table), but is more widely applicable to a variety

of flame conditions.

28

3. Model Development Methodology

This chapter outlines the overall methodology employed in developing a simplified soot model for

application in natural gas fuelled compression ignition engines. The goal is to begin model development

and validation with simple laboratory flame conditions and move to progressively more engine-like

conditions. This idealized workflow is illustrated in Figure 3.1.

Figure 3.1 – Workflow diagram of project

3.1. Experimental Datasets for Model Validation

A review of detailed soot measurements for methane and natural gas flames in the literature was

conducted in order to determine the availability of validation data. The full review can be found in

Appendix A of this thesis. While validation data is widely available for more heavily sooting simple fuels

such as ethylene, such as the study by Santoro et al. [76] and other studies detailed in [77], detailed

soot data for methane flames is relatively limited. This is due in large part to the lower tendency of

soot formation in methane combustion in atmospheric laboratory flame conditions. Nonetheless,

datasets with soot measurements were found for coflow laminar diffusion flames, coflow turbulent

diffusion flames, counterflow laminar diffusion flames, shocktubes, and laminar premixed flames. In

addition, soot measurement data was also available for natural gas fuelled engines, but limited to

exhaust measurements. Unfortunately, exhaust-only measurements are not particularly useful for

developing a robust soot model as the solution for a "correct" exhaust soot prediction is not a unique

one. For example, a model could have an inception rate that is too high, but is absolved by having an

29

oxidation rate that is also too high. For this reason, spatially resolved measurements of soot are more

useful – particularly in applications where there are clearly defined soot formation/oxidation regions.

As discussed in Section 2.4 earlier, turbulent combustion modelling presents many of its own

challenges that would exceed the scope of this work and as such, development and validation for the

data found was not considered. Furthermore, the combustion behaviour of premixed flames and

shocktubes is fundamentally different from the combustion behaviour of the non-premixed

combustion found in direct injection compression ignition engines. Hong et al. [50] used a similar

approach in developing their soot model by using shock tube data from various experiments to

calibrate their model and then later applying it to a diesel engine simulation. However, while the model

gave reasonable qualitative results, the soot predictions quantitatively were under-predicted. Thus, it

was also determined that model development and validation for these cases would not be particularly

useful in serving our target application and were also neglected. Consequently, the focus for model

development and validation in the work presented focuses on the coflow laminar diffusion flame setup

(discussed in further detail in Section 2.3). Fortunately, the body of literature investigated for coflow

laminar diffusion flames provided the most volume of soot measurement data of all the types of

experimental investigations on soot in methane and natural gas combustion.

3.2. Comparison to Sectional Detailed Soot Chemistry Model

Despite the fact that there are several studies on methane/air coflow flames, the majority of the

measurements made in these studies are limited to spatially resolved measurements of soot volume

fraction only. Spatially resolved measurements of other important soot characteristics, such as number

density (primary/aggregate) or soot particle diameter (primary/aggregate) is limited to a study by

Schittkowski et al. [78] and is subject to a high degree of uncertainty due to the measurement

techniques used. In addition, there exists no data set where measurements were made on the rate of

soot formation/oxidation. In order to avoid creating a "curve-fitted" model that is only applicable in

30

one or two applications, it is desired to have some avenue of validation for the rates of soot

formation/oxidation as well.

One strategy to carry out this validation is to use a detailed sectional soot chemistry model that has

been developed in parallel in another study [20]. Since this model has been well-validated in a variety

of applications [5],[20],[42],[44] we can use this existing model to generate a rich dataset of numerical

"measurements" that can fill in the gaps found in literature. Thus, numerical experiments using the

sectional detailed soot chemistry model will be run and data from these numerical experiments can be

used to supplement existing measurement data in the development and validation of our simplified

soot model. This adapted workflow is outlined in Figure 3.2. However, it is important to note that the

employed sectional detailed soot model was developed and validated for ethylene/air flames, not

methane/air flames. Nonetheless, the similarity between the two fuels and their combustion behaviour

(chemical kinetic mechanisms for the two fuels are often interchangeable) should allow for reasonable

results from the model to be obtained.

Figure 3.2 – Adapted workflow diagram of project

31

3.3. Experimental Cases Considered

Data from studies by Smooke et al. [63], Thomson et al. [79] and Joo and Gülder [80], and

Schittkowski et al. [78] was used to validate the developed model over a variety of operating

conditions. In the study by Smooke et al. [63], spatially resolved measurements of soot volume fraction

were made in the laminar flame using a technique called thermocouple particle densitometry (TPD).

With TPD, soot volume fractions are determined by introducing a clean thermocouple into a sooting

area of the flame and subsequently relating the measured temperature history to an expected

temperature history using thermophoretic mass transfer formulations. The flame in the Smooke et al.

[63] case was placed inside a cylindrical chimney enclosure located at the edge of the coflow air inlet.

Thomson et al. [79] and Joo and Gülder [80] collected spatially resolved measurements of soot volume

fraction but at higher pressures using two non-intrusive techniques - Spectral Soot Emission diagnostics

(SSE) and Line-of-Sight-Attenuation (LOSA) - to make their measurements. The details and theory of SSE

and LOSA are discussed elsewhere in [81] and [82] respectively. The flame was surrounded with a

chimney composed of three flat windows to facilitate optical access and the burner assembly was

placed inside a high-pressure assembly which is illustrated in Figure 3.3. Thomson et al. [79] and Joo

and Gülder [80] used the same experimental setup but conducted their measurements in separate

facilities; however, Joo and Gülder [80] conducted some tests at higher pressures but also only

reported measurements using the SSE diagnostics technique.

32

Figure 3.3 – Schematic of high pressure combustion rig taken from [79]. Finally, in the study by Schittkowski et al. [78], spatially resolved measurements of soot volume

fraction, primary particle radius, and primary particle number density were made using a laser-induced

incandescence (LII) system, the details of which can be found within the aforementioned study. Figure

3.4 summarizes the burner geometry and operating conditions of the studies investigated in this work.

Data Set Geometry (cm)

Flow Rates (g/s) Operating Pressure (atm)

Smooke et al. [63]

Methane = 0.16 Air = 24.64

Argon = 0.004

1

Thomson et al. [79]

Methane = 0.00055 Air = 0.4

10 – 40

Joo and Gülder [80]

Methane = 0.00055 Air = 0.4

10 – 60

Schittkowski et al. [78]

Methane = 0.00326 Air = 1.172

1

(a) (b)

Figure 3.4 – (a) Table summarizing burner geometry and operating conditions of experiments (b) Diagram of coflow burner defining and

3.4. Considerations for use in Westport Simulations

In addition to ensuring the developed simplified model could well reproduce the sooting behaviour

observed in the aforementioned experiments, it was important to consider some implications for

future application in turbulent engine simulations such as those at Westport Innovations.

33

3.4.1. Coupling of Soot Model to Gas Phase Species Consumption

As discussed in Section 2.4, many of the current methods employed to calculate turbulent

combustion chemistry reduce computational costs by pre-calculating reaction rates, species

compositions, and temperatures in a table as a function of various parameters like mixture fraction ( )

or scalar dissipation ( . This approach is evident in the flamelet library approach (steady and

unsteady) as well in some aspects of the ILDM approach. While this method is an effective way to

include combustion chemistry without drastically reducing the chemical mechanism, it presents a

major problem for soot modelling. As the species conservation equations are not explicitly solved in

these approaches, it is not possible to couple the consumption of gas phases species to the

formation/oxidation of soot in these types of simulations. This can potentially cause some

overprediction in soot concentration levels as soot formation/growth can continue indefinitely. To

investigate the effect that this can have on the performance of the proposed soot model, simulations

with and without gas phase species consumption were run and their results were compared.

3.4.2. Coupling to Radiation Heat Transfer

Another important feedback loop that is difficult to implement with the flamelet library and ILDM

approach is the radiation heat loss due to soot particles in the flame. In these tabulated approaches to

solving for flame temperatures indirectly, it is difficult to incorporate a feedback mechanism by which

the presence of soot particles decreases the local temperature. Since a lowered temperature will

usually result in reduced reaction rates, neglecting the coupling of radiation heat transfer can result in

overpredicted flame temperatures and hence, overpredicted soot levels. Similar to the approach

described in Section 3.4.1, simulations with and without coupled radiation heat transfer were run and

their results were compared.

34

4. Mathematical Formulation

This chapter describes the theory and mathematical models employed within the presented work.

In particular, the conservation equations solved for mass, momentum, species, and energy for the

laminar coflow flame are described as well as the formulations of the simplified soot model.

4.1. Computational Domain

The flames considered in this study were all axi-symmetric co-flow laminar diffusion flames and as

such, the 3-dimensional flame is reduced to a 2-dimensional computational domain, which reduces

computational costs. A cylindrical coordinate system is constructed in the radial ( ) and axial ( )

directions and a representative diagram of the computational domain is shown in Figure 4.1.

Figure 4.1 – Schematic of the computational domain (greyed out area) super-imposed on a diagram of a typical laminar coflow diffusion flame. Also illustrated is the orientation of the coordinate system used. Note that the illustration is not to scale. Taken from [77].

4.2. Governing Equations

The 2-D laminar coflow diffusion flame code used in this thesis solves fully-coupled elliptical

conservation equations for mass, momentum, species, and energy. The laminar flame code used in this

study has been widely applied to other laminar coflow diffusion flame studies such as the recent efforts

by Guo et al. [83], Zhang et al. [42], Dworkin et al. [20], and Eaves et al. [5].

35

4.2.1. Conservation of mass

The equation for conservation of mass is as follows:

(4.1)

Where is the mixture density and and are the axial and radial velocities respectively.

