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Numerical Investigation of Sooting Propensity ofEthanol at Elevated Pressures
by
Vishal Singh
A thesis submitted in conformity with the requirementsfor the degree of Masters of Applied Science
Graduate Department of Aerospace EngineeringUniversity of Toronto
Copyright © 2014 by Vishal Singh
Abstract
Numerical Investigation of Sooting Propensity of Ethanol atElevated Pressures
Vishal Singh
Masters of Applied Science
Graduate Department of Aerospace Engineering
University of Toronto
2014
The effects of pressure on the combustion properties of ethanol were investigated. One-dimensional
analysis was employed to evaluate the fundamental combustion properties of ethanol as a func-
tion of equivalence ratio and pressure. Two mechanisms were used in the one-dimensional
simulation. The predicted results displayed good agreement with experimental data. Further,
a state of the art computational framework was used to simulate the two-dimensional, ax-
isymmetric, co-flow laminar diffusion flames of ethanol. The framework is a highly-scalable
combustion modelling tool designed specifically for use on large multi-processor parallel com-
puter systems. The flame structure predictions were obtained for ethanol. The soot volume
fraction had an annular structure and maximum concentrations occured in the annular region.
In agreement with the past theoretical and numerical measurements for other gaseous fuels,
the current work showed a constant flame height and an increasing trend of soot yield with
increasing pressure.
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Acknowledgements
I would like to sincerely thank Professor C.P.T. Groth for giving me the chance to study at the
University of Toronto’s Insitute for Aerospace Studies under his direction. His guidance and
support through the course of my studies has been greatly appreciated.
Furthermore, I would like thank Dr. Marc Charest in helping me understand the code. I would
also like to thank Dr. Scott Northrup in helping me with the conjugate heat transfer develope-
ment. I have learned a great deal in object-oriented programming and parallel programming
from him.
I would also like to thank my friends at UTIAS CFD lab who have made my experience here
truly enriching. I particular I would like to thank Adam, Martin, Sandipan and Shahriar.
Lastly, I would like to thank my parents, sister and brother for their support and prayers.
Toronto, 2014 Vishal Singh
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Contents
Abstract ii
Acknowledgements iii
Contents iv
List of Figures vi
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Ethanol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Soot Formation, Oxidation and Modelling . . . . . . . . . . . . . . . . . 3
1.2.3 Effects of Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.1 Numerical Investigation of One-Dimensional Laminar Premixed Flames . 6
1.3.2 Numerical Simulation of Two-Dimensional Axisymmetric
Laminar Co-Flow Diffusion Flames . . . . . . . . . . . . . . . . . . . . . . 7
2 Numerical Solution Methods 10
2.1 One-Dimensional Laminar Premixed
Flame Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
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2.1.1 Cantera Solution Method for Flame Speed
and Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Axisymmetric Laminar Co-Flow Diffusion
Flame Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.2 Finite-Volume Solution Discretization Method . . . . . . . . . . . . . . . 14
2.2.3 Low-Mach-Number Preconditioning . . . . . . . . . . . . . . . . . . . . . 15
2.2.4 Round-Off Error Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.5 Inviscid Flux Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.6 Higher-Order Spatial Accuracy for Inviscid Fluxes . . . . . . . . . . . . . 18
2.2.7 Viscous Flux Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.8 Steady State Relaxation Method . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.9 Radiation Transfer Equation (RTE) . . . . . . . . . . . . . . . . . . . . . 20
2.2.10 Discrete Ordinates Method (DOM) . . . . . . . . . . . . . . . . . . . . . 22
2.2.11 Parallel Block-Based Solution Scheme for RTE . . . . . . . . . . . . . . . 23
2.2.12 Overall Solution Algorithm for RTE . . . . . . . . . . . . . . . . . . . . . 24
2.2.13 Soot Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.14 Soot Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 Results I: One-Dimensional Laminar Premixed Flames 27
3.1 Benchmark Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Laminar Flame Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.1 Effects of Chemical Kinetic Mechanism . . . . . . . . . . . . . . . . . . . 28
3.2.2 Effects of Stoichiometry and Pressure . . . . . . . . . . . . . . . . . . . . 34
3.3 Adiabatic Flame Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4 Results II: Laminar Co-Flow Diffusion Flames for Ethanol 39
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4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3 Discretization and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 40
4.4 Solution Procedure for Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.6 Predicted Temperature Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.6.1 Contours of Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.6.2 Radial Temperature Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.7 Prediction of Flame Heights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.8 Prediction of Flame Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.9 Concentrations of Intermediate Species . . . . . . . . . . . . . . . . . . . . . . . . 49
4.9.1 Acetylene Mass Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.9.2 Carbon Monoxide Mass Fraction . . . . . . . . . . . . . . . . . . . . . . . 49
4.10 Predicted Soot Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.10.1 Contours of Soot Volume Fraction . . . . . . . . . . . . . . . . . . . . . . 52
4.10.2 Radial Profiles of Soot Concentration . . . . . . . . . . . . . . . . . . . . 52
4.10.3 Sooting Propensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5 Conclusions and Future Research 56
5.1 Conclusions I: One-Dimensional Laminar
Premixed Flames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.2 Conclusions II: Axisymmetric Laminar Co-Flow
Diffusion Flames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.3 Recommendations for Future Research . . . . . . . . . . . . . . . . . . . . . . . . 57
References 65
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List of Figures
1.1 UITAS high-pressure laminar co-flow diffusion flame burner apparatus. . . . . . . 7
2.1 Schematic diagram showing generic 2D quadrilateral computational cell. . . . . . 15
2.2 Schematic diagram showing diamond path viscous flux reconstruction stencil for
a quadrilateral computational cell. . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1 Comparison of experimental data of laminar flame speed to predicted values from
Cantera for ethanol using the Dryer mechanism at 1 atm and 5 atm. . . . . . . . 28
3.2 Comparison of Dryer and Reduced Dryer mechanism predicted flame speed values
from Cantera at 1 atm and 5 atm. . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Comparison of Dryer and Reduced Dryer mechanism predicted flame speed values
from Cantera at 10 atm and 15 atm. . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Comparison of Dryer and Reduced Dryer mechanism predicted flame speed values
from Cantera at 20 atm and 25 atm. . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.5 Comparison of Dryer and Reduced Dryer mechanism predicted flame speed values
from Cantera at 1 atm and 25 atm. . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.6 Comparison of calculated values of laminar flame speed from Cantera for pres-
sures between 1 atm and 25 atm for Dryer and Reduced Dryer mechanism. . . . 35
3.7 Laminar flame speed as a function of pressure comparison with theory. . . . . . . 36
3.8 Adiabatic flame temperature of Dryer and Reduced Dryer mechanism as a func-
tion of pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
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4.1 University of Toronto Institute for Aerospace Studies (UTIAS) high-pressure
laminar co-flow diffusion burner schematic. . . . . . . . . . . . . . . . . . . . . . 40
4.2 Schematic of computational domain. . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3 2D computational grid showing the clustering of computational cells. The mesh
contains 61,080 cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4 Typical convergence history for ethanol-air flames. The normalized L2–norm of
the continuity equation is shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.5 Experiemental photographs of co-flow diffusion flames of approximately 5 mm in
height for ethanol at pressures of 1 atm and 5 atm as obtained by Karatas [14]. . 44
4.6 Effect of reaction mechanism and pressure on flame structure for ethanol at
pressures of 1, 5, 10 and 15 atm. Peak temperature is indicated on the bottom
right and left corners. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.7 Radial temperature profiles at three axial locations for ethanol for pressures 5,
10 and 15 atms using Dryer mechanism. . . . . . . . . . . . . . . . . . . . . . . . 47
4.8 Flame radius as a function of pressure for ethanol using Dryer mechanism. . . . . 49
4.9 Effect of reaction mechanism and pressure on acetylene mass fraction for ethanol
at pressures of 5, 10 and 15 atmospheres. Peak mass fraction is indicated on the
bottom right and left corners. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.10 Effect of reaction mechanism and pressure on carbon monoxide mass fraction for
ethanol at pressures of 5, 10 and 15 atmospheres. Peak mass fraction is indicated
on the bottom right and left corners. . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.11 Effect of reaction mechanism and pressure on soot volume fraction for ethanol
at pressures of 5, 10 and 15 atmospheres. Peak soot volume fraction is indicated
on the bottom right and left corners. . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.12 Radial soot volume fraction at three axial locations for ethanol for pressures of
5, 10 and 15 atm using Dryer mechanism. . . . . . . . . . . . . . . . . . . . . . . 54
4.13 Effect of pressure on the maximum carbon conversion factor for ethanol using
Dryer mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
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Chapter 1
Introduction
1.1 Motivation
The majority of practical combustion devices such as industrial furnaces, gas turbine combustors
and internal combustion engines burn high carbon-content fossil fuels. However, the current
sources of fossil fuels are rapidly declining [1]. In addition, there are serious concerns regarding
the emissions such as carbon dioxide, carbon monoxide, soot and other particulates from the
use of fossil fuels that are harmful to human health and the environment. As the global demand
for energy rises, there is an increasing concern for fuel security and environmental impacts that
result from the use of fossil fuels to meet the demand [2]. Thus, there is an urgent need to
identify renewable fuel sources to meet rising global demand. Biofuels are an attractive option
since they are renewable and are potentially more environmentally friendly [3].
For optimal efficiency and weight, combustion in practical gas turbines is turbulent and is
carried out at high pressures. Operating at elevated pressures is desired since the combustion
intensity (energy released per unit volume) is proportional to the square of pressure. However,
the pressure can have an impact on engine emissions. In particular, pressure has significant
effect on the overall soot production by directly affecting the production and oxidation rates
of soot in turbulent diffusion flames of typical aviation gas turbine engines. These flames are
difficult to characterize due to experimental limitations related to optical accessibilty, complex
flame geometries and different time and length scales involved. For these reasons, laminar
flames are often studied since they provide easily controlled conditions and the results can be
projected to practical turbulent flames using the flamelet approach [4].
Focusing on soot, these carbonaceous particles that are emitted in the upper atmosphere from
1
Chapter 1. Introduction 2
aircraft engines affect the global thermal balance by either absorbing sunlight or by contribut-
ing to contrail formation [5]. The presence of soot also adversely affects the performance of
propulsion systems. Soot radiation is the dominant mechanism for the spread of unwanted fires
and is a major heat load on combustor components causing reliability issues. Finally, soot itself
can be toxic to human life and soot particles are strongly associated with detrimental health
effects. In the Earth’s atmosphere, soot contributes to the entrapment of solar radiation that
is believed to lead to global warming [6].
While obviously a detrimental pollutant, there are very few fundamental studies on soot forma-
tion and there is no complete framework to charactize its behaviour under different conditions.
In recent years, there have been some studies that have been carried out under atmospheric
conditions but they do not represent accurately the conditions inside practical gas turbines.
Some recent experimental studies have been carried out at University of Toronto Institute for
Aerospace Studies (UTIAS) combustion lab for gaseous and liquid fuels at elevated pressures.
