© copyright by pankaj kumar 2012

Low-dimensional Models for Real-time Simulation of

Internal Combustion Engines and Catalytic

After-treatment Systems

A Dissertation

Presented to

the Faculty of the Department of Chemical and Biomolecular Engineering

University of Houston

In Partial Fulfillment

of the Requirements for the Degree

Doctor of Philosophy

in Chemical Engineering

by

Pankaj Kumar

May 2012

This dissertation is dedicated to my

Daddi, Mummy and Daddy

Acknowledgements

I would like to thank my advisors Prof. Balakotaiah, Prof. Franchek and Prof.

Grigoriadis for their support and guidance. I was lucky enough to have three ad-

visors which provided breadth to my dissertation work. I could not have imagined

having better advisors for my PhD study. Beside my advisors I would also like to

thank Prof. Mike Harold and Prof. Dan Luss for their valuable feedback as my

committee members and discussions during the group meetings. I would also like

to express my thanks to the support from Ford motor company, both financially and

technically. The two summer internship at Ford Motor Company was very beneficial

for my research work. The regular technical discussions with Imad Makki, Steve

Smith and James kerns from Ford ensured the continual progress. Mike Uhrich

helped a lot in conducting experiment at Ford research laboratory and generously

shared his technical expertise.

I am indebted to many of my colleagues for their support. Special thanks are

due to Ram Ratnaker for stimulating technical and spiritual discussions and without

whom the last four year would not have been this much fun. Santhosh Reddy was

generous enough to share the thesis template which saved me lots of effort and

have been the best resource for any software or technical help. I am thankful to

Saurabh, Divesh, Nitika, Pratik, Pranit, Arun, Bijesh, Richa and Priyank for all their

support and the technical discussions we had. Pratik, Pranit and Arun were the

best roommates, I could wish for and I am very thankful for their support. Specially,

Pratik has been my roommate for almost my entire PhD stay in Houston and I

really cherish his friendship. I am also thankful to my karate instructor Sensei

Deddy Mansyur, as his teachings helped me in dealing with stresses and kept me

physically and mentally healthy. Lastly, I would like to express my gratitude towards

my brother, sister and my parents for their constant support and encouragement

without which this thesis would not have been possible.

vi

Low-dimensional Models for Real-time Simulation of

Internal Combustion Engines and Catalytic

After-treatment Systems

An Abstract

of a

Dissertation

Presented to

the Faculty of the Department of Chemical and Biomolecular Engineering

University of Houston

In Partial Fulfillment

of the Requirements for the Degree

Doctor of Philosophy

in Chemical Engineering

by

Pankaj Kumar

May 2012

vii

Abstract

The current trend towards simultaneously increasing fuel-to-wheels efficiency

while reducing emissions from transportation system powertrains requires sys-

tem level optimization realized through real-time multivariable control. Such an

optimization can be accomplished using low-order fundamentals (first-principles)

based models for each of the engine sub-systems, i.e. in-cylinder combustion

processes, exhaust after-treatment systems, mechanical and electrical systems

(for hybrid vehicles) and sensor and control systems.

In this work, we develop a four-mode low-dimensional model for the in-cylinder

combustion process in an internal combustion engine. The lumped parameter or-

dinary differential equation model is based on two mixing times that capture the

reactant diffusional limitations inside the cylinder and mixing limitations caused by

the input and exit stream distribution. For given fuel inlet conditions, the model pre-

dicts the exhaust composition of regulated gases (total unburned HC’s, CO, and

NOx) as well as the in-cylinder pressure and temperature. The results show good

qualitative and fair quantitative agreement with the experimental results published

in the literature and demonstrate the possibility of such low-dimensional model for

real-time control.

In the second part of this work, we propose a low-dimensional model of the

three-way catalytic converter (TWC) for control and diagnostics. Traditionally, the

TWC is controlled via an inner-loop and outer-loop strategy using a downstream

and upstream oxygen sensor. With this control structure, we rely on the oxygen

sensor voltage to indicate whether the catalyst has saturated. However, if the oxi-

dation state of the catalyst could be estimated, than a more pro-active TWC control

strategy would be feasible. A reduced order model is achieved by approximating

the transverse gradients using multiple concentration modes and the concepts of

internal and external mass transfer coefficients, spatial averaging over the axial

viii

length and simplified chemistry by lumping the oxidants and the reductants. The

model performance is tested and validated using data on actual vehicle emissions

resulting in good agreement. The computational efficiency and the ability of the

model to predict fractional oxidation state (FOS) and total oxygen storage capacity

(TOSC) make it a novel tool for real-time fueling control to minimize emissions and

diagnostics of catalyst aging.

ix

Table of Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Internal combustion (IC) engines . . . . . . . . . . . . . . . . . . . . 2

1.2 Three way converter . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Legislations and development in automotive powertrain control . . . 6

1.4 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4.1 IC engine modeling . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4.2 Three-way catalytic converter modeling . . . . . . . . . . . . 12

1.5 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

In-cyliner Combustion Modeling . . . . . . . . . . . . . . . . . . . . 14

Chapter 2 Spark Ignited IC Engine Combustion Modeling . . . . . . . . . 15

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Model development . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.1 Derivation of the low-dimensional in-cylinder combustion model 20

2.2.2 Species balance for in-cylinder combustion . . . . . . . . . . 27

2.2.3 Derivation of energy balance for in-cylinder combustion . . . 32

2.2.4 Energy balance for in-cylinder combustion . . . . . . . . . . . 34

2.2.5 Heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2.6 Fuel composition and global reaction kinetics models . . . . . 36

x

2.3 Simulation of IC engine behavior and emissions using the low-dimensional

model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.3.1 Effect of Air to fuel ratio . . . . . . . . . . . . . . . . . . . . . 47

2.3.2 Effect of fuel blending . . . . . . . . . . . . . . . . . . . . . . 50

2.3.3 The effect of engine load and speed . . . . . . . . . . . . . . 55

2.3.4 Sensitivity of the model . . . . . . . . . . . . . . . . . . . . . 58

2.4 Extensions to the low-dimensional combustion model . . . . . . . . . 64

2.4.1 Extensions to the combustion chamber model . . . . . . . . . 65

2.4.2 Torque model . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.4.3 Controller design . . . . . . . . . . . . . . . . . . . . . . . . . 67

2.5 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 68

Chapter 3 Homogeneous Charge Compression Ignition . . . . . . . . . . 70

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.2 Model equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.3 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Three-way Catalytic Converter Modeling . . . . . . . . . . . . . . 77

Chapter 4 Low-dimensional Three-way Catalytic Converter Modeling with

Detailed Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.2 Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.3 Kinetic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.3.1 Experimental Set-up . . . . . . . . . . . . . . . . . . . . . . . 94

4.4 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Chapter 5 Low-dimensional Three-way Catalytic Converter Modeling with

Simplified Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

xi

5.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.2 Kinetic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.3 Experimental Validation of the Low-dimensional Model . . . . . . . . 112

5.3.1 Modeling Results . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.3.2 Model Updating for Diagnostics . . . . . . . . . . . . . . . . . 117

5.3.3 Model Validation on FTP Cycle . . . . . . . . . . . . . . . . . 120

5.4 Comparison of Green and Aged Catalyst Performance . . . . . . . . 121

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Chapter 4 Spatial-temporal Dynamics in a Three-way Catalytic Converter 128

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.2 Kinetic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.3 Model 1: Low-dimensional Model . . . . . . . . . . . . . . . . . . . . 128

6.3.1 Discretized Model . . . . . . . . . . . . . . . . . . . . . . . . 132

6.3.2 Experimental Validation . . . . . . . . . . . . . . . . . . . . . 134

6.4 Model 2: Validation with Detailed Model . . . . . . . . . . . . . . . . 140

6.4.1 Discretized model . . . . . . . . . . . . . . . . . . . . . . . . 141

6.4.2 Case 1: Single reaction . . . . . . . . . . . . . . . . . . . . . 142

6.4.3 Case 2 Multiple reaction including ceria kinetics . . . . . . . . 144

6.5 Effect of design parameters on catalyst light-off and conversion effi-

ciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.5.1 Effect of change in washcoat thickness . . . . . . . . . . . . . 146

6.5.2 Non-uniform catalyst activity . . . . . . . . . . . . . . . . . . . 148

6.5.3 Effect of cell density . . . . . . . . . . . . . . . . . . . . . . . 152

6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Chapter 7 Conclusions and Recommendations for Future Work . . . . . 163

7.1 In-cylinder combustion modeling . . . . . . . . . . . . . . . . . . . . 163

7.1.1 Summary and conclusions . . . . . . . . . . . . . . . . . . . . 163

xii

7.1.2 Recommendations for future work . . . . . . . . . . . . . . . 165

7.2 Three-way catalytic converter modeling . . . . . . . . . . . . . . . . 166

7.2.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . 166

7.2.2 Recommendations for future work . . . . . . . . . . . . . . . 168

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

xiii

List of Figures

Figure 1.1 Schematic of four stroke SI internal combustion engine (En-

cyclopedia Britannica inc., 2007) . . . . . . . . . . . . . . . . 4

Figure 1.2 Schematic of inner and outer control loop in partial volume

catalyst . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Figure 1.3 Schematic of inner and outer control loop in full volume cat-

alyst . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Figure 1.4 FTP -75 cycle . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Figure 2.1 Intake manifold pressure variation with throttle position as a

function of time. . . . . . . . . . . . . . . . . . . . . . . . . . 41

Figure 2.2 In-cylinder temporal variation of (a) Total unburned hydro-

carbon concentration, (b) Unburned oxygen concentration,

(c) Pressure and (d) Temperature with time . . . . . . . . . . 42

Figure 2.3 Temporal variation of in-cylinder (a) CO2 concentration, (b)

NOx concentration, (c) CO concentration and (d) Hydrogen

concentration . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Figure 2.4 In-cylinder variation of (a) Pressure and (b) Temperature

during a complete cycle after a periodic state is attained . . 45

Figure 2.5 Variation of exhaust gas concentrations with time (a) Un-

burned hydrocarbon, (b) Exhaust oxygen, (c) Exhaust NOx

and (d) Exhaust CO . . . . . . . . . . . . . . . . . . . . . . . 46

Figure 2.6 Normalized reaction rate as a function of temperature over

a cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Figure 2.7 Hydrocarbon conversion with crank angle for λ=1 . . . . . . 49

xiv

Figure 2.8 Effect of change in air/fuel ratio on peak temperature and

pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Figure 2.9 Variation of regulated exhaust gases with air fuel ratio (a)

Predicted from low-dimensional model and (b) Experimen-

tally observed [12] . . . . . . . . . . . . . . . . . . . . . . . . 51

Figure 2.10 (a) Impact of blending on emissions and (b) In-cylinder tem-

perature and pressure under constant air/fuel ratio of λ=1 . . 53

Figure 2.11 Comparison of reaction rate during a cycle for E0 and E50

(a) Normalized reaction rate for 50% ethanol (vol% ) blended

gasoline, (b) Normalized reaction rate for gasoline, (c) CO

oxidation rate and (d) NOx formation rate . . . . . . . . . . . 54

Figure 2.12 Impact of blending on (a) Emissions and (b) In-cylinder tem-

perature and pressure at constant flowrate (λ goes leaner) . 56

Figure 2.13 (a) Impact of change in load on engine emissions as pre-

dicted by model, (b) Experimentally observed variation in

emissions (Heywood, 1988), (c) In-cylinder peak tempera-

ture variation with load and (d) Effect of load on in-cylinder

peak pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Figure 2.14 Impact of engine speed on (a) NOx and HC emission as

predicted by model, (b) In-cylinder peak temperature and

pressure, (c) Experimentally reported NOx with change in

speed (Celik, 2008) and (d) Experimentally reported HC and

NOx with change in engine speed (Heywood, 1988) . . . . . 58

Figure 2.15 Influence of in-cylinder dimensionless mixing time on (a)

Emissions and (b) In-cylinder tempeature and pressure . . . 60

xv

Figure 2.16 Impact of change in crevice volume on (a) Hydrocarbon emis-

sion, (b) In-cylinder temperature, (c) CO emission and (d)

NOx emission for τmix,1 =0 and τmix,2 =0 . . . . . . . . . . . 62

Figure 2.17 Impact of change in inlet temperature on (a) Hydrocarbon

emission, (b) NOx emission, (c)In-cylinder temperature and

(d) In-cylinder pressure . . . . . . . . . . . . . . . . . . . . . 65

Figure 3.1 In-cylinder pressure and temperature for a HCCI engine . . 72

Figure 3.2 In-cylinder emissions for a HCCI engine . . . . . . . . . . . 74

Figure 3.3 Simulated exit emissions for an HCCI engine . . . . . . . . 75

Figure 4.1 Schematic diagram of inner and outer loop control strategy . 80

Figure 4.2 Three-way catalytic converter schematic . . . . . . . . . . . 83

Figure 4.3 Total Carbon balance in terms of mole fractions at TWC inlet

and exit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Figure 4.4 Sensors location schematic . . . . . . . . . . . . . . . . . . 95

Figure 4.5 Operating condition in terms of feed gas air-fuel ratio and

vehicle speed . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Figure 4.6 Comparision of model predicted and experimental CO con-

version for lean to rich step change at a constant vehicle

speed of 30 mph . . . . . . . . . . . . . . . . . . . . . . . . 97

Figure 4.7 Comparision of model predicted and experimental HC con-


speed of 30 mph . . . . . . . . . . . . . . . . . . . . . . . . 97

Figure 4.8 Comparision of model predicted and experimental NO con-


speed of 30 mph . . . . . . . . . . . . . . . . . . . . . . . . 98

Figure 4.9 Fractional oxidation state of the catalyst during lean to rich

step change . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

xvi

Figure 4.10 Catalyst wall (brick) and feedgas temperature for a lean to

rich step change experiment . . . . . . . . . . . . . . . . . . 99

Figure 4.11 comparision of model predicted vs experimentally observed

CO emission at constant vehicle speed of 60 mph . . . . . . 100


NO emission at constant vehicle speed of 60 mph . . . . . . 101


HC emission at constant vehicle speed of 60 mph . . . . . . 102


CO2 emission at constant vehicle speed of 60 mph . . . . . 103


O2 emission at constant vehicle speed of 60 mph . . . . . . 104

Figure 5.1 Operating condition: Feed gas A/F (λ) and the inlet feed

temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Figure 5.2 Comparison of model predicted vs experimentally observed

(a) oxidant emission and (b) reductant emission at vehicle

speed of 30 mph for a green catalyst . . . . . . . . . . . . . 115

Figure 5.3 Fractional oxidation state of the catalyst . . . . . . . . . . . . 116

Figure 5.4 Comparison of model predicted vs experimentally observed

(a) oxidant emission and (b) reductant emission for vehicle

speed of 30 mph with an aged catalyst . . . . . . . . . . . . 118

Figure 5.5 Comparison of model predicted vs experimental (a) oxidant

and (b) reductant emissions for an idle operation (speed=0

mph) with an aged catalyst . . . . . . . . . . . . . . . . . . 119

Figure 5.6 Comparision of (a) oxidant and (b) reductant emissions with

threshold catalyst over a FTP cycle . . . . . . . . . . . . . . 122

Figure 5.7 Change in FOS over bag one and two of a FTP cycle . . . . 123

xvii

Figure 5.8 Light-off behavior of green and aged catalyst with 1.5% re-

ductant in feed under stoichiometric operation . . . . . . . . 124

Figure 5.9 Impact of washcoat diffusion on conversion in a TWC: Bi-

fucation plot for 1.5% reductant feed under stoichiometric

operation at (a) u=1m/s and (b) u=10m/s . . . . . . . . . . . 126

Figure 6.1 Experimental validation for oxidant emission at idle vehicle

speed with an aged catalyst . . . . . . . . . . . . . . . . . . 135

Figure 6.2 Experimental validation for reductant emission with an aged

catalyst at idle vehicle speed. . . . . . . . . . . . . . . . . . 136

Figure 6.3 Oxidant emission for first 300s of FTP (ac=0.3) . . . . . . . 137

Figure 6.4 reductant emission for first 300s of FTP (ac=0.3) . . . . . . . 137

Figure 6.5 Observed lambda as computed using chemical composition 139

Figure 6.6 Model comparision of internal mass transfer concept with

detailed model for a single reaction . . . . . . . . . . . . . . 143

Figure 6.7 Model comparision of internal mass transfer concept with

detailed model for a single reaction . . . . . . . . . . . . . . 144

Figure 6.8 Steady state axial temperature for different inlet feed tem-

perature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Figure 6.9 Bifurcation plot for uniform activity at u=1 m/s for 1.5% re-

ductant at stoichiometry . . . . . . . . . . . . . . . . . . . . 148

Figure 6.10 Effect of change in washcoat thickness on catalyst light-off . 149

Figure 6.11 Effect of change in washcoat thickness on exit conversion

efficiency transient . . . . . . . . . . . . . . . . . . . . . . . 150

Figure 6.12 Effect of change in washcoat thickness on exit temperature

transient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Figure 6.13 Steady state temperature profile for non-uniform catalyst load-

ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

xviii

Figure 6.14 Effect of change in loading profile on conversion transient at

constant feed temperature of T=550K . . . . . . . . . . . . . 153

Figure 6.15 Effect of change in loading profile on conversion transient at

constant feed temperature of T=650K . . . . . . . . . . . . . 154

Figure 6.16 Effect of change in cell density in ceramic substrate for con-

stant space velocity (45868 hr−1) and constant feed temper-

ature (T=550K) . . . . . . . . . . . . . . . . . . . . . . . . . 156

Figure 6.17 Effect of change in cell density in ceramic substrate for con-


ature (T=550K) . . . . . . . . . . . . . . . . . . . . . . . . . 157

Figure 6.18 Effect of change in cell density in metallic substrate for con-


ature (T=550K) . . . . . . . . . . . . . . . . . . . . . . . . . 158

Figure 6.19 Effect of change in cell density in metallic substrate for con-


ature (T=550K) . . . . . . . . . . . . . . . . . . . . . . . . . 159

Figure 6.20 Comparision of ceramic and metallic substrate for constant

feed temperature of 550 K and constant space velocity and

composition . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Figure 6.21 Comparision of ceramic and metallic substrate for constant

feed temperature of 550 K and constant space velocity and

composition . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Figure 6.22 Comparision of metallic and ceramic substrate for same cat-

alyst loading . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

xix

List of Tables

Table 1.1 California emission standards for passenger cars . . . . . . . . 8

Table 2.1 Woschni’s correlation parameters . . . . . . . . . . . . . . . . 36

Table 2.2 Global kinetics for propane and ethanol combustion . . . . . . 37

Table 2.3 kinetic constants for propane and ethanol combustion (units in

mol, cm3, s, cal) . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Table 2.4 System parameters . . . . . . . . . . . . . . . . . . . . . . . . 40

Table 4.1 LEV II Emission standards for passenger cars and light duty

vehicles under 8500 lbs, g/mi [CEPA, 2011] . . . . . . . . . . . 79

Table 4.2 Numerical constants and parameters used in TWC simulation 92

Table 4.3 Global reaction in Three way catalytic converter . . . . . . . . 92

Table 4.4 Brick dimensions and loading . . . . . . . . . . . . . . . . . . . 94


Table 5.2 Global reaction kinetics . . . . . . . . . . . . . . . . . . . . . . 110

Table 5.3 Kinetic parameters for a Pd/Rh based TWC with specifications

shown in Table 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Table 5.4 Brick dimensions and loading of catalyst in FTP test . . . . . . 120

Table 5.5 kinetic parameters for a threshold 70 g/ft3 Pd/Rh based TWC . 120

Table 6.1 Kinetic parameters for a Pd/Rh based TWC . . . . . . . . . . . 128


Table 6.3 Nominal properties of standard and thin walled Cordierite sub-

strate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Table 6.4 Nominal properties of standard and thin-wall metallic substrate 154

Table 6.5 Physical properties of washcoat, ceramic and metallic substrate155

xx

Nomenclature

Part I

a crevice flow parameter

Ath throttle area (m2)

B cylinder bore (m)

〈C〉 volume averaged concentration (mol/m3)

Ccr concentration within crevice region (mol/m3)

Cd drag coefficient

Cm flow averaged concentration (mol/m3)

Cx vehicle drag coefficient

E experimentally determined constant

fr friction coefficient

F exit molar flow rate (mol/s)

F in inlet molar flow rate (mol/s)

Fcr crevice exchange flow rate (mol/s)

Fv force on the vehicle wheel (N)

g acceleration due to gravity (m/s2)

hc,c coolant side heat transfer coefficient (W/(m2 K))

hc,g gas side heat transfer coefficient (W/(m2 K))

H inj molar enthalpy of component j at inlet conditions (J/mol)

Hj molar enthalpy of component j at exit conditions (J/mol)

4HR,T heat of reaction computed at temperature T (J/mol)

i reaction index

I moment of inertia of the powertrain (Kg m2)

iD reduction ratio of the differential

xxi

iG reduction ratio of the gearbox

j component number

k thermal conductivity of wall (W/(m K))

l cylinder wall thickness (m)

.mf fuel mass flow (kg/s)

Mv mass of the vehicle (kg)

Nc total number of components

NR total number of reactions

P in-cylinder pressure (Pa)

Pcrevice crevice pressure (Pa)

Pout downstream pressure (Pa)

Q inlet volumetric fuel flowrate ( m3/s)

Qfuelin inlet volumetric fuel flowrate ( m3/s)

Qairin inlet volumetric air flowrate ( m3/s)

.