4.2.2. Conservation of momentum

The equation for the conservation of axial momentum is as follows:

(4.2)

Here, represents the local pressure, is the dynamic viscosity of the gaseous component of the

mixture, and is the gravitational acceleration, assumed to be solely in the axial direction. Similarly,

the equation for the conservation of radial momentum is as follows:

(4.3)

4.2.3. Conservation of species

The conservation equations for gaseous species mass fractions are defined as:

(4.4)

is defined as the mass fraction of the species, and are the diffusive velocities in the axial

and radial directions respectively of the species, is the molecular weight of the species, and

is the molar production rate of the species per unit volume. is denoted as the total number

of gaseous species present in the chemical mechanism. An important point to note is that also

includes contributions/consumption due to soot formation/oxidation chemical processes. When

36

gaseous species consumption due to soot formation/oxidation were decoupled, only included

contributions from chemical reactions between gaseous species only.

4.2.4. Conservation of energy

The equation for the conservation of energy is formulated as:

(4.5)

Where is the constant pressure specific heat of the mixture, is the local temperature, is the

thermal conductivity of the mixture, is the constant pressure specific heat of the species, is

the specific enthalpy of the species, is the constant pressure specific heat of soot, is the

mass fraction of soot, and are the radial and axial thermophoretic velocities of soot particles,

is the specific enthalpy of soot, is the molecular weight of soot, which is considered to be the

same as carbon in this study, is the molar production rate of soot per unit volume, and is the

total radiation heat transfer from both gaseous species and soot. In specific simulations where

radiation heat transfer was decoupled, the term was neglected. Zhang [42] found that only H2O,

CO2, and CO contributed significantly to radiation heat transfer compared to the rest of the other

gaseous species. Thus, only these species were accounted for (along with radiation from soot particles)

in the term. Thermal properties of soot were assumed to be identical to that of graphite – the

thermal properties of which can be obtained from JANAF thermochemical tables [84].

4.3. Diffusivity of Gaseous Species

The diffusion velocity of gaseous species as noted in Equation (4.4) is calculated in the present

work using the following relation:

37

(4.6)

In the above equation, is the ordinary diffusion velocity of the species, and similarly, is

the thermal diffusion (i.e. thermophoretic) velocity of the species. is a correction diffusion

velocity term which becomes necessary due to the use of mixture-averaged diffusion coefficients. With

the use of a mixture-averaged diffusion coefficient, the net diffusion flux may not sum to zero, so the

correction term is applied [85].

and are in turn calculated by using an approximated mixture-averaged formulation

[77]:

(4.7)

(4.8)

Here is the mole fraction of the species, is the thermal diffusion ratio of the species, and

is the mixture diffusion coefficient of the species. is defined as:

(4.9)

Here is defined as the binary diffusion coefficient. In this study, thermal diffusion was neglected for

all species except for H2 and H.

4.4. Radiation Heat Transfer

Radiation heat transfer in the laminar coflow flame code is calculated using the Discrete Ordinate

Method (DOM) and is only briefly described here. The DOM radiation heat transfer model used in this

work was first developed by Liu et al. [86] and further details on the DOM model can be found in the

respective study. The DOM method is advantageous in that the accuracy of the model is comparable to

more computationally intensive approaches such as a Monte-Carlo approach and that it does not

require any a priori assumptions about the optical thickness of the flame being modelled [77].

38

Radiation Transfer Equations (RTEs) are numerically solved in an axisymmetric cylindrical coordinate

system and the angular and spatial coordinates are discretized using a T3 quadrature system [86]. This

was coupled with a statistical narrow-band-based correlated-absorptivity ( ) model in order to

determine the absorption coefficients of the gas phases species H2O, CO2, and CO. The spectral

absorption of soot was determined to be:

(4.10) Here, is the spectral absorption of soot and is the wavenumber of the spectral band. This

formulation was based on experimental measurements by Buckius and Tien [77],[87].

4.5. Simplified Soot Model

The semi-empirical soot model employed in this study is based on previous work done by

Fairweather et al. [46] and later updated by Woolley et al. [47]. It adds two equations that track soot

mass and soot number density and reduces computational costs by calculating averaged soot particle

diameters per control volume instead of tracking the soot particle size distributions and aggregate

structures. As a semi-empirical model, some simplifications have been made based on the current

understanding of the fundamentals of soot formation and oxidation (reviewed by Frenklach in [21]).

This two-equation approach has been shown to have some success in predicting soot in both laminar

and turbulent diffusion flames with the majority of work being done at lower pressures (See Section

2.2.3).

Soot inception (or nucleation), is typically understood in the literature to be the combination of

molecules known as polycyclic aromatic hydrocarbons (PAH) that condense to the solid phase [20].

However, the incipient species in the simplified model is instead chosen to be acetylene, as it is itself a

precursor to the formation of PAHs and removes the necessity to include large chemical kinetic

mechanisms, which can add to the computational costs of turbulent engine simulations. The chemical

formula for soot inception used in the present model is given as:

39

(4.11) Originally, Fairweather et al. [46] based inception solely on acetylene, while an update by Woolley et al.

[47] also included inception via benzene. Early efforts were made to use benzene in the present work

as well; however, as a relatively slow forming species, it was found to be problematic with the ILDM

approach of handling turbulent combustion. In addition, well validated chemical kinetic mechanisms

for methane combustion that included the formation of benzene were not available in the literature.

As Westport already employs a modified GRI-mechanism by Huang et al. [88] that works well at

predicting other pollutants (like CO), it was also not desired to change the chemical kinetic mechanism

to accommodate benzene formation. Thus, a soot inception mechanism solely through acetylene was

employed in this study.

As with inception, acetylene is assumed to be the only species that contributes to the surface

growth of existing soot particles and is represented by the following chemical reaction:

(4.12) As discussed earlier in Section 2.1.3, this is a simplification of the currently understood and accepted

mechanism of soot particle surface growth known as HACA. Oxidation in the simplified soot model is

considered via surface reactions with O2 and OH only and is represented as:

(4.13)

(4.14) While Fairweather et al. [46] originally neglected soot oxidation via OH, this was later included in the

two equation model in an update by Woolley et al. [47]. This better corresponds to the currently

accepted understanding that O2 and OH are dominant contributors to soot oxidation. The rate

constants for OH oxidation are taken from the study by Fenimore and Jones [28]. The soot chemical

reactions (4.11) through to (4.14) are governed by the following equations:

(4.15)

40

(4.16) (4.17)

(4.18) Where denotes a reaction rate determined by a typical Arrhenius rate expression, is the soot

particle surface area per unit volume of the mixture, and , , and represent the

concentrations of C2H2, O2, and OH respectively in units of

. represents the collision efficiency of

the OH molecules, which is set to 20% in this study. The initial values used for the Arrhenius reaction

rates in equations (4.15) to (4.18) are listed in Table 4.1. Rates for to are taken from Fairweather

et al. [46] while is derived from the rate determined by Fenimore and Jones [28].

Table 4.1 – Summary of reaction rate constants in the Arrhenius form

, where units are in g,

cm, mol, s, K.

1.35E6 0 20 365

5.0E4 0 12 080

1.78E6 0 19 630

864 0.5 0

As noted in equations (4.16), (4.17), and (4.18), there is a functional dependence on the soot particle

surface area for surface growth and oxidation. In addition, Fairweather et al. [46] suggested that this

functional dependence can simply be represented as a linear function such that:

(4.19) The linear dependence assumption is reasonable as it follows that as the soot particle surface area

increases, the tendency of surface reactions occurring would also increase. On the other hand, Liu et al.

[89] suggested that the functional dependence would be better represented as a proportional relation

to the square root of the soot particle surface area such that:

(4.20)

The rationale behind this change is related to the phenomenon of “soot surface ageing” (see Section

2.1.3), where the rate of surface growth for a soot particle tends to decline as it undergoes further

41

surface growth. With Liu and coworkers' implementation, the dependence of surface growth on

surface area is reduced as the soot particle grows larger. This in effect, according to Liu et al. [89],

reproduces an ageing effect of the soot particle. While both functional dependencies hold merit, Liu et

al. [89] noted that similar results for both assumptions could be attained if model constants for surface

growth were adjusted. Thus, the model in this work uses the linear dependence defined in equation

(4.19).

The soot aggregate structure in this simplified approach is neglected, and it is assumed that all soot

particles are spherical in shape. Thus, the available surface area of soot per unit volume of the mixture

is given by:

(4.21)

Here, is the soot aggregate number density defined as the number of soot particles per unit mass of

mixture. , which is the diameter of the representative sphere of soot calculated by using the relation:

(4.22)

where is the mass fraction of soot, and is the density of soot, taken to be 1.9 [g/cc].

In order to solve for the soot mass fraction and soot number density , two additional source

term equations are solved:

(4.23)

(4.24)

Where is the mass of soot, is the molar mass of carbon taken to be 12.011 [g/mol] in this study,

is Avogadro’s number and is Boltzmann’s constant. represents the minimum number of

carbon atoms found in a soot particle, which in a sense, determines the minimum diameter of a soot

particle. In this study, was set to 90,000 which translates into an incipient soot particle size of

42

approximately 12 nm. The 12 nm size was kept in part due to the minimum detection size of soot

particles in common experimental apparatuses. The rationale is that any soot particles smaller than this

would not be detected in experiments and therefore not play a role in model validation. Furthermore,

previous studies found that the predictions generated by the two equation model were relatively

insensitive to the value of selected [45]. is an agglomeration constant that determines the rate

at which smaller soot particles combine. For this study, was initially set to 3, but is known to vary

between 3 to 9 [45],[46].

4.5.1. Soot transport equations

The transport of soot in the work presented is calculated in a similar manner to the methods

presented by Zhang et al. [42], Eaves et al. [5], and Chernov et al. [44] in similar sooting laminar coflow

flame studies. If one accounts for normal diffusion, thermopohresis, and soot formation and oxidation,

the soot transport equations for soot mass and number density is as follows:

(4.25)

(4.26)

Thermophoretic velocity in this work is calculated according to the definition provide by Talbot et al.