Fuels investigated for its sooting propensity include methane [7], biogas [8], ethylene [9] and n-
heptane [10]. In addition, the computational framework used in this study has been previously
applied to investigate the sooting propensity for several gaseous fuels and biogas [9,11–13]. Soot
formation and oxidation processes are dependant on several parameters. As a result, systematic
fundamental studies of simple small-scale non-premixed laminar flames are essential in order to
develop accurate models neccessary to study high-pressure turbulent flames.
Though experimental and numerical studies on the sooting propensity of methane, biogas and
other fuels [11] have been studied in the past, no such equivalent study has been done for
ethanol. This thesis therefore includes a numerical investigation of the sooting propensity of
ethanol as a function of pressure using laminar co-flow diffusion flames as the case study. The
numerical results are then compared qualitatively to the partial results obtained previously
by Karatas [14] within UTIAS combustion lab. However, due to high uncertainties of the
experimental setup to handle liquid fuels, no direct measurements of soot formation and flame
were available for quantitative comparisons to the numerical predictions.
1.2 Background
1.2.1 Ethanol
Ethanol is a very important energy carrier that can be produced from renewable energy sources.
It can be used as a fuel extender, octane enhancer and as an alternative fuel to replace reformu-
Chapter 1. Introduction 3
lated gasoline [15]. Recent developments suggest that ethanol can be derived more efficiently
from renewable sources thus reducing dependencies on fossil fuel energy sources [16].
There is a growing interest worldwide to find out new and cheap carbohydrate sources for
production of ethanol [17]. Currently, a heavy focus is on bio-fuels made from crops, such as
corn, sugar cane, and soybeans, for use as renewable energy sources. Though it may seem
beneficial to use renewable plant materials for bio-fuel, the use of crop residues and other
biomass for bio-fuels raises many concerns about major environmental problems, including
food shortages and serious destruction of vital soil resources [18]. For a given production line,
the comparison of the feedstocks includes several issues. This includes chemical composition of
the biomass, cultivation practices, vailability of land and land use practices, use of resources,
energy balance, emission of greenhouse gases, acidifying gases and ozone depletion gases. In
addition, absorption of minerals to water and soil, injection of pesticides, water requirements
and water availability are some other concerns which need to be addressed.
Ethanol feedstocks can be divided into three major groups. Which are sucrose-containing
feedstocks (e.g. sugar cane, sugar beet, sweet sorghum and fruits), starchy materials (e.g. corn,
milo, wheat, rice, potatoes, cassava, sweet potatoes and barley), and lignocellulosic biomass
(e.g. wood, straw, and grasses).
In the short-term, the production of ethanol as a vehicular fuel is almost entirely dependent on
starch and sugars from existing food crops [19]. The drawback in producing ethanol from sugar
or starch is that the feedstock tends to be expensive and demanded by other applications as
well [20]. Any ethanol project attacks seven major national issues: This includes sustainability,
global climate change, biodegradability, urban air pollution, carbon sequestration, national
security, and the farm economy. Lignocellulosic biomass is envisaged to provide a significant
portion of the raw materials for ethanol production in the medium and long-term due to its
low cost and high availability [18].
1.2.2 Soot Formation, Oxidation and Modelling
The complex nature of hydrocarbon-based reacting flows that involve many different physical
processes prevent a complete understanding of the soot formation and oxidation process. Cur-
rently, the least understood aspects of soot formation are the intermediate steps leading up to
the formation of the first soot particles [11]. However, the formation and destruction of soot
are relatively more well defined.
During combustion, a hydrocarbon fuel is pyrolyzed to form a smaller hyrocarbon, for example
Chapter 1. Introduction 4
acetylene, which react and form small aromatic compounds. Primary soot particles first appear
after the formation of soot precursors such as polyacetylenes and polycyclic aromatic hydro-
carbons (PAH). These particles react at the particle surface, increasing in size, colliding with
each other and forming new larger particles called agglomerates. During the formation of soot
particles, precursors and soot are also oxidized by oxygen-containing species. The oxidation
process continues until the soot particles are completely oxidized or diffused out of the flame
envelope [11]. The whole conversion process from gas to solid is largely chemistry controlled
and occurs over several miliseconds [21].
The interactions of different physical processes which occur over a wide range of spatial and
temporal scales pose significant challenges to the modelling of soot in flows. Detailed models of
soot formation includes three main components: gas phase kinetic mechanism, soot chemistry
sub-model and an aerosol dynamics model. The first model describes the formation of soot
precursors and large cyclic hydrocarbon molecules. The second model describes gas-particle
conversion which includes nucleation of primary soot particles and any reactions which occur
between the gas and particle surface. Finally the aerosol dynamics model, is used to describe
the inter-particle collision as well as any heat and mass transfer between the gas and the solid
phase [11]. Due to detailed nature of soot modelling, empirical soot models are adopted in
numerical simulations to remain computationally tractable. The semi-empirical model that has
been adopted in this study is an acetylene-based soot model [22]. Such models use a simple
kinetic mechanism based on acetylene as the sole precursor responsible for the nucleation and
growth of soot particles. Surface growth was assumed first order in acetylene concentration and
is dependant upon some function of the aerosol surface area. Two transports equations for soot
mass fraction and soot particle number density are solved in this semi-empirical model [23–25].
Soot formation and oxidation strongly affects the structure and stability of laminar diffusion
flames by enhancing radiation and altering local temperature fields. Reaction rates are highly
dependant upon temperature and therefore local gaseous species concentrations are strongly
influenced by the presence of soot. Thus, fully understanding the soot formation process in
laminar diffusion flames is essential for various engineering applications [12]. Previous experi-
mental and numerical studies on soot formation that were conducted for various fuels include
methane-air flames at pressures from 10 to 60 atm [26], syngas-air, syngas-methane and biogas-
air flames from 5 to 20 atm [27] and ethylene-air flames from 0.5 to 5 atm [12]. Some important
observations were made regarding the validity of the semi-empirical soot model adopted in the
numerical simulations. Overall, the predictions for soot volume fraction and temperature dif-
fered significantly from measurements. However, the soot model was still able to capture the
effects of pressure and predict the correct trends with reasonable accuracy. The discrepancies
Chapter 1. Introduction 5
were mainly attributed to the errors introduced by the acetylene-based soot model and uncer-
tainties in the boundary conditions. For instance, applying an adiabatic boundary condition
which matched the experimental results the closest, over-predicted the temperatures near the
tube wall and resulted in high soot concentrations low in the flame [11].
1.2.3 Effects of Pressure
Soot formation and oxidation processes are highly dependant on the flame environment. One
of the most important enviroment parameter is pressure. Pressure affects the flame structure
and other flame properties in both premixed and non-premixed flames. Though most practical
combustion devices operate at elevated pressures, our understanding of pressure’s influence on
the flame structure is still inadequate [28].
In laminar premixed flames, density of the mixture is proportional to the pressure while diffusiv-
ity is inversely dependant on pressure. Thus, an increase in pressure limits diffusive processes
while increases the density of the mixture. As a result, the flame speed of the fuel is re-
duced [29,30]. The adiabatic flame temperature however is rather independent of pressure and
is therefore unaffected, except near stoichiometric conditions where the temperature can go up
by approximately 50-100 K for a significant increase in pressure.
Conversely for laminar diffusion flames, pressure affects the shape and structure of the flame.
This is because pressure directly affects the chemical kinetics and buoyant forces [31]. Since
buoyancy induced acceleration is proportional to the pressure squared, an increase in pressure
will change the shape of the flame. In high pressure laminar diffusion flames, the streamlines
contract towards the center, due to buoyancy effects, thereby decreasing the diameter of the
flame [32]. It has also been suggested that the decrease in flame diameter at elevated pressures
could be due to increases in reaction rates, which in turn are caused by higher temperatures
and steeper concentration gradients [32,33].
Previous theoretical studies have shown that to a first-order approximation, flame height should
be independent of pressure and should only depend on mass flow rate [34, 35]. It was thought
that residence time, and thereby the axial velocity along the flame centreline was independent
of pressure, based on experimental measurements [33] and numerical predictions [36]. The
predictions were based on findings that the flame radius is proportional to p−0.5 [37,38], which
implies that centreline velocity and thereby residence time should be independent of pressure.
If the residence time is indeed not dependent on pressure, then flame height should be constant
at all pressures. While many previous experimental studies have indicated that this may not be
Chapter 1. Introduction 6
true, Joo [26] and Gulder [26] and Bento et al. [39] have clearly demonstrated that the visible
flame heights are indeed constant across a wide range of pressures. Numerically, Charest et
al. [6, 11] as well as Liu et al. [36] have also shown that flame heights are essentially constant
with pressure for fixed fuel mass flow rates.
1.3 Objectives
1.3.1 Numerical Investigation of One-Dimensional Laminar Premixed Flames
Before investigating the sooting propenstity of ethanol, it is important to first determine some
key fundamental combustion properties of the fuel. A preliminary investigation of ethanol
properties was carried out herein in which solutions of unstrained, planar, one-dimensional (1D),
laminar, premixed flames were computed and analyzed using an open source software package
known as Cantera [40]. Chemical kinetic mechansims of varying levels of detail designed for
modelling the combustion of ethanol fuel were considered, as described in Chapter 2. The
premixed flames solutions were determined and numerical solution of the laminar flame speeds
and adiabatic flame temperatures were obtained as a function of equivalence ratio and pressure.
The flame speed and temperature are key combustion properties that reveal much about nature
of the fuel [31].
Knowledge of flame speed gives an overall summary of some of the important characteristics of
the fuel such as reactivity, diffusivity and exothermicity and is critical in the design of practical
combustion devices [41]. In addition, flame speed also reveals the susceptibility of a flame fed
by a particular fuel mixture to flashback or extinguish [42].
The adiabatic flame temperature is the equilibrium temperature of the combustion products
assuming that no work is done at constant pressure [31]. It is an important parameter in quan-
tifying the heat release associated with the combustion of the fuel. The flame temperature can
also be a useful indicator of pollutant formation. In addition, the adiabatic flame temperature
also has an influence on flame propagation and extinction. Fuels with high flame temperatures
have higher heating values, which consequently raises the flame speed of the fuel [30].
Numerical simulations of the 1D premixed flames and the resulting flame speeds and tempera-
tures also provide an excellent way of examining the chemical kinetic mechanisms, both detailed
and reduced, and validating them against experimental data as well as other previous, more es-
tablished, kinetic mechanisms [43]. Converged solutions for the 1D premixed flame simulations
provided values for the following properties as a function of equivalence ratio and for a range
Chapter 1. Introduction 7
Figure 1.1: UITAS high-pressure laminar co-flow diffusion flame burner apparatus.
of pressures from 1 to 25 atm for ethanol using full Dryer [16] and Reduced Dryer [44] chemical
kinetic mechanism:
• flame speed;
• adiabatic flame temperature;
Cantera was used in evaluating all of these quantities and its usage is further detailed in Chapter
2 while the results of the numerical computations for the premixed flames described above are
presented and discussed in Chapter 3.
1.3.2 Numerical Simulation of Two-Dimensional Axisymmetric
Laminar Co-Flow Diffusion Flames
Though most combustion devices involve high pressure turbulent flames, there is no complete
framework of understanding flames in this regime due to experimental limitations, complex
flame geometries, complex physical and chemical interactions and large disparity in time and
length scales [6]. For these reasons, laminar flames are often studied. Since the turbulent flames
can be thought of as interactions of infinitesimal laminar flame structure, understanding the
interactions in laminar case is crucial to the understanding of the complex interactions for the
turbulent case [31].