Q energy added by spark (J/s)

Qcr crevice volumetric flow rate ( m3/s)

Qv heating value of fuel (J/Kg)

R universal gas constant (8.314 J/(K mol) or (m3Pa)/(K mol))

R ratio of connecting rod length to crank radius

R(C) reaction rate at concentration C (mol/(m3s))

rc compression ratio

Rw radius of the wheels (m)

S frontal vehicle surface (m2)

t instantaneous time (s)

T average bulk gas temperature (K)

Tc coolant temperature (K)

Tw,c coolant side wall temperature (K)

xxii

Tw,g gas side wall temperature (K)

Te effective toque (N m)

Tl load torque (N m)

tmix,i ith mixing time

V instantaneous volume of cylinder (m3)

Vc clearance volume (m3)

Vcr crevice volume (m3)

Vd displaced or swept volume (m3)

VR total volume of the cylinder (m3)

vv vehicle velocity (m/s)

w average cylinder gas velocity (m/s)

xe ethanol mole fraction

λ ratio Air/ fuel actual to that at stoichiometry

θ crank angle (rad)

φ slope of road (rad)

Ω angular speed (rad/s)

νij stoichiometric coefficient of ith reaction and jth component

ρ density of in-cylinder gaseous mixture (kg/m3)

ρa density of air (kg/m3)

γ heat capacity ratio

σ Stefan -Boltzman constant W/ (m2K4)

ε emissivity

ηe effective efficiency

ηm power transmission efficiency

xxiii

Part 2

Symbols Definition

A pre-exponential factor (mol m−3 s−1)

a pore radius (m)

Cp specific heat capacity (J kg−1 K−1)

D diffusivity (m2s−1)

E activation energy (J mol−1)

h heat transfer coefficient (W m−2K−1)

kme external mass transfer coefficient (m s−1)

kmi internal mass transfer coefficient (m s−1)

kmo overall mass transfer coefficient (m s−1)

L TWC brick length (m)

N number of species

Nr number of reactions

r vector of reaction rate (mol m−3 s−1)

R gas constant (J mol−1K)

RΩ hydraulic radius of monolith channel (m)

t time (s)

T temperature (K)

TOSC total oxygen storage capacity (mol m−3)

〈u〉 feed gas velocity (m s−1)

X mole fraction

Xfm cup-mixing mole fraction in fluid phase

〈Xwc〉 volume averaged mole fraction in washcoat

Xs mole fraction at gas-solid interface

xxiv

Greek symbols

θ fractional oxidation state of catalyst

εw void fraction (porosity) of washcoat

τ tortuosity

λ normalized air/fuel ratio

ν matrix of stoichiometric coefficient

ρ density (kg/m3)

δw effective wall thickness (m)

δc washcoat thickness (m)

Subscripts

i reaction index

j gaseous component index

f fluid phase

s solid phase

w wall / washcoat

Superscripts

in inlet condition

0 initial condition

xxv

Chapter 1 Introduction

Future automotive engines will have to achieve extremely demanding diagnos-

tics and feedback control requirements to drastically minimize consumed fuel (re-

cent regulations dictate a 54.5 mile per gallon fuel economy by 2025) and keep

harmful emissions to practically zero. Automotive engines are complex electro-

mechanical systems with multiple subsystems (air handling, turbocharger, complex

in-cylinder thermo-fluid and combustion dynamics, etc.). Additionally, the engine

exhaust after-treatment system involves catalytic chemistry and reaction dynamics

that at the meso- and microscopic level determine the removal of pollutants. The

corresponding chemical and mass/heat transfer processes span multiple spatio-

temporal scales. The two systems, engine and exhaust after-treatment, are highly

interdependent based on coupled operational constraints and interconnected dy-

namics with conflicting objectives. Therefore, diagnostics and feedback control

of the overall system are defined and constrained by the complexity of the cou-

pled electro-mechanical, thermo-fluid, and chemical processes, and the limited

on-board computational capabilities.

The current trend towards simultaneously increasing fuel-to-wheels efficiency

while reducing emissions from transportation system powertrains requires system

level optimization realized through real-time multivariable control. Such an opti-

mization can be accomplished by using low-order fundamentals (first-principles)

based models for each of the engine sub-systems, i.e. in-cylinder combustion

processes, exhaust after-treatment systems, mechanical and electrical systems

(for hybrid vehicles) and sensor and control systems. The combustion process and

catalytic after-treatment systems can be described by the fundamental conserva-

tion laws (species, momentum and energy) of diffusion-convection-reaction type.

Such a description consisting of many partial differential equations with coupling

1

between the transport process and the complex chemistry is extremely demanding

computationally and has limited utility for system level optimization studies. For

online optimization and real-time control, these physics based models must be

low-dimensional, retain the qualitative features of the system, and have sufficient

quantitative accuracy. In our view, the bottleneck for attaining real-time onboard

system level optimization is the lack of accurate low-dimensional models for the

internal combustion (IC) engine. To achieve the project goal we will address the

development of a low-dimensional model for in-cylinder combustion and three-way

catalytic converter (TWC).

1.1 Internal combustion (IC) engines

Automobile engines are the major source of urban pollution. Emissions from

the individual cars are usually low but with millions of vehicles on road the total

emission adds up. The year 2010 will approach 800 million passenger cars with

an annual worldwide production of new cars approaching 100 million. Pollution by

an automobile is contributed by the combustion of fuel and also the evaporation of

fuel itself. In an ideal or a desired engine, the combustion of fuel will result in rela-

tively harmless CO2, H2O and N2 as the end products. However, due to incomplete

combustion several other by-products like carbon monoxide (CO) and unburned

hydrocarbons (HC) are also emitted. The high temperature within the cylinder also

facilitates the Zeldovich mechanism in which nitrogen and oxygen in air reacts to

form nitrogen oxides, collectively called as NOx. The NOx and HC are precursors

to the formation of ground level ozone, a major component of smog. Ground level

ozone can lead to health problems such as breathing difficulty, lung damage and

reduced cardiovascular functioning. Presence of NOx also contribute to the forma-

tion of acid rain. CO is also highly toxic gas that combines with hemoglobin to form

carboxyhemoglobin that leads to reduced flow of oxygen in the bloodstream. The

relative proportion of different emissions and the amount depend on the fuel type

2

(gasoline, diesel, biofuel) as well as engine design. Spark ignited gasoline engines

remain the most common form of internal combustion engines. We present here

a brief introduction, the similarities and difference between the three engines from

the modeling perspective.

1. Spark ignition engines (Gasoline engine)

In a conventional spark ignited (SI) internal combustion engine air and fuel are

usually pre-mixed before being injected into the cylinder using a carburetor or fuel

injection system. Shown in Fig 1.1 is the schematic of the four-stroke SI engine.

The fuel mixture are compressed in the compression stroke and a spark is initiated

just before the end of compression stroke that leads to the flame front propagation.

The moving flame compresses the unburned gas (also known as end gas) ahead

of the flame, which may sometime lead to such an increase in the end gas tempera-

ture that the mixture auto ignites. This causes high frequency pressure oscillations

inside the cylinder that produce a sharp metallic noises referred to as a knock.

Knocking is one of the major reasons that limit the higher compression ratio that

can be used in SI engines. The fuel used is characterized by its octane number,

which is a measure of the resistance to auto ignition. By definition, normal heptane

(n-C7H16) has a value of zero octane number and isooctane (C8H18) has an octane

number of 100. The gasoline engines are usually operated around stoichiometry

as they use three-way catalytic converters (TWC) for emission abatement, whose

perform optimaly around stoichiometry.

2. Compression ignition (Diesel engine)

In diesel engines, only air is injected through the inlet valve, the fuel is injected

directly in the cylinder just before the start of a compression stroke. The load is

controlled by changing the amount of fuel injected in each engine cycle, keeping

the air flow at a given engine speed almost constant. The compression ratios in

diesel engines is much higher than that observed in SI engines. The typical values

3

Figure 1.1: Schematic of four stroke SI internal combustion engine (Encyclopedia

Britannica inc., 2007)

for SI engines are of the order of 8-12, while CI engines can have compression ra-

tios of 12-14. Also, the diesel engines are usually run slightly lean of stoichiometry

and require exhaust aftertreatment units such as LNT and SCR to meet emission

norms.

3. Stratified charge engines :

This system tries to combine the best features from both SI and CI engines.

1)The fuel is injected directly into the combustion chamber during compression

stroke and thereby avoids the knock problem that limits the conventional SI engine

with premixed feed. 2) The fuel mixture is ignited with a spark plug that provides

direct control of the ignition and thereby avoids the fuel ignition quality requirement

of the diesel. 3) The engine power level can be controlled by controlling the amount

of fuel injected in each engine cycle and thereby avoids the pumping loss by un-

throttled air flow. In stratified charge engines, the air/fuel ratio varies with position

within the cylinder.

4

1.2 Three way converter

The TWC is a monolith that comprises of multiple parallel channels (400-900

cpsi) with catalysts loaded around the wall surface refered to as washcoat. The

monolithic arrangement have several advantages over the pellet kind of arrange-

ment. The most important being the low pressure drop across the channel over

high flow rates. The monoliths have large open frontal area and straight parallel

channels that lead to low flow resistance. The other advantage being the ability to

make compact reactors, freedom in reactor orientation and good thermal and me-

chanical shock property. The earlier three-way catalytic converter were only used

for oxidation and as such the platinum group metal (PGM) consisted of Pt and Pd.

With the introduction of legislation for NOx emission, Rh was added which enabled

TWC to reduce NOx along with oxidizing CO and HC. The typical alumina wash-

coat has a loading of 1.5 to 2.0 g per in3. The ceria content can range up to 10%

. The PGM loading varies in range 10-100 g/ft3. Most close coupled catalyst have

high Pd for high temperature durability.

Traditionally, the TWC is controlled based on catalyst monitor sensor (CMS) set

points (Fiengo et al., 2002, Makki et al., 2005). Shown in Fig.1.2 is the schematic

representation of a typical inner and outer loop TWC control strategy (Makki et al.,

2005). A TWC unit, usually consists of two bricks separated by a small space.

In partial volume catalyst (Fig.1.2 ) the HEGO sensor is located in between the

two bricks, while in a full volume catalyst (Fig.1.3) the HEGO is placed after the

second brick, i.e., at the exit of the TWC. The advantage of using a partial volume

system is that it provides fueling control in a delayed system, i.e., even if there is

a breakthrough detected after the first brick, it can be treated in the second brick.

With an OBD requirement to monitor the entire catalyst performance, and also for

design and cost consideration, a full volume catalyst has to be used. Typically,

UEGO is placed after the engine for more accurate A/F measurement while HEGO

5

Figure 1.2: Schematic of inner and outer control loop in partial volume catalyst

is preferred to measure A/F after the TWC because of its lower cost and faster

response time. The inner loop controls the A/F to a set value while the outer loop

modifies the A/F reference to the inner loop to maintain the desired HEGO set volt-

age (around 0.6-0.7 V, depending on design and calibration) to achieve the desired

catalyst efficiency. With this arrangement we rely on the emissions breakthrough

at the HEGO sensor, to determine if the catalyst is saturated with oxygen or not

and as such it imposes a limitation on the controller design particulary for the low

emission vehicles.

1.3 Legislations and development in automotive powertrain con-

trol

Environmental regulations limit the amount of carbon monoxide (CO), hydrocar-

bons (HC) and nitrogen oxides (NOx) that can be released from an automobile’s

6

Figure 1.3: Schematic of inner and outer control loop in full volume catalyst

tailpipe. From its inception in 1970 when the US congress passed the Clean Air

Act (CAA), the stringent legislation has been a driving force to reduce fuel con-

sumption and engine emissions. A federal test procedure (FTP) simulating the

average driving condition in the US was established in 1975 by the Environment

Protection Agency (EPA). Shown in Fig.1.4 is the FTP75 drive condition. The to-

tal emissions from each of the three phases is collected separately in a bag and

a weighing factor is used to compute total emission. As a point of reference, the

pre-1968 unregulated vehicle would produce emission of about 83-90 g/mile CO,

13-16 g/mile HC and 3.5-7 g/mile NOx when tested in the present US Federal test

cycle. Also established around the same time as the EPA was the California Air

Resource board (ARB). California is the only state that has the authority to adopt

its own emission regulation. The other state have the choice to either adopt the

ARB norms or the federal norms. Generally, the California norms are much more

7

Table 1.1: California emission standards for passenger cars

category Durability basis (miles) NMOG g(/miles) CO (g/miles) NOx (g/miles)

Tier 1 50,0000 0.25 3.4 0.4

100,000 0.31 4.2 0.6

TLEV 50,000 0.125 3.4 0.4

120,000 0.156 4.2 0.6

LEV 50,000 0.075 3.4 0.05

120,000 0.09 4.2 0.07

ULEV 50,000 0.04 1.7 0.05

120,000 0.055 2.1 0.07

SULEV 120,000 0.01 1 0.02

PZEV 150,000 0.01 1.0 0.02

ZEV -0- -0- -0- -0-

stringent than Federal norms. Shown in Table 1.1 (Heck and Farrauto, 2002) are

the California emission standards. After 2003, Tier1 and TLEV standards were

removed as available emission categories. Prior to the 1990 amendment to the

CAA, the catalyst was supposed to maintain performance for 50,000 miles. After

the amendment the catalyst have been required to last 100,000 miles for the model

year 1996 onward.

During the early implementation of CAA (1976-1979), the NOx standards were

relaxed and as such the catalysts used were only required to oxidize CO and HC.

Running the engine rich and using exhaust gas recirculation (EGR) was sufficient

to reduce the NOx formation to meet the legislation requirement. However, running

rich increased the CO and HC emission from the engine so secondary air was

pumped into the exhaust gas to provide sufficient O2 for the oxidation of CO and

unburned HC. Thus, the early TWC had a Pt and Pd based catalyst only, with a

stabilized alumina washcoat. The use of the converter spurred development in

other fields as well. Due to the fact that lead poisons the catalyst, the year 1975

saw the widespread introduction of unleaded gasoline. This resulted in a significant

reduction in ambient lead levels and alleviated many serious environmental and

health concerns associated with lead.

8

With new stringent legislation around 1979-1986, the NOx emission in automo-

bile exhaust was limited to less than 1g/mile and the ’first generation’ TWC could

no longer meet the legislative requirement. Different catalyst like Ru and Rh were

added to the TWC to reduce NOx. Ru formed a volatile oxides at the temperature

condition encountered in the automobile and was dropped from further considera-

tion. Rh was added to TWC which enabled it to reduce NOx along with oxidizing

CO and HC. It was observed that with the Pt, Pd and Rh based catalyst, if the

engine could be operated around stoichiometric, i.e., air-to-fuel ratio (λ) = 1, all

three pollutant could be simultaneously converted. For lean feed the CO and HC

conversion is high, however the NOx emission increases while with rich feed the

NOx could be properly reduced at the expense of high CO and HC emission.

A key development in this area was the introduction of Heated Exhaust Gas

Oxygen Sensors (HEGO) that made the close loop control around TWC feasible.

At the same time the development in the fuel quality, with lesser sulphur made the

Pd based close coupled catalyst sustainable. Pt shows higher degree of sintering

with temperature as compared to Pd, however Pd is much more susceptible to

sulphur poisoning. A cold start emission is known to be the major contributor in

the total observed emission and with the close-coupled catalyst earlier light-off is

achieved that has lead to a significant drop in cold start emission. As shown in

Table 1.1, over the years there has been a significant reduction in the allowable

emission and in particular NOx. Partial zero emission vehicles (PZEV) have the

same emission requirements as super ultra lean emission vehicle (SULEV) with

an additional requirement of zero evaporative loss. A purge cannister containing

activated carbon is usually used to limit evaporative loss. The ZEV will probably be

battery operated.

The CO2 emissions are not regulated directly, however they are controlled

through fuel mileage requirement defined by the Corporate Average Fuel Economy

9

Figure 1.4: FTP -75 cycle

(CAFE). CAFE is the annual sales weighted average fuel economy, expressed in

miles per gallon (m.p.g.). The manufacturer pays a penalty, if the average fuel

economy of the manufacturer fleet falls below the defined standard. The program

was established by the Energy policy and Conservation Act of 1975 in response

to the 1973-74 Arab oil embargo and was the main force behind a 52% increase

in new vehicle fuel economy between 1978 (18 m.p.g.) and 1985 (27.5 m.p.g.)

(NHTSA, 2003). Since 1985, however, the CAFE standards for passenger cars

have not increased and stayed constant at 27.5 m.p.g. from 1990 to 2010. The

policy is becoming stringent again with a current legislation requiring an average

fuel economy standard of 35.5 m.p.g. by model year 2016 and an average of 54.5

miles/gal for cars and light duty trucks by year 2025.

Apart from emissions, the 1990 amendment to the Clean Air Act, also requires

vehicles to have built-in On-Board Diagnostics (OBD) system. The OBD is a

computer based system designed to monitor the major engine equipment used

to measure and control the emissions. OBD regulations ensure compliance with

emission standards by setting requirements to monitor selected emission system

10

components (e.g., catalytic converters) or in-use emission levels, and to alert the

driver/operator—such as by a dashboard-mounted malfunction indicator light—

when a problem is detected. One such requirement is to raise a flag when the

catalyst activity falls below a threshold. Such requirements have motivated the

growth of physics based models for control and diagnostics.

1.4 Literature review

1.4.1 IC engine modeling

Different kind of models have been developed for SI engine modeling, mostly

by mechanical engineers and, as such, the major emphasis has been given to flow

distribution and power output as compared to emissions. In general the modeling

approach can be divided into two main groups. The fluid dynamic model and the

thermodynamic model.

The fluid dynamic model is also called the multidimensional model and involves

partial differential equation of mass, energy and momentum balance in spatial co-

ordinates and time. One of the earliest work in Internal combustion modeling has

been reported by Blumberg et al. (1979), Griffin et al., (1979) and Heywood et al

(1988). Griffin et al., (1979) considered a three-dimensional inviscid flow in which

combustion was modeled by constant volume heat addition. Multidimensional flow

field calculation in internal combustion engine have also been reported in others

work such as Gupta et al., 1980 and Diwakar et al., 1981. Carpenter and Ramos

(1984) used two equations (k/ε) model of turbulence and axis symmetric two di-

mensional mass, momentum and energy balance equation to numerically study

turbulent flow fields in a four stroke homogeneous charge SI engine. Spark was

modeled as a constant energy source. A single one step irreversible propane

oxidation reaction was used to model combustion. Similar model with slight mod-

ification are still commonly used. Dinler and Yucel (2010) also used a similar k-ε

turbulence model to study the effect of air to fuel ratio on combustion duration.

11

They also used just a single irreversible reaction to model combustion.

Thermodynamic models are derived using the first law of thermodynamics for

an open system. In such models, the combustion is usually modeled using two

different approachs. In the first approach, the Wiebe function, or the cosine burn

rate (Heywood, 1988), is used to empirically compute the burning rate. While in the

second approach, a mathematical model of the turbulent flame propagation (Hey-

wood, 1988; Bayraktar and Durgun, 2003), is used to estimate the mass burning

rate.

The other common form of modelling is called the mean value modeling (MVM).

This model is intermediate between large cycle simulation model and simple phe-

nomenological transfer function models. This method predict the mean value of

gross external engine variables (such as crank shaft speed, engine output torque)

and the gross internal engine variables (thermal and volumetric efficiency) dynam-

ically in time. As such they are computationally less expensive and are often used

for control application. Hendricks and Sorenson (1990) described MVM model con-

sisting of three differential equation to model SI engine. The model was validated

with steady state experiments.

1.4.2 Three-way catalytic converter modeling

A lot of work has been published on TWC, or monolith modeling, including dif-

ferent levels of detail. Voltz et al (1973) developed the global kinetic mechanism for

CO oxidation on platinum catalysts. Most of the modeling work since then involv-

ing global reactions, use the kinetic expression form as proposed by Voltz. Oh and

Cavendish, (1982) studied the response to step change in feed stream temperature

on catalytic monoliths transient. They used four global kinetic reaction mechanism

and a two phase model to simulate TWC performance. A pseudo-steady state as-

sumption was used for solid phase concentration neglecting washcoat diffusional

effects. Siemund et al. (1996) used four global reactions and compared the model

12

with experimental work. They used similar rate kinetics equation as used by Oh

and Cavendish, (1982) but also included NO reduction. They used quasi-steady

state assumption for mass balance and gas phase energy balance and included

the transient term for only solid energy balance. Dubien et al., (1998) extended the

kinetic model to include water gas shift and steam reforming reactions, compris-

ing a total of nine global reactions. Total hydrocarbons were split as fast and slow

burning hydrocarbons. Pontikakis et al., (2004) included the global reactions for

ceria kinetic and used the same mathematical model as in Siemund et al. (1996).

1.5 Outline of Thesis

The main objective of this work is to develop a fundamental based low-dimensional

model of IC engine and three-way catalytic converter that can be used for opti-

mization and control. In the first chapter a general introduction of the problem, the

legislative requirements and how it spurred the growth of the automobile industry

is discussed. In the second chapter a fundamental based low dimensional model

of the SI internal combustion engine is developed. This model is used to predict

combustion characteristics under various operating conditions. In chapter three,

the extension of the low dimensional model for the HCCI engine is discussed. In

chapter four the development of a low-dimensional model for a three-way catalytic

converter is discussed. A detailed kinetic is used to simulate catalyst performance

under varied operating conditions. In chapter five, a simplified kinetic model is pre-

sented to study the oxygen storage dynamics in TWC. In chapter six, the spatial

temporal dynamics in a TWC are studied using the simplified kinetics model. In

chapter seven, the results are summarized and the recommendations for future

work are provided.

13

Part I

In-cyliner Combustion Modeling

14

Chapter 2 Spark Ignited IC Engine Combustion Mod-

eling

In this work, we develop a four-mode low-dimensional model for the in-cylinder

combustion process in an internal combustion engine. The lumped parameter ordi-

nary differential equation model is based on two mixing times that capture the reac-

tant diffusional limitations inside the cylinder and mixing limitations caused by the

input and exit stream distribution. For given fuel inlet conditions, the model predicts

the exhaust composition of regulated gases (total unburned HC’s, CO, and NOx)

as well as the in-cylinder pressure and temperature. The model is able to cap-

ture the qualitative trends observed with change in fuel composition (gasoline and

ethanol blending), air/fuel ratio, spark timing, engine load and speed. The results

show good qualitative and fair quantitative agreement with the experimental results

published in the literature and demonstrate the possibility of such low-dimensional

model for real-time control. Improvements and extensions to the model are dis-

cussed.

2.1 Introduction

The plethora of information that a combustion model provides can help under-

stand the complex sub-processes occurring in an internal combustion engine and

especially the various interdependencies between these processes. The combus-

tion process is of prime importance as it couples directly with the engine operating

characteristics, power and efficiency as well as emissions. Thus it is imperative

to have a good physics based model of combustion process in order to satisfy

the current trend toward simultaneously increasing fuel to wheels efficiency while

reducing emissions. The detailed computational fluid dynamics (CFD) based mod-

els, although good for physical understanding of the process, are not good for opti-

15

mization and parametric studies as they are computationally very expensive; while

the empirical zeroth order models need re-calibration with changes in operating

conditions.

Thus in this work we propose to extend the recently developed low-dimensional

model for combustion process that retains all the essential physics of the system

and is yet computationally very efficient so as to be solved in real time (Kumar,

et al. 2010). The low-dimensional model was derived by spatially averaging the

detailed three-dimensional convection-diffusion-reaction (CDR) model employing

the Lyapunov-Schmidt (LS) technique of classical bifurcation theory which retains

all the parameters of the original equations in the low-dimensional models, and all

the qualitative features subsequently (Bhattacharya et al 2004, Kumar et al.2010).

Several models have been developed in the literature to simulate the spark igni-

tion engine cycle (Heywood et al., 1988, Blumberg et al. 1979 and Verhelst et. al.,

2009). These can be broadly categorized as fluid dynamic models or thermody-

namic models, depending on whether the governing equation is derived from the

detailed fluid flow or by considering the thermodynamics laws. The fluid dynamics

based models are also popularly termed as ‘multi-dimensional models’ as they can

give the detailed solution including the spatial variations inside the cylinder. Here,

the governing equations are obtained by using the species, momentum and energy

balances resulting in partial differential equations in time and space which makes

these models computationally very demanding (in terms of memory and speed).