[90] and is given by the equation:

(4.27)

4.6. Numerical Method

The numerical method used in this work is similar to the methods used in numerical studies of

other laminar flames [5],[20],[42],[44],[77]. As such, the reasoning and subsequent development of

these approaches is not part of the scope of this work. Further details can be found in [77].

Finite-volume method discretizations were used in order to solve the above-mentioned

conservation equations and soot equations. A staggered mesh was employed in order to avoid

43

calculated pressure gradients that are independent of the local control volume’s pressure. In order to

solve the discretized equations, a semi-implicit scheme was used to solve the coupled pressure and

velocity as well as the discretized governing equations [91]. A second order central difference scheme

is used to discretize the diffusive terms while a power law scheme was used to discretize the

convective terms [91]. Momentum and continuity (which is converted to a pressure correction

equation) equations are solved independently using a Tri-Diagonal Matrix Algorithm (TDMA). Next, the

gaseous species equations are solved simultaneously in each control volume in order to deal with the

overall stiffness of the system. Finally, the soot transport equations and energy equation are solved in a

segregated manner with the TDMA approach.

An arbitrary initial guess for the system is used (typically a temperature of 1900 K and ambient air)

and pseudo-time stepping is used to arrive at a converged steady state solution. Chemical reaction

rates for gaseous species as well as thermal properties are calculated using subroutines from the open-

source CHEMKIN-II [92],[93] libraries. Transport properties of gaseous species including mixture

averaged values for fluid viscosities, thermal conductivities, and diffusion coefficients were calculated

using TPLIB [85],[94]. Two different chemical mechanisms were used: a C1/C2 mechanism originally

developed by Slavinskaya and Frank [7] and a modified GRI-mechanism by Huang et al. [88] developed

for low temperature high pressure methane/air combustion. A modified version of the Slavinskaya and

Frank [7] mechanism with enhanced PAH growth was also employed in certain simulations. The

modifications to this mechanism are described in the work by Dworkin et al. [20] and Slavinskaya et al.

[95].

4.6.1. Mesh and boundary conditions

The properties of the mesh used for finite-volume discretizations varied depending on the

experiment and the geometric properties of each mesh can be found in Table 4.2. A non-uniform mesh

in all cases is employed in order to resolve the large gradients of temperature, species, velocity, etc.

near the flame while reducing computational cost in areas farther away from the flame. Generally

44

speaking, the mesh size was kept at a constant small size until a distance of approximately three times

the flame height and flame radius upon which a stretching factor was employed to allow the mesh size

to grow. This effect can be observed in Figure 4.2.

Data Set Number of Control Volumes

Size of domain [cm x cm]

Initial (stretch start)

{stretch factor} [cm, cm, unitless]

Initial (stretch start)

{stretch factor} [cm, cm, unitless]

Smooke et al. [63] 192 x 92 12.29 x 5.41 0.02 (1.00)

{1.071}

0.05 (6.70)

{1.0205}

Schittkowski et al. [78]

224 x 130 19.23 x 4.37 0.02 (1.99)

{1.075}

0.05 (7.80) {1.03}

Thomson et al. [79] and Joo and

Gülder [80]

240 x 130 3.25 x 1.304 0.0035 (0.31)

{1.075}

0.0077 (1.30) {1.03}

Table 4.2 – Summary of geometric properties of meshes used in laminar coflow flame simulations.

A diagram of the non-uniform mesh is presented in Figure 4.2. As previously mentioned in Section

4.1, the computational domain considered is a 2-dimensional slice of the axisymmetric coflow flame. As

a result, in addition to the normal inlet and outflow boundary conditions employed, there are

additional constraints for the boundary conditions at the axis of symmetry and outer radial boundary of

the computational domain. The inlet condition in all cases was considered to be uniform in both

temperature and velocity. The outflow boundary was defined as a zero gradient condition given as:

(4.28)

Similarly, zero gradient conditions are also imposed on the axis of symmetry:

(4.29)

The outer radial boundary condition varied depending on whether or not it was an open air flame (i.e.

free-slip) or a flame within a chimney (i.e. no-slip). For a free-slip outer radial boundary the following

conditions were used:

45

(4.30)

On the other hand, for a no-slip outer radial boundary condition, the following conditions were used:

(4.31)

Figure 4.2 – Diagram of typical non-uniform mesh employed in the simulations presented. Adapted from [77].

4.6.2. Parallel Computation

The laminar coflow flame code employed in this work takes advantage of a distributed-memory

parallelization with strip-domain decomposition method in order to make calculations tractable and

complete within reasonable time limits. Further details on the development of the parallel flame code

can be found in [77]. The computational domain is divided uniformly by assigning each row of control

volumes perpendicular to the z-axis to an individual CPU. The Message Passing Interface (MPI) library

[96] is used to facilitate the communication and distribution of workload between CPUs. Calculations

were performed on the General Purpose Cluster (GPC) at the SciNet supercomputer centre using a 8-

core Intel Xeon E5540s with 2.53 GHz chip speeds and InfiniBand network connections.

46

4.7. Detailed Sectional Soot Model

As outlined in Section 3.2, a detailed sectional soot model will be employed to run numerical

experiments in order to complement the existing experimental data found in literature and facilitate

the validation of the simplified soot model presented in this study. The detailed sectional soot model is

the result of an ongoing parallel research program being conducted at the CRL at the University of

Toronto and more details on the model can be found in the work presented by Zhang et al. [42],[77].

The major differences between the detailed sectional model and the simplified model are illustrated in

Table 4.3. The biggest simplification in the simplified model is the absence of a soot particle size

distribution, followed by the simplified inception pathway through acetylene, and the surface growth

rate with a linear dependence on soot surface area.

Mechanism Simplified Model Detailed Sectional Model

Soot Inception Nucleation of soot particles based on acetylene

Nucleation of soot particles based on the PAH molecule pyrene (A4)

Surface Growth Reaction rate based on acetylene and a linear dependence on soot surface area

HACA surface reaction scheme with an empirical parameter to correct for deficiencies in model

Oxidation Reaction rates based on oxidation models from Lee et al. (O2) [31] and Fenimore and Jones (OH) [28] and a linear dependence on soot surface area

Oxidation via O2 and OH accounted for as part of the HACA surface reaction scheme. OH oxidation based on collisional frequency and O2 oxidation based on Frenklach and Wang [26] model based on Nagle and Strickland-Constable [30] rate

Soot aerosol dynamics Soot particle size distribution is neglected and all soot particles are assumed to agglomerate into spheres

Soot particle size distribution is modeled using a sectional approach with 35 sections to track primary particles and an additional 35 sections to track aggregate structures. The smallest bin size 0.86 nm and the largest bin size is 13867 nm

Table 4.3 – A summary of the major differences between the employed simplified soot model and a previously developed detailed sectional model.

47

The detailed sectional model was run using an identical flame code with almost the same governing

equations, the exception being the soot transport equations where each section required its own

transport equation. The DOM radiation model was used along with the same mesh, boundary

conditions, and solvers. An empirical parameter, , is used in the detailed model as a correction factor

to account for the actual number of reactive surface sites relative to the number predicted by the

current implementation of the HACA model. A value of is typically selected for a simulation such that

reasonable predictions of soot volume fractions are made. More details on can be found in [77] and

in recent investigations by Dworkin et al. [20] and Eaves et al. [5].

48

5. Development and Validation of Soot Model

The simplified soot model developed in this work was based on the two equation approach

demonstrated by Fairweather et al. [46]. The model was initially calibrated for a turbulent natural gas

flame and as such, the initial model constants are used as a basis for initial investigations on the

model’s performance in the previously mentioned experimental studies chosen for model validation (in

Section 3.3). However, the original Fairweather et al. [46] model lacked OH oxidation, so the Fenimore

and Jones [28] OH oxidation model was also included. The initial model constants have been listed

earlier in Section 4.5.

5.1. Chapter Outline

The work described in this chapter will begin with a sensitivity analysis of the model parameters in

the simplified soot model in Section 5.2. Calculations and results from the simplified model and

detailed sectional model are subsequently presented in Sections 5.3 - 5.6. Improvements and

modifications to the simplified model are discussed in Section 5.7 and updated simulation findings are

presented in Section 5.7.2. The effect of uncoupling gas phase species consumption and radiation heat

transfer from the rest of the model is then subsequently investigated in Section 5.8. Finally, the

computational cost between the simplified model and the detailed sectional model is compared in

Section 5.9.

5.2. Sensitivity Analysis of Parameter Terms in Simplified Model

The parameters used in the simplified two equation soot model consist of the pre-exponential term

and activation energy in each Arrhenius rate expression for equations (4.15) to (4.18) in Table 4.1.

Activation energies for each rate equation have been investigated in previous studies [45],[46] and are

considered to be constants in this study and hence kept at their original values listed in Table 4.1. This

appears to be a reasonable approach as Woolley et al. [47] also kept the original activation energies

when adapting and updating the two equation model for a different combustion application. Additional

parameters in the two equation model also consist of the and terms used in the soot number

49

density source term equation (4.24). The major parameters investigated in this study are summarized

in Table 5.1. The other parameters in the simplified model that were not investigated in this study were

the activation energies, , for the inception, growth, and O2 oxidation Arrhenius rate expressions in

equations (4.15) to (4.17).

Parameter Note

Pre-exponential term in Arrhenius rate expression for soot inception.

Pre-exponential term in Arrhenius rate expression for soot surface growth.

Pre-exponential term in Arrhenius rate

expression for soot oxidation via O2.

Pre-exponential term in Arrhenius rate expression for soot oxidation via OH.

Determines the minimum number of carbon atoms in an incipient soot particle and hence determines the minimum diameter of a soot particle.

Determines the rate at which soot particles agglomerate.

Table 5.1 – List of major parameters in simplified two equation soot model.