As a part of this thesis, two-dimensional (2D) laminar coflow diffusion flames simulations of
the UTIAS high-pressure co-flow burner (shown in Figure 1.1) were be carried out using the
Chapter 1. Introduction 8
in house Computational Framework for Fluids and Combustion (CFFC) code. In particular,
the code will use the 2D flame solution method for primitive variables, Flame2Dp which was
developed by Charest et al. [11,45] which in turn was built upon the previous work of Northrup
and Groth [46]. This computational framework is a highly-scalable combustion modelling tool
which has been developed specifically for use on large multi-processor, high-performance and
parallel computer systems. The numerical framework uses a parallel-implicit time marching
scheme with a Newton-Krylov iterative solution procedure, a second order finite volume spatial-
discretization scheme and a block-based adaptive mesh refinement (AMR). The aim is to solve
the unmodified compressible form of the Navier-Stokes equations governing multi-species reac-
tive flows on 2D axisymmetric domains using multi-block, body fitted, quadrilateral meshes.
Piecewise linear limited reconstruction and Riemann-solvers’ flux evaluation, applicable to all
flow speeds, are used. An added challenge is faced due to large differences between spatial and
temporal scales which leads to numerical stiffness. The stiffness is also enhanced significantly at
very low Mach numbers when the disparities between acoustic and convective velocities become
large. When upwind schemes are applied in low Mach number cases, excessive artificial dissi-
pation is introduced, thus affecting the overall solution accuracy [47]. To overcome the stiffness
introduced by low speed flows, low-Mach-number preconditioning is adopted. In addition, the
framework features detailed treatments of chemical kinetics, radiation transport and soot for-
mation and transport. Radiation is modelled using the discrete ordinates method (DOM) [48]
to solve the radiative transfer equation and spectral absorption coefficients for gas band ab-
sorption are approximated using the wide-band model developed by Liu et al. [49] which is
based on the statistical narrow-band correlated-k (SNBCK) method [49, 50]. Furthermore, a
simplified semi-empirical model is employed to predict chemistry, nucleation, surface growth,
coagulation, and oxidation of soot particles [51, 52]. The latter assumes that acetylene is the
primary precursor leading to the formation of soot particles. Charest et al. [7,11–13], have ap-
plied this semi-empirical soot model to numerically investigate gaseous fuels such as methane,
ethylene and biogas.
In this thesis, numerical simulations of two-dimensional laminar co-flow diffusion flames were
performed for ethanol at pressures of 1, 5, 10 and 15 atm using two chemical kinetic mechanisms.
The full Dryer [16] and Reduced Dryer [44] chemical kinetic mechanisms were both considered.
The numerical solutions for the axisymmetric flames were compared against partial experimen-
tal results from UTIAS combustion lab to assess and quantify the differences between the model
and actual experiments. Though only experimental results were available for 1 atm and par-
tially for 5 atm, this numerical investigation is a useful first step and helpful in understanding
the general trends of diffusion flame structure and sooting propensity of ethanol. Solutions
Chapter 1. Introduction 9
provide predictions of the following properties for ethanol:
• flame structure (height, width etc.);
• temperature contours; and
• soot concentration;
The computational framework for laminar flames and its usage are discussed in further detailed
in Section 2.2, while the data and results of the laminar co-flow diffusion flame computations
are presented in Chapter 4.
Chapter 2
Numerical Solution Methods
2.1 One-Dimensional Laminar Premixed
Flame Modelling
As mentioned in Chapter 1 of this thesis, Cantera was used to evaluate the flame speeds and
adiabatic temperatures of ethanol liquid fuel as a function of pressure. Cantera is a suite
of object-oriented software tools for problems involving chemical kinetics, thermodynamics,
and/or transport processes [40]. It is a C++ based code with interfaces for python, Matlab, C,
and Fortran 90.
In the present study, two different existing mechanisms were considered when determining the
solutions of laminar premixed flames for ethanol. The mechanisms that were considered herein
are as follows:
• the Dryer mechanism [16];
- 238 elementary reactions
- 39 species
• the Reduced Dryer [44];
- 154 elementary reactions
- 29 species
- reduced from the Dryer mechanism
The Dryer mechanism was developed for combustion of ethanol and blends of ethanol with stan-
dard gasoline in gasoline engines. The mechanism includes low and intermediate temperature
chemistry under knock-prone conditions [16]. While the reduced Dryer mechanism is based on
10
Chapter 2. Numerical Solution Methods 11
the Dryer mechanism and developed for reduced computational cost [44].
2.1.1 Cantera Solution Method for Flame Speed
and Temperature
Cantera solves the steady state form of the conservation equations for a reactive gaseous mixture
in one-dimension (1D) to compute the laminar flame speed [29]. The conservation equations
of interest are continuity, species conservation and energy conservation equations. For steady
planar one-dimensional flows, the conservation equations can be expressed as:
dm′′
dx= 0, (2.1a)
m′′dYidx
+d
dx(ρYivi,diff ) = ωiMWi, for i = 1, 2, .., N species, (2.1b)
m′′cpdT
dx+
d
dx(−kdT
dx) +
N∑i=1
ρYivi,diffcp,idT
dx= −
N∑i=1
hiωMWi. (2.1c)
Here m′′i is the mass flux, Yi is the species mass fraction, vi,diff is the species diffusional velocity,
ωi is the species reaction rate and lastly MWi is the species molar mass. The flame speed Sl is
related to m′′ by using Sl = m′′/ρ.
In order to close the governing equations, Cantera employs the ideal-gas equation of state
and temperature dependent polynomials for species properties. In addition, a chemical kinetic
mechanism is required to provide the values for source term ωi. Feature of the numerical method
emplyed by Cantera is that the discretized form of the governing equations are solved using a
hybrid Newton/time-stepping algorithm to accelerate convergence [53].
Starting with a one-dimensional mesh, the mesh is authomatically refined locally in the region
with high gradients until an accurate steady-state premixed flame solution is achieved. In this
study, the two mechanisms for ethanol required 1000 grid points to obtain sufficiently accurate
converged solutions.
In the discretized form of the equations, the nth equation of the jth point in the 1D mesh is in
the form
F j,n(φ) = 0, (2.2)
where F j,n only depends on the solution at the points j-1, j, j+1. This produces a non-
linear coupled system of algebraic equations. The resulting residual equations are solved with
a modified Newton’s method. The initial step in solving the system of equations employs the
Chapter 2. Numerical Solution Methods 12
classical Newton’s method, which linearises around an initial solution estimate, φ0, and is given
by
F 0lin,n = Fi(φ
0) +∑j
∂Fi∂φj
∣∣∣∣φ=φ0
(φj − φ0j ), (2.3)
then the linear problem is solved to generate a new estimate for φ given by
F lin(φ1) = 0, (2.4a)
φ1 = φ0 − [J0]−1F 0, (2.4b)
where J is the system Jacobian, ∂Fi/∂φj .
If the Newton iterations fail to find a steady-state solution to the premixed flame of interest,
an attempt is made to solve a pseudo-transient problem, which potentially contains a larger
domain of convergence [53]. The pseudo-transient problem is created by adding transient terms
in each conservation equation, where physically reasonable, as shown below:
Adφ
dt= F (φ), (2.5a)
F (φn+1)−Aφn+1 − φn
∆t= 0. (2.5b)
The quantity A in Equation (2.5a) is a diagonal matrix with entries of 1 on the diagonal for
the equations with a transient term, and 0 for the constraint equations.
In summary, the algorithm of the C++ program in Cantera that solves for the one-dimensional
laminar flame speed and temperature proceeds as follows :
1. input the chemical mechanism, fuel-oxidizer mixture and environment condition data;
2. calculate unburned mixture properties;
3. determine equilibrium conditions and burned gas properties;
4. create the grid;
5. use provided initial guess of temperature and velocity profiles;
6. call inbuilt newton solver subroutine to iteratively improve upon the initial guess;
7. refine grid and call solver subroutine again if necessary; and
8. output to Tecplot (commercial plotting tool) for plotting of final solution data.
Chapter 2. Numerical Solution Methods 13
2.2 Axisymmetric Laminar Co-Flow Diffusion
Flame Modelling
The numerical solution procedure used in this study to simulate laminar reacting flows with de-
tailed chemistry, non-gray radiative heat transfer and soot is based on the numerical framework
developed by Charest et al. [6, 11]. This computational framework is a highly-scalable com-
bustion modelling tool which has been developed specifically for use on large multi-processor
high-performance parallel computer systems. The framework solves the conservation equation
for multi-species compressible reacting flows with soot on axisymmetric domains using multi-
block, body-fitted, quadrilateral meshes. Soot was modelled using a simplified semi-empirical
model approach proposed by Liu et al. [22]. The model describes the chemistry, nucleation,
surface growth, coagulation, and oxidation of soot particles [51, 52]. The latter assumes that
acetylene is the only precursor leading to the formation of soot particles.
The governing conservation equations are solved numerically utilizing a finite-volume scheme
developed by Groth and co-workers [54, 55] using the semi-discrete approach. Piecewise linear
limited reconstruction and Riemann-solver-based flux functions are used to evaluate inviscid
fluxes [56]. Viscous fluxes were determined using a second-order diamond path integral method
developed by Coirier and Powell [57]. Low-Mach-number preconditioning of both the inviscid
flux and the temporal derivative are applied in order to permit the efficient and accurate solution
of the conservation equations for low-speed flows associated with laminar flames [47]. The non-
linear, coupled ordinary differential equations (ODEs) were then solved to steady-state using
the block-based parallel implicit time-marching scheme in conjunction with a Newton-Krylov
iterative method [54].
In modelling radiation emitted and absorbed by soot and some participating gases, a discrete
ordinates method (DOM) coupled with point-implict finite volume approach was employed [58].
The spectral absorption coefficients for gas band absorption are approximated using the wide-
band model developed by Liu et al. [49] which is based on the statistical narrow-band correlated-
k method [49,50]. In addition, Cantera was employed to evaluate thermodynamics and transport
properties along with gas-phase kinetic rates.
The computational framework adopted in this study for laminar flames was previously applied
succesfully to study the effects of both high pressure and low gravity on flame structure and
sooting propensity for several gaseous fuels and biogas [9,11–13]. For the purposes of complete-
ness, the key aspects of the computational framework for laminar flames of Charest et al. [6,11]
are summarized below.