Thermodynamic models are developed using the first law of thermodynamics

together with mass balance (Heywood et. al., 1988 and Baraktar et al. 2003). As

spatial variation is not considered in these models, they form a set of ODE’s in

time only and are thus also called zero-dimensional models. The most commonly

used thermodynamic model is the ‘two-zone’ model, where the complete cylinder

is treated as a single zone for the entire engine cycle other than combustion during

16

which it is divided into two zones known as burned and unburned zones separated

by a thin ignition film. Before combustion, all the mass is assumed to be un-reacted

as unburned zone and after combustion the whole mass is treated as burned zone,

with two separate zones during combustion.

To model the combustion part, usually two different approaches are used. In

the first method, a pre-defined empirical mass burning relations like the cosine

burn rate or the Wiebe function is used (Heywood et. al., 1988). These relations

require combustion start time as well as combustion duration to be provided as

input. As these properties depend strongly on the engine operating conditions

(such as air/fuel ratio, fuel composition, etc.), the method has limitations in terms

of extension to different operating regimes. In the second approach, combustion is

modeled using a turbulent flame propagation model (Heywood et. al., 1988). How-

ever, ignition of the cylinder charge is not modeled, rather the start of combustion is

initialized by assuming instantaneous formation of the ignition kernel at or shortly

after the ignition timing (Verhelst et. al., 2009). Also, the flame propagation speed,

which is determined empirically, will be a function of the operating conditions.

The ‘zero-dimensional’ model lacks the effect of spatial variation while the mul-

tidimensional fluid dynamic model is computationally very expensive. In this work,

we develop a low-dimensional model for an IC engine cycle by using a spatially

averaged three-dimensional detailed convection-diffusion-reaction (CDR) model

(Bhattacharya et at., 2004). The derived model includes the relevant physics and

chemistry occurring at different times and length scales but is in the form of a few

ordinary differential equations so that it can be used for parametric studies and

real-time optimization and control. The combustion of gasoline is modeled by us-

ing the global reaction kinetics. Thus, the model does not require pre-specification

of combustion time as it can automatically predict ignition caused by the rise in

temperature after the spark discharge. The model also predicts the composition of

17

the exhaust gases and the effect of various design and operating variables on the

exhaust gas composition.

We describe the low-dimensional model in some detail in the next section. The

model is used to predict the influence of various operating variables on the exhaust

gas composition and the in-cylinder temperature and pressure. The predictions are

compared to available experimental data in the literature. We also discuss briefly

some possible extensions or further improvements to the model and how it may be

integrated with exhaust after-treatment models.

2.2 Model development

The spark ignition (SI) engine cycle consists of 4 consecutive steps: intake,

compression, power (combustion and expansion) and exhaust. It is an open sys-

tem with time dependent control volume (function of crank angle). The instanta-

neous volume of the cylinder V (t) as a function of crank angle is given by (Heywood

et. al., 1988),

V (t)

Vc= 1 +

1

2(rc − 1)

[R + (1− (cos θ(t)))−

(R2 −

(sin2 θ(t)

))1/2], (2.1)

where, Vc is the clearance volume, rc is the compression ratio, R is the ratio of

connecting rod length to crank radius and θ(t) is the crank angle at any time t.

Differentiating Eq. 2.1 w.r.t. time we get,

dV

dt= Vc

1

2(rc − 1)

sin θ +sin θ cos θ(

R2 − sin2 θ)1/2

Ω, (2.2)

where, Ω = dθdt

is the angular speed. In the present work, Ω is kept constant.

The combustion in SI engine is initiated by spark discharge which in turn raises the

temperature around the spark plug which ignites the gases leading to the flame

front propagation. This phenomenon of gas combustion can best be analyzed by

18

considering the reaction kinetics instead of using empirical mass burn relations.

However, the detailed chemical kinetics for gasoline combustion will involve more

than 500 different intermediate species with thousands of reactions as shown by

Curran et al.(2002). The global reaction kinetics used have been shown to be able

to capture the relevant trends quite accurately (Westbrook et al., 1981, Jones et.

al., 1988 and Marinov et. al., 1995). Gasoline is a complex chemical mixture of

several hundred hydrocarbons. For simplicity we represent gasoline as composed

of 80% fast burning hydrocarbon and 20% slow burning. As shown later, these two

lump representations are the simplest that can properly predict the exit unburned

hydrocarbon as well as the temperature maxima for slightly richer condition. The

peak temperature occurs at around stoichiometry for one lump while it shifts to a

richer side with two lumps which agrees with the trend reported in literature for

gasoline (Heywood et. al., 1988). Next, to model the combustion process, the

simplest approach will be to model the combustion cylinder as an ideal (perfectly

mixed) single compartment with uniform concentration and temperature throughout

the cylinder. While this simplest model may predict the fuel blending and stoichio-

metric effects (such as the NOx maximum at slightly leaner conditions) as well as

the in-cylinder temperature and pressure with reasonable accuracy, it gives errors

in predicting hydrocarbon conversion as it omits the importance of crevice effect,

which is considered one of the major reason for unburned hydrocarbon emissions

(Heywood et. al., 1988). Thus, the next simplest model, includes the crevice ef-

fect, the large difference in the temperature of the in-cylinder gases and the outer

wall of the cylinder, and the mixing effects within the cylinder due to flow field and

molecular or turbulent diffusion.

In this work, we focus on this next simplest non-trivial model and assume that

the combustion chamber can be modeled as comprised of two control volumes

where the cylinder contents exchange species and energy with relatively very small

19

volume of the crevice (aggregated as single block), whose temperature can be

taken as the same as the wall temperature. Further, we do not assume infinitely

fast mixing in the combustion chamber, but introduce two mixing times that account

for the effect of finite mixing between reactants and products. Our model reduces

to the ideal combustion chamber model in the limit of these mixing times tending

to zero. The model formulation is discussed below.

2.2.1 Derivation of the low-dimensional in-cylinder combustion model

We extend the recently developed two-mode species balance model by Bhat-

tacharya et al. (2004) for constant volume system, to model the variable volume

IC engine combustion chamber. We refer to Appendix A for details and explain

here only the main concepts. The combustion cylinder is divided into N number of

smaller compartments interacting with each other [Remark: The number N could

be arbitrarily large but in practice, it is sufficient to use four to six compartments].

The detailed convection-diffusion-reaction (CDR) model is used for each compart-

ment followed by Lyapunov-Schmidt (LS) technique to develop a low-dimensional

model in two modes using the cup-mixing (or flow weighted) concentration Cm and

the volume averaged concentration 〈C〉. We first show the derivation for a simpler

case involving only one control volume and later extend the concept for the case of

two control volume, the main cylinder and crevice as used in SI engine modeling.

In recent work, Bhattacharya et al. (2004) have developed a low-dimensional

model for homogeneous stirred-tank reactors by averaging the full three dimen-

sional convection-diffusion-reaction (CDR) equation for the isothermal case. In the

first step of their approach, the reacting volume is divided into N cells (where N

can be arbitrarily large) and the Liapunov-Schmidt technique is used to coarse-

grain the CDR equation at the meso scale over each cell. In the second step, this

interacting cell model is further reduced to a two-mode model consisting of a sin-

gle differential equation and an algebraic equation relating the two concentration

20

modes (the cup-mixing concentration, Cm, and the volume averaged concentra-

tion, 〈C〉). For example, for cases in which the inlet and exit flow rates and reactor

volume are independent of time, the two-mode model may be written as

d 〈C〉dt′

+R (〈C〉) =1

τ

(Cinm − Cm

),

Cm − 〈C〉 =1

τ

(t′mix,2C

inm − t′mix,1Cm

),

where τ is the residence time, t′mix,1 is the overall mixing time in the tank, which

depends on the local variables (such as local velocity gradients, local diffusion

length, diffusivity) as well as reactor scale variables while t′mix,2 captures the effect

of nonuniform feeding of the reactants. When both mixing times are zero, Cm = 〈C〉

and the above model reduces to the classical ideal CSTR model. It was shown by

Bhattacharya et al. that when 0 <tmix,2τ 1, the above two-mode model has the

same qualitative features as the full CDR equation. In general, the mixing times

t′mix,1 and t′mix,2 depend on molecular properties as well as the flow field and reactor

geometry and other factors (such as the locations of inlet and exit streams, baffle

positions, stirrer speed, etc.) and may be expressed as

t′mix,1 = τmα1︸︷︷︸micromixing

+ τMα2︸︷︷︸macromixing

, (2.3)

t′mix,2 = τmα3︸︷︷︸micromixing

+ τMα4︸︷︷︸macromixing

, (2.4)

where τmis the characteristic local scale mixing (also called micromixing) time

present within the tank (which depends on the molecular properties such as species

diffusivities), while τM is the characteristic large scale (or macromixing) time in the

tank (which depends on the flow field and other macro variable mentioned above].

The numerical coefficients αi, i = 1, 2, 3, 4 depend on reactor geometry as well as

feed and exit stream distributions. It should be noted that it is the overall mixing

21

times that enter the final averaged model and not the individual (micro and macro)

contributions. However, based on the typical characteristic values of τmand τM ,

both micro and macro mixing contributions may be important for liquid phase reac-

tions while macromixing may be dominant for gas phase reactions.

We extend the above approach for the case of IC engines with the following

assumptions: (i) N interacting cells (ii) the exchange or circulation flow between

cells is much larger than inlet or exit flow at any time (iii) even though the total

volume varies with time, the relative volume fractions of the cells remain constant.

Based on operational conditions IC engine cycle is divided into 3 stages

1. only intake valve open

2. both valves closed

3. exit valve open

Case 1 and 3 represent semi-batch condition, with only inlet and exit flow re-

spectively. While case 2 corresponds to the batch operation condition. Model

development is first discussed for a general case and then special cases are dis-

cussed.

[Note: Bold letters represent vectors/matrices while scalers are written in nor-

mal font]. In a matrix form we can write mass balance for N number of interacting

perfectly mixed cells as,

d

dt(VR(t)C(t)) = Qin(t)Cin(t)−Qe(t)C(t) + Q(t)C(t)−VR(t)R(C), (2.5)

where,

VR =

Vi , i = j

0 , i 6= j

Q =

−Σnj=1,i Q

cij , i = j

Qcji , i 6= j

Qin =

Qini , i = j

0 , i 6= j

and

22

Qe =

Qexi , i = j

0 , i 6= j

,

where, VR ε RN×N and Vi is the volume of ith cell, C = [C1 C2.....]T ε RN×1 where

Ci is the spatially averaged concentration of each cell. Q, Qin and Qe ε RN×N , Qcij is

the circulation flowrate from cell i to cell j, Qini and Qex

i are the inlet and exit flow

from ith cell, respectively. The reaction rate vector R(C) ε RN×1. The total volume

of the reactor is given by, VR=∑

Vi, and the fractional volume are defined as αi=Vi

/VR. Assume that although total volume is a function of time, each cell volume

changes proportionately such that the relative volume fraction remains constant.

Thus VR(t)=VR(t) α. Rearranging Eq. 2.5 we get,

Q(t)C(t) =d

dt(VR(t)αC(t)) + VR(t)αR(C(t)) + Qe(t)C(t) −Qin(t)Cin(t). (2.6)

It may be noted that Q is a symmetric matrix with zero row and column sum. Thus

at any time t,

Q(t)y0 = 0, (2.7)

with y0 = [1 1 1 1 .....]T . Similarly adjoint eigenvector is given by

vT0Q(t) = 0 , (2.8)

with vT0

= [1 1 1 1 .....] . Let ε = 1/||Q(t)||, then for the limit ε −→ 0 (i.e. very fast

circulation flowrate), from Eqs. 2.6 and 2.7 we get C = 〈C〉y0 i.e. when circulation

flow rate is very high, all the cells are in perfect communication and the concentra-

tion is uniform, given by 〈C〉 . For small but finite ε (ε << 1), there exist a deviation

23

from equilibrium state, the concentration then is given by

C = 〈C〉y0 + C′, (2.9)

where,

C′ = εw1 + ε2w2 + ... (2.10)

Lets define the inner product as

〈x,y〉 ≡ yTαx, (2.11)

where x ε RN×1 and y ε RN×1 and α ε RN×N is the volume fractions. The volume

averaged concentration is defined as

〈C〉 = 〈C,v0〉 ≡ vT0αC =

∑ni=1 ViCi∑ni=1 Vi

. (2.12)

Taking the inner product of Eq. 2.9 with v0 and using result from Eq. 2.12 gives

the solution to⟨C′,v0⟩

as ⟨C′,v0

⟩= 0. (2.13)

Multiplying Eq. 2.6 by vT0

on LHS and using Eq. 2.8 we get

0 = vT0

(d

dt(VR(t)αC(t)) + VR(t)αR(C(t)) + Qe(t)C(t) −Qin(t)Cin(t)

). (2.14)

Now simplifying,

Term 1: VR(t)vT0αC(t) = VR(t) 〈C〉y0,

Term 2: vT0 α R(C) = vT0α(R( 〈C〉y0)+ R

′( 〈C〉y0)C′

)= R(〈C〉)+R′(〈C〉) 〈C′,v0〉+O (ε2) = R(〈C〉)+O (ε2) ,

Term 3: (v0)T Qe(t)C(t) = qe(t)Cm(t),

where, cup mixing concentration Cm ≡ vT0QeC∑Ni=1 Q

exi

=∑Ni=1 Q

exi Ci∑N

i=1Qexi

,

24

Term 4: (v0)T Qin(t)Cin(t) = qin(t)Cm,in(t),

where, Cm,in ≡ vT0QinC∑Ni=1Q

ini

=∑Ni=1 Q

ini Ci,in∑N

i=1Qini

,

Also qin(t) =∑

Qini and qe(t) =

∑Qexi is the total cumulative inlet and exit

flowrates respectively. Substituting above simplification in Eq. 2.14 we get

d

dt(VR 〈C〉) + VRR( 〈C〉 )− qin(t)Cm,in(t) + qe(t)Cm = 0. (2.15)

The above equation relates bulk measurable quantities like cup-mixing and volume

averaged concentration and total flow. For the case where ||Q(t)|| >> 1, i.e. very

fast circulation, the concentration will be uniform and equal to 〈C〉. Thus for zeroth

order, the model equation reduces to

d

dt(VR 〈C〉) + VRR( 〈C〉 )− qin(t)Cm,in(t) + qe(t) 〈C〉 = 0. (2.16)

Eq. 2.15 can be solved to get temporal evolution of average concentration within

the reactor provided there exist a closure relation, relating Cm and 〈C〉 . To obtain

that a local equation will be derived, substitute Eq. 2.9 in Eq. 2.6 and keep only

the leading order terms we get,

QC′=d

dt(VRα 〈C〉 )y0 + VRαR(〈C〉 )y0 + Qe(t) 〈C〉 y0 −Qin(t)Cin(t). (2.17)

Now if we assume that although the total volume is a function of time, volume

fractions α are constant, i.e., each cell is varying at constant rate. Then, we can

rewrite Eq. 2.17 as

QC′=

[d

dt(VR 〈C〉 ) + VRR(〈C〉 )

](αy0) +Qe(t) 〈C〉 y0 −Qin(t)Cin(t). (2.18)

Substituting Eq. 2.16 into Eq. 2.18 we get

25

QC′= [qin(t)Cm,in(t)− qe(t) 〈C〉 )] (αy0) +Qe(t) 〈C〉 y0 −Qin(t)Cin(t). (2.19)

Rearranging Eq. 2.19 we get,

C′= inv(Q)

[qin(t)αy0 −

Qin(t)Cin(t)

Cm,in(t)

]Cm,in(t)− inv(Q) [qe(t)α−Qe(t)] 〈C〉 y0.

(2.20)

It may be noted that matrix Q, having a zero eigenvalue, is not invertible. However,

the inverse can be defined using the constraint given by Eq. 2.13. From Eq. 2.9

Cm and 〈C〉 can be related by

Cm =vT0QeC

qe=〈C〉 vT0Qey0 + vT0QeC

′

qe, (2.21)

Cm = 〈C〉 +vT0QeC

′

qe(2.22)

Substituting the result obtained in Eq. 2.20 after regularization we get

Cm = 〈C〉 +

(vT0Qe

qe

)inv(Q)

[qin(t)αy0 −

Qin(t)Cin(t)

Cm,in(t)

]Cm,in(t) (2.23)

−(

vT0Qe

qe

)inv(Q) [qe(t)α−Qe(t)] 〈C〉 y0 ,

Cm = 〈C〉+ τmix,2Cm,in − τmix,1Cm, (2.24)

where τmix,1, τmix,2 are the dimensionless mixing time given by

26

τmix,2 =

(vT0Qe

qe

)inv(Q)

[qin(t)αy0 −

Qin(t)Cin(t)

Cm,in(t)

], (2.25)

and

τmix,1 =

(vT0Qe

qe

)inv(Q) [qe(t)α−Qe(t)] y0. (2.26)

Eqs. 2.25 and 2.26 can be solved together with Eq. 2.15 to obtain temporal

evolution of concentration within the reactor.

Special cases : IC engine

1. Intake stroke: Assuming no valve overlap, during an intake stroke only inlet

valve is open, which implies Qe = 0. Thus from Eq. 2.25-2.26, we get τmix,1 =

τmix,2 = 0.

2. Compression and power stroke: During this period of engine cycle, both

the valves are closed, there is no flow in or out of the system. Thus Qe = 0 and

τmix,1 = τmix,2 = 0.

3. Exhaust stroke: Only exhaust valve is open. As there is no inflow τmix,2 = 0

but τmix,1 6= 0. For the case where valve overlap takes place, during the overlap

period both mixing times will be non zero.

2.2.2 Species balance for in-cylinder combustion

The combustion cylinder is divided into two zones, the main combustion cylinder

and the crevices. The crevices are the small regions between the piston and the

wall where the unburned gases escape during compression and is one of the main

reasons for HC emission. Also, the crevice volume is assumed to be constant and

does not vary much with piston movement. With these assumptions and using

model Eq 2.16, the volume averaged species balance equation in the two-mode

27

form is given by

d(〈Cj〉)dt

=1

V

[F inj − Fj +

NR∑i=1

νijRi(〈C〉)V − 〈Cj〉dV

dt− Fj,cr

], (2.27)

d(Ccr,j)

dt=

1

Vcr

[Fj,cr +

NR∑i=1

νijRi(Ccr)Vcr

], (2.28)

Cm,j − 〈Cj〉 = tmix,2Cinm,j − tmix,1Cm,j , (2.29)

Fj,cr = Qcr(aCm,j − (1− a)Ccrj), (2.30)

where the suffix i and j stands for the reaction number and the gaseous compo-

nent, respectively. Here, NR is the total number of reactions, Cm,j and 〈Cj〉 is the

flow (velocity) weighted concentration and volume averaged concentration of the

j-th component, respectively. F inj and Fj are the molar flow rates in and out of the

cylinder, respectively. νij gives the stoichiometric coefficient defined in standard

notation as negative for reactant and positive for products. Ri(〈C〉) is the rate of

the ith reaction evaluated at the volume averaged concentrations and in-cylinder

temperature. Similarly, Ri(Ccr) represents rate of reaction evaluated at crevice

concentration and wall temperature condition. Eqs. 3.1 and 3.2 gives the overall

species balance for the j-th component within the cylinder and inside the crevice

region, respectively. The species balance accounts for the change in species con-

centration within the cylinder due to mass flow in and out, generation by chemical

reaction, volume change and crevice flow effect.

Eq. 3.3 represents the interaction between two averaged concentrations Cm,j

and 〈Cj〉 , expressed in terms of dimensionless mixing times tmix,1 and tmix,2 which

accounts for the non-uniformity in the cylinder. The time tmix,1 depends on the diffu-

28

sivities of the reactant species, engine speed, swirl ratio and the velocity gradients

in the cylinder, i.e. it captures the mixing limitations inside the cylinder. The mixing

time tmix,2 accounts for the feed distribution (pre-mixed or unmixed) effect as well

as the mixing between the feed and the products. The dependence of feed stream

distribution like the number of valves, flows through each valve etc. is captured

by the inlet cup mixing concentration(Cinm,j

). As discussed in the earlier section

and in more detail elsewhere (Bhattacharya et al., 2004), in the limit of both mixing

times vanishing, the spatial gradients become negligible and the model reduces to

the classical ideal or perfectly mixed (CSTR) model. When the mixing times are

small but finite, the LS procedure retains the same accuracy and relevant qualita-

tive features as the detailed CDR model (Bhattacharya et al., 2004). The important

point to note is that the conversion of the reactants (fuel) is determined by the flow

weighted concentration Cm, while the reaction (combustion) rates are evaluated at

the volume averaged concentration 〈C〉. If the feed (air and fuel) is pre-mixed and

there is no overlap between inlet valve and exit valve timing, the dimensionless

time tmix,2 becomes zero.

In Eq. 3.4, Qcr and Ccr are the exchange flow rate (between the main flow and

the crevice) and the concentration in the crevice region, respectively. The crevice

is modeled as an isolated zone within the cylinder with a total volume (Vcr) equal

to 3.5% of the clearance volume and characterized by very high surface/volume

ratio such that its temperature can be assumed to be the same as the cylinder wall

temperature. [Since the crevice volume is small, no distinction is made between the

cup-mixing and volume averaged concentrations in the crevice]. The parameter ‘a’

in Eq. 3.1 determines the direction of flow. When the in-cylinder pressure is higher

than the crevice pressure, the flow is into the crevice (a = 1) and when the flow

is out of the crevice and into the cylinder a = 0. The rate of flow into or out of the

crevice is modeled using flow through a valve given by

29

Qcr = Qcr,0

√|P − Pcrevice|, (2.31)

where Qcr,0 is the function of crevice area and flow drag coefficient. The average

pressure inside the combustion cylinder and the pressure within the crevice region

are given by P and Pcrevice respectively, which are obtained using an ideal gas law

as

P =

(Nc∑j

Cm,j

)RT (2.32)

and

Pcrevice =

(Nc∑j

Ccr,j

)RTwall (2.33)

In the above expressions R=8.314 J/K mol, is the universal gas constant and Nc is

the total number of gaseous components. The total inlet volumetric flow rate is the

cumulative sum of the air (Qairin ) and fuel flow rates (Qfuel

in ). The air flow rate can

be computed using the first principles based air path dynamic model (Franchek et.

al., 2006) as follows

Qairin = ηvol

Vd2

Ω

2π. (2.34)

Here, Vd is the engine displacement (Vd = (rc−1)Vc). The volumetric efficiency ηvol

of the induction process is given by

ηvol = Ek − 1

γ+rc −

(PambPman

)γ (rc − 1)

, (2.35)

where E is an experimentally determined quantity, γ is specific heat ratio, Pamb and

Pman are the ambient and inlet manifold pressures, respectively. The inlet concen-

tration of air can be computed using the ideal gas law at ambient temperature and

30

manifold pressure condition. For a given intake the valve throttle position i.e., for a

constant air flow-rate, the amount of fuel to be injected should decrease with an in-

crease in blending (for example, xe, the mole fraction of ethanol in the fuel) as well

as for an increase in desired λ (air/fuel

(air/fuel)s). Thus for a given air flow, the required

fuel flow rate can be computed as

Qfuelin =

1

λ (10.6− 7.6xe)Qairin . (2.36)

[The numerical factors 10.6 and 7.6 follow from the stoichiometry of fuel combus-

tion with oxygen]. Next, the concentration of gases entering the cylinder can be

calculated using the ideal gas law at manifold pressure and ambient temperature

condition and with the mole fraction chosen to satisfy the λ requirement.