The Smooke et al. [63] data set was used as a baseline case upon which to investigate the

sensitivity of soot volume fraction, soot aggregate averaged diameters, soot number density, soot

inception rate, soot surface growth rate, and soot oxidation rate (henceforth collectively referred to as

“sooting behaviour”) to the parameters listed in Table 5.1. Simulations were run where a single

parameter was modified while all other parameters were held constant. Then, the effect on the

aforementioned soot details relative to the baseline case with initial parameters listed in Table 4.1 and

Section 4.5 was recorded. This process was repeated for each parameter listed in Table 5.2. Peak values

of sooting behaviour were recorded in both the wing and the centreline of the flame (illustrated in

Figure 5.1).

50

Figure 5.1 – Diagram of the “wing” and “centreline” regions of a typical flame.

Figure 5.2 – Sensitivity of sooting behaviour to the pre-exponential value of soot inception, . Solid

lines show peak values at the wing of the flame and dashed lines show values in the centreline of the flame. A logarithmic scale is applied to the Y-axis. Figure 5.2 shows the sensitivity of sooting behaviour on the pre-exponential value of . The

legend shown in Figure 5.2 is used for all subsequent graphs in 5.2. Here, “Pre-exponential Factor

Increase” is defined as:

Pre-exponential Factor Increase

(5.1)

51

A similar definition for subsequent “Pre-exponential Factor Increase” is used in Figure 5.3 to Figure 5.7.

Similarly, “Change relative to baseline” is defined as:

(5.2)

Not surprisingly, Figure 5.2 demonstrates that all sooting behaviour terms are nearly linearly

dependent on the inception rate (a log scale is applied since underpredictions share the same weight as

overpredictions, which is not obvious on a linear scale). An increase in inception rate leads to more

soot particles being created and hence a higher soot particle number density and soot volume fraction.

In addition, the increased soot particle surface area means that surface growth and oxidation

mechanisms become stronger as they are linearly related to the soot particle surface area. On the

other hand, the inception rate does not seem to largely affect the soot aggregate averaged diameters

as the additional soot mass gained due to a higher surface growth rate (due to a higher surface area

from a higher number of particles) that is effectively cancelled out by a higher oxidation rate.

52

Figure 5.3 – Sensitivity of sooting behaviour to the pre-exponential value of soot surface growth, . Solid lines show peak values at the wing of the flame and dashed lines show values in the

centreline of the flame. A logarithmic scale is applied to the Y-axis. Legend is in Figure 5.2. The surface growth rate sensitivity analysis, which is illustrated in Figure 5.3, shows that all sooting

behaviour aspects are very sensitive to the pre-exponential . The behaviour is less pronounced in

the centreline of the flame, and in fact, the soot inception rate and soot number density actually drops

in the centreline relative to the baseline case. This can be explained by the fact that inception and

surface growth both compete for acetylene, so the increased surface growth rate will reduce the

inception rate, thereby lowering the soot particle number density.

53

Figure 5.4 – Sensitivity of sooting behaviour to the pre-exponential value of soot oxidation via O2,

.

Solid lines show peak values at the wing of the flame and dashed lines show values in the centreline of the flame. A logarithmic scale is applied to the Y-axis. Legend is in Figure 5.2. An analysis of the sooting behaviour sensitivity on soot oxidation via O2, shown in Figure 5.4, revealed

that the O2 oxidation model by Lee et al. [31] has a limited effect on sooting behaviour in the flame

compared to surface growth and inception. Only in the extreme case of removing O2 oxidation entirely

does one start to observe significant changes to sooting behaviour characteristics in the wing of the

flame. Removing O2 oxidation creates the asymptote observed in Figure 5.4 as the logarithmic value of

zero is undefined. It is worth noting that the sooting behaviour in nearly all cases is unaffected by

changes in the O2 oxidation rate. One exception to this is the O2 oxidation rate itself, which actually

increases faster in the centreline than it does in the wing with increasing O2 oxidation rate.

54

Figure 5.5 – Sensitivity of sooting behaviour to the pre-exponential value of soot oxidation via OH, . Solid lines show peak values at the wing of the flame and dashed lines show values in the centreline of the flame. A logarithmic scale is applied to the Y-axis. Legend is in Figure 5.2. The sooting behaviour as affected by the OH oxidation rate, shown in Figure 5.5, is very similar to

effects observed with changes to the O2 oxidation rate described above. As before, the removal of OH

oxidation by setting the pre-exponential value of AOH to zero creates the asymptote observed. For the

most part, there are minimal changes to the soot volume fraction, soot number density, and soot

diameter predictions in the centreline and wing. This is in contrast to the changes to the O2 oxidation

rate where changes in the sooting behaviour in the wings of the flame were much more noticeable.

Thus, it appears that the O2 oxidation model is the dominant oxidizing mechanism in this particular

simplified model and flame setup. The competing behaviour between O2 oxidation and OH oxidation is

once again observed – as OH oxidation is increased, the O2 oxidation rate decreases and vice versa.

Interestingly, increasing the OH oxidation pre-exponential did not significantly increase the overall peak

OH oxidation rate. As the peak OH concentration in the cases investigated did not vary, this suggests

55

that in the particular flame investigated, OH concentration is the limiting factor, and not the pre-

exponential of the oxidation rate.

Figure 5.6 – Sensitivity of sooting behaviour to the selected incipient particle diameter, with the default value at 12 nm. Solid lines show peak values at the wing of the flame and dashed lines show values in the centreline of the flame. A logarithmic scale is applied to the Y-axis. Legend is in Figure 5.2. The sensitivity of sooting behaviour on the parameter, which affects the incipient particle’s

diameter, is shown in Figure 5.6. It can be observed that the soot particle number density is most

affected by the choice of incipient particle diameter, which is not surprising as the source term for soot

number density [Equation (4.24)] is heavily influenced by the choice of the parameter. The

smaller is, the less inception reactions [Equation (4.11)] are required to occur before a new soot

particle is created (and vice versa). This change in soot number density will in turn affect the available

soot surface area, and as such, affects the soot surface growth and oxidation rates in a similar manner.

However, the overall change in measureable sooting characteristics other than soot number density

(i.e. soot volume fraction and soot aggregate diameters) is minimal.

56

Figure 5.7 – Sensitivity of sooting behaviour to the selected agglomeration rate, . Solid lines show peak values at the wing of the flame and dashed lines show values in the centreline of the flame. A logarithmic scale is applied to the Y-axis. Legend is in Figure 5.2. Sooting behaviour was found to be largely unaffected by the agglomeration parameter, with the

exception of soot aggregate diameters and soot particle number density, as discerned in Figure 5.7.

This is not too surprising as the agglomeration parameter, only plays a role in the soot number

density source term in Equation (4.24). As increases, the tendency of soot particles to combine and

agglomerate increases, effectively decreasing the soot particle number density. As this does not affect

the soot mass fraction, it follows that the averaged soot particle diameter will also increase.

5.3. Model Development at 1 atmosphere using the Smooke et al. [63] data set

Computations were completed based on the experiments performed by Smooke et al. [63] as

described in Section 3.3. A modified GRI mechanism developed by Huang et al. [88] was used for

simulations that employed the simplified two equation soot model while a detailed mechanism with

PAH formation developed by Slavinskaya and coworkers [7],[20],[95] was used for simulations that

utilized the sectional soot model. An value of 1 was selected for the simulations conducted with the

57

detailed model for this flame. In the following discussion, the term “simplified model” refers to the

“base simplified model” with constants as outlined in Table 5.5 in Section 5.7.

The radial profiles of soot volume fraction measurements and calculated soot volume fractions

were compared at four different axial heights in the flame and are shown in Figure 5.8. The detailed

sectional code predicted a peak soot volume fraction of 0.49 ppm in the wing of the flame, which

matches the experimental peak measurement of 0.49 ppm in the wing of flame. However, the detailed

sectional code predicted the peak at a slightly higher axial position in the flame – approximately 0.6 cm

higher (of which the peak flame height was approximately 6 cm). This is postulated to be the result of

delayed PAH formation due to some deficiencies in the chemical mechanism used. Since the detailed

code uses PAHs as a direct precursor to soot inception, this delay in sooting behaviour is likely due to

the aforementioned slow PAH formation. The improvement of PAH formation behaviour in chemical

mechanisms is currently the focus of other ongoing studies and further investigation is outside the

scope of this work and instead, the delay was accounted for in the comparisons by simply making the

comparisons at an axial offset of 0.6 cm. On the other hand, the simplified model predicted a peak soot

volume fraction of 0.24 ppm, which underpredicts by slightly more than a factor of two, just over the

reported measurement uncertainty of . However, unlike the detailed sectional model, the peak

soot volume fraction is predicted at the correct axial height in the flame since the soot inception is

based on acetylene and does not suffer a delay. In both cases, the overall computed radial soot volume

fraction profiles capture the general trends found in the experimental measurements. However, at

higher axial heights (namely Z3 and Z4), the soot volume fraction predictions from both models are

shifted outward in the radial direction, with noticeable underprediction in the centreline. This

behaviour has been observed in other studies [5],[20],[63] and appears to be independent of the soot

model used. It is theorized that the centreline underprediction is related to deficiencies in the chemical

mechanisms' ability to predict the correct amount of soot precursor growth within the centre of the

flame. Nonetheless, the qualitative and quantitative results obtained are promising compared to the

58

available experimental data. Recent studies by Eaves and coworkers at CRL have also shown that this

behaviour might also be due to the fact that the model neglects pre-heating of the fuel tube caused by

the flame sitting near the inlet. At the time of writing, this work has not yet been published.

Figure 5.8 – Soot volume fraction profiles at different axial heights above the burner. Z1, Z2, Z3, and Z4 correspond to heights of 2.0, 2.25, 2.5, 2.75 cm for experimental measurements and computations from the simplified model for the Smooke et al. [63] flame. For computations from the sectional model, 0.6 cm was added to each axial height to account for the delay in PAH formation.