Chapter 2. Numerical Solution Methods 14
2.2.1 Governing Equations
As mentioned above, the computational framework used in this study solves the compressible,
2D axisymmetric form of the Navier-Stokes equations for a multi-component of thermally-
perfect reactive gaseous mixture. The governing equations in this case are as follows:
∂ρ
∂t+∇ · (ρv) = 0, (2.6a)
∂
∂t(ρv) +∇ · (ρvv + pI) = ∇ · τ + ρg, (2.6b)
∂
∂t(ρe) +∇ · [ρv(e+
p
ρ)] = ∇ · (v · τ )−∇ · q + ρg · v, (2.6c)
∂
∂t(ρYk) +∇ · [ρYk(v + V k)] = ωk, for k = 1, ..., N , (2.6d)
where t is the time, v is the velocity vector, ρ is the density of the mixture, p is the mixture
pressure, e is the mixture energy, τ is the fluid stress tensor, g is the gravitational acceleration
vector, q is the heat source flux vector, Yk is the mass fraction of the kth species, Vk is the
diffusional velocity of the kth species and ωk is the time rate of change of the kth species. The
ideal gas law is used to calculate the mixture density. The heat flux vector, q, is given by
q = −κ∇T + ρ
N∑k=1
hkYkV k − qrad, (2.7)
where κ is the thermal conductivity of the mixture, h and Y are the individual species enthalpy
and mass fractions respectively while qrad is the radiation heat flux. The gas phase diffusion
velocity vector for the kth species is given by
V k = −Dk
Yk∇Yk, (2.8)
where Dk is the species averaged diffusion coefficient.
2.2.2 Finite-Volume Solution Discretization Method
The coupled system represented by Equations (2.6a) - (2.6d) of the reactive gas mixture are
solved numerically using the parallel, implicit and finite-volume based scheme developed pre-
viously by Groth et al. [54, 55] and subsequently improved by Charest et al. [6, 11]. In this
approach, the domain of the physical problem is discretized into a number of finite-size compu-
tational cells and the integral form of the conservation equations is then applied to and solved
for the cells. For a general quadrilateral cell, (i, j), in a two-dimensional domain as shown in
Chapter 2. Numerical Solution Methods 15
Figure 2.1: Schematic diagram showing generic 2D quadrilateral computational cell.
Figure 2.1, the semi-discrete, non-linear and coupled form of the governing equations for the
primitive, cell-averaged solution quantities, W , are as follows:
dW i,j
dt=∂W
∂U
∣∣∣∣ij
·
− 1
Aij
∑face,k
(F k · nk∆lk)ij + Sij
(2.9)
where U ij and W ij are the cell averaged conserved and primitive representation of the solution
vectors. In Equation (2.9), Aij is the cell area, nk and ∆lk are the normal vector and edge
length for the kth face respectively, and Sij is the source term that includes contributions
from chemistry, gravity and boundary conditions. The flux, F k, includes both the inviscid and
viscous fluxes contributions (F and F v respectively).
2.2.3 Low-Mach-Number Preconditioning
Obtaining a solution of the non-linear system of ODEs of Equation (2.9) above resulting from
the finite-volume spatial discretization at low mach numbers has an added difficulty due to
the stiffness induced by the disparate velocity scales of the acoustic and convective waves [59].
Hence preconditioning is employed to reduce the numerical stiffness of the problem. Precondi-
tioning replaces the physical time derivatives with artificial derivatives in order to better match
the magnitudes of the acoustic and convective waves. The computational framework used
herein adopts the preconditioning scheme developed by Weiss and Smith [47]. The modified
Chapter 2. Numerical Solution Methods 16
conservation equations for an axisymmetric co-ordinate system can be expressed as:
Γ∂W
∂t+∂F
∂r+∂G
∂z=∂Fv
∂r+∂Gv
∂z+ S, (2.10)
where Γ is the preconditioning matrix as developed by Weiss and Smith. The eigenvalues of
the preconditioned Jacobian matrix in the r-direction, Γ−1 ∂F∂W , are as follows:
λ = [u′ − a′, u, u, u′ + a′, u, ..., u]T , (2.11)
for which
u′ = u(1− α), (2.12a)
a′ =√α2u2 + V 2
p , (2.12b)
α = 0.5(1− βV 2p ), (2.12c)
β = ρp +ρT (1− hp)
ρhT. (2.12d)
The subscripts in the expressions above represent partial derivatives of the quantities of interest.
For a perfect gas, ρT = −ρ/T , ρp = 1/(RT ), hp = 0, and hT = cp. The so-called preconditioned
sound speed, Vp, can be defined as
Vp = min[max(Vinv, Vpgr, Vvis,Mref · a), a], (2.13)
where a is the sound of speed and Mref is a reference Mach number used to prevent singularities
at stagnation points. A value of 10−4 for Mref was used in all the simulations conducted herein.
The remaining subscripted terms are the inviscid, pressure-gradient, and viscous velocity scales
as derived in [47] and given by
Vinv =√u2 + v2, (2.14a)
Vpgr =
√|∆p|ρ
, (2.14b)
Vvis =µ
ρ∆x, (2.14c)
where ∆p is the pressure gradient within the cell and ∆x is the length of the computational
cell.
2.2.4 Round-Off Error Control
Previous study by Charest et al. [6,11] determined that below Mach numbers of 10−3, machine
round-off errors began to contaminate the solutions for pressure. In order to fix this issue,
the solution method was modified to use the procedure developed by Choi and Merkle [60].
Chapter 2. Numerical Solution Methods 17
A reference pressure, p0, is used to reduce the influence of round-off errors in pressure at low
Mach numbers. Due to the modification, the new overall pressure, p, is then given by
p = p0 + p′, (2.15)
where p0 is the reference pressure and p′ is the deviation from the local pressure from p0. The
reference pressure is subtracted from Equation (2.6b) and p is replaced by p′ in the numerical
solution vector, W .
2.2.5 Inviscid Flux Evaluation
A second-order upwind Godunov scheme is employed to evaluate the inviscid numerical fluxes
at cell interfaces. Godunov’s method assumes that the solution within each computational
cell is piecewise constant and that the solution at the cell interface can be approximated via
upwinding [61]. Upwinding also preserves the monotonicity of the solution.
Given left and right solutions states, WL and WR, the flux at a cell interface is defined in
terms of a function that involves the solution of a Riemann problem, in a direction along the
cell interface normal, n. The numerical flux is given by
F · n = F(WL,WR, n), (2.16)
where the evaluation of F requires the solution of the Riemann problem. Roe’s approximate
Riemann solver [62] is used in the current solution method to solve the Riemann problem and
evaluate the fluxes. Harten’s correction [63] is also employed with the solver to rectify violations
of the entropy condition at sonic points. The flux in one direction at a cell interface is given by
F (R (WL,WR)) =1
2(FR + FL)− 1
2|A|∆W , (2.17)
where FL and FR are the left and right fluxes as functions of the left and right primitive solution
states WL and WR, ∆W = WR - WL. |A| = R|Λ|R−1 where R is a matrix composed of
primitive right eigenvectors and Λ is the eigenvalue matrix. The matrix A is the linearised flux
Jacobian evaluated at a reference state W . The reference state is typically chosen such that
Roe’s conditions are relaxed for reacting flows [64]. Therefore, the Roe-averaged flow variables
are given by
u =
√ρRuR + ρLuL
ρR + ρL, (2.18)
where uR and uL are either of the flow variables u, v, h, Yk. Additionally, ρ, the Roe averaged
density, is given by√ρRρL.
Chapter 2. Numerical Solution Methods 18
To control the dissipation at low Mach numbers that results with the use of upwind methods,
Equation (2.17) can be re- derived by using the pre-conditioned wave speeds as shown by Weiss
and Smith [47]. The linearised flux Jacobian term (with the |A| matrix) in Equation (2.17) is
modified as shown below.
|A|∆W ' A∆W = Γ
(Γ−1
F
W
)∆W = Γ|AΓ|∆W , (2.19)
where |AΓ| = RΓ|ΛΓ|RΓ−1. The subscript Γ shows that the matrix eigenvectors and eigenval-
ues are derived using the preconditioned system.
2.2.6 Higher-Order Spatial Accuracy for Inviscid Fluxes
Godunov’s method is first-order accurate and as a result the scheme can be excessively dissipa-
tive. To extend Godunov’s method to second or higher orders can be challenging since schemes
with constant coefficients will have non-monotonic behaviour near solution discontinuities [56].
In the computational framework of Charest et al. [6,11] the solution at the cell interface between
two adjacent cells is reconstructed by a piecewise linear function to achieve second-order spatial
accuracy. Slope limiters are employed to ensure that monotonic (non-oscillatory) behaviour of
the solution is preserved at or near discontinuities.
The slope limiters function by locally reducing the reconstructed solution to first-order, thereby
damping out any over- or under-shooting behaviour at discontinuities. The reconstructed left
and right solution values at a cell interface given by the piecewise linear limited representation
for an interface (i + 12 , j) are given by
WL = W ij + φij
[∂W
∂r
∣∣∣∣ij
(ri+ 12,j − rij) +
∂W
∂z
∣∣∣∣ij
(zi+ 12,j − zij)
], (2.20)
WR = W i+1,j + φi+1,j
[∂W
∂r
∣∣∣∣i+1,j
(ri+ 12,j − ri+1,j) +
∂W
∂z
∣∣∣∣i+1,j
(zi+ 12,j − zi+1,j)
], (2.21)
where φ is the slope limiter. Slope limiting is performed using methods designed especially
for multiple dimensions [56]. The cell gradients are calculated using linear reconstruction from
Green-Gauss theory as discussed by Barth and Jespersen [65].
2.2.7 Viscous Flux Evaluation
The computational framework in this study employs the centrally-weighted diamond-path
method, as developed by Coirier and Powell [57], to evaluate the viscous fluxes at the boundaries
Chapter 2. Numerical Solution Methods 19
Figure 2.2: Schematic diagram showing diamond path viscous flux reconstruction stencil for a
quadrilateral computational cell.
of each computational cell. The viscous flux for a cell face is given by
Fv · n = G(W ,∇W ,n), (2.22)
where G is the viscous flux function.
Figure 2.2 shows the diamond-path method. Gradients at each face are calculated by applying
the divergence theorem along the path. The solution is known at the cell-centred vertices,
however the solution state at the vertices (nodes) of the cell must be interpolated. A weighting
scheme developed by Zingg and Yarrow [66] that linearly reconstructs nodal data using cell-
centred solution data of the neighbouring cells is used.
2.2.8 Steady State Relaxation Method
The computational framework for laminar flames uses Newton’s method to numerically solve
the non-linear algebraic equations resulting from the spatial discretization of the steady form of
the governing conservation equations given above. In particular, the semi-discrete form of the
governing equations are iteratively relaxed to converge to a steady-state solution, W , satisfying
R(W ) =dW
dt= 0. (2.23)
Chapter 2. Numerical Solution Methods 20
Charest et al. [6,11] developed the Newton algorithm which is employed in this computational
framework for laminar flames. Their approach follows the scheme originally developed by Groth
and Northrup [67]. The implementation of the algorithm makes use of a Jacobian matrix free,
inexact Newton method along with an iterative Krylov subspace linear solver.
Starting from an initial estimate for the solution, W 0, successively improved estimates of the
solution to the equation above can be found by solving the linear system for ∆W n.(∂R
∂W
)n∆W n = J(W n)∆W n = −R(W n), (2.24)
where J is the residual Jacobian and n is the step number. The improved solution at the next
step, n+ 1, is found by using
W n+1 = W n + ∆W n. (2.25)
This iterative process is repeated until a suitable reduction in the residual norm is achieved
and the condition ||R(W n)|| = ε||R(W 0)|| is satisfied; ε is a tolerance that is set to 10−7 in
the solution method.