The exit volumetric flow rate is modeled as the flow through an orifice and is

expressed as

Q = CdAth√

2 (P − Pout) /ρ, (2.37)

where Cd is a drag coefficient, Ath is the exit valve area, P is the cylinder pressure,

Pout is the downstream pressure (assumed constant) and ρ is the instantaneous

density of gaseous mixture inside the combustion chamber. No backflow of gases

from the exit manifold to the cylinder was allowed and thus for the duration when

the exit valve is open the flow is either out, given by Eq. 2.37, or is zero. The

combustion reactions are highly exothermic, so there exists a huge variation in

the temperature during a single combustion cycle. In the following section, we

formulate the energy balance equation to capture the thermal effects.

31

2.2.3 Derivation of energy balance for in-cylinder combustion

The First Law of Thermodynamics for an open system is given by

·Qheat −

·Ws +

Nc∑j=1

F inj H

inj −

Nc∑j=1

FjHj =∂E

∂t, (2.38)

where·Qheat is the net rate of flow of heat to the system,

·Ws is the rate of shaft work

(which includes the work done by piston movement) done by the system on the

surrounding, H inj and H j are the molar enthalpies of the jth gaseous component at

cylinder inlet and exit conditions, respectively. Here, E is the total internal energy

of the system given by

E =m∑j=1

NjEj, (2.39)

where Nj is the moles of jth component. The energy Ej is sum of internal

energy Uj and the kinetic energy and the potential energy and the total energy is

given by E. Neglecting changes in K.E and P.E we have,

E =Nc∑j=1

NjEj

=

Nc∑j=1

Nj (Hj − PVj)

=Nc∑j=1

Nj (Hj)− PNc∑j=1

NjVj

=

Nc∑j=1

Nj (Hj)− PVR , (2.40)

where Hj, Vj are the molar enthalpy and molar volume respectively. Differenti-

ating the above equation w.r.t. time, we get

32

dE

dt=

Nc∑j=1

Nj

(dHj

dt

)+

Nc∑j=1

Hj

(dNj

dt

)− d

dt(NTotRT ) , (2.41)

where NTot =Nc∑j=1

Nj.

Also from the mass balance equation we have

dNj

dt= F in

j − Fj +

NR∑i=1

νijRi(〈C〉)VR, (2.42)

substituting Eq. 2.42 in Eq. 2.41 we get,

dE

dt=

Nc∑j=1

NjCpj

(dT

dt

)−

Nc∑j=1

NjRdT

dt−RT

Nc∑j=1

dNj

dt+

Nc∑j=1

Hj

(F inj − Fj +

NR∑i=1

νijRi(〈C〉)VR

),

(2.43)

substituting above result in energy balance equation, we get

·Qheat −

.

WS +Nc∑j=1

F inj H

inj −

Nc∑j=1

FjHj =Nc∑j=1

NjCpj

(dT

dt

)−

Nc∑j=1

NjRdT

dt−RT

Nc∑j=1

dNj

dt

+Nc∑j=1

Hj

(F inj − Fj +

NR∑i=1

νijRi(〈C〉)VR

),(2.44)

which simplifies to

·Qheat −

.

WS +Nc∑j=1

F inj

(H inj −Hj

)−

NR∑i=1

Ri(〈C〉)VR ∗ (4HR,i)T +RTNc∑j=1

dNj

dt

=Nc∑j=1

NjCpj

(dT

dt

)−

Nc∑j=1

NjRdT

dt, (2.45)

where (4HR,i)T =∑Nc

j=1 νijHj. Assuming that the shaft work is equal to the work

33

done by the piston (PVR) we get,

dT

dt=

1

VR∑Nc

j=1 〈Cj〉(Cpj −R

) .

Qspark −.

Qcoolant − PVR +∑Nc

j=1 Finj

(H inj −Hj

)+(∑NR

i=1Ri(〈C〉)VR (−4HR,i)T

)+RT

∑Ncj=1

dNjdt

.(2.46)

The above equation gives the temporal variation of temperature inside the com-

bustion cylinder.

2.2.4 Energy balance for in-cylinder combustion

Eq. 2.46 along with the species balance Eq. 3.1 can be simplified to obtain the

energy balance equation as

dT

dt=

1(Nc∑j

〈Cj〉 V(Cpj −R

))[·Qspark −

·Qcoolant − P

·V +

Nc∑j=1

F inj

(H inj −H j

)(2.47)

+R T∑ d(〈Cj〉V )

dt+

NR∑i

Ri(〈C〉) V (−4HR,iT ) +Qcr (1− a)Nc∑j=1

Ccrj(Hcrj −H j

)].

In the above equation,·Qspark is the rate of energy added by the spark and is mod-

eled as an external energy source (Carpenter et. al., 1985) that adds sufficient

energy to ignite the system.·Qcoolant is the heat transferred from the bulk gas to

the wall and (4HR,i)T gives the heat of ith reaction at temperature T [Remark: For

exothermic reactions, (−4HR,i) is positive]. Cpj is the specific heat of gas at con-

stant pressure. Eqs. 3.1 and 3.5 describe the general species and energy balance

that are valid for the entire cycle of the IC engine. However, depending upon the

stage of the IC engine cycle, the different terms entering the species and energy

balance equations need to be properly assigned. For example, the inlet flow rate

will be non-zero only during the intake stroke. The physical properties of the gases

used in Eq. 3.5, are calculated assuming ideal gas behavior. The term contain-

34

ing·Qcoolant in Eq. 3.5 is determined using a pseudo steady state assumption as

explained in the next section.

2.2.5 Heat transfer

Heat is transferred from gases inside the combustion cylinder to the chamber

wall by convection and radiation and through the chamber wall by conduction. In

addition, there is convection from outside the cylinder wall. For a steady one-

dimensional heat flow through a wall, the following equations relate the heat flux(·q =

·Qcoolant/A

)and the temperatures.

Heat transfer from the bulk gas to the cylinder wall is given by,

qg = hc,g (T − Tw,g) + σε(T 4 − T 4

w,g

), (2.48)

where hc,g is gas side heat transfer coefficient, T and Tw,g is the average bulk gas

temperature and gas side wall temperature respectively, ε is the emissivity and σ

is the Stefan -Boltzman constant 5.67 × 10−8W/ (m2K4) . Heat transfer within the

cylinder wall is given as,

qw =k (Tw,g − Tw,c)

l, (2.49)

where Tw,c is the coolant side wall temperature, l is the wall thickness and k is

the thermal conductivity of wall. Heat transfer from the cylinder wall to the engine

coolant is given as,

·qC = hc,c (Tw,c − Tc) , (2.50)

where hc,c is coolant heat transfer coefficient and Tc is coolant temperature. As-

suming pseudo steady state and neglecting radiation, the above three equations

can be combined to give the wall heat flux as

·q =

(T − Tc)(1hc,g

+ lk

+ 1hc,g

) . (2.51)

35

Table 2.1: Woschni’s correlation parameters

Engine cycle period C1 C2

Gas exchange period 6.18 0

Compression period 2.28 0

Combustion and expansion period 2.28 3.24×10−3

To compute the convective heat transfer coefficient on the gas side, Woschni’s

correlation (Heywood et. al., 1988) is employed

hc,g = 3.26B−0.2p0.8T−0.55w0.8. (2.52)

Here, B is cylinder bore, p is the instantaneous cylinder pressure measured in

KPa and w is the average cylinder gas velocity given by

w = C1Sp + C2VdTrprVr

(p− pm) , (2.53)

where pr, Vr and Tr are reference pressure, volume and temperature, respectively.

In this work, the reference value was chosen to be the condition during the closing

of the intake valve. The parameters p and pm represents the in-cylinder pressure

and the motored pressure, respectively. The motored pressure is computed as-

suming that the cylinder pressure equilibrates with the inlet pressure at bottom

dead center (V = rcVc).

pm =

(rcVcV

)γpin (2.54)

The constants C1 and C2 for Eq. 2.53 are given in table 2.1

2.2.6 Fuel composition and global reaction kinetics models

As stated earlier, gasoline is a complex mixture of over 500 different hydrocar-

bons that have between 5 to 12 carbon atoms. To represent gasoline combustion

properly, one would need to use a lumped species model to correctly include the

characteristic combustion behavior of all the major contributing groups involved.

36

Table 2.2: Global kinetics for propane and ethanol combustion

no reaction orders of reaction ref

1 C8H18+172O2 −→ 3CO + 4H2O [C8H18]0.25 [O2]1.5 Westbrook et al., 1981

2 (CH2)n+nO2 −→ nCO + nH2O [(CH2)n]0.1 [O2]1.85(modified)

3f CO+12O2 → CO2 [CO] [H2O]0.5 [O2].25

Westbrook et al., 1981

3b CO2 → CO+12O2 [CO2] Westbrook et al., 1981

4f CO +H2O → CO2+H2 [CO] [H2O] Jones et al., 1988

4b CO2+H2 → CO +H2O [CO2] [H2]

5f N2+O2 → 2NO [O2]0.5 [N2] Heywood et al., 1988

5b 2NO → N2+O2 [NO]2 [O2]−0.5

6 H2+12O2 −→ H2O [H2] [O2]0.5 Marinov et. al., 1995

7 C2H5OH + 2O2−→ 2CO + 3H2O [C2H5OH]0.15 [O2]1.6 Westbrook et al., 1981

For example, if four-lumps are to be used then species from aliphatic, straight

chain, branched and cyclic compounds and aromatic compounds can be included.

Also if blended fuel is used, ethanol should also be included as another lump to

represent the fuel composition. For simplicity, in the present work we use a two-

lump model comprising of fast burn and a slow burning hydrocarbon group to model

gasoline. It was observed that a minimum of two lumps was required for gasoline

model to correctly predict phenomena such as the attainment of peak tempera-

ture for slightly richer condition . This preliminary two lump model can easily be

extended to multiple lumps and this will be considered in future work.

Similar to the fuel composition complexity, the combustion reaction kinetics can

also be very complicated. The detailed chemical kinetics for hydrocarbon com-

bustion involves more than 500 different intermediate species with thousands of

reactions (Westbrook et al., 1981), but they are not suitable for low-dimensional

modeling. Global kinetics becomes critical in that case as it can describe the sys-

tem behavior with relatively fewer equations, in terms of final species only. The

global reaction schemes shown in Table 2.2 were considered for the work.

In the simplified scheme considered in this work, reactions 1,2 and 7 represent

the partial combustion of fuel. Reaction 3 is further oxidation of CO to CO2. Reac-

tion 4 is known as a water-gas shift reaction and is the major path for production

37

Table 2.3: kinetic constants for propane and ethanol combustion (units in mol, cm3,

s, cal)

Reaction no A β Ea1 5.7×1011 0 300002 1.2×1011 300003f 3.98*1014 0 400003b 5× 108 0 400004f 2.75× 1012 0 200004b

5f 6× 1016 −0.5 1372815b

6 1.8× 1013 0 345007 1.8× 1012 0 30000

of H2 in combustion. Reaction 5 is representative of NOx formation while reaction

7 is ethanol combustion which occurs for ethanol blended gasoline fuel. Finally,

reaction 6 represents the combustion of hydrogen produced by the water-gas shift

reaction.

Before presenting the simulation results, we consider briefly the manifold dy-

namics. Shown in Figure 2.1 is the variation of manifold pressure with change in

throttle plate angle at constant rpm. The first principle model commonly used in

the literature (Franchek et. al. 2006) is used to simulate intake manifold dynamics.

As the throttle angle is increased, the intake valve cross-sectional area increases.

This increases the intake air flowrate and thus the manifold pressure goes up. As

can be observed from Figure 2.1, the manifold pressure attains a steady state very

quickly. Thus, a pseudo steady state assumption was applied and the intake man-

ifold pressure was assumed constant at 0.8 atm in further computations using the

low-dimensional combustion model.

The low-dimensional model for combustion as described above consists of ordi-

nary differential equations(ODE’s) representing the various species balances and

the energy balance equation. For the two-lump gasoline blends considered in this

work, the model equations consists of 10 species (C8H18, (CH2)n , O2, CO2, H2O,

38

N2, CO, NO, H2, C2H5OH) balances each for the crevice and combustion cylin-

der and an energy balance equation making a total of 21 ODE’s [Note: Since the

cup-mixing and volume averaged concentrations in the combustion chamber are

related linearly through the algebraic relation, Eq. 3.3, the former is not counted as

unknown]. It is assumed that the feed (air) enters the intake manifold at ambient

conditions, atmospheric pressure and 298K. The cylinder temperature is also ini-

tialized to the ambient value and thus the first few cycles in the simulations shown

in Figure 2.2 represents the cold start of the engine and then the system attains

a periodic steady state. It should be pointed out that the set of ODE’s are highly

stiff and hence require standard stiff-ODE solvers, which are readily available in

MATLAB or FORTRAN (such as ODE15s, ODE23s etc. in MATLAB or LSODE in

FORTRAN).

The model equations contain several design parameters and operating vari-

ables. Since our goal here is to validate the model by comparing the model predic-

tions with available experimental data and examine some key trends (and not an

exhaustive parametric study), we have fixed the values of some of these parame-

ters as shown in Table 2.4.

2.3 Simulation of IC engine behavior and emissions using the

low-dimensional model

Shown in Figures 2.2 and 2.3 are the variation of the combustion products con-

centrations, the temperature and pressure inside the combustion chamber under

stoichiometric (λ = 1) conditions, as predicted by the low-dimensional combustion

model. The following observations follow from these plots: (i) as expected, the IC

engine attains a periodic steady-state very quickly (within a fraction of a second),

even from a cold-start condition (ii) the unburned hydrocarbon concentration (Fig-

ure 2.2a) rises during intake and compression stage, reaching its maximum value

just before the combustion and then drops sharply after the ignition. There exists

39

Table 2.4: System parameters

System parameters Value

Bore diameter, B 7.67 cmClearance length 1.27 cmrpm 1500Compression ratio, rc 9

Crank length/Rod, R 4Cylinder wall thickness, l 1 cmCoolant thermal conductivity,hc,c 750 W/m2KWall thermal conductivity, k 54 W/mKAmbient temperature 298 KAmbient pressure, Pamb 1 atmExhaust manifold pressure, Pout 1.1 atm

Inlet manifold pressure, Pman 0.8 atm

Coolant temperature, Tc 373 KCd 0.1

E 0.6γ 1.33tmix,1 0.2tmix,2 0

a corresponding peak in temperature and pressure caused by the heat released

by the highly exothermic combustion reactions. These sharp gradients in temper-

ature, pressure and concentrations are the reason for the stiffness of the model.

(iii) From Figure 2.3, we can also conclude that among all currently regulated gas

emissions considered here, the NOx emission is most sensitive to temperature, es-

pecially to the peak temperature (as can be expected). While all other emissions

attain fairly steady state value after just the first cycle, the NOx emission shows the

most noticeable change before its value is stabilized after around 4-5 cycles, which

is the same amount of time for the temperature to achieve its steady state value.

Shown in Figure 2.4 is the temporal variation of pressure and temperature in the

cylinder over a single cycle, after a periodic steady-state is attained. The complete

cycle comprises of two revolution or 4π crank angle rotation. At θ = 00, the intake

valve opens and the the premixed air-fuel mixture enters the system. The feed

gases (air charge) are at a lower temperature (298K) compared to the gases left in

40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Tim e (s )

Inta

ke m

anif

old

pres

sure

(atm

)

I n cre as in gth rott le angle

Figure 2.1: Intake manifold pressure variation with throttle position as a function of

time.

the cylinder after the previous combustion cycle, thus we see a dip in temperature

initially. Similarly, for the pressure during the intake stroke, the volume is increasing

and so is the number of moles of gases in the cylinder. Thus, from Figure 2.4 it

appears that initially the volume increase dominates the moles added and there is

a drop in pressure but eventually a pseudo steady state is reached and pressure is

almost constant during the intake stroke. From θ = 1800, when the piston reaches

the bottom dead center (BDC) the compression stroke begins. The compres-

sion work done by the piston leads to an increase in the in-cylinder temperature

41

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2x 104

Time (s)

Unb

urne

d hy

droc

arbo

n (p

pm)

(a)

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5x 105

Time (s)

oxyg

en (

ppm

)

(b)

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

Time (s)

pres

sure

(at

m)

(c)

0 0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500

3000

3500

Time (s)

Tem

pera

ture

(K

)

(d)

Figure 2.2: In-cylinder temporal variation of (a) Total unburned hydrocarbon con-

centration, (b) Unburned oxygen concentration, (c) Pressure and (d) Temperature

with time

42

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14x 104

Time (s)

CO

2 (pp

m)

(a)

0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

6000

7000

8000

Time (s)

NO

con

c (p

pm)

(b)

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5x 104

Time (s)

CO

(pp

m)

(c)

0 0.2 0.4 0.6 0.8 10

200

400

600

800

1000

1200

1400

1600

Time (s)

Hyd

roge

n (p

pm)

(d)

Figure 2.3: Temporal variation of in-cylinder (a) CO2 concentration, (b) NOx con-

centration, (c) CO concentration and (d) Hydrogen concentration

and pressure. In Figure 2.4, the crank angle of 120 to 8.50 before top dead center

(BTDC) represent the time period during which the spark was activated. Then,

after a small delay combustion starts leading to a sharp rise in the temperature

and pressure. The model predicts a smaller delay after ignition, as compared to

a real system. This happens because unlike the present model where the whole

mass ignites at once in a real system ignition occurs through flame front propa-

gation introducing a delay to obtain peak temperature. This phenomenon can be

43

captured by using a multi-compartment type low–dimensional model and will be

considered in future work. It can be observed that the rise in temperature, due to

combustion, is much higher as compared to the temperature or pressure rise due

to energy provided by the spark. For the power stroke, we see a drop in pressure

and temperature as the cylinder volume increases. After θ = 5400, the exhaust

valves were opened and the gases were expelled out.

Shown in Figure 2.5 is the concentrations of various exhaust gases coming

out of the combustion chamber. As the exhaust valve opens periodically, we see

pulses of exhaust gas concentration at the exit. Although the reaction is almost

over by the time the exhaust valve opens, we still observe change in concentration

at the exit, because of the change in volume of the reactor and also because of

moles exiting the reactor. As shown in Figure 2.5, the model predicts an average

NOx concentration of around 920 ppm, CO around 0.25%, unburned hydrocarbon

around 320 ppm and around 0.25% oxygen in the exhaust. These numbers agree

qualitatively with results presented in the literature (Heck et. al., 2002), where

the relative ranges of exhaust concentrations suggested are: NOx 100-3000 ppm

and unburned hydrocarbons (HC) 500-1000 ppm. The unburned hydrocarbon pre-

dicted is on the lower side because we do not include the converstion of gasoline to

the intermediate hydrocarbons and also because of the absence of valve overlap

in this model, which contributes to a significant amount of unburned hydrocarbon.

Also the CO prediction is lower than the experimentally reported value of around 1-

2%, as the H:C ratio in the fuel considered is higher than that observed in gasoline

(' 1.865). The CO prediction decreased when the fuel was changed from octane

to propane, which has a relatively lower C:H ratio. A second possible reason may

be that, the CO oxidation kinetics used in this study was obtained under fuel lean

condition and may need to be modified to accommodate the fuel rich condition.

Shown in Figure 2.6 is the normalized reaction rates as a function of temperature

44

0 100 200 300 400 500 600 7000

5

10

15

20

25

Crank angle (deg)

Press

ure (

atm)

(a)

0 100 200 300 400 500 600 7000

500

1000

1500

2000

2500

Crank angle (deg)

Temp

eratu

re(K)

(b)

Figure 2.4: In-cylinder variation of (a) Pressure and (b) Temperature during a com-

plete cycle after a periodic state is attained

45

0 0.2 0.4 0.6 0.8 1330

340

350

360

370

380

390

400

410

Time (s)

HC

con

c (p

pm)

(a)

0 0.2 0.4 0.6 0.8 1

1500

2000

2500

3000

Time (s)

O2 c

onc

(ppm

)

(b)

0 0.2 0.4 0.6 0.8 11000

2000

3000

4000

5000

6000

Time (s)

CO

con

c (p

pm)

(d)

0 0.2 0.4 0.6 0.8 1500

1000

1500

2000

2500

3000

Time (s)

NO

con

c (p

pm)

(c)

Figure 2.5: Variation of exhaust gas concentrations with time (a) Unburned hydro-

carbon, (b) Exhaust oxygen, (c) Exhaust NOx and (d) Exhaust CO

during a single cycle. The rate of oxidation of fast burning hydrocarbon, CO and

N2 are shown in Figure 2.6a. As expected, the CO oxidation reaction starts after

the hydrocarbon is oxidized to CO. Around the peak temperature, the HC rate is

almost zero, implying that most of the reactants are converted by the time the peak

temperature is reached. It can be observed from the plot that the NOx formation

requires a very high temperature and reaches a maximum value when the system

temperature is maximum.

46

The reaction rate for the water gas shift reaction (WGS) and hydrogen oxidation

shows similar trends because of coupling of CO and H2 through the WGS reaction.

When WGS is high, more H2 will be produced leading to a higher reaction rate for

H2 oxidation and vice versa. Shown in Figure 2.7 is conversion as a function of

crank angle over a cycle. The fast burn hydrocarbon goes to almost a complete

conversion. Due to the presence of slow burning hydrocarbon the total conver-

sion is around 97% w.r.t. total hydrocarbon. The burn duration (xb=0 to xb ≈1) as

predicted by the model is around 450 which is in close agreement with the values

reported in the literature (Heywood et. al., 1988). This supports our assumption

that the global kinetic models used are sufficient to represent the detailed com-

plex combustion mechanism for predicting the regulated gas emissions. Unlike the

Wiebe function based model where burn duration and ignition delay are the prede-

fined parameters, the major advantage of using kinetics to represent combustion

is that these parameters are computed automatically based on species reactivity

and system temperature.