5.3.1. Comparisons to numerical data from detailed sectional soot model

While soot volume predictions from the simplified soot model are promising, the computed values

can be improved by adjusting some of the model parameters. However, as previously stated in Section

3.2, it is not desired to simply adjust the parameters at random since arriving at a reasonable

prediction is not a unique solution. Instead, the detailed sectional soot model has been employed to

produce a numerical data set upon which further details on sooting behaviour can be validated. In the

case of Smooke et al. [63], only a few soot volume fraction measurements were made at four heights

in the flame, which were not adequate to reproduce a reasonable looking contour of soot volume

59

fraction. Thus, contours of soot volume fraction could only be compared amongst the simplified code

and detailed sectional model, illustrated in Figure 5.9.

Figure 5.9 – Contours of soot volume fraction (ppm) generated using the simplified model (left) and detailed sectional model (right) for the Smooke et al. [63] flame. The contours of soot volume fraction compare favourably between the simplified model and the

detailed model, with the major difference being the peak soot volume fractions predicted and the

slight axial shift in soot as previously observed in Figure 5.8.

60

Figure 5.10 – Contours of soot number density of aggregates (#/cc) generated using the simplified model (left) and detailed sectional model (right) for the Smooke et al. [63] flame. Computed soot number density contours are shown in Figure 5.10 – unlike the contour of soot volume

fraction, there are significant differences in the predictions of soot number density between the two

models. Firstly, the peak soot number density is about 40% higher in the simplified model than in the

detailed model, with the peak in the wing instead of in the centreline of the flame. The axial shift in the

location of soot is once again observable in the contours. Based on the current implementation of the

simplified model, the higher soot number density can be attributed to either an inception rate that is

too high, or an agglomeration rate that is too low. It’s worth noting that in order to make the

comparison reasonable, particles smaller than 12 nm (the incipient particle size in the simplified model)

were neglected for the purposes of calculating soot number density in the detailed model. However,

particles smaller than 12 nm were still used in the calculation of mass averaged aggregate diameters in

the following comparison. Overall, the discrepancy between the two models can also likely be

attributed to the lack of soot aerosol dynamics in the simplified model (such as fragmentation of

particles) and is a given limitation of the simplified model as it neglects soot aggregate structure. The

soot number density is also sensitive to the incipient particle diameter; however, in order to reduce

61

soot number density, the incipient particle diameter would have to be increased, making it unrealistic

(as most experimental apparatuses start to detect soot particles over a size of ~10 nm).

Figure 5.11 – Contours of soot aggregate mass averaged diameters (nm) generated using the simplified model (left) and detailed sectional model (right) for the Smooke et al. [63] flame. The contours of mass averaged diameters of soot aggregates, shown in Figure 5.11, were compared to

get a better idea of which soot kinetic mechanisms (i.e. inception, surface growth, oxidation,

agglomeration) needed improvement and also to evaluate the ability of the model to predict soot

diameters. As the minimum size of a soot particle was prescribed to be 12 nm, it is not surprising to see

a relatively flat distribution of particle diameters predicted by the simplified model. This is due to the

fact that even a miniscule amount of soot calculated by the model outside the core of the flame will be

reported to have a diameter of 12 nm. The peak size of diameters between the two models differ by

about a factor of two, which suggests the simplified model needs a higher surface growth rate and/or a

higher agglomeration rate to produce particles of a larger size. It is also possible the oxidation rates

may be too high in the model, which could also result in smaller than expected particles.

62

In order to better understand how the model parameters in the simplified model should be

modified in order to better reflect fundamental sooting behaviour as calculated by our detailed model,

inception, surface growth, and oxidation rates were plotted along the pathline of maximum soot, which

can be seen in Figure 5.14. This tracks the evolution of a soot particle that passes through the point of

peak soot volume fraction as it travels through the flame. An example of a soot pathline, which is

calculated considering both the bulk fluid velocity and the soot thermophoretic velocity, is illustrated

below in Figure 5.12.

Figure 5.12 – Example of a pathline of maximum soot. In order to be comparable to the simplified code, the inception rate for the sectional model was

calculated by summing the cumulative contributions to soot mass until it reached the incipient soot

particle size of the simplified model, which is highlighted in Figure 5.13. Essentially, the inception rate

in the simplified code has all the particle growth and agglomeration “built-in” from the gaseous phase

to its incipient particle size of 12 nm. As such, the inception rate in the sectional code needs to be

combined with the particle growth and agglomeration rates that occur between its incipient size of

about 0.9 nm to a size of approximately 12 nm.

63

Figure 5.13 – Diagram of methodology used to compare inception mechanisms between the simplified code and the detailed code.

Figure 5.14 – Calculated plots of soot inception rate, surface growth rate, and oxidation along the pathline of maximum soot for the Smooke et al. [63] flame. The Y-axis is plotted on a logarithmic scale. The cumulative inception rate in the detailed model is similar in terms of its peak value compared to

the inception rate calculated by the soot model, as seen in Figure 5.14. While the peak inception rate is

higher in the simplified model, the major difference is the fact that significant amounts of soot

inception happen earlier in the flame than in the sectional model. This can likely be attributed to earlier

64

abundance of acetylene in the simplified model and the aforementioned delay in PAH formation in the

detailed model. In fact, the shape of the inception rate curves for the detailed and simplified models

closely mirror plots of pyrene (PAH) and acetylene concentration, respectively (not shown here). The

rates of surface growth and oxidation were observed to be higher in the detailed model compared to

the simplified model. However, the simplified model again showed higher rates of surface growth and

oxidation lower in the flame, which is not surprising since inception is calculated to occur lower in the

flame as well. Similarly, the integrated sum of the rates (i.e. the area under each rate curve) also shared

the same comparisons: the integrated inception rate was higher in the simplified model and the

surface and oxidation integrated rates were higher in the detailed model.

Overall, initial results from the simplified and detailed model of the Smooke et al. [63] flame show

that although the simplified model is nearly within the experimental error of measured peak soot

volume fraction, modifications to the surface growth rate and agglomeration rate of the simplified

model may be warranted. This conclusion arises from the fact that soot number densities appear to be

overpredicted in the simplified model while soot aggregate mass averaged diameters appear to be

underpredicted, relative to the detailed soot model. In addition, an analysis of the inception, surface

growth, and oxidation rates along the pathline of maximum soot reinforce the idea that the current

surface growth rate of the simplified model is insufficient.

5.4. Model Development at 1 atmosphere using the Schittkowski et al. [78] dataset

Nearly identical computations and analyses were performed using the experimental configuration

and data provided by Schittkowski et al. [78] as those described in Section 5.3. Calculations for the

simplified model were completed with a modified version of the GRI mechanism by Huang et al. [88].

The detailed model employed a recently developed improved PAH mechanism that is largely the same

as the mechanism by Slavinskaya and Frank [7] with a few modifications to some chemical reaction

rates. This new PAH mechanism attempts to address some of the deficiencies in PAH formation

observed earlier in Section 5.3 and in other ongoing studies. An value of 0.65 was used in the

65

detailed model in order to match the calculated peak soot volume fraction to the experimental peak

soot volume fraction in the wing of the flame. As before, “simplified model” refers to the “base

simplified model” with constants as outlined in Table 5.5 in Section 5.7.

(a)

(b)

Figure 5.15 – Contours of soot volume fraction side by side with experimental measurements in the Schittwkowski et al. [78] flame. Experimental measurements are on the left side of the flame and computations are on the right side of the flame. Results of the simplified model are shown in (a) and results of the detailed model are shown in (b). Computed soot volume fraction contours from the simplified and detailed models were compared to

the experimental measurements made by Schittwkowski et al. [78]. Unfortunately, the raw data from

the Schittwkowski et al. [78] flame is no longer available and only the black and white images of the

contours from the paper could be used for comparisons. In addition, the method of using an LII

diagnostic technique introduces many uncertainties into the measurements made as the LII

66

measurement technique currently requires several assumptions about flame and soot properties as an

input – further details on this process can be found in a study by Will et al. [97]. Since no measurement

uncertainty was reported by Schittwkowski et al. [78] for their techniques, an uncertainty of

was assumed for the purposes of comparison with calculated results. As the parameter in the

detailed model was adjusted to fit the experimental results, it is not surprising that the calculated peak

of 0.50 ppm closely matches the apparent measured peak of 0.55 ppm, both in the wings of the flame.

However, the location of the calculated peak soot volume fraction is once again slightly shifted upward

in the axial direction. On the other hand, the simplified model underpredicted the peak soot volume

fraction by about a factor of 4, giving a calculated maximum soot volume fraction of 0.14 ppm. The

predicted tip of the flame (indicated by the location of the soot volume fraction contour) also does not

reach the same height in the simplified model as observed in the experimental measurements. This

suggests that the soot oxidation rates in the flame may be too high, or that inception and growth rates

in the model are too low.

67

(a)

(b)

Figure 5.16 – Contours of soot particle diameter in the Schittkowski et al. [78] flame. (a) shows the experimental measurements of primary particle diameter on the left and the calculated contour of primary particle diameter on the right from the detailed model. (b) shows the calculated contour of mass averaged aggregate particle diameter from the detailed model on the left and the simplified model on the right. Due to the limited experimental measurements made, only primary particle size of soot particles were

reported by Schittkowski et al. [78]. Since primary particles in the simplified model are not tracked (due

to the lack of a soot aerosol dynamics model), only a comparison to the detailed sectional model could

be made, which is shown in Figure 5.16. Experimental measurements show a predicted peak primary

particle size of approximately 20 nm in the flame, which compares favourably with the detailed model's

prediction of a peak primary particle size of 13 nm. In addition, the spatial distribution of primary

particle size predicted by the detailed model is a good match to the experimental results. In order to

have some insight on the particle sizes being predicted by the simplified model, the mass averaged

aggregate diameters are compared amongst the two models. A peak of about 42 nm was predicted by

the detailed model and a peak of 20 nm. This reinforces the earlier findings from Section 5.3.1 that

68

suggest the growth rate and/or the agglomeration rate predicted by the simplified model is too low

and/or the oxidation rates are too high.