Newton’s method, at each step, requires the solution of the linear system Jx = b where
x = ∆W and b = −R(W ). For a system of non-linear, coupled equations, the Jx = b
system is large, sparse and asymmetric. Such a system can be solved by using the generalized
minimal residual (GMRES) technique as developed by Saad and Schultz [68, 69]. GMRES is
an Arnoldi-based solution method which generates orthogonal bases of the Krylov subspace
to construct the solution. GMRES only uses matrix vector products at each step to create
the new trial Krylov vectors thereby greatly reducing the storage requirements for forming
J [70]. The GMRES iterations can be terminated by solving the system to some previously
specified tolerance, ||Rn + Jn∆W n)|| < χ||R(W n)||, where χ is set to 0.1 in the solution
method. Finally, to further lower memory requirements, the computational framework uses
a modified version of GMRES, GMRES(m), that restarts and refreshes the search directions
every m iterations.
2.2.9 Radiation Transfer Equation (RTE)
The radiation transfer equation (RTE) is the conservation law for the spectral intensity applied
to a monochromatic beam of light [71]. It is derived by applying an energy balance to a beam
of photons which are confined to a infinitesimal solid angle element and passing through an
infinitesimal volume of participating media [11]. In cylindrical coordinates, the steady form of
the RTE is a function of two dimensions(r,z) and two angular coordinates(θ,ψ) is
Chapter 2. Numerical Solution Methods 21
µ
r
∂
∂r(rIv)−
1
r
∂
∂ψ(ηIv) + ξ
∂
∂z(Iv) = −βvIv + kvIbv +
σsv4π
∫4πIv(s)Φv(s, s)dΩ, (2.26)
where Iv is the spectral intensity, Ibv is the blackbody radiative intensity, s is a unite vector,
wn is the wavenumber, kv is the absorption coefficient, σv is the scattering coefficient, and Φwn
is the scattering phase function. Also, β = kv + σsv is the extinction coeffient, θ and ψ are the
polar and azimuthal angles and µ, η and ξ are the direction cosines. These values define the
the unit vector s in the Cartesian coordinates which is given by:
s = µi+ ηj + ξk. (2.27)
In the absence of scattering, Equation (2.26) reduces to first-prder linear-hyperbolic differential
equations.
The next important quantity in solving for the heat flux term in Equation (2.6c) is the spectral
radiation flux vector, qv. It is defined as the net flow of radiant energy due to radiation from
all directions per unit area, time and wavenumber interval [11]. This is given by
qv =
∫ 4π
0sIv dΩ. (2.28)
Next, rearranging the RTE equation and integrating over Ω (all the solid angles), the divergence
of the heat flux vector can be expressed as
∇ · qv = kv[4πIbv −∫ 4π
0Iv dΩ] = kv[4πIbv −Gv], (2.29)
where Gv is the total incident radiation and is given by
Gv =
∫ 4π
0Iv dΩ. (2.30)
Since the radiative properties vary with v the wavenumber, Equation (2.29) must be integrated
over the entire spectrum. Thus the divergence of the the total radiation heat flux, ∇ · qrad in
the source term of Equation (2.6c) can be determined.
In this study the solution of the RTE is done using the DOM method which provides an excellent
balance between computational efficiency and accuracy. However, evaluating ∇ · qrad using the
DOM is costly due to the large number of unknowns associated with non-gray radiation. As
a result, solution of the RTE is decoupled from the gas-particle flow equations and solved
sequentially in a loosely-coupled fashion at each iteration. Refer to the study of Charest for
further details on the radiation modeling [11].
Chapter 2. Numerical Solution Methods 22
2.2.10 Discrete Ordinates Method (DOM)
Equation (2.26) is solved using the space-marching finite-volume approach outlined by [58].
The approach yields
µml (AEIE,ml −AW IW,ml) + ξml (ANIN,ml −ASIS,ml)
− (AE −AW )(αm,l+1/2IP,m,l+1/2 − αm,l−1/2IP,m,l−1/2
)/wml
= ∆V κP (IbP − IP,ml) ,
where the subscript P denotes quantities at the cell center, and the subscripts E, W , N and S
refer to quantities evaluated at the respective cell faces. For the sake of clarity, the quadrature
and band indices have been removed.
The cell volume, ∆V , and face areas, A, are as follows:
AN = AS = π(r2E − r2W ), (2.31a)
AE = 2π∆zrE , (2.31b)
AW = 2π∆zrW , (2.31c)
∆V = π(r2E − r2W )∆z, (2.31d)
where ∆r and ∆z are the cell-sizes in the r- and z-directions, respectively. For the purpose of
reducing the number of unknowns, the cell-edge intensities are related to the volume-averaged
intensity by the following expression:
IP,ml = γsIN,ml + (1− γs)IS,ml,= γsIE,ml + (1− γs)IW,ml, (2.32a)
IP,ml = γaIP,m,l+1/2 + (1− γa)IP,m,l−1/2, (2.32b)
where γs and γa are the spatial and angular differencing parameters respectively. For both
parameters, a value of 1 corresponds to upwind differences and 0.5 corresponds to central
differences. Central differences were used for both the spatial and angular discretization in all
of the computations.
Substituting into Equations (2.32a) and (2.32b) rearranging for the nodal intensity gives∆V κP + µmlAE/γs + ξmlAN/γs −
(AE −AW )αm,l+1/2
wmlγa
IP,ml =
∆V κP Ibp + µmlAEWIW,ml/γs + ξmlANSIS,ml/γs
− (AE −AW )[αm,l−1/2 + (1− γa)αm,l+1/2/γa
]IP,m,l−1/2/wml, (2.33)
Chapter 2. Numerical Solution Methods 23
where
AEW = (1− γs)AE + γsAW , (2.34a)
ANS = (1− γs)AN + γsAS . (2.34b)
Numerical solution of Equation (2.31a) proceeds as follows. First, the surface intensities and
internal source terms are estimated everywhere in the domain. The lower left corner of the
domain is chosen as a starting point so that all outgoing directions lie in the first quadrant
(i.e. µml > 0 and ξml > 0). Since the west and south faces of the control volume in this corner
are part of the enclosure surface, their intensities are specified by the boundary conditions.
From these known face values, IP,ml is computed using and the downstream intensities IE,ml
and IN,ml are determined from Equation (2.32a). One by one, the first-quadrant intensities
are calculated for all volumes in the enclosure. This procedure is repeated three more times
starting from the remaining corners of the enclosure and covering the other three quadrants
of directions. After sweeping all directions for IP,ml, the boundary values and radiative source
terms are updated. This procedure is repeated until convergence is met [11].
Solutions were deemed converged when the maximum change in the cell-averaged intensity
everywhere in the domain from one iteration to the next was less than a specified tolerance.
Throughout this thesis, the following convergence criterion was used:
∣∣In+1P − InP
∣∣ ≤ 10−12 for all P , m, l (2.35)
where the superscripts n and n+ 1 denote the iteration number.
2.2.11 Parallel Block-Based Solution Scheme for RTE
The DOM was solved in a parallel fashion at each time-step on the multi-block mesh along with
Equations (2.6a–2.6d) and Equations (2.36a–2.36b) by simultaneously sweeping all directions
on the domain local to each processor. Solution content was shared among the processors by
exchanging the state at the face-center of cells aligned with the block boundaries. Changes
in mesh resolution were handled by linearly interpolating the coarse-mesh solution onto the
fine-mesh and averaging the fine-mesh solution onto the coarse-mesh. Since the radiation solver
employs a space-marching technique, repeated numerical iterations are required to propagate
information from upstream boundaries to downstream blocks. As a result, a penalty in terms of
parallel efficiency was incurred because the number of iterations required to solve the radiation
field increased with the number of blocks.
Chapter 2. Numerical Solution Methods 24
2.2.12 Overall Solution Algorithm for RTE
The overall algorithm is summarized as follows:
1. Set initial conditions for W and I everywhere in domain.
2. Compute the spectral absorption coefficient for mixture.
3. Solve the RTE using the DOM described in Section 2.2.10 .
4. Update ∇ · qrad.
5. Solve Equation (2.23) for the gas/soot mixture, performing n Newton iterations.
6. Update primitive solution state W .
7. If not converged, return to step 2. The convergence criteria is defined in Section 2.2.8.
Throughout this study only one Newton iteration (n = 1) for was performed before updating
the radiation intensity field. Larger values of n up to five were tested but found to deteriorate
the performance of the overall solver. As n is increased, the CPU time required to advance the
solution a fixed interval ∆t in the computational domain decreases. However, increasing n also
increases the number of iterations required to obtain a converged, coupled solution.
2.2.13 Soot Modelling
Mathematical models describing the soot particles and their interactions with the gaseos mixture
are complex. In order to remain computationaly tractable, the mathematical representation is
simplified. This study simplies the aerosol representation by adopting a model based on the
solution of just two transport equations [23–25]. The transport equations are solved in terms
of number of soot particles per unit mass, NS and soot mass fraction Ys are
∂
∂t(ρYs) +∇ · [ρYs(v + VY )] = SY , (2.36a)
∂
∂t(ρNs) +∇ · [ρNs(v + VN )] = SN , (2.36b)
where v is the mixture velocity vector, VY and VN are soot diffusion velocities which include
contributions from both thermophoresis and Brownian motion. The variables SY and SN are the
source terms for Ys and Ns respectively due to nucleation, growth, oxidation and coagulation.
These source terms are defined in the following section.
Chapter 2. Numerical Solution Methods 25
The integration of these simplified transport equations will enable computation of the total
number concentration and volume fraction of the soot particles [49]. The simplified aerosol
description makes several assumptions. It assumes that the soot particles are perfectly spher-
ical and upon collision coalesce to form a new spherical particle that has the same equivalent
mass [49]. In addition, the spherical soot particles are also assumed to have constant composi-
tion and density.
However this approach has several shortcomings. The main shortcoming is the assumption that
the particles coalescing instantaneously upon collision. This has an affect on the soot particle
growth and oxidation rates since these quantities are highly dependant on particle surface area.
However more sophisticated treatment of the soot aerosol dynamics is beyond the scope of this
thesis.
2.2.14 Soot Chemistry
As mentioned before, the simplied soot kinetics as described by Liu et al. [49] model the
soot formation and oxidation through four steps - nucleation, surface growth, coagulation and
oxidation. Additionally acetylene is assumed to be the sole precursor for the presence of soot.
The resulting reactions describing these steps for soot are as follows:
C2H2 −−→ 2 C(s) + H2,
C2H2 + n · C(s) −−→ (n + 2 ) · C(s) + H2,
C(s) + 12 O2 −−→ CO,
C(s) + OH −−→ CO + H,
C(s) + O −−→ CO,
n · C(s) −−→ Cn(s).