2.3.1 Effect of Air to fuel ratio

We now use the low-dimensional model to study the effect of air to fuel ra-

tio (λ) on the exhaust gas composition. Shown in Figure 2.8 is the variation of

peak temperature and pressure with change in λ. The peak temperature occurs

for a slightly richer condition which is in agreement with the trends observed in

the literature (Heywood et. al., 1988). The two peaks observed correspond to

the fast and slow burning component, respectively. It was observed that with one

lump for gasoline, the peak temperature occurred around stoichiometry. The peak

temperature will shift toward the richer side with an increase of the slow burning

component. This happens because although the overall λ may be rich, it becomes

close to stoichiometry just w.r.t. fast burn component, which can react with oxygen

first because of higher reactivity, thereby showing the maxima. Shown in Figure

47

1000 1500 2000 2500

0

0.2

0.4

0.6

0.8

1

Temperature (K)

Nor

mal

ized

reac

tion

rate

HC oxidationCO oxidationN2 oxidation

1000 1500 2000 2500

0

0.2

0.4

0.6

0.8

1

Temperature (K)

Nor

mal

ized

rea

ctio

n ra

te

slow burn HCWGSH2 oxidation

Figure 2.6: Normalized reaction rate as a function of temperature over a cycle

48

300 320 340 360 380 400 420 440 460 480

0

0.2

0.4

0.6

0.8

1

crank angle (degree)

conv

ersi

on

fast burn HCslow burn HCTotal HC

Figure 2.7: Hydrocarbon conversion with crank angle for λ=1

2.9 is the variation of average mole fractions (ppm) of exhaust gases with change

in the air-fuel ratio at constant rpm =1500 and the throttle plate position (constant

inlet pressure). The flow rate of air and fuel was kept constant at the same value

as stoichiometry, only the composition (mole fraction) of inlet feed was manipu-

lated to change λin. We can observe that the leaner mixture gives lower emissions

in terms of unburned hydrocarbon and carbon monoxide. If we make the mixture

too lean, the combustion quality becomes poor and eventually misfire will occur.

As expected, for the rich operating conditions, CO and HC emissions rise sharply.

The NOx concentration shows a maxima w.r.t. air to fuel ratio λ. This is observed

because the NOx formation is a strong function of temperature and oxygen con-

centration (nitrogen is always in excess). From the reaction rate vs temperature in

Figure 2.6, it is obvious that the NO formation starts after the HC oxidation. Thus

49

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.252300

2350

2400

2450

2500

2550

λ

Tem

pera

ture

(K)

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.2520

20.5

21

21.5

22

22.5

Pres

sure

(atm

)

Rich λ=1 Lean

Figure 2.8: Effect of change in air/fuel ratio on peak temperature and pressure

at very rich conditions, not enough oxygen is present to oxidize (or react with) the

nitrogen while at a very lean condition, the temperature is low for the NO formation

reaction to occur. The peak temperature occurs at a slight rich condition (Figure

2.8), however there is not enough oxygen present then for NO formation. Thus,

as the mixture is leaned out, the initial decrease in temperature is offset by the in-

crease in oxygen concentration and the peak for NO concentration is observed at a

slightly leaner condition, around λ = 1.05. The results observed agree qualitatively

with those reported in the literature (Heck et. al., 2002, Heywood et. al., 1988).

2.3.2 Effect of fuel blending

It is well known that the CO and HC concentration decreases with the ethanol

blending. However for NOx emissions, there is a slight ambiguity as seen in some

work in the literature; (Najafi et al. 2009 and Bayraktar H., 2005) has shown NOx

to increase with blending while other works, (Celik et. al., 2009; Koci et. al. 2009),

present a decreasing trend. The difference arises from whether the air fuel ratio

50

0.85 0.9 0.95 1 1.05 1.1 1.15 1.20

1000

2000

3000

4000

5000

6000

mol

e fra

ctio

n pp

m

0.85 0.9 0.95 1 1.05 1.1 1.15 1.20

200

400

600

800

1000

1200

NO

x ppm

Normalized air to fuel ratio

HC

CO

Rich

λ=1

Lean

(a)

0.7 0.8 0.9 1 1.1 1.2 1.30

2000

4000

6000

8000

mol

e fr

actio

n pp

m (H

C a

nd N

Ox)

Normalized air to fuel ratio0.7 0.8 0.9 1 1.1 1.2 1.3

0

2

4

6

8

CO

%

CO

HC

NOX

(b)

Figure 2.9: Variation of regulated exhaust gases with air fuel ratio (a) Predicted

from low-dimensional model and (b) Experimentally observed [12]

51

is kept constant while changing fuel properties or not. Both of the cases were

simulated with blended fuel i.e. one with constant air/fuel ratio and the other by just

keeping the total flow rate constant.

Shown in Figure 2.10, is the simulated results obtained with different ethanol-

gasoline blends. The total volumetric flowrate (air + fuel) was kept constant and

the fuel composition changed as the blending percentage increased from 0 to 100.

The (Air/Fuel)stoichiometry for ethanol is 8.95, while that for gasoline is 14.6. The fuel

mixture used to simulate gasoline in the present work has the corresponding ratio

value of 15.03. Thus, if pure gasoline is blended with ethanol, the amount of air

required for stoichiometric burning reduces. So as to compensate, the fuel flowrate

need to be increased and air flow reduced to obtain the same total flowrate. There-

fore, from Figure 2.10b, it is observed that the peak temperature is almost constant

(0.1% variation), even though fuel composition is changed from 0 to 100% ethanol.

This is because of the presence of more moles of fuel, which compensates for

low heat of reaction as the blending % increases. The model predicts the qual-

itative trends correctly. The NOx, HC and CO emissions decrease continuously

with increase in blending. From the plot of normalized reaction rate vs temperature

(Figure 2.11a), it can be observed that the ethanol reaction rate being higher than

other hydrocarbon is consumed first, even before peak temperature is reached.

Thus, CO is formed earlier in the ethanol blended fuel which consumes the oxygen

present to get oxidized. Hence, compared to pure gasoline combustion, nitrogen

has relatively less oxygen available when the required temperature for NO forma-

tion is reached. Figure 2.11 confirms that the CO oxidation rate has increased

and the N2 oxidation rate reduced, leading to a decrease in NOx formation and

decrease in unburned hydrocarbon and CO emission.

Shown in Figure 2.12 is the effect of blending on emissions for the case where

air and fuel flowrate was unaltered, while increasing the ethanol blending from 0

52

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

200

400

600

800

1000

mol

e fr

actio

n pp

m

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12000

2100

2200

2300

2400

2500

CO

ppm

Ethanol volume fraction

CO

HC

NO xNO x

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12475

2480

2485

Tem

pera

ture

(K)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 121.4

21.5

21.6

Pres

sure

(atm

)


(b)

Figure 2.10: (a) Impact of blending on emissions and (b) In-cylinder temperature

and pressure under constant air/fuel ratio of λ=1

53

1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0

0

0 .2

0 .4

0 .6

0 .8

1

Tem pera ture (K)

Nor

mal

ized

reac

tion

rate

C8H

8 oxid.

C2H

2 oxid.

CO oxid.

N2 oxid

C2H

5OH oxid

(a )

1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0

0

0 .2

0 .4

0 .6

0 .8

1

Tem pera ture (K)

Nor

mal

ized

reac

tion

rat

e

C8H

8 oxid.

C2H

2 oxid.

CO oxid.

N2 oxid

(b)

3 4 0 3 5 0 3 6 0 3 7 0 3 8 0 3 9 0 4 0 0 4 1 0 4 2 0 4 3 0

0

1

2

3

4

5

6

7

8

9

1 0x 1 0

5

cra nk a ng le(deg )

CO

oxi

datio

n ra

te

E 0E 5 0

(c)

3 4 0 3 5 0 3 6 0 3 7 0 3 8 0 3 9 0 4 0 00

1 0

2 0

3 0

4 0

5 0

C ra nk ang le (deg )

NO

x rea

ctio

n ra

te

E0E5 0

(d)

Figure 2.11: Comparison of reaction rate during a cycle for E0 and E50 (a) Nor-

malized reaction rate for 50% ethanol (vol% ) blended gasoline, (b) Normalized

reaction rate for gasoline, (c) CO oxidation rate and (d) NOx formation rate

54

to 100%. The simulated results agrees qualitatively with the experimental work

presented in the literature (Bayraktar H., 2005) where it is shown that HC and

CO decrease while NOx increases for the range 0 to 12% blending with ethanol.

The present model shows NOx increases up to about 10% ethanol blending, after

which the NOx starts to decrease. As the stoichiometric air/fuel ratio required for

ethanol is much lower than gasoline, the system becomes leaner if we increase

the blending ratio without changing the air and fuel flowrate. Thus, we observe

the same kind of trend as exhibited by leaner air-fuel mixtures. The CO and total

HC emission decreases with an increase in the blending percentage while NOx

shows a maxima as obtained with a change in air-fuel ratio. At very high blending,

there is a slight increase in HC concentration which may be due to misfire. The

peak temperature and pressure decrease with blending because of lower heat of

combustion of ethanol as compared to gasoline.

2.3.3 The effect of engine load and speed

Shown in Figure 2.13 is the simulation result of the effect of change in load on

engine emission, temperature and pressure at a constant rpm and other engine

parameters. The hydrocarbon reduces slightly while the NOx emission increases

with an increase in load. The results obtained match with the trends reported in

the literature (Heywood et. al., 1988). As the engine load increases, the amount

of air and fuel entering the cylinder increases (for constant λ). This leads to an

increase in in-cylinder temperature and pressure leading to an increase in NOx

and a drop in hydrocarbon emission. Simulations were also performed keeping

the load constant and varying rpm (Figure 2.14). The NOx emission was observed

to decrease while the HC emission increased. While this is in contrast to what is

generally observed, it may be because we are keeping other parameters constant

irrespective of the change in rpm. As the rpm increases, it can be noted from

Eq. 2.34 that the volumetric flow rate into the cylinder increases, and at the same

55

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

200

400

600

800

1000

1200

1400

mol

e fr

actio

n pp

m

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

500

1000

1500

2000

2500

3000

CO

ppm


COHC NO

x

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11000

1500

2000

2500

3000

Tem

pera

ture

(K)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 114

16

18

20

22

Pres

sure

(atm

)


P

T

(b)

Figure 2.12: Impact of blending on (a) Emissions and (b) In-cylinder temperature

and pressure at constant flowrate (λ goes leaner)

56

250 300 350 400 450 500 550 600 650 7001000

1500

2000

2500

3000

3500

4000

4500

imep, Kpa

conc

entr

atio

n (p

pm) NO x

HC

(b)

Figure 2.13: (a) Impact of change in load on engine emissions as predicted by

model, (b) Experimentally observed variation in emissions (Heywood, 1988), (c)

In-cylinder peak temperature variation with load and (d) Effect of load on in-cylinder

peak pressure

time an increase in rpm implies a reduced residence time. Thus, we see a drop in

temperature as rpm increases, and this leads to a decrease in NOx emission. For

higher rpm, however, the model predicts increase in temperature and decrease in

HC emission with rpm, which occurs because at a higher rpm the time available for

heat transfer per cycle by the coolant reduces, leading to an increase in in-cylinder

temperature which compensates for increase in air-fuel flowrate.

57

1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0 3 0 0 0 3 5 0 0 4 0 0 00

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0

NO

x ppm

rpm (rev /m in)1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0 3 0 0 0 3 5 0 0 4 0 0 0

2 5 0

3 5 0

4 5 0

5 5 0

6 5 0

HC

ppm

(a )

1000 1500 2000 2500 3000 3500 40002200

2300

2400

2500

2600

2700

rpm (rev/min)

Tem

pera

ture

(K)

1000 1500 2000 2500 3000 3500 400020

24

28

32

Pres

sure

(atm

)

(b)

1200 1400 1600 1800 2000 2200 24001000

1500

2000

2500

3000

3500

Speed rev/min

NO

x ppm

(c)

1000 1200 1400 1600 1800 20001500

2000

2500

3000

Speed rev/min

mol

e fr

acti

on p

pmNOxHC

(d)

Figure 2.14: Impact of engine speed on (a) NOx and HC emission as predicted by

model, (b) In-cylinder peak temperature and pressure, (c) Experimentally reported

NOx with change in speed (Celik, 2008) and (d) Experimentally reported HC and

NOx with change in engine speed (Heywood, 1988)

2.3.4 Sensitivity of the model

Sensitivity (∂Xi∂pj

) is defined as how a desired output (Xi : peak temperature ,

peak pressure, exit concentration of hydrocarbon, CO and NOx) varies with sys-

tem parameters (pj) like dimensionless mixing time (τmix,1), crevice volume, spark

timing and duration, compression ratio, feed composition, reaction kinetics etc. In

this section, we examine briefly the sensitivity of the cycle simulation results to the

values of selected parameters.

58

Sensitivity to mixing time

In the base case model considered in this work there is no valve overlap. Thus,

τmix,2=0, for all of the stages in the engine cycle and τmix,1 is non-zero during the

exhaust stroke and was assigned a constant value of 0.2. It can be seen from Eq.

2.24, shown in Appendix A, that cup mixing concentration is lower than volume

averaged concentration as τmix,1 is a positive parameter. Increasing the mixing

time τmix,1, implies that the concentration inside the cylinder is higher as compared

to the fluid leaving the system. This also agrees with what is expected intuitively

as it will take finite time for the reactant to mix uniformly. So near the exit port

as the gases leave the reactor, concentration should drop and will become lower

than the averaged concentration in the reactor. The effect of increasing τmix,1,

is similar to that of increasing the internal exhaust gas recirculation (EGR) since

higher mixing time implies more gases are left behind in the cylinder. The com-

bustible leftover gases act like a diluent (or inert), increasing the specific heat of

the system thereby reducing the temperature inside the cylinder. Since NO for-

mation is very sensitive to temperature change, the concentration of NOx drops as

the mixing time increases. The HC conversion is a weak function of τmix,1, and

reduces a little, while CO is almost unchanged. The peak temperature drops as

mixing time increases, because of the increase in EGR fraction, while the peak

pressure remains almost unchanged as a result of the two competing effects of a

decrease in the temperature and an increase in reactant moles. Figure 2.15 shows

the influence of mixing time τmix,1 on emissions, as well as, peak temperature and

pressure.

It may be noted that in the limit τmix,1 → 0 the model reduces to a classical

one-mode ideal combustion chamber model with Cm = 〈C〉. Though this one-

mode model can also be used to predict the basic trends qualitatively, it predicts

peak temperature and NOx emission that are higher than observed, and thus a

59

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

400

800

1200

1600

2000

2400

τmix1

mol

e fr

actio

n pp

m

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35500

1000

1500

2000

2500

2800

CO

ppm

CO

HC

NO x(a)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.352300

2400

2500

2600

2700

τmix1

Tem

pera

ture

(K)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.3510

15

20

25

Pres

sure

(atm

)

Tmax

Pmax

(b)

Figure 2.15: Influence of in-cylinder dimensionless mixing time on (a) Emissions

and (b) In-cylinder tempeature and pressure

60

two mode model is required for better prediction. The extension of the model to

include the valve overlap case, where τmix,2 6= 0 will be examined in future work.

Sensitivity to crevice volume

The crevice is one of the main reasons for unburned hydrocarbon, other reason

being wall quenching and incomplete combustion (Heywood et. al., 1988). Due

to its high surface to volume ratio the temperature in the crevice is close to the

wall temperature, which is much cooler as compared to the gases in the reactor.

During the compression stroke and combustion period, when the reactor pressure

is high, some of the gases escape into the crevice avoiding primary combustion.

Increasing the crevice volume increases its capacity to store unburned gases, thus

leading to an increase in unburned hydrocarbon emission. By trapping some of

unburned hydrocarbon it also reduces the peak temperature and leads to a small

drop in NOx and CO emissions. Shown in Figure 2.16, is the sensitivity of model

prediction as the crevice volume is increased from 0 to 5% of the clearance volume.

Similarly, reducing the crevice flow rate coefficient (Qcr,0), reduces the unburned

hydrocarbon at exit.

Sensitivity to spark duration and timing

In our model the spark is activated between 12 to 8.5 degree BTDC with a rate

of 1.4×105 J/s which approximates to an energy input of 54.76 J for an engine

rotating at 1500 rpm. The above value for the heat rate would be much lower if a

compartment type model is considered, because in that case, the spark is required

to ignite just the nearby gases and then the flame front will propagate. However, in

the present model, a higher energy input by the spark is required as all the mass

ignites at once. Still the amount of heat added by the spark is a small fraction

as compared to the amount of heat produced by combustion. Keeping the total

amount of energy added by the spark constant and reducing the duration and en-

ergy rate accordingly leads to an earlier start of ignition as compared to the slow

61

0 0 .5 1 1 .5 2 2 .5 3 3 .5 4 4 .5 53 8 0

3 9 0

4 0 0

4 1 0

4 2 0

4 3 0

4 4 0

4 5 0

4 6 0

crev ice v o lu m e (% o f clea ra n ce v o lu m e)

HC

ppm

(a)

0 1 2 3 4 5 6 7 8 9 1 02 6 1 0

2 6 2 0

2 6 3 0

2 6 4 0

2 6 5 0

2 6 6 0


Tem

pera

ture

(K)

(b)

0 0 .5 1 1 .5 2 2 .5 3 3 .5 4 4 .5 51 9 5 0

2 0 0 0

2 0 5 0

2 1 0 0

2 1 5 0

2 2 0 0

2 2 5 0

2 3 0 0

2 3 5 0


CO

ppm

(c )

0 0 .5 1 1 .5 2 2 .5 3 3 .5 4 4 .5 52 0 0 0

2 0 5 0

2 1 0 0

2 1 5 0

2 2 0 0

2 2 5 0

2 3 0 0

2 3 5 0

2 4 0 0


NO

x ppm

(d)

Figure 2.16: Impact of change in crevice volume on (a) Hydrocarbon emission, (b)

In-cylinder temperature, (c) CO emission and (d) NOx emission for τmix,1 =0 and

τmix,2 =0

62

spark, and the unburned hydrocarbon decreases and NOx emission increases. A

3.5 times shorter ignition duration resulted in around 10% reduction in hydrocar-

bon emission and around 5% increase in NOx. Spark timing also influences the

peak temperature and pressure. For example, by advancing the spark timing so

that spark is ignited 10 earlier resulted in a slight increase in temperature and led

to around 2.5% increase in NOx and simultaneous decrease of around 3-4% in

hydrocarbon emission. The CO emission was almost insensitive. Similarly, for the

spark retard i.e., ignition time moved closer to top dead center (TDC) lead to a

decrease in NOx and an increase in hydrocarbon emission. The trend observed

matches the one reported in the literature (Heywood et. al., 1988).

Sensitivity to reaction kinetics

As can be expected, emissions are a strong function of the combustion kinetics

used. When the gasoline is represented as a single lump, although the NOx emis-

sion, temperature and pressure trend could be predicted quite accurately using

the low dimensional model developed here; the hydrocarbon emission was under-

determined. Thus a two lump gasoline model was selected with 80% fast burning

and 20% slow burning. As stated in the introduction, the prediction of hydrocarbon

emission could be improved by using a five or six lump model of gasoline and a

more detailed kinetic model for combustion of each class of hydrocarbon. However,

as the goal of the study was to use a simple non-trivial model, we lump hydrocar-

bon as fast burning and slow burning and use the kinetics available in the literature

for prediction. Iso-octane being the typical representation for gasoline, was used

as the major component. As for slow burning, the combustion kinetics slower than

octane combustion was used. A 10% increase in the rate of slow burning com-

ponent results in a decrease of unburned hydrocarbon emission by approximately

10%, while it has very little effect on NOx emission, decreasing its value by just

around 1%. Small change (±5%) in rate of reaction for the fast burning component

63

does not influence emissions much.

Sensitivity to feed inlet temperature

Shown in Figure 2.17 is the effect of inlet temperature on emission and in-

cylinder temperature and pressure. The in-cylinder temperature increases, as ex-

pected, but the sensitivity is low. This is because the feed gas (air+fuel) enters the

cylinder at a temperature in the order of 300K while the gases left within the cylin-

der from an earlier combustion cycle are at a much higher temperature around

1000 K. Also since the inlet temperature is higher, keeping the heat supplied by

spark constant, leads to early ignition (ignition delay time reduces). The hydro-

carbon emission increases by around 3% for 10% increase in feed temperature

(compared to ambient conditions) and the NOx emission shows around 4% in-

crease. The peak in-cylinder pressure decreases with an increase in temperature

because at constant inlet pressure condition, an increase in inlet temperature leads

to a decrease in inlet concentration.

2.4 Extensions to the low-dimensional combustion model

The main goal of this work was to provide a first principles based low-dimensional

in-cylinder combustion model so that it may be coupled with an exhaust after-

treatment model and control schemes for real-time simulation and optimization of

the overall system. Thus, we have presented only the simplest non-trivial model

that retains the main qualitative features of the in-cylinder combustion process. The

model presented here can be extended to homogeneous charge compression igni-

tion (HCCI), gasoline direct injection (GDI) or variable valve timing (VVT) engines.

In addition, the model predictions can be improved (at the expense of increased

complexity and computational time) by relaxing the various assumptions. A few of

these extensions are discussed below in more detail.

64

290 300 310 320 330 340 350335

340

345

350

355

360

Tin K

Unb

urne

d hy

droc

arbo

n (p

pm)

(a)

290 300 310 320 330 340 350930

940

950

960

970

980

990

1000

1010

Tin

K

NO

x con

cent

ratio

n (p

pm)

(b)

290 300 310 320 330 340 3502484

2486

2488

2490

2492

2494

Tin

K

Tem

pera

ture

(K)

(c)

290 300 310 320 330 340 35019

19.5

20

20.5

21

21.5

22

Tin K

Pres

sure

(atm

)

(d)

Figure 2.17: Impact of change in inlet temperature on (a) Hydrocarbon emission,

(b) NOx emission, (c)In-cylinder temperature and (d) In-cylinder pressure

2.4.1 Extensions to the combustion chamber model

In the preliminary model studied here, after averaging the model is reduced to

a single compartment. Thus, we do not see the ignition delay, which will appear

if we extend the single compartment model to a multi-compartment model or use

multiple-temperature and concentration modes to account for the spatial variations.

This extension will also improve the model prediction for ignition delay.

In the present work, hydrocarbon oxidation kinetics considered here, takes con-

65

version of hydrocarbon directly to CO. But to capture the hydrocarbon and CO

emission more properly we need to extend the kinetics to include the reactions

involving the conversion of heavier hydrocarbons to intermediate lighter smaller

hydrocarbons. This will increase the number of ODE’s to be integrated and re-

quires the kinetics of oxidation of the different lumps, but can improve the CO and

HC emission predictions.

Also, in the present model, the valve overlap was not considered. The presence

of valve overlap is expected to reduce NOx emission and increase hydrocarbon

prediction. This is because, valve overlap leads to some flow of combustion gases

to intake manifold, leading to an effect similar to internal EGR and thus should

lead to a decrease in system temperature. To include this extension two more con-

trol volumes (exhaust and inlet manifold) will need to be clubbed with the present

combustion model.