(a)

(b)

Figure 5.17 – Contours of soot particle number density in the Schittkowski et al. [78] flame. (a) Shows the experimental measurements of primary particle number density on the left and the calculated primary particle number density on the right from the detailed model. Different scales are used for each half of the flame. (b) Shows the calculated contour of aggregate particle number density from the detailed model on the left and the simplified model on the right. Similar comparisons were made with respect to particle number density, as seen in Figure 5.17. For the

same reasons described previously, only the detailed model could be used to compare with the

experimental data. Here, we observe the first real problem that the detailed model exhibits as the peak

value of soot primary particle density is overpredicted by an order of magnitude. This is likely due in

part to the detailed model's handling of coalescence, which describes the process in which small,

liquid-like soot particles combine into a new primary particle sphere instead of forming a typical soot

aggregate structure. Without proper consideration of coalescence, the predicted primary particle

number densities can be expected to be overpredicted, as observed. The improvement of this

69

coalescence model in the detailed code is ongoing in a parallel research program and is not part of this

study. On the other hand, the predicted aggregate particle number densities appear to compare

favourably across the simplified and detailed model. Once again, the simplified model predicts a higher

peak value by about a factor of 1.5, suggesting again that either the inception rate is too high, or the

agglomeration rate is too low.


Using a similar process outlined earlier in Section 5.3.1, soot inception, surface growth, and

oxidation rates were compared along the pathline maximum soot volume fraction between the

simplified model and the detailed model. The resulting plots, shown in Figure 5.18, illustrate once again

that although the peak inception rates predicted by both models are comparable, the peak growth rate

and subsequent oxidation rates are too low in the simplified model. This is not too surprising as the

earlier comparisons to experimental and numerical data from the detailed model showed that the

simplified model underpredicted soot volume fractions and particle diameters. As before, the

integrated sum of the rates (i.e. the area under each rate curve) also shared the same results with the

resulting integrated inception rate calculated higher in the simplified model and the resulting surface

and oxidation integrated rates calculated higher in the detailed model.

70

Figure 5.18 – Calculated plots of soot inception rate, surface growth rate, and oxidation along the pathline of maximum soot for the Schittkowski et al. [78] flame. The Y-axis is plotted on a logarithmic scale.

5.5. Results at elevated pressures

The flames of Thomson et al. [79] and Joo and Gülder [80] were simulated at pressures of 10 atm to

40 atm in order to investigate the performance of the simplified model at higher pressures.

Calculations for the simplified model were completed with a modified version of the GRI mechanism by

Huang et al. [88]. The detailed model employed the improved PAH mechanism that is based on the

mechanism by Slavinskaya and Frank [7] with a few modifications to some chemical reaction rates. An

value of 0.10 was used in the detailed model at all pressures investigated as it gave reasonable

predictions of peak soot volume fraction for each case investigated. Even better predictions could have

been obtained by specifically adjusting the value for each pressure, but it was deemed an

unnecessary use of computational resources due to the limited benefits and scientific merit it would to

the study. Finally, “simplified model” refers to the “base simplified model” with constants as outlined in

Table 5.5 in Section 5.7.

71

(a) (b)

(c)

Figure 5.19 – Contours of soot volume fraction (ppm) with experimental measurements made by Joo and Gülder [80]. Experimental measurements are on the left and calculated contours from the simplified model are on the right. Operating pressures are at 10 atm (a), 20 atm (b), and 40 atm (c). Calculated contours of soot volume fraction from the simplified model are compared with

experimental measurements in Figure 5.19. For the most part, differences in measurements made

between LOSA and SSE in Thomson et al. [79] and by SSE in Joo and Gülder [80] were found to be

negligible, with the largest differences occurring between LOSA and SSE measurements in Thomson et

al. [79] at higher pressures. Further details on the discussions of the cause of these discrepancies can

be found in their respective papers, but in the interest of saving the reader the trouble of looking at

three nearly identical measurement data sets, only the most recent results from Joo and Gülder [80]

are discussed. At all pressures, the spatial distribution of soot volume fraction is well reproduced in the

simplified model; however, there is some overall underprediction in the peak soot volume fraction

values. Notably, there is significant underprediction on the centreline of the flame, but the values in

the wing of the flame are all within experimental uncertainty. This can be attributed in part to the large

degree of uncertainty of SSE measurements in the core of the flame, where accuracy is limited due to

uncertainty of the temperature measurements and optical limitations due to the thin flame [79]. In

72

addition, the general trends observed in the experiment as pressure increases (thinning of flame,

higher soot volume fractions) are also well reproduced by the simplified model.

(a)

(b)

(c)

Figure 5.20 – Contours of soot volume fraction (ppm) with experimental measurements made by Joo and Gülder [80]. Experimental measurements are on the leftt and calculated contours from the detailed model are on the right. Operating pressures are at 10 atm (a), 20 atm (b), and 40 atm (c). The results of the detailed sectional soot model are compared with experimental results in Figure 5.20.

The delayed upward axial shift in soot volume fraction predictions are once again observed, but the

predicted values in the wing are well within experimental uncertainty. The centreline soot volume

fraction is within experimental uncertainty at 10 atm, but slightly over-predicted at higher pressures.

As mentioned earlier, there is some additional uncertainty of the measurements made in the core of

the flame. Another possible source of discrepancy is the performance of the refined PAH mechanism,

to which the sooting behaviour in the centreline is highly sensitive. The continued development of the

PAH mechanism, including its validation at higher pressures, is unfortunately outside the scope of this

work.

73


While the detailed sectional model has been extensively investigated at atmospheric conditions, it

has not been as thoroughly validated at higher pressures – especially with respect to particle number

densities and soot particle diameters since there is no available experimental data for these soot

characteristics. Nonetheless, a brief comparison between the two models can provide some insight to

the computed sooting behaviour.

10 atmospheres 20 atmospheres 40 atmospheres

Detailed Model 131 254 309

Simplified Model 63 106 176

Table 5.3 – Calculated peak mass averaged aggregate particle diameters (nm) in the Joo and Gülder [80] flames.

10 atmospheres 20 atmospheres 40 atmospheres

Detailed Model 4.9E10 1.3E11 2.8E11

Simplified Model 1.3E11 1.6E11 2.0E11

Table 5.4 – Calculated peak aggregate particle number densities (#/cc) in the Joo and Gülder [80] flames. Table 5.3 and Table 5.4 list the calculated peak values of soot particle diameters and number densities

from the investigated high pressure flames. Of note is that the detailed model appears to indicate that

the soot diameters calculated by the simplified model should be larger, which is consistent with the

results in Section 5.4 and 5.5. The calculated peak soot number densities are similar, with the simplified

model having a higher soot number density at 10 atmospheres, and the detailed model having a higher

soot number density at 40 atmospheres. It is unclear why this change in behaviour is observed, but it is

likely due to the change in aggregation and coalescence behaviour of soot as pressure increases. Since

the employed detailed sectional soot model uses aggregation and coalescence models originally

developed for one atmosphere, it is possible that these models need to be adjusted for increased

pressures. As aggregation/coalescence behaviour can greatly affect the calculated number densities

[98], it is not surprising that there are some observed inconsistencies in the particle number densities.

Nonetheless, it is still interesting that the detailed model suggests that predicted particle diameters

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should be higher in the simplified code, which once again implies that the growth rate and/or

agglomeration rate and/or oxidation rate should be adjusted.

5.6. Preliminary Results in Engine Simulations

The developed simplified model and some of its iterations were applied to an engine code at

Westport. The details of the code and the engine are unfortunately proprietary and cannot be

disclosed, but some general comments about the performance of the developed soot model can be

made. The simplified soot model was used in a turbulent reacting flow simulation with a moving mesh

that attempts to model the combustion behaviour in a reciprocating compression ignition engine. The

results from using the simplified model with the initial parameters showed that soot mass fractions

being predicted were about an order of magnitude too low, as well as soot diameters being an order of

magnitude too low, compared to experimental results. A possible reason for this is an inception rate

that is too low, a growth/agglomeration rate that is too low, and/or an oxidation rate that is too high.

As seen in the sensitivity analysis in Section 5.2, an increase in any of the above mentioned parameters

can remedy the underpredictions in soot mass and diameters. An odd result from initial runs of the two

equation soot model showed that the calculated soot diameters in the exhaust were smaller than the

incipient diameter of 12 nm, peaking at 5 nm. This can happen in a scenario where soot particles

undergo so much soot oxidation that they shrink to sizes below their incipient size. This suggests that

the soot oxidation mechanisms in the simplified model are too strong and that they result in oxidation

of far more soot than is realistic. In addition, the results from Section 5.3 to 5.5 would seem to suggest

that the growth rate and agglomeration rate needs to be improved.

5.7. Model Improvements

The cumulative simulation results from Section 5.3 to 5.6 were used to direct the modification of

parameters in the simplified model, with the goal of having a single set of parameters that performed

reasonably well in all conditions investigated.

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5.7.1. Oxidation Mechanism of Soot Model

The oxidation mechanisms of soot, in comparison to the mechanisms of inception and surface

growth, have received far less attention in the literature. Most of the existing oxidation models have

been employed in non-smoking flames; however, this is not a thorough test of the accuracy of the

model as correct soot volume fractions can still be obtained as long as all the soot has been oxidized. In

other words, the soot oxidation models can overpredict the level of soot oxidation occurring in the

model and still correctly predict the peak soot volume fraction within the flame. In order to properly

test the performance of a soot oxidation model, it is desired to test its performance on both a smoking

and non-smoking flame. The improvement of oxidation models has been investigated before by Liu et

al. [29] and is also the subject of current investigation for the further development of the detailed

sectional model. As the refinement of oxidation mechanisms in soot modelling is a project of its own

right, only a brief look at utilizing some of the suggestions at improving current oxidation mechanisms

are investigated.