From the chemical kinetic mechanism above, the source term in Equation (2.6c) can be ex-
pressed as
SY = 2MS(R1 +R2)− (R3 +R4 +R5)As, (2.38)
where MS is the molar mass of soot assumed to be equal to molar mass of carbon and R3, R4
and R5 are the soot oxidation reaction rates for O2, OH and O respectively. The terms R1 and
R2 are the soot nucleation and surface growth rates which is expressed as
Chapter 2. Numerical Solution Methods 26
R1 = k1[C2H2], (2.39a)
R2 = k2A0.5S [C2H2], (2.39b)
where rate constants k1 and k2 are [49]
k1 = 1000exp(−16103/T ), (2.40a)
k2 = 1750exp(−10064/T ). (2.40b)
While As is the soot surfaqce area per unit volume which describes the dependence of soot
surface growth. This quantity is related to soot mass and number density by this following
relation:
As = π(6
π
1
ρs
YsNs
)2/3(ρNs), (2.41)
where here ρs is the density of soot particles and is taken to be 1900kg/m3. The oxidation
rates R3, R4 and R5 are modelled by using [49]
R3 = 120(kap
1 + kz), (2.42a)
R4 = φOHk4T−12 pOH , (2.42b)
R5 = φOk5T−12 pO. (2.42c)
The source term in Equation (2.39a) describes the production and oxidation of soot particle
number density which includes nucleation and agglomeration. This source term is expressed as:
SN =2
CminNAR1 − 2Ca(
6Ms
πρs)1/6(
6kBT
ρs)[C(s)]1/6(ρNs)
11/6, (2.43)
where NA is Avogadro’s number, Cmin is the number of carbon atoms in the initial soot particle,
CA is the rate constant for agglomeration and C(s) is the molar concentration of soot. In this
study, coalescense was neglected by setting CA to zero and Cmin is set to 700 based on the
recommendations of Liu et al. [49].
Chapter 3
Results I: One-Dimensional Laminar
Premixed Flames
3.1 Benchmark Case
Before proceeding to the numerical study of some of the fundamental important combustion
properties of ethanol as a function of pressure, the validity of the chemical kinetic mechanisms
for ethanol were first explored by comparing the predicted laminar flame speeds to the existing
experimental data of Mansour et al. [72]. The experimental measurements for were conducted
flame speed data was done for pressures ranging from 1 atm and 5 atm for an initial mixture
temperature of 380 K at equivalence ratios from 0.7 to 1.4. The numerical predictions of laminar
flame speed were obtained using the Dryer [16] mechanism and Cantera as discussed in Chapter
2.
The comparison of the predicted laminar flame speeds from Cantera to the experimental mea-
surements of Mansour et al. [72] is depicted in Figure 3.1 for the Dryer mechanism. In general,
the predictions given in Figure 3.1 for Dryer mechanism are in rather good agreement with
experimental measurements. The mechanism is able to capture the correct trends for flame
speed as a function of equivalence ratio. The difference between the measured and calculated
value for laminar flame speed is within 10%. Accounting for experimental error, the calculated
values are in good agreement with the experimental values. In general for fuel lean conditions
conditions, the discrepancy is the greatest at around 10%. While for fuel rich conditions, the
experimental and calculated values closely match. The preceeding results are an indicator of
the level of accuracy of the mechanisms for ethanol considered here.
27
Chapter 3. Results I: One-Dimensional Laminar Premixed Flames 28
Figure 3.1: Comparison of experimental data of laminar flame speed to predicted values from
Cantera for ethanol using the Dryer mechanism at 1 atm and 5 atm.
3.2 Laminar Flame Speed
3.2.1 Effects of Chemical Kinetic Mechanism
As mentioned in Chapter 2, two mechanisms for ethanol are considered in this work. Which are
the Dryer mechanism [16] and Reduced Dryer mechanism [44]. The reduced Dryer mechanism is
obtained by using the Directed Relation Graph (DRG) method proposed by Law et al [73]. This
systematic approach for mechanism reduction generates a skeletal mechanism by applying the
theory of DRG to identify unimportant species which in turn removes the reactions involving
the unimportant species. The theory quantifies the direct influence of coupling among the
species [73]. The reduced mechanism used here is generated using ignition delay calculations
as the benchmark [44].
Chapter 3. Results I: One-Dimensional Laminar Premixed Flames 29
Comparison of the predicted flame speed calculations from Cantera for ethanol using two dif-
ferent mechanisms are shown in Figures 3.2, 3.3, 3.4 and 3.5. The figures show the predicted
flame speeds for equivalence ratios between 0.5 – 1.8 or 2.0 and at all pressures from 1 to 25
atm using the Dryer [16] and Reduced Dryer mechanism [44]. Note that all of the premixed
flame simulations were conducted with an initial unburned mixture temperature of 300 K.
Overall, both of the mechanisms for ethanol seem do a reasonable job of predicting the laminar
flame speed, and provide generally similar results. The results differ to within 15% of each other
at the intermediate pressures and differ about 5% at the extreme pressures at 1 atm and 25
atm. Another notable feature of the reduced mechanism is that it consistently over-predicts the
flame temperature as compared to the Dryer mechanism. The difference is especially greatest
at fuel lean conditions.
Chapter 3. Results I: One-Dimensional Laminar Premixed Flames 30
Figure 3.2: Comparison of Dryer and Reduced Dryer mechanism predicted flame speed values
from Cantera at 1 atm and 5 atm.
Chapter 3. Results I: One-Dimensional Laminar Premixed Flames 31
Figure 3.3: Comparison of Dryer and Reduced Dryer mechanism predicted flame speed values
from Cantera at 10 atm and 15 atm.
Chapter 3. Results I: One-Dimensional Laminar Premixed Flames 32
Figure 3.4: Comparison of Dryer and Reduced Dryer mechanism predicted flame speed values
from Cantera at 20 atm and 25 atm.
Chapter 3. Results I: One-Dimensional Laminar Premixed Flames 33
Figure 3.5: Comparison of Dryer and Reduced Dryer mechanism predicted flame speed values
from Cantera at 1 atm and 25 atm.
Chapter 3. Results I: One-Dimensional Laminar Premixed Flames 34
3.2.2 Effects of Stoichiometry and Pressure
Based on the results for the benchmark validation study discussed above and the comparisons
of the predicted flame speeds for ethanol using the two different mechanisms, it is evident
that both mechanisms yield rather similar results. Next the effects of fuel/air stoichiometry
and pressure on the predicted laminar flame speeds are evaluated for both mechanisms. The
predicted laminar flame speeds for equivalence ratios ranging from 0.5 to 1.8 and at pressures
of 1, 5, 10, 15, 20 and 25 atm are all shown in Figure 3.6.
It is evident from the Figure 3.6 the influence of stoichiometry on the flame speeds of ethanol.
In general, the flame speed increases with equivalence ratio from lean and reaches peak values
under slightly fuel rich conditions close φ = 1.1. As the mixture becomes fuel rich, flame speed
decreases.
The effects of pressure and equivalence ratio on the thermal diffusivities are evident for the
range of fuel compositions of interest. In theory, increased pressure results in a significant
reduction in the thermal diffusivity. This would account for with the observed reduction in
flame speed with an increase in pressure as seen in Figure 3.6.
Theoretical considerations suggest that the laminar flame speed, sL, is proportional to p−0.5 for
conventional hydrocarbon fuels [29]. To assess the validity of this result for the fuels in question,
the laminar flame speed data has been re-plotted as a function of pressure using log-log scales
for the plots. The results are shown in Figure 3.7. A line representing the expected theoretical
behaviour that sL ∝ p−0.5 is also shown in each plot for reference purposes.
As evident from the plots of the flame speeds for ethanol as a function of pressure depicted
in Figure 3.7, the expectation that the flame speed is indeed proportional to p−0.5 would also
seem valid here. The numerical curves for each equivalence ratio in these cases are in relatively
good agreement with the expected theoretical result. The agreement is best represented for
pressures between 1–10 atm and for leaner flames. At high pressures and equivalence ratios,
the exponent seems to become slightly higher for all of the fuels, which has been observed in
past experimental research by Okajima et al. [74].
Chapter 3. Results I: One-Dimensional Laminar Premixed Flames 35
Figure 3.6: Comparison of calculated values of laminar flame speed from Cantera for pressures
between 1 atm and 25 atm for Dryer and Reduced Dryer mechanism.
Chapter 3. Results I: One-Dimensional Laminar Premixed Flames 36
Figure 3.7: Laminar flame speed as a function of pressure comparison with theory.
Chapter 3. Results I: One-Dimensional Laminar Premixed Flames 37
3.3 Adiabatic Flame Temperature
Finally, the effect of stoichiometry and pressure on the adiabatic flame temperature is shown
in Figure 3.8 for the two mechanisms of ethanol. In general, the maximum flame temperature
occurs for slightly fuel rich conditions for all compositions with values ranging from 2200 K
to 2300 K at 1 atmosphere. This is explained by taking into account the balance of energy
required to increase the temperature of the combustion products (depending on the molar
specific heat of the products) and the energy lost during dissociation of the products at high
temperatures [30]. If dissociation were ignored, the peak of the flame temperature curve would
be at the stoichiometric point. Since dissociation is inhibited at high pressures, the temperature
peak at 25 atmospheres pressure corresponds with the stoichiometric point, as shown in Figure
3.8. In general, the flame temperature increase with pressure is minimal even at 25 atmospheres;
the increase in temperature was noted to be less than 100 K.
Chapter 3. Results I: One-Dimensional Laminar Premixed Flames 38
Figure 3.8: Adiabatic flame temperature of Dryer and Reduced Dryer mechanism as a function
of pressure.
Chapter 4
Results II: Laminar Co-Flow
Diffusion Flames for Ethanol
4.1 Introduction
The objective of the simulations described in this chapter was to investigate via a numerical
approach the sooting propensity of fully vaporized ethanol. This was accomplished by per-
forming simulations of laminar co-flow diffusion flames for this liquid biofuel for a range of
pressure conditions. The numerical simulations were carried out using a state-of-the-art lami-
nar combustion modelling framework developed by Charest et al. [6,11]. Numerical results were
obtained for both the reduced Dryer [44] and Dryer [16] chemical kinetic mechanisms. However,
due to current limitations of the experimental setup to vaporize ethanol, stable experimental
flames were only possible in the previous research by Karatas [14] for 1 atm and flickering
flames were obtained for 5 atm. For this reason, no experimental measurements of soot fraction
or temperature were available for comparison to the simulated results for the cases of interest
here. Only qualitative comparisons of a visual nature were possible between the experimental
and numerical results. This work simulates ethanol which is diluted with nitrogen by 70% by
volume. The fuel was also pre-heated to 400 K to mimic the experimental setup examined by
Karatas [14].
39
Chapter 4. Results II: Laminar Co-Flow Diffusion Flames for Ethanol 40
Figure 4.1: University of Toronto Institute for Aerospace Studies (UTIAS) high-pressure lami-
nar co-flow diffusion burner schematic.
4.2 Experimental Setup
The experimental setup employs a laboratory-scale, high-pressure axisymmetric co-flow burner
developed at UTIAS [26, 38, 39]. The UTIAS burner consists of a central fuel tube with 3 mm
inner diameter and a concentric tube of 25.4 mm inner diameter that supplies the air as seen in
Figure 4.1. The outer surface of the stainless steel fuel nozzle is tapered to reduce the formation
of wakes behind the tube walls to improve flame stability [14]. Sintered metal foam is inserted
in the fuel and air nozzles to straighten the flow and provide as close as possible a uniform
velocity profile at the nozzle exit.