2.4.2 Torque model

In the present work, the engine speed is assumed constant. However, in a real

system the engine speed is a function of mass air flow or the engine load. To

quantify this, we can use the torque balance given by (Saerens et. al., 2009),

IdΩ (t)

dt= Te(t)− Tl(t), (2.55)

where Te and Tl represents the effective torque (toque measured on the engine

shaft) and load torque respectively. The engine torque has been modeled in litera-

ture (Saerens et. al., 2009) as

Te(t) = ηe

.mfQv

2πΩ, (2.56)

where Qv is the heating value of gasoline and ηe is an experimentally determined

value to represent combustion and torque effective efficiency. The load torque can

66

be determined from the transmission and driveline model,

Fvvv = 2πΩTlηm, (2.57)

where Fv is the force acting on the wheel of the vehicle and vv is the vehicle velocity

and ηm is the power transmission efficiency,

vv = Rw2πΩ

iDiG, (2.58)

where Rw is the radius of the wheels, iD and iG are the reduction ratio of the

differential and the reduction ratio of the gearbox, respectively. The vehicle model

is

Fv = Mvdvvdt

+ SCxρav

2v

2+ frMvg cosφ+Mvg sinφ, (2.59)

where Mv is the mass of the vehicle, S is the frontal surface of the vehicle, Cx is

the drag coefficient of the vehicle, ρa is the density of air, fr is the friction coeffi-

cient and φ is the slope of the road. By combining the above torque model with

the combustion model, the emissions produced over a specific drive cycle can be

simulated. This will be pursued in future work.

2.4.3 Controller design

Most gasoline engines are controlled by throttling the air into the intake mani-

fold. The control over which the driver has direct control is the throttle angle plate.

We can implement our first principal based low-dimensional model in the current

controller scheme. The system input will be the throttle angle and speed based

on which the model will compute the required fuel flowrate and exhaust gas com-

position after combustion. This will give us λ, which can be used for closed loop

controller design.

67

2.5 Summary and Discussion

As stated in the introduction, the main goal of this article was to develop a fun-

damentals based low-dimensional in-cylinder combustion model that can predict

the composition of the regulated exhaust gases as a function of the various design

and operating variables. The low-dimensional model developed involves a total of

10 different species and consist of mass balance for each species in crevice and

cylinder and an energy balance for a total of 21 ODEs. The model is then veri-

fied for different operating conditions and was observed to agree qualitatively with

the results reported in the literature. The basic findings and assumptions can be

summarized as follows: (a) Results: (i) Out of all the regulated emissions, NOx for-

mation is most sensitive to peak temperature and occurs at a very high temperature

(above 1800 K) (ii) CO and hydrocarbon emission decreases with an increase in

air-fuel ratio (λ), while the NOx exhibits a maxima occurring for slightly leaner mix-

ture conditions (iii) Ethanol blending decreases CO and hydrocarbon emissions

while NOx emission may be higher or lower depending on the mode of operation,

(iv) Reducing the crevice volume can reduce the unburned hydrocarbon emissions,

(v)Advancing the spark timing will lead to an increase in NOx emissions; Assump-

tions: (a) All of the fuel injected is assumed to enter the cylinder and undergoes

combustion: This is the simplification of the real system in which not all of the fuel

evaporates and some stick to the inlet valves, while some leftover from the earlier

cycle may evaporate, adding purge, (b) Fuel and air mixture are treated as an ideal

gas, (c) Fuel and air gets premixed before entering the cylinder (d) Engine speed

is assumed constant, (e) There is no valve-overlap, and hence backflow of gases

from cylinder to the intake manifold.

We have demonstrated that the model presented here, though preliminary, is

the simplest non-trivial model that has the correct qualitative features. As dis-

cussed above, the quantitative predictions of the model can be improved by ex-

68

tending the model and relaxing some of the assumptions. Based on the sensitivity

results presented, the quantitative features of the model can also be fine tuned to

any specific IC engine design or specific mode of operation. For simplicity, this

work considered only the case of port injection with pre-mixed feed. However,

the present approach may be extended to include mixing limitations outside of the

cylinder (before the air-fuel mixture enters the in-take valve). The model can also

be extended to direct injection and other such operating conditions.

69

Chapter 3 Homogeneous Charge Compression Igni-

tion

3.1 Introduction

In a Homogeneous charge compression ignition (HCCI) system the air and fuel

are mixed together before entering the cylinder. The mixture is compressed until

the spontaneous ignition takes place. A traditional spark ignition is used when the

engine is started cold to generate heat within the cylinder and quickly heat up the

catalyst. Thus it combines features and advantage from both the SI engine (air

and fuel premixing) and the diesel engine (compression ignition). Also as the fuel

is distributed uniformly and thus in a relatively lower concentration as compared to

direct injection, the soot formation is not significant. Another advantage of homo-

geneous combustion is that it leads to lower combustion temperature compared

to localized burning by flame front propagation in SI engines and thus leads to

reduction in NOx formation.

The basic problem with the HCCI engines is in it’s difficulty to control the ignition

timing. If the ignition does not begin when the piston is positioned for power stroke,

the engine will not run properly and is one of the major deterring factor from the

widespread commercialization of HCCI engines. However, with the advancement

of technology, such as variable compression ratio, variable induction temperature,

variable exhaust gas percentage and variable valve actuation, the HCCI engines

is becoming a reality. General Motors (GM) demonstrated the combustion process

for the first time in two drivable concept vehicles, a 2007 Saturn Aura and Opel

Vectra. It is claimed that HCCI provides up to 15% fuel saving, while meeting

current emission standards (GM press release,2007).

The model described in the previous chapter is more appropriate for the case

of HCCI as compared to the SI engine. In the SI engine the combustion is initiated

70

by spark discharge and then flame front propagates, while in the HCCI engine a

uniform mixture of air and fuel is injected into the cylinder and then a spontaneous

homogeneous combustion occurs due to compression. Thus a homogenous model

with a detailed kinetic model is appropriate. In this chapter we will extend the model

to HCCI engines.

3.2 Model equation

In HCCI engines, the air and fuel are pre-mixed before being injected into the

cylinder. The volume averaged species balance equation in the two-mode form is

given by

d(〈Cj〉)dt

=1

V

[F inj − Fj +

NR∑i=1

νijRi(〈C〉)V − 〈Cj〉dV

dt− Fj,cr

], (3.1)

d(Ccr,j)

dt=

1

Vcr

[Fj,cr +

NR∑i=1

νijRi(Ccr)Vcr

], (3.2)

Cm,j − 〈Cj〉 = tmix,2Cinm,j − tmix,1Cm,j , (3.3)

Fj,cr = Qcr(aCm,j − (1− a)Ccrj). (3.4)

The energy balance equation is modified by omitting the heat added by spark.

The modified energy balance equation for HCCI system is given by,

dT

dt=

1(Nc∑j

〈Cj〉 V(Cpj −R

))[−·Qcoolant − P

·V +

Nc∑j=1

F inj

(H inj −H j

)(3.5)

+R T∑ d(〈Cj〉V )

dt+

NR∑i

Ri(〈C〉) V (−4HR,iT ) +Qcr (1− a)

Nc∑j=1

Ccrj(Hcrj −H j

)].

71

0 100 200 300 400 500 600 7000

10

20

30

Crank angle (deg)

Pres

sure

(at

m)

0 100 200 300 400 500 600 7001000

1500

2000

2500

3000

Crank angle (deg)

Tem

pera

ture

(K)

Figure 3.1: In-cylinder pressure and temperature for a HCCI engine

The heat loss by radiation is neglected, to get total heat loss through the engine

wall to coolant as

·q =

(T − Tc)(1hc,g

+ lk

+ 1hc,g

) . (3.6)

72

3.3 Simulation results

The same parameter as used in the SI engine simulation were used. The same

kinetic model was used as well. Although, to ensure auto ignition the compression

ratio was increased from 9 to 12. The spark is activated only for the cold start

duration (0.6s) after which the model is switched to HCCI mode. The energy bal-

ance equation was modified and the heat loss by radiation is neglected. Shown

in Fig 3.1, is the in-cylinder pressure and temperature for an HCCI engine for a

stoichiometric operation. Compared to the SI engine, with HCCI the temperature

rise is more gradual in the absence of point energy source as spark. To start with,

a spark is needed and after a few engine cycles, the spark can be switched off

and the rise in temperature during compression will be sufficient enough to cause

ignition. Shown in Fig. 3.2 is the in-cylinder CO2, NO, CO and H2 emission.

Shown in Fig 3.3 is the exit emissions with an HCCI engine. A clear trend in

HC and NO emission can be seen as the system is switched from SI to HCCI

mode at time t=0.6s. As the temperature is relatively lower in HCCI, it leads to

a reduction in NOx emission. However, the unburned hydrocarbon emission may

increase. As the NOx emission is relatively lower than compared to SI engines, HC

conversion can be improved by using a lean burn, which is also known to improve

fuel efficiency.

3.4 Conclusion

It has been demonstrated that the assumptions used in deriving the low-dimensional

model for SI engines are more closely valid for the HCCI engine. The model as-

sumes the air and fuel to be pre-mixed before injection and the non-uniformity in

the concentration within the cylinder is accounted by mixing times which is true for

HCCI engines. Also, in the SI engine simulation the spark was modeled as a con-

stant energy source, which was assumed to add energy uniformly. While in a real

73

0 1 2 30

5

10

x 104

Tim e (s)

CO

2 (ppm

)

0 1 2 30

2000

4000

6000

8000

Tim e (s)

NO

con

c (p

pm)

0 1 2 30

1

2

3

x 104

Tim e (s)

CO

(ppm

)

0 1 2 30

500

1000

Tim e (s)

Hyd

roge

n (p

pm)

Figure 3.2: In-cylinder emissions for a HCCI engine

74

1 2 320

10

0

10

20

30

Time (s)

HC

con

c (p

pm)

0 1 2 32000

4000

6000

8000

Time (s) C

O c

onc

(ppm

)

0 1 2 31000

1500

2000

2500

3000

Time (s)

NO

con

c (p

pm)

0 1 2 313.8

13.85

13.9

13.95

14

Time (s)

H2O

con

c (%

)

Figure 3.3: Simulated exit emissions for an HCCI engine

75

system the spark will only ignite the gas in the vicinity of the spark plug and then

the flame front propagates. The error in modeling the spark is not present in the

HCCI system, this increases the model accuracy and validity in using the averaged

concentration within the cylinder. The simulation results have shown the capability

of the model to simulate auto-ignition in an absence of spark.

76

Part 2

Three-way Catalytic Converter Modeling

77

Chapter 4 Low-dimensional Three-way Catalytic Con-

verter Modeling with Detailed Kinetics

In this chapter, we propose a low-dimensional model of the three-way catalytic

converter (TWC) that would be appropriate for real-time fueling control and TWC

diagnostics in automotive applications. The model reduction is achieved by ap-

proximating the transverse gradients using multiple concentration modes and the

concepts of internal and external mass transfer coefficients, spatial averaging over

the axial length and simplified chemistry by lumping the oxidants and the reduc-

tants. The model performance is tested and validated using data on actual vehicle

emissions resulting in good agreement.

4.1 Introduction

Automobile emissions such as carbon monoxide (CO), hydrocarbons (HC) and

nitrogen oxides (NOx) are regulated through the Clean Air Act. Shown in Table

4.1 is the LEV II emissions standards as followed by California Air Regulation

Board (CARB). LEV III, to be phased-in over 2014-2022 introduces even stricter

emissions standards. Apart from emissions, the 1990 amendment to the Clean

Air Act, also requires vehicles to have built-in On-Board Diagnostics (OBD) sys-

tem. The OBD is a computer based system designed to monitor the major engine

equipment used to measure and control the emissions. Having an optimal fuelling

controller for the three-way catalytic converter (TWC) utilizing a transient physics

based model for the TWC will play a major role in satisfying future low emission

and OBD guidelines.

The TWC is a reactor used to simultaneously oxidize CO and HC to CO2 and

H2O while reducing NOx to N2. The air-fuel mixture entering the TWC is often

78

Table 4.1: LEV II Emission standards for passenger cars and light duty vehicles

under 8500 lbs, g/mi [CEPA, 2011]

Category 50,000 miles/ 5 years 120,000 miles / 11 years

NMOG CO NOx PM HCHO NMOG CO NOx PM HCHO

LEV 0.075 3.4 0.05 - 0.015 0.09 4.2 0.07 0.01 0.018

ULEV 0.040 1.7 0.05 - 0.118 0.055 2.1 0.07 0.01 0.011

SULEV - - - - - 0.01 1.0 0.02 0.01 0.004

quantified using the normalized air to fuel ratio (A/F), defined as

λ =(A/F )actual

(A/F )stoichiometry.

Thus, λ > 1 corresponds to a (fuel) lean operation while λ < 1 corresponds to a

rich operation. It is well known that there exists a narrow zone around stoichiometry

(λ = 1) where the TWC efficiency is simultaneously maximum for all the major pol-

lutants (Heywood, 1988; Heck et al., 2009). Thus, gasoline engines are normally

controlled to operate around stoichiometry. However, in real world operating condi-

tions, slight excursions from the stoichiometric condition are often observed. Thus,

ceria stabilized with zirconia is added in the TWC to act as a buffer for oxygen stor-

age, among other reasons (Kaspar et al., 1999), and to help curb the breakthrough

of emissions.

The TWC is controlled based on catalyst monitor sensors (CMS) set points

(Fiengo et al., 2002, Makki et al., 2005), specifically universal exhaust gas oxy-

gen sensor (UEGO) and heated exhaust gas oxygen sensor (HEGO) set points.

An overview of oxygen sensor working principles can be found in Brailsford et al.

(1997); Riegel et al. (2002); Baker and Verbrugge (2004). Both UEGO and HEGO

sensors measure the air-to-fuel ratio (A/F). However, HEGO is a switch type oxy-

gen sensor with sharp transition around stoichiometry, UEGO can be used to mea-

sure A/F over a wider range. Shown in Fig. 4.1 is a block diagram representation

of a typical inner and outer loop TWC control strategy (Makki et al., 2005). A TWC

79

Figure 4.1: Schematic diagram of inner and outer loop control strategy

unit, usually consists of two bricks separated by a small space. In a partial vol-

ume catalyst, the HEGO sensor is located in between the two bricks, while in a

full volume catalyst the HEGO is placed after the second brick i.e., at the exit of

the TWC. The advantage of using a partial volume system is that it provides fuel-

ing control in a delayed system, i.e., even if there is a breakthrough detected after

brick one, the second brick will still reduce emissions. Due to the design consid-

eration and manufacturing cos a full volume catalyst is desirable. Typically for a

air/fuel control, UEGO is placed after the engine for a more accurate A/F mea-

surement, while HEGO is preferred to measure A/F after the TWC because of its

lower cost and faster response time. The inner loop controls the A/F to a set value

while the outer loop modifies the A/F reference to the inner loop to maintain the

desired HEGO set voltage (around 0.6-0.7 V, depending on design and calibration)

to achieve the desired catalyst efficiency. With this arrangement we rely on emis-

sions breakthroughs at the HEGO sensor to determine if the catalyst is saturated

(lean) or depleted (rich) of oxygen storage and as such it imposes a limitation on

the controller design.

If the true oxidation state of the catalyst can be measured or modeled, then a

80

model based approach to tighter control on breakthrough emissions would be fea-

sible. Emission control then would be less dependent on sensor location and thus

applicable for both partial and full volume catalyst systems. This can be achieved

using a physics based model for the TWC. In the literature, most of the models

for TWCs are represented by a set of partial differential equations (PDEs) in time

and space (Oh and Cavendish, 1982; Siemund et al., 1996; Auckenthaler et al.,

2004; Pontikakis et al., 2004; Joshi et al., 2009) and as such their discretization

results in several hundreds of ordinary differential equations (ODEs) depending

upon the number of grid points used for describing spatial variations and species

considered. Although such models provide a good description of the actual sys-

tem, they are computationally expensive for on-board implementation. On the other

hand, the over-simplified control based oxygen storage models (Muske et al., 2004;

Brandt et al., 1997) treat the TWC as a limited integrator and are usually empiri-

cally designed. Such models may not be accurate over a wide range of operating

conditions encountered in a real system and are inadequate for tight emissions

control.

In this work, we present a low-dimensional TWC model that would be appropri-

ate for real-time on-board fueling control and TWC diagnostics. The reduced order

model thus obtained retains the essential features and gives high fidelity with re-

spect to oxygen storage and is yet computationally efficient enough for implemen-

tation in the control algorithm. The model predicts the fractional oxygen storage

(FOS) level (or “bucket level”) and the total oxygen storage capacity (TOSC) (or

“bucket size”) of the TWC. These quantities directly impact the ability to regulate

the state of the catalyst and the prediction of aging resulting in accurate fueling

control and TWC diagnostics, respectively. The model performance is tested us-

ing actual vehicle emissions resulting in good agreement. The model development

and its validation are discussed in the following sections.

81

4.2 Model Development

The TWC is a monolith that comprises of multiple parallel channels (400-900

cpsi) with the catalyst loaded around the wall surface called washcoat. Shown in

Fig. 4.2 is a schematic representation of a close-coupled three-way catalytic con-

verter and the physical phenomena occurring over a single channel. The TWC can

be modeled as a three-dimensional system involving convection-diffusion and re-

action with variations in radial and axial directions. Assuming azimuthal symmetry,

reduces the system to a two-dimensional model. Using a low-dimensional method

and utilizing the effective mass transfer coefficient concepts, the two-dimensional

model can be further reduced to a one-dimensional model with variation along the

axial direction alone (Joshi et al., 2009). However, the above models are still rep-

resented by PDEs along the length and time, and as such are difficult for real-time

implementation. In this work, we further simplify the one-dimensional model by ax-

ially averaging to obtain a zero-dimensional model, represented by a set of ODEs.

The axially averaged model, referred in the literature as the ‘Short Monolith Model’

is known to have the same qualitative features of the full PDE model (Gupta and

Balakotaiah, 2001).

In this work, a single channel is assumed to be the representative of the entire

catalyst and can be calibrated to satisfy this assumption. Each channel is divided

into two phases: the fluid or bulk phase and the solid or washcoat region. The

feed gas enters the channel mainly by convection and is transported to the wall

through diffusion. We use internal and external mass transfer coefficient concepts

to capture the transport in the radial direction (Balakotaiah, 2008). The reactions

only occur in the washcoat where the catalyst is present and not in the bulk gas

phase. The product and unreacted species are transported back to bulk gas phase

through diffusion from where they are carried to the exhaust by convection. The

model equations are derived using species and energy balances for the fluid and

82

Figure 4.2: Three-way catalytic converter schematic

the solid phase (Joshi et al., 2009) and is commonly called two-phase model.

The species balance in the fluid phase (for gas phase species) is given by

∂Xfm

∂t= −〈u〉 ∂Xfm

∂x− kmoRΩ

(Xfm − 〈Xwc〉

). (4.1)

The species balance in the washcoat (for gas phase species) is

εw∂〈Xwc〉∂t

=1

CTotalνT r +

kmoδc

(Xfm − 〈Xwc〉

). (4.2)

The energy balance in the fluid phase is

ρfCpf∂Tf∂t

= −〈u〉 ρfCpf∂Tf∂x− h

RΩ

(Tf − Ts

), (4.3)

and the energy balance for the washcoat is

δwρwCpw∂Ts∂t

= δwkw∂2Ts∂x2

+ h(Tf − Ts

)+ δcr

T (−∆H) . (4.4)

with the boundary conditions given by

83

Xfm,j(x) = X0fm,j(x) @t = 0 (4.5)

〈Xwc,j(x)〉 =⟨X inwc,j(x)

⟩@t = 0 (4.6)

Tf (x) = T 0f (x) @t = 0 (4.7)

Ts(x) = T 0s (x) @t = 0 (4.8a)

Xfm,j(t) = X infm,j(t) @x = 0 (4.9)

Tf (x, t) = T inf (t) @x = 0 (4.10)

∂Ts∂x

= 0 @x = 0 (4.11)

∂Ts∂x

= 0 @x = L (4.12)

For control application the above model was simplified by averaging along the

axial direction. Let’s define the length averaged variables as

Xfm =1

L

L∫0

Xfmdx. (4.13)

Similar defination were used for other variables as 〈Xwc〉 , Tf and Ts. We as-

sume Xfm(L) = Xfm, i.e the exit concentration is assumed to be same as the

84

concentration within the reactor, a continuous stirred tank reactor (CSTR) assump-

tion. With the above assumption and using the definition (Eq.4.13) we integrate

Eq.4.1 from x = 0 to x = L and use boundary condition (Eq.4.5) to derive the

averaged species balance in the fluid phase as

dXfm

dt= −〈u〉

L

(Xfm −Xin

fm(t))− kmoRΩ

(Xfm − 〈Xwc〉) . (4.14)

It is assumed that the average rate of reaction is equal to the reaction evaluated

at average concentration. This assumption will become exact for linear kinetics.

Integrating Eq.4.2 from x = 0 to x = L we get the averaged species balance in the

washcoat (for gas phase species) as

εwd 〈Xwc〉dt

=1

CTotalνT r +

kmoδc

(Xfm − 〈Xwc〉) . (4.15)

The overall mass transfer coefficient matrix (kmo) is given by

k−1mo = k−1

me + k−1mi, (4.16)

where kme and kmi are the external and internal mass transfer coefficient matrices.

The averaged energy balance in the fluid phase is

ρfCpfdTfdt

= −〈u〉 ρfCpf

L

(Tf − T inf (t)

)− h

RΩ

(Tf − Ts) , (4.17)

and the average energy balance for the washcoat is

δwρwCpwdTsdt

= h (Tf − Ts) + δc

Nr∑i

ri (−∆Hi) . (4.18)

It may be noted that Eq.5.8 does not involve the conductivity terms. This is because

the term gets cancelled because of the boundary condition Eqs.4.11 and 4.12.

85

Here, δc is the washcoat thickness and δw represents the effective wall thickness

(defined as sum δs + δc, where δs is the half-thickness of wall) , ρw and Cpw are the

effective density and specific heat capacity, respectively, defined as δwρwCpw =

δcρcCpc + δsρsCps, where the subscript s and c represent the support and catalyst

washcoat, respectively.

The model developed in Joshi et at., (2009) did not include ceria kinetics. To

quantify the oxygen storage on ceria, we define the fractional oxidation state (FOS),

θ of ceria as,

θ =[Ce2O4]

[Ce2O4] + [Ce2O3]. (4.19)

It may be noted that the denominator in Eq.4.19 gives the total concentration of

ceria, which may be assumed constant. As each molecule of Ce2O3 stores half a

mole of oxygen, the total oxygen storage capacity (TOSC) will be half that of total

ceria capacity, or in other words, total ceria concentration is double of TOSC. It may

be noted here that by storage we mean the short term oxygen storage capacity or

the sites accessible for oxygen storage during the fast transients. From Eq. 4.19

and definition of TOSC, the rate of change of θ is proportional to the rate of change

of Ce2O4. Thus,

dθ

dt=

1

2TOSC(rstore − rrelease) , (4.20)

where rstore and rrelease are the rate of formation and concumption of Ce2O4, re-

spectively.