Unfortunately, there exists only one smoking/non-smoking flame dataset in the literature where

detailed measurements of soot were taken. This investigation was done by Santoro et al. [76] for

several ethylene/air flames, the details of which can be found in their respective paper. Of interest is

the flame denoted as “F2”, which is the non-smoking flame, and “F4”, which is the smoking flame. Liu

et al. [29] found that their employed oxidation model (Nagle and Strickland-Constable [30] for O2 and

Fenimore and Jones [28] for OH) combined with a semi-empirical two equation model could not

properly replicate the non-smoking and smoking behaviour of the F2/F4 flames simultaneously. To

rectify this, Liu et al. [29] introduced a correction factor of the form:

(5.3)

where and are parameters of the correction factor formula. The purpose of this correction

factor, which ranges from 0 to 1, is to introduce a sharp cutoff in the oxidation rates predicted by the

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employed models in regions where the temperature is too low to facilitate oxidation. Liu et al. [29]

found that many experimental measurements showed that the cutoff for soot oxidation behaviour is

around 1300-1400 K and that the models employed overpredicted oxidation behaviour above these

temperatures. By multiplying the correction factor to the oxidation models used in their soot model,

Liu et al. [29] was able to reproduce the observed experimental non-smoking and smoking behaviour of

the flames.

This process was repeated for the simplified soot model in this study. An identical mesh to the one

used for the Smooke et al. [63] simulations was employed and the modified GRI mechanism by Huang

et al.[88] was utilized. , which determines the threshold at which the correction factor drops off

was set to 1525 K and , which controls the steepness of the drop-off was set to 60 K. These differ

slightly from the parameters Liu et al. [29] proposed as his parameters were specifically tuned to their

model. The employed correction factor alongside the original rates is shown in Figure 5.21 along with

an illustration of its effects on the oxidation rates calculated by the simplified model.

(a)

(b)

Figure 5.21 – (a) Graph illustrating the correction factors employed. (b) The effect of the correction factors on the rates predicted by the utilized oxidation models. By using the modified correction factors, the F2 and F4 flame conditions from Santoro et al. [76] were

simulated using the simplified model. As we were only interested in matching the soot volume

fractions predicted by the model and replicating non-smoking and smoking behaviour, the growth rate

of the simplified model was tuned (lowered to 70% of the initial pre-exponential value in Table 4.1) to

77

give the correct peak soot volume fractions. The results of these simulations are summarized in Figure

5.22.

(a)

(b)

Figure 5.22 – Integrated soot volume fraction (ppm · cm2) profiles of the Santoro et al. [76] flames of the F2 Non-smoking flame in (a) and the F4 smoking flame in (b).

78

Integrated soot volume fraction, which considers the total amount of soot at each axial height

throughout the flame, is defined as:

(5.4)

Here, represents the integrated soot volume fraction at a given height in the flame. As one can

observe from Figure 5.22, without using the correction factors, the smoking behaviour is not

reproduced by the simplified model as integrated soot volume fraction drops to zero after an axial

height of 10 cm in Figure 5.22b. However, with the correction factors employed, some soot “escapes”

the flame and the smoking behaviour is thus observed. In addition, the overall peak integrated soot

volume fractions remain the same in both the non-smoking and smoking cases. This correction factor

was then applied to the other methane/air flame simulations and negligible changes to predicted soot

volume fraction, soot diameter, and soot number density was observed. Thus, it was determined that

the correction factors for the oxidation models should be employed in future iterations of the

simplified model.

5.7.2. Updated Model Parameters and Improved Results

Model parameters were adjusted using a few differing strategies. One early strategy was to

attempt to match the global peak inception, surface growth, and oxidation rates predicted by the

detailed model to rates predicted the simplified model. While this proved successful for the specific

operating condition in which the constants were tuned, results were not favourable when the model

was applied to other operating conditions. A better strategy was to attempt to match the peak rates

along the pathlines of maximum soot by considering the peak rate predicted and the integrated sum of

the rates (i.e. the area under rate curves). This method, along with the conclusions made in Sections

5.3 to 5.6 helped guide the iterative process of improving the model predictions across all investigated

conditions. The effect of changing certain parameters on sooting behaviour could also be predicted by

using the results of the sensitivity analysis in Section 5.2. The final version of the proposed parameters

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along with the original parameters in the simplified model is shown in Table 5.5. The author concedes

that while the parameters shown are improvements over the base simplified model, further

refinement is possible (especially with respect to model performance in Westport's engine simulations)

and is still being investigated at the time of writing. Further details on the approach used can be found

in Appendix B.

Parameter Fairweather et al. [46] Model

Base Simplified Model Improved Simplified Model

1.35E6 1.35E6 1.35E6

5.0E4 5.0E4 1.0E5

1.78E6 1.78E6 1.78E6

No OH oxidation model 864 864

90 000 90 000 90 000

3 3 9

Table 5.5 – Summary of parameters in simplified model. See Equation (5.3) for definition of correction factor (identical for both O2 and OH). Changes are highlighted in red and underlined. The results of the improved simplified model are summarized in the following figures. The rationale for

these changes were outlined in Sections 5.3 to 5.6 and Section 5.7.1 but are repeated briefly again for

the reader. The observed inception behaviour between the simplified model and the detailed model

were similar, so no changes were made. However, analysis indicated that both soot volume fraction

and mass averaged aggregate diameters needed to be increased, so the surface growth rate was

increased. Increasing the agglomeration rate ( ) also increased soot diameter while reducing .

Finally, a correction factor was applied to the oxidation rates in order to account for smoking and non-

smoking behaviour observed in the Santoro et al. [76] flame.

As the spatial distribution of predicted sooting behaviour did not change significantly in any of the

investigated conditions, only the peak values in the wing of the flame and the centreline of the flame

are reported for brevity. The unmodified model from Fairweather et al. [46] is also shown for

comparison. As one can observe, the behaviour of the unmodified model is very similar to the base

simplified model.

80

(a)

(b)

Figure 5.23 – Peak values of soot volume fraction predicted by the simplified model compared to experimental results in the centreline (a) and the wing (b) of the Smooke et al. [63] and Schittkowski et al. [78] flames. Figure 5.23 shows the changes in soot volume fraction predictions with the results of the improved

model compared to the original parameters and experimental measurements at atmospheric

conditions. In both cases the original model was underpredicting soot volume fractions in the wing of

the flame, but with the improved model, the predictions are now within experimental error. An

81

improvement in the unmodified Fairweather et al. [46] model is also seen in the prediction for peak

soot volume fraction in the wing of the Schittkowski et al. [78] flame.

(a)

(b)

Figure 5.24 – Peak values of mass averaged aggregate particle diameters predicted by the simplified model compared to the detailed model in the centreline (a) and the wing (b) of the Smooke et al. [63] and Schittkowski et al. [78] flames. The mass averaged aggregate particle diameters calculated by the simplified model are compared to

the diameters calculated by the detailed model in Figure 5.24 since no experimental results are

82

available for this particular soot characteristic. A numerical uncertainty of was assumed for the

detailed model. The improved model increases the diameter predictions in both cases in the wing of

the flame; however, there is a slight overprediction in the centreline of the Smooke et al. [63] flame.

This is not too concerning since there is great uncertainty in the behaviour of all soot models in the

centreline of the flame to begin with.

83

(a)

(b)

Figure 5.25 – Peak values of aggregate particle number density predicted by the simplified model compared to the detailed model in the centreline (a) and the wing (b) of the Smooke et al. [63] and Schittkowski et al. [78] flames. Similarly, the aggregate particle number densities computed by the simplified model are compared to

those calculated by the detailed model in Figure 5.25. The improved model reduces the overprediction

of soot number density in the Schittkowski et al. [78] flame while maintaining a reasonable prediction

84

in the Smooke et al. [63] flame in both the centreline and wing of the flames. Overall, the improved

simplified model is shown to rectify the deficiencies observed in Section 5.3 and 5.4 with regards to

soot volume fraction, particle diameter, and number density predictions.

(a)

(b)

Figure 5.26 – Peak values of soot volume fraction predicted by the simplified models and detailed model compared to experimental results from Joo and Gülder [80] in the centreline (a) and wing (b) of the flame.

Results for the high pressure cases are also presented, beginning with the calculated peak soot

volume fractions in Figure 5.26. The improved simplified model was still able to predict soot volume

85

fractions in the higher pressure flames within the prescribed experimental uncertainty. No significant

change in centreline peak soot volume fractions were observed, but predicted soot volume fractions in

the wing of the flame increased throughout all pressures. This reinforces the observation so far that the

centreline sooting behaviour is relatively insensitive to changes in the soot model. Also once again

observed is the overprediction in soot volume fraction in the centreline of the flame by the detailed

model. As stated earlier in Section 5.5, it is difficult to ascertain the significance of this overprediction

due to the many uncertainties regarding both the experimental measurements made in the centreline

and the performance of the soot model in the core of the flame.

86

(a)

(b)

Figure 5.27 – Peak values of mass averaged aggregate diameter as predicted by the simplified models and detailed model in the centreline (a) and wing (b) of the Joo and Gülder [80] flame. An uncertainty of is assumed for the detailed code calculations. While the particle diameters predicted by the detailed model are not validated at higher pressures,

they appear to reinforce the findings at lower pressures that the initial simplified model underpredicts

particle diameter size. The improved model, as seen in Figure 5.27, improves particle diameter

predictions in both the centreline and wing of the flame relative to the detailed code. When combined

87

with the earlier demonstrated improvements at one atmosphere for particle diameter, this does seem

to indicate the improved simplified model better predicts particle diameters.

5.8. Effect of Coupling – Gas Phase Species Consumption and Radiation

As previously mentioned in Section 3.4, the effect of gas phase species coupling and radiation

coupling to soot formation and oxidation was investigated. Computations using the original model

parameters were repeated with coupling disabled for all conditions in order to investigate the

importance of coupling soot formation/oxidation to the depletion of their respective gas phase species

and also the importance of the DOM radiation model on soot production with the results presented.