4.3 Discretization and Boundary Conditions
The computational domain for the co-flow diffusion flames in this work is shown in Figure 4.2.
The fuel jet is emitted from the center of the annular ring while the air mixture is emitted
from the outer ring. The computational domain extends radially outwards 20 mm and 25 mm
downstream of the fuel nozzle exit. The far-field boundary was treated using a free-slip condition
which neglects any shear imparted to the co-flow air by the burner walls. The modelled domain
is also extended 9 mm upstream into the fuel and air tubes to account for the effects of fuel
preheating observed by Guo et al. [75] and to allow uniform velocity distribution to be specified
at the inlet to better represent the inflow velocity distribution at the nozzle exit. Thus, uniform
Chapter 4. Results II: Laminar Co-Flow Diffusion Flames for Ethanol 41
Figure 4.2: Schematic of computational domain.
velocity and temperature profiles were specified for both the fuel and air inlet boundaries. At
the outlet, temperature, velocity and species mass fractions are extrapolated while the pressure
is held fixed.
A simplified representation of the fuel tube geometry was used in simulating the flame. The
tapered edge of the fuel tube in the experimental setup was approximated by a tube with 0.4
mm thick walls. The three surfaces that lie along the tube wall were modelled as adiabatic
walls with zero-slip conditions in this work. The computational domain was divided into 192
cells and 8 blocks in the radial and 320 cells and 64 blocks in the axial direction. The resulting
computational mesh (as shown in Figure 4.3) which contains 61,080 structured cells and non-
uniformly stretched. These computational cells were stretched towards the burner exit to
capture interactions close to the fuel tube walls and towards the centreline to capture the
high temperature gradients in the core flow of the flame. Charest et al. [6, 11] have previously
demonstrated that this level of mesh resolution is more than sufficient to accurately represent
the laminar co-flow diffusion flames associated with the UTIAS high-pressure burner of interest
here.
Chapter 4. Results II: Laminar Co-Flow Diffusion Flames for Ethanol 42
Figure 4.3: 2D computational grid showing the clustering of computational cells. The mesh
contains 61,080 cells.
4.4 Solution Procedure for Simulations
To obtain the numerical solutions for ethanol laminar co-flow flames using the computational
framework developed by Charest et al. [6, 11], the following steps were followed:
Step 1: The computational domain was initialized with cold air, fuel at 400 K, and a small
rectangular region above the fuel exit plane is used as an igniter to initiate the chemical reac-
tions. The igniter region is initialized with an unburnt fuel-air mixture temperature of 1800
K.
Step 2: Ignition can be a rather chaotic that includes a complex transient phase. A semi-
implicit relaxation scheme was used within the computational framework [46] to partially solve
Equations (2.6a) – (2.6d) and find an improved initial starting estimate for Newton’s method.
Due to the rather long transient chaotic transient phase for the ethanol flames, longer period
of semi-implicit iterations (approximately 600 000 iterations) was required in order to obtain a
suitable initial estimate for the implicit solver.
Chapter 4. Results II: Laminar Co-Flow Diffusion Flames for Ethanol 43
Thousands of Iterations
Den
sity
L2-
Nor
m
686 688 690 692 69410-4
10-3
10-2
10-1
100
101
102
103
104
105
Step 2
Step 3
Step 4
Step 5
Figure 4.4: Typical convergence history for ethanol-air flames. The normalized L2–norm of the
continuity equation is shown.
Step 3: The solution from step 2 was used as an initial guess to solve Equations (2.6a) – (2.6d)
using the fully implicit scheme as described in Section 2.2. As the solution converges to the final
solution, the value for Mref was decreased to 10−4 while the CFL value was slowly increased
to between 1–5.
Step 4: Using the fully converged solution from step 3 as the initial guess and the DOM
radiation model is employed without soot, the Equations (2.6a) – (2.6d) are solved once again.
Step 5: Obtain the final solution with soot. This results in another chaotic non-linear phase.
This phase was nursed by reducing the CFL to between 0.1–0.5 and then slowly increasing it
back up to 1–3.
Convergence was said to be achieved when the mass, momentum and energy residual L2–
norms were reduced by at least five orders of magnitude. Figure 4.4 shows sample of a typical
convergence history for the ethanol-air laminar co-flow flames, illustrating each of the simulation
phases outlined above.
Chapter 4. Results II: Laminar Co-Flow Diffusion Flames for Ethanol 44
Figure 4.5: Experiemental photographs of co-flow diffusion flames of approximately 5 mm in
height for ethanol at pressures of 1 atm and 5 atm as obtained by Karatas [14].
4.5 Experimental Results
Karatas [14] attempted to conduct the experiments for ethanol flames of interest; however, was
unsuccesful due to issues with the fuel vaporizer used in the experiments. Karatas was however
partially successful in that a stable flame was obtained at 1 atm and a slightly flickering flame
was observed at 5 atm. Visually, the flame did not soot at 1 atm while the flame produced
soot at a pressure of 5 atm. The experimental photographs of these two flames are shown
in Figure 4.5, with estimated flame heights of approximately 5 mm based on the luminous
regions of the flame. Due to the high uncertainties in the experimental setup for ethanol,
no measurements of soot volume fraction nor temperature were conducted. The experimental
values of 0.5 mg/s for fuel mass flow rate and 0.11 g/s for air mass flow rate were matched in the
numerical simulations to allow a qualitative comparison of the simulations to the experimental
images of the flames.
Chapter 4. Results II: Laminar Co-Flow Diffusion Flames for Ethanol 45
4.6 Predicted Temperature Field
4.6.1 Contours of Temperature
The predicted temperature contours for ethanol using two different mechanisms are shown in
Figure 4.6. In general both mechanisms predict approximately the same maximum temperature
and similiar temperature contours especially at higher pressures. In general the differences
between the two mechanisms for maximum temperature is less than 1.5%. Also, the predicted
maximum temperature of Reduced Dryer is consistently lower than Dryer mechanism. Referring
to Figure 4.6 again, the flame structure varies with increase in pressure. Flames at 1 atm exhibit
greater bulging while flames at higher pressures are narrower and more conical.
4.6.2 Radial Temperature Profiles
Three axial locations were chosen to compute the radial temperature profiles as seen in Fig-
ure 4.7 for the Dryer mechanism. The three locations chosen are: low in the flame where soot
undergoes nucleation, middle of the flame where it is close to the maximum soot volume fraction
and higher in the flame where soot is oxidized. In general, the peak temperatures moves closer
toward the centreline across all pressures.
Chapter 4. Results II: Laminar Co-Flow Diffusion Flames for Ethanol 46
Radius (mm)
Hei
gh
t (m
m)
-10 -5 0 5 10
-5
0
5
10
15
20
25T(K)
1900180017001600150014001300120011001000900800700600500400
DryerReduced Dryer
1 atm
1862.35 K1833.22 K
Radius (mm)
Hei
gh
t (m
m)
-10 -5 0 5
-5
0
5
10
15
20
25T(K)
1900180017001600150014001300120011001000900800700600500400
DryerReduced Dryer
5 atm
1973.01 K1972.07 K
Radius (mm)
Hei
gh
t (m
m)
-10 -5 0 5
-5
0
5
10
15
20
25T(K)
1900180017001600150014001300120011001000900800700600500400
DryerReduced Dryer
10 atm
2011.61 K 2013.87 K
Radius (mm)
Hei
gh
t (m
m)
-5 0 5
-5
0
5
10
15
20
25T(K)
1900180017001600150014001300120011001000900800700600500400
DryerReduced Dryer
2041.91 K
15 atm
2038.17 K
Figure 4.6: Effect of reaction mechanism and pressure on flame structure for ethanol at pressures
of 1, 5, 10 and 15 atm. Peak temperature is indicated on the bottom right and left corners.
Chapter 4. Results II: Laminar Co-Flow Diffusion Flames for Ethanol 47
Radius (mm)
Tem
per
atu
re (
K)
0 1 2 3 4
500
1000
1500
2000
0.5 mm1.0 mm1.5 mm
5 atm
Radius (mm)
Tem
per
atu
re (
K)
0 0.5 1 1.5 2 2.5 3 3.5 4
500
1000
1500
2000
0.5 mm1.0 mm1.5 mm
10 atm
Radius (mm)
Tem
per
atu
re (
K)
0 0.5 1 1.5 2 2.5 3 3.5 4
500
1000
1500
2000
0.5 mm1.0 mm1.5 mm
15 atm
Figure 4.7: Radial temperature profiles at three axial locations for ethanol for pressures 5, 10
and 15 atms using Dryer mechanism.
Chapter 4. Results II: Laminar Co-Flow Diffusion Flames for Ethanol 48
4.7 Prediction of Flame Heights
In this work, the stoichiometric mixture fraction was used to estimate the flame height of the
flames from numerical solutions. Given a steady state solution, the mixture fraction of the fuel
can be calculated for the entire domain by using the following equation [76]:
Z =
YC−YC,2
mWC+
YH−YH,2
nWH+
YO−YO,2
αWO
YC,1−YC,2
mWC+
YH,1−YH,2
nWH+
YO,1−YO,2
αWO
, (4.1)
where Y and W are the mass fraction and atomic weights of the carbon, hydrogen and oxygen
atoms, m and n are the number of carbon and hydrogen atoms in the fuel, 1 and 2 signify the
fuel and air inlets, and α is the stoichiometric fuel-air ratio.
From the given equation, the predicted flame height has been imposed on the temperature
contour plots in Figure 4.6. The peak temperature of the contours are indicated by the values
on the lower left and right corners as seen in Figure 4.6. In general, the flame height remains
constant across all pressures for both mechanisms at about 3.15 mm. As previously mentioned,
experimentally, the flame height at 1 and 5 atm was found to be approximately 5 mm, which
is very close to the predicted flame heights obtained in the present numerical simulations.
4.8 Prediction of Flame Radius
Previous experimental and numerical research with sooting diffusion flames has shown that the
flame radius proportional to p−0.5 for conventional hydrocarbon fuels. This effect leads to the
observation that flame height should more or less be constant with increasing pressure.
The effect of pressure on flame radius was investigated by measuring the radius of the flames
shown in Figure 4.8 in an effort to quantify the exponent on the pressure. As Figure 4.8
demonstrates, the computed flame radii for ethanol do not agree the expected theoretical be-
haviour for hydrocarbon but instead appear to be more proportional to p−0.35. The overall
slope is lower than expected but this can probably be attributed to the presence of hydroxide
in ethanol. Further investigation is required to fully assess this influence on the laminar co-flow
flame structure.
Chapter 4. Results II: Laminar Co-Flow Diffusion Flames for Ethanol 49
Pressure (atm)
Fla
me
Rad
ius
(mm
)
2 4 6 8 10 12 14 1618
0.6
0.8
1
1.2
1.4
1.6
1.8
2
DryerTheory
slope = -0.35
Figure 4.8: Flame radius as a function of pressure for ethanol using Dryer mechanism.