Eq. 5.1 represent the species balance in the fluid phase and accounts for the

change in species concentration in the fluid phase due to convection and mass

transfer to the washcoat. Here, the column vectors, Xfm and Xinfm(t) ∈ RN , repre-

sent the exit and inlet mole fractions of the species in the fluid phase, respectively.

The column vector, r ∈ RNr where each element ri ∀ i ∈ [1, Nr], represents the

rate of the ith reaction. The parameters N and Nr represent the total number

86

of gaseous species and reactions, respectively. The stoichiometric matrix, ν ε

RNr×N , is a matrix of stoichiometric numbers with rows representing the reaction

index while the columns represents species index. The average feed gas velocity,

〈u〉 , is computed using the measured air mass and known A/F ratio (or λ). The

total concentration (CTotal) is computed at the channel inlet using the ideal gas law

CTotal =P

RT inf (t). (4.21)

Here, P represents the total gas pressure, assumed constant as one atm. By ex-

pressing Eq. 5.1 and 5.2 in mole fractions, we inherently assume that CTotal is

constant over the length of the channel. This assumption is easily validated by

performing the total carbon mole balance at the catalyst inlet and exit. Shown in

Fig.4.3 is the comparison of the total carbon balance at inlet (solid red curve) and

exit (dotted (blue) curve) of the catalyst in terms of mole fractions, as observed ex-

perimentally. As the total carbon balance holds even in terms of mole fractions, we

can conclude that the total concentration is almost constant, or there is negligible

pressure drop along the length of the reactor. The total carbon is computed using

the relation ,

Total Carbon = [CO] + [CHy] + [CO2]

The gradients in the transverse direction are accounted by the use of internal and

external mass transfer coefficients, computed using the Sherwood number (Sh)

correlations. The external mass transfer coefficient matrix kme ∈ RN×N is defined

by

kme =Df Sh

4RΩ

. (4.22)

Here, Sh is a diagonal matrix given by, Sh =Sh∞ I, where I ∈ RN×N is the identity

matrix. The asymptotic value of Sh∞ depends on the flow geometry as well as

the kinetics. Here, we use a constant value corresponding to the fast reaction

87

1800 1850 1900 1950 2000 2050 2100

0.125

0.13

0.135

0.14

Time (s)

Mol

e fr

actio

n

inletexit

Figure 4.3: Total Carbon balance in terms of mole fractions at TWC inlet and exit

asymptote (ShT ) with a numerical value of 3.2 corresponding to a rounded square

shaped flow area (Bhattacharya et al., 2004). Assuming the gases to be diluted in

nitrogen, the gas phase diffusivity matrix, Df ∈ RN×N is also a diagonal matrix with

the ith diagonal element representing the diffusivity of the ith species in nitrogen.

To compute the diffusivity as a function of temperature, we use the Lennard-Jones

calculation for molecular diffusivity (Bird et al., 2002) and then correlate this as

Df = 1.4813 10−9T 1.68f [in m2s−1].

As both species (reductant and oxidant) have almost similar molecular mass, a

single value of diffusivity is used, i.e. Df = Df I. The concentration gradient within

the washcoat and diffusional effect is captured using the internal mass transfer

coefficient matrix (kmi) (Balakotaiah, 2008).

kmi =DsShiδc

. (4.23)

88

For the washcoat, because of the smaller pore size, the effective diffusivity will be

dominated by Knudsen diffusion. This is calculated as a function of the catalyst

temperature (Ts) as shown in Eq.4.24. As each species diffuses independent of

each other in the Knudsen regime, the washcoat diffusivity (Ds) matrix becomes a

diagonal matrix with the diagonal elements (Dsi) representing diffusivity of the ith

species, as follows:

Dsi =εwτ

97a

√TsMi

, (4.24)

where Mi is the molecular mass of the ith species. Here, εw is the washcoat poros-

ity, τ is the tortuosity, a is the mean pore size and Ds is in m2s−1. The molecular

mass of the reductant is taken as 28 g mol−1 while that for the oxidant is 32 g mol−1.

The internal Sherwood number matrix, Shi ∈ RN×N , is evaluated as a function of

Thiele matrix (Φ) as follows (Balakotaiah, 2008)

Shi = Shi,∞ + (I + ΛΦ)−1ΛΦ2. (4.25)

[Remark: The above result is an extension of the result derived by Balakotaiah

(2008) for linear kinetics to non-linear kinetics by replacing the matrix of rate con-

stants keff by the Jacobian of the rate vector evaluated at the washcoat-gas inter-

facial conditions]. For the case of a square channel with a rounded square flow

area, the internal asymptotic Sherwood matrix is given by, Shi,∞ = Shi,∞ I,where

Shi,∞ = 2.65 and the constant Λ = 0.58 (Joshi et al., 2009). The Thiele matrix, Φ2

∈ RN×N , is defined as

Φ2 = δ2c(Ds)

−1

(− 1

CTotal

d (R(X))

dX

)X=XS

= δ2c(Ds)

−1(−J). (4.26)

The Jacobian, J= 1CTotal

dR(X)dX

is the derivative of the rate vector w.r.t concentration,

evaluated at gas-washcoat interfacial concentrations. Here, R(X) = νT r(X), i.e.

89

R(X) ∈ RN and represents the overall reaction rate for each species. For non-

linear kinetics with multiple species, the Jacobian might become a non-diagonal

matrix. This happens because of the coupling between the species due to reac-

tions. Thus, it may be noted that although the external mass transfer coefficient

matrix is diagonal, the internal mass transfer coefficient matrix, in general, is a

non-diagonal matrix. Further, the Jacobian matrix J is evaluated at the solid-gas

interfacial concentrations given by the expression,

Xs = (kme + kmi)−1 (kmeXfm + kmi 〈Xwc〉) . (4.27)

For computational simplification, J can be evaluated at bulk (fluid phase) condi-

tions. From Eq. 6.11, it can be seen that in the limiting case of fast reactions

or thick washcoat or low values of washcoat diffusivity (i.e., ‖Φ‖ >> 1),Shi ap-

proaches Φ. The above procedure for calculating kmi is valid only if the matrix Φ2

has positive eigenvalues. For auto-catalytic kinetics or the case of reactant/product

inhibition where the rate goes through a maximum, the washcoat diffusion-reaction

problem may have multiple solutions. In such cases, Φ2 can have negative eigen-

values and the kmi can be multi-valued (as discussed by Gupta and Balakotaiah

(2001) for the analogous external mass transfer problem and Joshi et al. (2009)

for the internal mass transfer problem). In such cases, the above procedure needs

to be modified. The simplest modification is to ignore the second term in Eq. 6.11

and use only asymptotic values for the internal mass transfer coefficients. How-

ever, this approximation may not be accurate when boundary layers exist at the

gas-washcoat interface. For example, for the case of a single reaction, the overall

mass transfer coefficient may be expressed as

1

kmo=

δcDsShi

+4RΩ

DfSh∞. (4.28)

90

Since Sh∞/4 is approximately unity but Shi can have values above two, the impor-

tance of internal and external mass transfer depends on the relative values of RΩ

Df

and δcDs

. In the present work, these values at 700K are 2.04 and 120.6, respec-

tively. Hence, boundary layer exists within the washcoat and the use of constant

Shi is not justified. A second possible modification is to define an effective rate

constant for each reactant species as ki,eff =(− 1CTotal

Ri(X)Xi

)X=Xfm

in which case,

Φ2 becomes a diagonal matrix with the diagonal terms defined as

Φ2ii =

δ2c

Ds,i

ki,eff , (4.29)

where Ri(X) represent the net rate of formation of the ith species (thus, for re-

actants −Ri is positive quantity) and Xi is the corresponding mole fraction. In

this work, the kinetic parameter used showed isothermal multiplicity and hence the

second approach (diagonal approximation) as described by Eq. 6.12 is employed

to compute the internal mass transfer coefficients. Similar to the external mass

transfer coefficient, h in Eqs. (5.7-5.8) represents the heat transfer coefficient and

is computed using the Nusselt number (Nu) correlation, as follows

h =Nu kf4RΩ

. (4.30)

An asymptotic value of Nu=Nu∞ = 3.2 for rounded square flow area was used in

this work (Bhattacharya et al., 2004). Eqs. 5.1, 5.2, 5.7, 5.8 and 5.10 form an initial

value problem with initial conditions given by Eq. 5.11:

Xfm,j = X0fm,j @ t = 0 j ∈ [1, .., N ]

〈Xwc,j〉 = X0fm,j @ t = 0

Tf = T 0f @ t = 0

Ts = T 0s @ t = 0

(4.31)

91

Table 4.2: Numerical constants and parameters used in TWC simulation

Constants Value

a 10× 10−9 mRΩ 181× 10−6 mδc 30× 10−6m2δs 63.5× 10−6 mkf 0.0386 Wm−1K−1

Cpf 1068 Jkg−1KCpw 1000 Jkg−1Kρw 2000 kg m−3

εw 0.41τ 8Sh∞ 3.2Nu∞ 3.2Shi,∞ 2.65Λ 0.58

sl.no. Reaction ∆H(J/mol)1 CO + 0.5O2 −→ CO2 -2.83e5

2 H2 + 0.5O2 −→ H2O -2.42e5

3 C3H6 + 4.5O2 −→ 3CO2 + 3H2O -1.92e6

4 NO + CO −→ CO2 + 0.5N2 -3.73e5

5 NO +H2 −→ H2O +N2 -3.32e5

6 CO +H2O CO2 +H2 -4.1e4

7 C3H6 + 3H2O 3CO + 6H2 3.74e5

8 CO + Ce2O4 −→ Ce2O3 + CO2 -1.83e5

9 19C3H6 + Ce2O4 −→ Ce2O3 + 1

3CO2 + 1

3H2O -1.14e5

10 Ce2O3 + 0.5O2 −→ Ce2O4 -1e5

11 Ce2O3 +NO −→ Ce2O4 + 0.5N2 -1.9e5

Table 4.3: Global reaction in Three way catalytic converter

These are solved using a semi-implicit and L- stable method (with no oscillations).

4.3 Kinetic Model

The global kinetic equation used for the modeling is shown in Table 4.3. As

the catalyst activity varies with catalyst loading, age and precious material com-

position, the proposed model needs to be adapted for a particular catalyst. The

parameters Ai and Ei for each of the reactions are the tunable parameters. The

kinetic parameter can be tuned manually, but with multiple parameters this be-

comes a very cumbersome and ineffective approach. The rate kinetic parameter

92

estimation can be modeled as an optimization problem and various methods such

as conjugate gradient (Montreuil et al., 1992), genetic algorithms (Pontikakis and

Stamatelos, 2004 ; Rao et al., 2009) etc., have been proposed in the literature.

In this work, a combination of genetic algorithm(GA) optimization and Levenberg-

Marquardt method is applied. The advantage of using a GA method is it’s a heuris-

tic search which involves a set of solutions instead of a single solution limited by

local minima. It mimics the natural evolution process and is a very effective tool

for systems involving multiple variables. Here, we start with a set of solutions and

then through operators like crossover and mutation evolve our solution to satisfy

our desired objective function (Goldberg, 1998; Pontikakis and Stamatelos, 2004;

Kumar et al., 2008). The objective function is defined as the minimization of total

error given by the mean error in computing A and O2.

errorTotal =errA + errO2

2, (4.32)

where the error in each species is defined as the root mean square (RMSE) of the

difference between predicted and actual conversion

erri = RMSE (ηpred − ηactual). (4.33)

Here, ηpred and ηact are the predicted and actual conversions, respectively. The

conversion is computed using the relation

ηpred =Xactualin −Xpred

exit

Xactualin

, (4.34)

and

ηact =Xactualin −Xactual

exit

Xactualin

. (4.35)

93

Table 4.4: Brick dimensions and loading

Description Values

SS dimension (in) 4.16x4.16x3.09

Washcoat PGM ratio (Pt:Pd:Rh) 0:10:1

Loading(g/ft3) 200

CPI/wall thick 900/2.5

Here, Xactualin is the measured concentration at the TWC inlet, while Xactual

exit and

Xpredexit are the measured and model predicted concentration at the TWC exit.

The conversion difference was chosen over the concentration difference to

compute the error because the conversion is a normalized parameter and thus

will not be biased towards a higher concentration reactant. Also, as the conver-

sion varies between zero and one, the maximum and minimum error will also vary

between zero and one. For the GA optimization the fitness of each species was

defined as

fitnessi = 1− erri (4.36)

The results obtained from GA were used as an initial guess for the MATLAB in-built

function "FMINCON", which solve the non-linear constrained optimization prob-

lems.

4.3.1 Experimental Set-up

The experimental data was collected using a 3.5L-4V-V6 2008 model year Mer-

cury. For our experiment, we sampled the right bank only. Table 4.4 lists the prop-

erties of the first brick in the catalyst. As it is a commercial catalyst, the exact ceria

loading was not known but was estimated to be between 15 and 20 weight per-

cent of the washcoat. Shown in Fig. 6.10 is the schematic of the sensors installed

to collect experimental data for model validation. Horiba MEXA- 7000 analyzers

were used to collect data for CO, HC, NOx and O2 at the feed gas, mid bed and

tailpipe locations. Separate mass spectrometer based H2 sensors were also used

to measure H2 at feed gas and mid bed positions. A FTIR apparatus gave water

94

Figure 4.4: Sensors location schematic

and ammonia measurements. Five thermocouples were also installed to measure

temperature at feed gas, brick, mid bed and tailpipe locations as shown in Fig.

6.10.

The operating condition is shown in Fig 4.5, the left axis shows the vehicle

speed while the right axis is feed gas air-fuel ratio. The vehicle was run at three

different speeds starting from idle to 30 then to 60 mph and then slowing down to

0 eventually. This was done to be able to collect data with a different temperature

and space velocity. Also by increasing and decreasing speed we got data with

the same space velocity but a different temperature. At each speed couple of

lean rich cycle was performed with step duration of 120s. Such long steps were

taken to ensure the catalyst gets saturated. Some fast oscillatory steps were also

considered involving step size of 10s. Till 4500s the vehicle was operated under

an open loop condition while afterward a close loop was also measured for λ=1

95

580 1000 1500 2000 2500 3000 3500 4000 4500 50000

1

2

3

4

time s

λ in

1000 2000 3000 4000 50000

20

40

60

80

Veh

icle

spee

d m

ph

Figure 4.5: Operating condition in terms of feed gas air-fuel ratio and vehicle speed

with speed varying in steps from 0 to 30 to 60 and then back to zero mph.

4.4 Simulation results

Shown in Fig.4.6 is the comparision of the model predicted vs the experimen-

tally observed CO converison. The blue curve represents the measured midbed

CO conversion, while the red curve represents the model predicted conversion.

The model predicts a slightly lower conversion as compared to experimentally ob-

served. This is because of the lumping of the axial coordinates.

Shown in Fig.4.7 and 4.8 are the comparision of hydrocarbon and NOx for the

same operating condition as in Fig.4.6. It can be observed that HC unlike CO or

NOx shows a gradual decrease in conversion. The model predicts a sharp drop fol-

lowed by gradual decay. The kinetic parameter for steam reforming had the highest

sensitivity in predicting this behavior. It may be noted that in Fig.4.8 the measure

conversion is negative, which implies that the NOx is formed in TWC. This could

possibly be because of the mesurement error or the time misalignment between

the feed and exit measurements. NOx formation follows the Zeldovich mechanism

96

1600 1620 1640 1660 16800

0.2

0.4

0.6

0.8

1

CO

con

vers

ion

time(s)

calcexpm

Lean Rich

Figure 4.6: Comparision of model predicted and experimental CO conversion for

lean to rich step change at a constant vehicle speed of 30 mph

1600 1620 1640 1660 1680

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

HC c

onve

rsio

n

time(s)

calcexpm

Figure 4.7: Comparision of model predicted and experimental HC conversion for


97

1600 1620 1640 1660 16800.2

0

0.2

0.4

0.6

0.8

1

1.2

NO c

onve

rsio

n

time(s)

calcexpm

Figure 4.8: Comparision of model predicted and experimental NO conversion for


and requires high temperature for its formation, which is unlikely in TWC environ-

ment. Shown in Fig.4.9 is the fractional oxidation state (FOS) or the bucket level

of the catalyst. A FOS of one represents a completely oxidized catalyst while a

FOS of zero represent completely reduced catalyst. The transition time for the

catalyst to move from the completely oxidized state (FOS=1) to the completely re-

duced state (FOS=0) (of the order of 8 sec), manifest as breakthrough delay in CO

emission (similarly, oxidant breakthrough is delayed for transition from rich to lean).

Shown in Fig.4.10 is the feed gas (dotted black curve) and brick (dashed blue

curve) temperature transient for a lean to rich step experiment. It is interesting to

note that although the feed temperature was essentially constant over the duration

of step change, the brick temperatue increased indicating the reduction of ceria by

CO to be an exothermic step. The model was validated for other operating condi-

tions as well. Shown in Fig 4.11-4.15 are the comparision of CO, NOx, HC, CO2

and O2 emissions, respectively for a constant vehicle speed of 60 mph. The dotted

98

1630 1632 1634 1636 1638 1640 16420

0.2

0.4

0.6

0.8

1

time s

frac

tiona

l oxy

gen

cont

ent

Figure 4.9: Fractional oxidation state of the catalyst during lean to rich step change

1600 1620 1640 1660 1680480

490

500

510

520

530

Tem

pera

ture

0 C

Time(s)

brickfeedgas

Figure 4.10: Catalyst wall (brick) and feedgas temperature for a lean to rich step

change experiment

99

2550 2600 2650 27000

2000

4000

6000

8000

10000

12000

14000

time

CO

con

cent

ratio

n pp

mcalc(midbed)meas(midbed)meas(feedgas)

Figure 4.11: comparision of model predicted vs experimentally observed CO emis-

sion at constant vehicle speed of 60 mph

(black) curve represents the feed gas composition while dashed (blue) and solid

(red) curves represent the experimentally observed and model predicted midbed

emission respectively. The model was tested on other operating conditions and

equally good results were observed.

4.5 Conclusion

A low-dimensional model of TWC for control and diagnostics is developed. In

developing such a model, we have used two main approximations. First, we have

simplified the problem of multi-component diffusion and reaction in the washcoat

and approximated the transverse gradients in the gas phase and washcoat by us-

ing multiple concentration modes and overall mass transfer coefficients. Second,

we have simplified the axial variations in temperature and concentration by using

averaging over the axial length scale. The model includes the oxygen storage ef-

100

2550 2600 2650 27000

200

400

600

800

1000

1200

1400

Time(s)

NO

con

cent

ratio

n pp

m

calc(midbed)meas(midbed)meas(feedgas)

Figure 4.12: comparision of model predicted vs experimentally observed NO emis-


fect because of ceria kinetics. The model is validated with the experimental result

and good agreement is observed. The model developed can be extended to either

include the detailed micro-kinetics or to include the simplified kinetics depending

on the desired objective.

101

2550 2600 2650 27000

100

200

300

400

500

600

Time(s)

HC

con

cent

ratio

n pp

m


Figure 4.13: comparision of model predicted vs experimentally observed HC emis-


102

2550 2600 2650 27001.22

1.24

1.26

1.28

1.3

1.32

1.34

1.36

1.38x 105

Time(s)

CO

2 con

cent

ratio

n pp

m calc(midbed)meas(midbed)meas(feedgas)

Figure 4.14: comparision of model predicted vs experimentally observed CO2

emission at constant vehicle speed of 60 mph

103

2550 2600 2650 27000

2000

4000

6000

8000

10000

12000

14000

Time(s)

O2 c

once

ntra

tion

ppm


Figure 4.15: comparision of model predicted vs experimentally observed O2 emis-


104

Chapter 5 Low-dimensional Three-way Catalytic Con-

verter Modeling with Simplified Kinetics

In this chapter we propose a simplified kinetics to be used with low-dimensional

model of the three-way catalytic converter (TWC) for real-time fueling control and

TWC diagnostics in automotive applications. Combining the low-dimensional three

way catalytic converter (TWC) model as described in the previous chapter, the re-

duced order model consists of seven ordinary differential equations and captures

the essential features of a TWC providing estimates of the oxidant and reduc-

tant emissions, fractional oxidation state (FOS), and total oxygen storage capacity

(TOSC). The model performance is tested and validated using data on actual ve-

hicle emissions resulting in good agreement for both green and aged catalysts

including cold-start performance. We also propose a simple catalyst aging model

that can be used to update the oxygen storage capacity in real time so as to cap-

ture the change in the kinetic parameters with aging. Catalyst aging is accounted

via the update of a single scalar parameter in the model. The computational effi-

ciency and the ability of the model to predict FOS and TOSC make it a novel tool for

real-time fueling control to minimize emissions and diagnostics of catalyst aging.

5.1 Mathematical Model

The model derivation is presented in an earlier chapter and the model equations

are summarized below. The species balance in the fluid phase (for gas phase

species) is given by

dXfm

dt= −〈u〉

L

(Xfm −Xin

fm(t))− kmoRΩ

(Xfm − 〈Xwc〉) . (5.1)

105


εwd 〈Xwc〉dt

=1

CTotalνT r +

kmoδc

(Xfm − 〈Xwc〉) . (5.2)


k−1mo = k−1

me + k−1mi, (5.3)


The external mass transfer coefficient matrix kme ∈ RN×N is defined by

kme =Df Sh

4RΩ

. (5.4)

Here, Sh is a diagonal matrix given by, Sh =Sh∞ I, where I ∈ RN×N is the identity

matrix. The asymptotic value of Sh∞ with a numerical value of 3.2 corresponding

to rounded square channel is used. The internal Sherwood number matrix, Shi

∈ RN×N , is evaluated as a function of Thiele matrix (Φ) as follows (Balakotaiah,

2008)

Shi = Shi,∞ + (I + ΛΦ)−1ΛΦ2. (5.5)

An effective rate constant for each reactant species as ki,eff =(− 1CTotal

Ri(X)Xi

)X=Xfm

,

in which case, Φ2 is a diagonal matrix with the diagonal terms defined as

Φ2ii =

δ2c

Ds,i

ki,eff . (5.6)


ρfCpfdTfdt

= −〈u〉 ρfCpf

L

(Tf − T inf (t)

)− h

RΩ

(Tf − Ts) , (5.7)

106


δwρwCpwdTsdt

= h (Tf − Ts) + δc

Nr∑i

ri (−∆Hi) . (5.8)

The oxygen storage on ceria, we define the fractional oxidation state (FOS), θ of

ceria as,

dθ

dt=

1


or

dθ

dt=

1

2TOSC(r2 − r3) , (5.10)

where r2 and r3 are reaction rates as defined in Table 5.2. Eqs. 5.1, 5.2, 5.7, 5.8

and 5.10 form an initial value problem with initial conditions given by Eq. 5.11:

Xfm,j = X0fm,j @ t = 0 j ∈ [1, .., N ]

〈Xwc,j〉 = X0fm,j @ t = 0

Tf = T 0f @ t = 0

Ts = T 0s @ t = 0

(5.11)

These are solved using a semi-implicit and L- stable method (with no oscillations).