Figure 5.28 – The effect of coupling on the predicted peak soot volume fractions in the wing and centreline (CL) of the Smooke et al. [63] flame and the Schittkowski et al. [78] flame. The results of simulations, as seen in Figure 5.28, show that neglecting coupling at the one atmosphere

low sooting conditions seems to have minimal effect on predicted soot volume fractions. The largest

degree of increase is noted to be approximately a factor of 2.2 – 2.5 in the wing and centreline of the

flames when the model is fully uncoupled. At low sooting conditions (i.e. a peak of ~0.50 ppm),

radiation coupling appears to be more important than species coupling. Species coupling appears to be

88

nearly negligible – this is is not surprising since such a low amount of soot is formed, barely any species

relative to the total concentration is consumed in the first place.

Figure 5.29 – Graph illustrating the factor of increase in calculated peak values of soot volume fraction in the wings and centreline (CL) of the high pressure Joo and Gülder [80] flames. From left to right, the 10, 20, and 40 atm cases are shown plotted with respect to the maximum soot volume fractions measured in the experiment. A logarithmic scale is applied to the Y-axis.

Similar calculations were made for the high pressure cases, with the effect on soot volume

fractions shown in Figure 5.29. Unlike the earlier comparisons at low sooting conditions, coupling

effects are very significant at these highly sooting, high pressure conditions. With both radiation and

gas phase species coupling removed, soot volume fractions are observed to jump up significantly -

more than 15 times higher in the wing, and 12 times higher in the centreline in the 40 atm case.

However, the removal of the DOM radiation model on its own does not appear to affect the soot levels

significantly (approximately a factor of 1.2 overprediction at 140 ppm). On the other hand, the absence

of species coupling on its own has a noticeable effect (approx. a factor of 4.5 overprediction at 140

ppm), but not nearly as significant as the case where both mechanisms are removed (approx. a factor

of 14.5 overprediction at 140 ppm). Another important result that should be highlighted is that the

effect of coupling on overprediction of soot appears to increase non-linearly as the sooting behaviour

of the flame increases. As the majority of soot modeling work for turbulent flames has been done at

89

lower pressures with low peak soot volume fractions, it is possible that these previous studies would

not have demonstrated the significance of coupling effects on soot predictions. This result has major

implications for future turbulent soot modeling studies at higher sooting levels (ex. high pressure

combustion devices) as many current methods in turbulent combustion modeling are incompatible

with the need to track the depletion of species due to soot formation/oxidation. The need to include

energy feedback due to radiation is also evidently important. An important caveat of this conclusion is

that the residence time of typical laboratory scale flames is usually an order of magnitude or more

larger than the typical residence time in a combustion application such as an ignition compression

engine. In this scenario, the shortened residence time may reduce the effect of uninhibited soot

inception and growth as soot particles will escape the fuel-rich high sooting areas more quickly.

Originally, calculations were attempted to determine the importance of coupling species and

radiation in the detailed sectional model as well. While radiation effects were similar, it was found that

the detailed sectional model could not converge (within a reasonable timeframe) when species

coupling was disabled. This is due to the fact that inception (and some aspects of surface growth) in the

detailed sectional model is dependent on the concentration of pyrene, which is typically orders of

magnitude smaller than acetylene. The relative effect of consuming pyrene is therefore much higher

since there is a smaller amount of it to begin with. As such, when species coupling is disabled, the

amount of soot formed jumps up to unrealistic levels, causing the simulation to become unstable. A

similar problem could also be expected for any other soot model where soot chemistry is based on

species that are relatively low in concentration compared to major species. This was one of the primary

reasons an inception rate incorporating benzene in the developed simplified soot model was not

considered.

5.9. Computational Cost Comparison

Ultimately, the goal of developing the simplified model in the first place was to reduce the

computational costs involved with soot modelling in order to make it feasible to include in more

90

complex simulations. Table 5.6 shows a representative comparison of the computational costs of

running the two differing models for the high pressure methane air flame by Joo and Gülder [80]. Both

models were run using an initial guess of 1900 K with air populating the entire computational domain.

Pseudo-timestepping for the calculations started at an interval of 1E-07 seconds and increased by a

factor of two every 500 iterations. If the code diverged, then the time step was decreased by a factor of

two and restarted. In the case of the simplified model, the timestepping interval could be increased up

to 2E-04 seconds, whereas in the detailed model, the interval could only be increased to 4E-05

seconds.

Model Average Time per iteration (sec)

Iterations needed for convergence

Total Simulation Time (min)

Detailed Model 5.3 40000 – 50000 3562 – 4454

Simplified Model 3.6 15000 – 20000 893 – 1191

Table 5.6 – Representative comparison of computational costs of running the 2-D laminar flame code with the detailed section soot model and the simplified two equation soot model. As one can see, the detailed model can require upwards of four times the computational time relative

to the simplified model. Another important distinction that is lost in the simplified comparison above is

the fact that the detailed model also requires the use of a more complicated chemical kinetic

mechanism with PAH formation whereas the simplified model works reasonably well with the much

simpler GRI-mechanism. The difference in computational cost can be explained by the fewer equations

needed to solve the less complicated chemical mechanism as well as the few equations needed to

solve soot (two for the simplified model versus 70 for the detailed sectional code).

91

6. Concluding Remarks

6.1. Conclusion/Summary

A two equation simplified soot model was developed and applied to simulations of laminar, co-flow

methane-air diffusion flames. Calculations were made for three different data sets at 1 atm and 10, 20,

and 40 atm. Initial comparisons to available experimental data demonstrated relatively good success at

predicting peak soot volume fractions and also replicating spatial distributions, with some

underprediction of peak soot volume fractions in the 1 atm cases. A detailed sectional soot model,

which is well validated at atmospheric conditions, was used to augment the available experimental

data as detailed measurements of particle diameter and number density are rare in the literature. It

was found that at atmospheric conditions, soot volume fractions and particle diameters were being

underpredicted by the simplified model relative to the detailed model. Particle number densities were

also slightly overpredicted relative to the detailed model. At higher pressures, soot volume fraction

predictions in the wing of the flame were predicted within experimental uncertainty, but soot

diameters were too small relative to predictions made by the detailed model. Along with comparisons

of the soot inception, growth, and oxidation rates between the two models, this seemed to support the

idea that the simplified model had a growth rate that was too low and/or an agglomeration rate that

was too low and/or oxidation rates that were too high. Preliminary application to Westport's engine

simulations reiterated many of the above mentioned problems – namely, total soot mass was

underpredicted as were the soot particle diameters.

An improved simplified soot model incorporating modifications including an increased growth rate,

agglomeration rate, and oxidation factors was developed and applied once again to the investigated

laboratory flames. Predictions of soot volume fraction and particle diameters at 1 atm were improved,

while particle number densities remained within experimental uncertainty. At higher pressure

conditions, the soot volume fraction predictions remained within experimental uncertainty and the

soot particle diameter calculations were also improved. Overall, the developed simplified model was

92

able to reasonably predict soot volume fraction, aggregate particle number densities, and mass-

averaged aggregate particle diameters across a wide range of operating conditions using a single set of

parameters. The thesis also provides the framework for a methodology in developing a simplified soot

model using augmented datasets through a more advanced detailed soot model. Unique in this

approach is the wide range of validation and different aspects of soot morphology, which also includes

the use of numerical data from a more detailed soot model. This is a considerable advancement over

past exercises in simply "tuning" a model to a specific set of conditions to give reasonable results.

The effects of coupling soot formation/oxidation to the depletion of gas phase species was also

investigated. The lack of coupling contributes to overprediction in soot levels, with increasing

overprediction as the sooting tendency of the flame increases. If radiation modeling is also neglected,

the amount of overprediction more than doubles, even though the lack of radiation modeling on its

own appears to have little consequence. This need for coupling has major implications for future

studies of turbulent highly sooting flame applications, as steady flamelet approaches (or similar

turbulent combustion modeling approaches) cannot properly take into account species depletion

effects due to soot formation/oxidation.

6.2. Future Work

The developed simplified model has been demonstrated to be a viable candidate for use in engine

simulations at Westport but it is still expected that further development will be necessary before full-

scale use is possible. Based on the work completed in this thesis, the following future work is

recommended:

1) Extensive testing in engine simulations at Westport is needed to verify the simplified model’s

ability to reproduce quantitatively the observed soot measurements from experiments as well

as the simplified model’s ability to reproduce trends observed throughout varying engine

conditions.

93

2) Very little work in the literature has focused on the development of improved soot oxidation

models and the brief investigation conducted in this study has already demonstrated a need for

improving the soot oxidation mechanisms employed. Preliminary tests in engine simulations

have also alluded to the poor performance of existing oxidation models. The difficulty in

properly validating soot oxidation models is largely due to the propensity to develop soot

models for non-smoking flames. Future development of soot models should be validated for

smoking and non-smoking flames, but this is hindered by a lack of experimental data for

smoking flames.

3) Investigate methods or approaches to address the importance of coupling species and

radiation heat transfer in highly sooting applications. In addition to its importance at highly

sooting conditions, it is possible that there is a threshold residence time before which the

effect coupling species and radiation heat transfer is less significant. While there are no

detailed soot measurements for flames with respect to residence time, it is possible that

numerical simulations can determine when/if there is a threshold residence time where

coupling can be neglected.

4) Detailed sectional model improvements, especially at higher pressures, are needed in order to

provide more valuable input to the development of the simplified model. Some of the

necessary improvements include: validation of soot particle diameters and number densities at

high pressures, improvement of coalescence model to properly track primary particles and

aggregate structures, improved HACA surface growth mechanism so that does not need to be

tuned for specific cases, validated PAH growth in chemical mechanism at higher pressures, and

incorporation of improved oxidation models. However, the further development of detailed

soot models are hindered by the lack of experimental data, especially with respect to soot

diameter and aggregate structure studies.

94

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