4.9 Concentrations of Intermediate Species
4.9.1 Acetylene Mass Fraction
The semi-empirical soot model employed in this work assumes acetylene as the sole precursor
to soot formation. Thus, this species is important in determining soot yield of ethanol. Since
the amount of acetylene present is an indicator of the soot yield, the acetylene mass fraction
contour is shown only for pressures which yields soot as seen in Figure 4.9. In general, both
mechanisms predicted the same amount of acetylene with amount of acetylene decreasing with
increasing pressure.
4.9.2 Carbon Monoxide Mass Fraction
Carbon monoxide (CO) is one of the products from the chemical reaction of acetylene. In
general, both mechanisms predicted the same amount of CO as shown in Figure 4.10.
Chapter 4. Results II: Laminar Co-Flow Diffusion Flames for Ethanol 50
Radius (mm)
Hei
gh
t (m
m)
-4 -2 0 2 4-5
0
5
10
y_C2H2
0.0220.01950.0170.01450.0120.00950.0070.00450.002
DryerReduced Dryer
0.02370.0233
5 atm
Radius (mm)
Hei
gh
t (m
m)
-4 -2 0 2 4-5
0
5
10
y_C2H2
0.0220.01950.0170.01450.0120.00950.0070.00450.002
DryerReduced Dryer
10 atm
0.0224 0.0229
Radius (mm)
Hei
gh
t (m
m)
-4 -2 0 2 4-5
0
5
10
y_C2H2
0.0220.01950.0170.01450.0120.00950.0070.00450.002
DryerReduced Dryer
15 atm
0.02170.0212
Figure 4.9: Effect of reaction mechanism and pressure on acetylene mass fraction for ethanol
at pressures of 5, 10 and 15 atmospheres. Peak mass fraction is indicated on the bottom right
and left corners.
Chapter 4. Results II: Laminar Co-Flow Diffusion Flames for Ethanol 51
Radius (mm)
Hei
gh
t (m
m)
-4 -2 0 2 4-5
0
5
10
y_CO
0.0350.031250.02750.023750.020.016250.01250.008750.005
DryerReduced Dryer
0.03880.0367
5 atm
Radius (mm)
Hei
gh
t (m
m)
-4 -2 0 2 4-5
0
5
10
y_CO
0.0350.031250.02750.023750.020.016250.01250.008750.005
DryerReduced Dryer
10 atm
0.0356 0.0383
Radius (mm)
Hei
gh
t (m
m)
-4 -2 0 2 4-5
0
5
10
y_CO
0.0350.031250.02750.023750.020.016250.01250.008750.005
DryerReduced Dryer
15 atm
0.03810.0349
Figure 4.10: Effect of reaction mechanism and pressure on carbon monoxide mass fraction for
ethanol at pressures of 5, 10 and 15 atmospheres. Peak mass fraction is indicated on the bottom
right and left corners.
Chapter 4. Results II: Laminar Co-Flow Diffusion Flames for Ethanol 52
4.10 Predicted Soot Formation
4.10.1 Contours of Soot Volume Fraction
For the case of 1 atm, the maximum soot volume fraction was determined to 0.0009 ppm
which is below what is measurable with the current experimental setup. Thus soot volume
fraction contours is not included for 1 atm. From Figure 4.11, the structure for predicted soot
distribution varied with increasing pressure. The soot volume fraction had an annular structure
and maximum concentrations occurred in the annular region. The annular structure was more
pronounced and thinner for increasing pressures.
4.10.2 Radial Profiles of Soot Concentration
Three axial locations have been chosen to compute the radial soot profiles as seen in Figure 4.12
for the Dryer mechanism. The three locations are the same the radial locations as mentioned
previously in Section 4.6.2. In general, the peak soot volume fraction moves towards the
centreline and soot yield increases as the pressure is increased. These similar trends are observed
for gaseous fuels as well.
Chapter 4. Results II: Laminar Co-Flow Diffusion Flames for Ethanol 53
Radius (mm)
Hei
gh
t (m
m)
-4 -2 0 2 4
-4
-2
0
2
4
6
8
10
12ppm
54.543.532.521.510.5
DryerReduced Dryer
5.48 ppm4.27 ppm
5 atm
Radius (mm)
Hei
gh
t (m
m)
-4 -2 0 2 4-5
0
5
10
ppm
2623.621.218.816.41411.69.26.84.42
DryerReduced Dryer
10 atm
27.26 ppm23.31 ppm
Radius (mm)
Hei
gh
t (m
m)
-4 -2 0 2 4-5
0
5
10
ppm
45403530252015105
DryerReduced Dryer
15 atm
47.44 ppm41.79 ppm
Figure 4.11: Effect of reaction mechanism and pressure on soot volume fraction for ethanol at
pressures of 5, 10 and 15 atmospheres. Peak soot volume fraction is indicated on the bottom
right and left corners.
Chapter 4. Results II: Laminar Co-Flow Diffusion Flames for Ethanol 54
Radius (mm)
So
ot
Vo
lum
e F
ract
ion
(p
pm
)
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
5.50
6.00
0.5 mm1.0 mm1.5 mm
5 atm
Radius (mm)
So
ot
Vo
lum
e F
ract
ion
(p
pm
)
0.0 0.5 1.0 1.50
5
10
15
20
250.5 mm1.0 mm1.5 mm
10 atm
Radius (mm)
So
ot
Vo
lum
e F
ract
ion
(p
pm
)
0 0.5 1 1.50
5
10
15
20
25
30
35
40
45
50
0.5 mm1.0 mm1.5 mm
15 atm
Figure 4.12: Radial soot volume fraction at three axial locations for ethanol for pressures of 5,
10 and 15 atm using Dryer mechanism.
4.10.3 Sooting Propensity
For a better measure of ethanol’s sooting propensity and its sensitivity to pressure, the variation
of the carbon conversion factor as a function of pressure was calculated rather than relying on
soot concentration alone. Previously, this parameter has been used by other researchers [77] to
quantify the effects of pressure on soot formation. The carbon conversion factor, ηs, is defined
as ms/mc where mc is defined as the carbon mass flow rate at the exit of the burner nozzle [78].
Chapter 4. Results II: Laminar Co-Flow Diffusion Flames for Ethanol 55
Pressure (atm)
Max
Fu
el C
arb
on
Co
nve
rsio
n t
o S
oo
t (%
)
4 6 8 10 12 14 16 18
0.005
0.01
0.015
0.02
Dryer
Slope = 2.3
Slope = 0.25
Figure 4.13: Effect of pressure on the maximum carbon conversion factor for ethanol using
Dryer mechanism.
The mass flux of soot through a horizontal cross-section is
ms = 2πρs
∫fvvr dr, (4.2)
where ρs = 1.9 g/cm3 is the density of soot [75], fv is the soot volume fraction and v is the
axial velocity. In calculating mc for ethanol, N2 was included since it is considered inert. As a
results, all the ethanol flames have the same carbon mass flow rate of 0.292 mg/s at the nozzle
exit. Referring to Figure 4.13, the maximum carbon conversion factor increased with pressure
but the exponent with pressure decreases as indicated by the value of slopes. Comparing with
the previous numerical study by Charest et al. [13] for methane, a pressure exponent of 2 was
found by Charest et al. for pressures in the range from 5 atm to 10 atm and a value of 0.5
was observed for the exponent in the range from 10 atm to 15 atm. In contrast, the results
depicted in Figure 4.13 indicate values for the pressure exponent of 2.3 and 0.25 respectively
for the same pressure ranges.
Chapter 5
Conclusions and Future Research
5.1 Conclusions I: One-Dimensional Laminar
Premixed Flames
In this work, one-dimensional laminar premixed flames of ethanol were simulated using the
Cantera software package. The converged solutions were then used to quantify the laminar
flame speed and adiabatic flame temperature as a function of equivalence ratio and pressure.
The numerical results were obtained using two different mechanisms. In general, the flame speed
calculated using both the mechanisms were within 10% of each other. The difference in adiabatic
flame temperature values between the two mechanisms were minimal. Since the difference
between the values obtained by the two mechanisms of ethanol are small, the reduced Dryer
mechanism can used to calculate various fuel properties of interest relatively quickly without
significant errors. This mechanism is expected to be particularly useful in more complicated
simulations.
The laminar flame speed and adiabatic flame temperature as a function of stoichiometry and
pressure was evaluated. In general, the flame speed increased with increasing equivalence ratio
and decreased after a particular equivalence ratio (slightly fuel rich conditions). While as the
pressure increases, the flame speed was found to decrease quite dramatically. When the pressure
was increased from 1 to 25 atmospheres, the flame speed generally decreased by at least 70%.
Stoichiometry effects were common for all the fuels in that the peak adiabatic flame temperature
was attained at an equivalence ratio of approximately 1.1 at 1 atmosphere pressure. This was
attributed to the occurrence of dissociation reactions that in effect that take away heat from the
reactions. This effect was noticeably smaller at higher pressures and flame temperature peaks
56
Chapter 5. Conclusions and Future Research 57
were found to be shift much closer to or at stoichiometric point at a pressure of 25 atmospheres.
The proportionality of the flame speed with p−0.5 was also investigated. The numerical results
were found to be in rather good agreement with this theoretical expectation.
5.2 Conclusions II: Axisymmetric Laminar Co-Flow
Diffusion Flames
Two-dimensional laminar axisymmetric co-flow diffusion flames were also simulated for ethanol
utilizing two different reaction mechanisms to assess the influence of pressure and chemical ki-
netic mechanism on flame structure and soot yield. For constant mass fuel flows, the predicted
flame heights essentially remained constant. The differences in stoichiometric predicted lami-
nar co-flow diffusion flame structure and heights obtained using the Dryer and reduced Dryer
chemical kinetic mechanisms were noted to be negligible. This is expected since the reduced
Dryer mechanism was derived from the full Dryer mechanism.
The effect on flame radius of pressure was also investigated for ethanol using the Dryer mecha-
nism. It was observed that the predicted flame radius from numerical simulations was propor-
tional to p−0.35. However, theory for conventional hydrocarbon fuels predicts p−0.5 [37]. This
difference was attributed to the presence of hydroxide which is a diatomic anion.
The influence of pressure on soot formation have similiar observation to other gaseous fuels [11,
12,36], i.e, soot yield increases with pressure. The maximum carbon conversion factor increased
with pressure but the exponent with pressure decreases as was shown in Figure 4.13 of the
previous chapter. This could be attributed to the presence of nitrogen dilution.
5.3 Recommendations for Future Research
The present numerical study of the combustion characteristics of ethanol includes some of the
first numerical research on the sooting propensity of this liquid fuel at elevated pressures. How-
ever, full experimental comparisons of soot volume fraction and temperature fields are required
to fully assess the numerical results for ethanol missing from this study. This research should
be extended by investigating other ethanol mechanisms such as Saxena [15] and Marinov [79]
for numerical simulations. There are several other attractive liquid fuels and biofuels such as
n-heptane, biodiesel, etc. Currently, the combustion laboratory at UTIAS has conducted ex-
periments for n-heptane which has a more complicated fuel structure as compared to ethanol.
Chapter 5. Conclusions and Future Research 58
Future work should include numerical investigation of the fuel-n-heptane which is also a suitable
surrogate for biodiesel [80] using the current numerical framework.
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