Thus, the model obtained consists of two species balance equations (Eqs. 5.1-

5.2) for each gaseous species, two energy balance equations (Eqs. 5.7-5.8) and a

balance equation for ceria (Eq. 5.10). For the kinetic model used in the work, the

final model involves two gaseous species and thus consists of seven ODEs only.

The constant parameters used in the simulation are shown in Table 6.2.

5.2 Kinetic Model

The kinetic behavior of a TWC has been described in the literature using ap-

proaches ranging from few global steps (Oh and Cavendish, 1982; Pontikakis et

al., 2004), on the order of 5-10 reactions, to several steps involving surface reaction

107


Constants Value



εw 0.41τ 8Sh∞ 3.2Nu∞ 3.2Shi,∞ 2.65Λ 0.58

mechanisms (Chatterjee et al., 2002) (on the order of 50 reactions). Depending on

the utility of the model or the level of details desired an appropriate kinetic model is

selected. In this work, we propose a simplified kinetic model to predict the oxygen

storage behavior of the catalyst for TWC control and diagnostics. As the desired

objective is to use the fractional oxidation state (FOS) and the total oxygen storage

capacity (TOSC) of the catalyst only and not the individual constituents species

emissions for the control design, the computational effort can be significantly re-

duced.

We combine all the chemical species into three different groups. We define the

net reducing agent ‘A’ as

[A] = (2 +y

2)[CHy] + [CO] + [H2] +

3

2[NH3], (5.12)

where [CHy] represents the general representation of hydrocarbon present in gaso-

line fuel. For the fuel used in this work, y=1.865. The net oxidizing group is defined

as

[Ox] = [O2] +1

2[NO]. (5.13)

108

From here on, unless specified otherwise, O2 will be used to represent total oxi-

dants. The oxidation products are defined as

[AO] = [CO2] + [H2O]. (5.14)

The constant coefficients appearing in Eqs.5.12-5.14 come from the stoichiomet-

ric number that is required for the complete combustion of the individual reactant

to the final products (CO2 , H2O and N2). Physically, the above equation implies

that one mole of CO is equivalent to one mole of H2 in terms of reducing capacity,

i.e., they require the same number of moles of oxygen for complete combustion.

The above model reduction is possible because most of the major reductants (CO,

HC, H2) show similar delay for breakthrough and oxygen is the common interacting

agent. A lumped kinetic model similar to the present one has also been used by

Auckenthaler (2005) in the modeling of oxygen storage in TWC. However, Aucken-

thaler’s model takes H2 as a separate lump and uses microkinetics.

It is should be noted that the above simplified kinetics model does not include

water either as a reductant or an oxidant, although water participates in the water

gas shift reaction and steam reforming. This is because the model assumes that

water does not contribute to a change in reductant concentration as those reac-

tions simply replace one reductant by an equivalent amount of other reductant. For

example, in the water-gas shift reaction, one mole of CO is replaced by one mole

of H2 keeping the total reductant concentration constant. [It should be pointed out

that both H2O and CO2 can oxidize ceria as shown by Möller et al. (2009). How-

ever, this occurs only at very high temperature (>800K) and large change in the

concentration of H2O and CO2 and absence of other reductants. For the conditions

considered in this work the variations in the H2O and CO2 concentrations are small

and hence we have neglected the oxidation of ceria by these species. Further, the

109

Table 5.2: Global reaction kineticssl.no. Reaction Reaction rate ( mol

m3.s) -∆H ( kJ

mol)

1 A+ 12O2 −→ AO r1 = ac

A1 exp(−E1RT

)XO2XA

Ts(1+Ka1XA)2 283

2 Ce2O3 + 12O2 −→ Ce2O4 r2=acA2 exp(−E2

RT)XO2(1− θ) TOSCgreen 100

3 A+ Ce2O4 −→ Ce2O3 + AO r3=acA3 exp(−E3

RT)XA θ TOSCgreen 183

Table 5.3: Kinetic parameters for a Pd/Rh based TWC with specifications shown in

Table 5sl.no. Reaction Ai (unit) Ei ( kJ

mol.K)

1 A+ 0.5O2 −→ AO 1.5× 1020 mol m−3s−1K 1052 Ce2O3 + 0.5O2 −→ Ce2O4 4.95× 1010 s−1 803 A+ Ce2O4 −→ Ce2O3 + AO 3.0× 107 s−1 75Absorption constant: Ka1 = Aa1 exp(−Ea/RT )Aa1 = 65.5 Ea = −7.99( kJ

mol K)

TOSCgreen = 200 molm3of washcoat

catalyst used by Möller et al., 2009 has platinum while our catalyst model assumes

no Pt].

Shown in Table 5.2 is the reaction kinetics used. Reaction one represents the

reductant oxidation while reactions two and three involve ceria oxidation and re-

duction, respectively. The rate expression for reductant oxidation is similar to that

used for CO oxidation by Voltz et al., (Voltz et al.,1973) while for ceria oxidation

and reduction the kinetics expression used is similar to kinetics commonly used for

NO2 adsorption on BaO for NOx trap (Bhatia et al., 2009). A similar rate expres-

sion form has also been used for CO oxidation and reduction by Pontikakis and

Stamatelos (2004).

The net rate of production of any species can be obtained by multiplying the

reaction rate with the corresponding stoichiometric numbers, i.e., R(X) = νT r(X).

110

Ordering the species as A,O2, AO,Ce2O3 and Ce2O4, we have

νT =

−1 0 −1

−12−1

20

1 0 1

0 −1 1

0 1 −1

and r =

[r1 r2 r3

]T

[Remark: In the present case, since the reactions involving the gas phase species

A and O2 are irreversible, we need to consider only the first two rows of νT in Eq.

5.2]. For example, the net rate of A production is -( r1+r3). Similarly, the net rate

of formation of Ce2O4 is r2 − r3, which is used to calculate the fractional oxidation

state (FOS) of ceria (θ), given by Eq. 4.19. The TOSC represents the total oxygen

storage capacity and is a function of aging. A green catalyst has a higher TOSC

value as compared to an aged one and this property can be used to determine

aging of the catalyst (TWC diagnostics). TOSC can be represented as

TOSC = acTOSCgreen, (5.15)

where ac is the normalized activity. For a green or a fresh catalyst ac = 1 and it

reduces as the catalyst ages. TOSCgreen is the maximum storage capacity for a

green catalyst. For a given catalyst age, it is assumed that TOSC remains con-

stant. It is assumed that the catalyst sintering, reduces both Pt/Pd/Rh and ceria

kinetics by a similar factor ac, as shown in Table 5.2. The heat of reaction values

are taken from the literature (Siemund et al., 1996; Yang et al., 2000; Rao et al.,

2009). It is interesting to note that using the heat of formation calculation the en-

thalpy change for ceria oxidation is 760 kJ/mol O2. However, Yang et al. (2000),

in their work based on a calorimetric method observed the heat of reaction to be

111

much smaller and we use the value reported of 200 kJ/mol of O2. Also we found

this value to be consistent with the experimental result observed in our work, where

ceria reduction (transition from Ce2O4 to Ce2O3) was found to be exothermic while

using thermodynamically calculated heat of reaction would predict it endothermic.

The heat of reaction 1 was taken to be the same as that observed for CO oxi-

dation. It may be noted that adding reactions 2 and 3 gives reaction 1; thus for

thermodynamic consistency, the heat of reaction for reaction 3 can be computed

by subtracting the heat of reaction 2 from that of reaction 1.

5.3 Experimental Validation of the Low-dimensional Model

An important step in parameter estimation is to choose the most representative

set of data for training. The entire data set cannot be used, because it will be

computationally too slow and also the events like deceleration fuel shut-off (DFSO)

(where lambda becomes large) will dominate the model error computation. Thus,

a smaller subset of 200-500s was selected. At a very high temperature, when the

reaction rate becomes too high the conversion is limited by the mass transfer and

as such the true kinetics cannot be estimated. Hence, the model was trained for

an intermediate temperature value (data collected from a vehicle running at idle

condition) and later the model is verified for other operating conditions. For any

given catalyst with an unknown age, parameter Ai, Ei and TOSC are the tuning

parameters. Then, for the same catalyst with different aging, the parameters Ai

assumes a fixed value, while just a single parameter ac needs to be updated.

Shown in Table 6.1 are the optimized kinetic parameters obtained using the GA

and Levenberg-Marquardt optimization method. For the green catalyst the TOSC

was estimated as 200 mol O2/m3 of the washcoat, while for an aged catalyst the

TOSC was estimated as 80 mol/m3 of the washcoat, respectively. A similar order of

magnitude (600 mol Ce/m3 of washcoat or TOSC=300 mol/m3), was earlier used by

Pontikakis et al., (2004) and Konstantas et al., (2007). Assuming a green catalyst

112

to be a reference state with ac = 1, from Eq. 5.15, the activity for aged catalyst

used is 0.4.

5.3.1 Modeling Results

A set of high and low frequency, lean-rich step change experiments were per-

formed at different mass flow rates. Of all the data collected, a small subset of data

is used for model validation. Shown in Fig.5.1 is the operating condition selected.

The solid (blue) curve represents the feed gas A/F as measured using UEGO sen-

sor, while the dotted (green) curve represents the feed gas inlet temperature. The

vehicle speed was maintained constant at 30 m.p.h. Shown in Fig.5.2 is the com-

parison of model predicted and experimentally observed total oxidant and reduc-

tant emission as a function of time. The dash-dotted (black) curves represents

the feed gas oxygen concentration while the dotted (blue) and solid (red) curves

represent the oxygen emission at the mid bed (after the first brick), as measured

by sensors and as predicted by the model, respectively. As expected, the oxygen

conversion is very low, around 25%, for the lean feed while it goes up to around

99% under rich phase. This is because for the extremely lean or rich feed, the

conversion observed is dictated by the limiting reagents concentration. There is

a delay in the breakthrough (difference in transition between dash-dot (black) and

dotted (blue) curve) of oxygen and reductant for the rich to lean and lean to rich

step changes, respectively. This happens because of the oxygen storage property

of ceria that occurs due to the transition of ceria from Ce3+ to Ce4+ and vice versa.

The model not only captures the steady state conversions properly but also pre-

dicts breakthrough accurately. In terms of fitness measure as defined in an earlier

chapter, the overall fitness was computed to be equal to 0.88. [The fitness value

ranges between zero and one with one being the best fit]. It may be noted, that

we define the error based on instantaneous emissions, which is more stringent,

compared to error computed based on the cumulative emissions since most of the

113

850 900 950 1000 1050 1100 1150 1200 1250 1300 13500.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

λ

850 900 950 1000 1050 1100 1150 1200 1250 1300 1350730

735

740

745

750

755

760

765

770

Time (s)

T in (K

)

Tinλ

Figure 5.1: Operating condition: Feed gas A/F (λ) and the inlet feed temperature

errors occur during the transient delay.

For control applications, the parameter of interest is the fractional oxidation

state (FOS) of the catalyst. An FOS of one represents a completely oxidized state

or that the "bucket level" is full while a FOS of zero represents a completely re-

duced state, i.e., the "bucket level" is empty. Shown in Fig. 5.3 is the FOS level

of catalyst for the lean-rich cycling experiment shown above. The dotted (green)

curve represents the feed gas UEGO response (λ) while the solid (blue) curve

gives the FOS of the catalyst. From t=875s to 910s, as the feed is lean, the cata-

lyst gets completely saturated with oxygen, i.e., FOS is one. Thereafter at t=910s,

following a step change from lean to rich, we see a sharp drop in the FOS level.

It may be noted that although the UEGO curve shows a step response, the FOS

curve takes a significant amount of time for the transition from completely oxidized

to completely reduced state ("emptying the bucket"). This time corresponds to the

delay observed in the reductant breakthrough. Similarly at t=1060 s, for the step

change from rich to lean feed the time taken for the complete oxidation ("filling the

bucket") is identical to the oxygen breakthrough delay. For the low frequency step

114

900 1000 1100 1200 13000

5000

10000

15000

Time(s)

O2 c

once

ntra

tion

(ppm

)

900 1000 1100 1200 13000

0.5

1

1.5

2

2.5

3x 104

Time(s)

redu

ctan

t con

cent

ratio

n (p

pm)


(a)

(b)

Figure 5.2: Comparison of model predicted vs experimentally observed (a) oxidant

emission and (b) reductant emission at vehicle speed of 30 mph for a green catalyst

115

900 1000 1100 1200 13000

0.2

0.4

0.6

0.8

11

Time (s)

FOS

900 1000 1100 1200 13000.96

0.98

1

1.02

1.04

1.06

1.08

1.11.1

λ

FOSλ

Figure 5.3: Fractional oxidation state of the catalyst

changes (t=875s to 1216s), the FOS saturates at one for the lean feed while it

saturates at zero for the rich feed. However, if the lean-rich oscillations are fast

enough the catalyst may never reach the saturation and it may oscillate at some

intermediate value as observed for the FOS value in the time span, t=1216s to

1308s. Additionally, if we compare Fig. 5.2 and 5.3, we can see that breakthrough

occurs when the catalyst is completely reduced or oxidized. This behavior is key

for controller design as it shows that if one can control the FOS, then emission

breakthrough can be monitored and controlled.

It may be noted that in an actual system the FOS will be a function of axial

coordinates, however for control applications a single axially averaged value as

predicted by the model is more meaningful for decision making. The FOS value

predicted by the simplified kinetic model was also compared with a more detailed

kinetic model and a good match was observed. To validate the robustness of the

model, the model performance was tested on various other operating conditions,

116

as well as on differently aged catalyst and is discussed in the following section.

5.3.2 Model Updating for Diagnostics

A big hurdle in the practical implementation of model based TWC control is the

requirement to update the model in real time. As the catalyst ages, the reactiv-

ity decreases because of the reduction of active surface area. Thus, the kinetic

parameters need to be updated over time. Typically, due to multiple species and

multi step reaction kinetics, several parameters need to be updated continuously

resulting in a challenging task for real time implementation. A major advantage of

the proposed modeling approach is that by just updating a single parameter ac, the

model can be used to predict emissions for differently aged catalyst.

Shown in Fig.5.4 are the comparison of the model predicted and experimen-

tally observed oxidant and reductant emissions at mid bed for an aged catalyst at

the same vehicle speed of 30 m.p.h. The storage capacity was found to have

decreased by 2.5 times as compared to the green catalyst. The same parameters

(activation energies and pre-exponential factors) as shown in Table 6.1 is used

with ac = 0.4. The model correctly predicts the breakthrough delays as well as

emissions. The overall fitness was 0.89. Comparing the reductant breakthrough

delay for lean to rich step, it can be observed that the delay has reduced from ap-

proximately 25s for the green catalyst to 10s for an aged catalyst, which is of the

same order of magnitude as the change in catalyst activity.

Similar tests were performed at other operating conditions like by increasing

and decreasing the vehicle speed, thereby changing the feed gas temperature.

Shown in Fig. 5.5 is the comparison for emission at idle vehicle speed. As ex-

pected, the delay observed in emission breakthrough increases as the gas flow

rate decreases and the model predicts it accurately. Interestingly, the model even

captures the small breakthrough as observed under a high frequency case with

idle speed from t= 1200s to 1300s, the overall fitness observed was 0.92.

117

1800 1850 1900 1950 2000 2050 21000

5000

10000

15000

Time(s)

O2 c

once

ntra

tion

(ppm

)calc(midbed)meas(midbed)meas(feedgas)

1800 1850 1900 1950 2000 2050 21000

0.5

1

1.5

2

2.5

3x 104

Time(s)

redu

ctan

t con

cent

ratio

n (p

pm)

(a)

(b)

Figure 5.4: Comparison of model predicted vs experimentally observed (a) oxidant

emission and (b) reductant emission for vehicle speed of 30 mph with an aged

catalyst

118

900 1000 1100 1200 13000

5000

10000

15000

Time(s)

O2 c

once

ntra

tion

(ppm

)

900 1000 1100 1200 13000

1

2

3

4x 104

Time(s)

redu

ctan

t con

cent

ratio

n (p

pm)


(a)

(b)

Figure 5.5: Comparison of model predicted vs experimental (a) oxidant and (b)

reductant emissions for an idle operation (speed=0 mph) with an aged catalyst

119

Table 5.4: Brick dimensions and loading of catalyst in FTP test

Description Values

SS dimension (in) 4.16x4.16x3.09

Washcoat PGM ratio (Pt:Pd:Rh) 0:69:1

Loading(g/ft3) 70

CPI/wall thick 900/2.5

Table 5.5: kinetic parameters for a threshold 70 g/ft3 Pd/Rh based TWC

sl.no. Reaction Ai (units) Ei ( kJMol.K

) TOSC( molm3 of washcoat

)

1 A+ 0.5O2 −→ AO 2.34× 1019 mol m−3s−1K 902 Ce2O3 + 0.5O2 −→ Ce2O4 1.16× 1011 s−1 80 103 A+ Ce2O4 −→ Ce2O3 + AO 3.10× 107 s−1 75

5.3.3 Model Validation on FTP Cycle

The model performance was also tested on the standard FTP cycle to evaluate

its cold start performance. Since the catalyst used in the tests had a different pre-

cious metal loading as shown in Table 5.4, the parameters were tuned using the

values shown in Table 6.1 as an initial guess. Shown in Table 5.5 are the tuned

parameters. The catalyst used was severely aged, and is classified as threshold

catalyst, thus the oxygen storage capacity was significantly reduced as shown by

the TOSC value in Table 5.5. Shown in Fig.5.6 are the comparisons of model pre-

dicted and experimentally observed oxidant and reductant emissions over the first

300s of a FTP cycle, respectively. The dash-dot (black) curve represents the feed

gas composition while the dotted (blue) and solid (red) curves represent the mea-

sured and model predicted emissions at TWC exit (after 1st brick). It may be noted

that for the first 40s, the model predicts almost zero conversion with the exit con-

centration curve overlapping with the inlet feed curve, however the experimental

observed value (dash (blue) curve) shows finite conversion. The difference arises

because of the model negligence of axial temperature gradient. Until the entire

catalyst reaches the ignition temperature, no conversion is expected in lumped

model, while due to the axial variation of temperature, the front part of the cata-

lyst may be above ignition leading to finite conversion observed at exit. Thus, the

120

lumped model is not accurate for the first 30-40s of cold start but thereafter the

model correlates well with the experimental result. The model is conservative and

predicts slightly higher emissions as compared to actual. Shown in Fig. 5.7 is the

FOS observed for the corresponding FTP cycle including both bag one and two.

The vehicle used was partial volume catalyst, i.e. HEGO located in-between the

two bricks and thus had outer loop controller designed based on HEGO set point.

As expected, the controller oscillates the feed gas A/F ratio in such a way as to

avoid having the catalyst saturated on either side (lean or rich) as seen by FOS

value which oscillates between zero and one. Now for the full volume catalyst, i.e.,

with HEGO placed at the end of two bricks, using HEGO as set point would lead

to breakthrough of emissions, which is not desirable. The model presented in this

work becomes useful in those cases, as one can replicate the same control be-

havior by using FOS as the set point, which is allowed to vary between pre-defined

upper and lower bound.

5.4 Comparison of Green and Aged Catalyst Performance

Having established the validity of the proposed model using experimental re-

sults as shown above, we use the model to study the effect of catalyst aging. The

kinetic parameters as shown in Table 6.1 are used with normalized activity of ac=1

for green catalyst and ac=0.4 for aged catalyst. From the experimental results

shown earlier it can be observed that: (i) The delay time for breakthrough is longer

in the green catalyst as compared to the aged catalyst. This happens because of

the reduction in storage capacity of ceria as it ages. (ii) Emissions are higher with

the aged catalyst because of the drop in catalyst activity. (iii) The breakthrough

occurs when the catalyst is completely reduced or oxidized and hence FOS can be

used as desired set point for TWC control.

Shown in Fig.5.8 are the steady state oxidant conversion and solid tempera-

ture, respectively, as a function of inlet feed temperature for a stoichiometric (i.e.,

121

0 50 100 150 200 250 3000

1

2

3

4

5

6

7x 104

Time(s)

O2 c

once

ntra

tion

(ppm

)


0 50 100 150 200 250 3000

2

4

6

8x 104

Time(s)

redu

ctan

t con

cent

ratio

n (p

pm)

(a)

(b)

Figure 5.6: Comparision of (a) oxidant and (b) reductant emissions with threshold

catalyst over a FTP cycle

122

0 200 400 600 800 1000 1200 14000

0.5

1

Time (s)

FOS

0 200 400 600 800 1000 1200 14000.95

0.975

1

1.025

1.051.05

λ

lambdaFOS

Figure 5.7: Change in FOS over bag one and two of a FTP cycle

λ = 1) feed mixture containing 1.5% reductant. The feed composition of 1.5% is

roughly the reductant concentration as can be seen from Fig.5.6 (dash-dot (black)

curve). The dashed (red) and solid (green) curves represent the aged and green

catalyst, respectively. The feed gas speed was kept constant at 1m/s and FOS

was initialized as one; however as these are steady state plots the initial value of

ceria considered does not influence the result. Shown in Fig.5.8a is the bifurcation

diagram for exit oxidant conversion. The well known ‘S’ shaped conversion curve

is obtained characterized by ignition and extinction points. It may be noted that, in

this preliminary simple model, we only account for change in active surface area

as the catalyst ages by changing the parameter ac or the effective pre-exponential

factors. Thus, the light-off curves can be observed to shift uniformly as the catalyst

ages, as is shown in Fig.5.8a. The reduction in active surface area is one of the

major change observed in aged catalyst, and is accounted in the model. However,

the washcoat structure may also change, leading to the change in washcoat diffu-

sional resistance and thus the observed activation energy, as is shown by Joshi et

123

400 450 500 550 6000

0.2

0.4

0.6

0.8

1

Tf,in (K)

Con

vers

ion

400 450 500 550 600400

450

500

550

600

650

700

750

Tf,in (K)

T S

greenaged

greenaged

(b)

(a)

Figure 5.8: Light-off behavior of green and aged catalyst with 1.5% reductant in

feed under stoichiometric operation

124

al. (2010). When the washcoat structure changes, the light-off will not only shift but

will also become more gradual as the catalyst ages (Heck et al., 2009, Joshi et al.,

2010) and these changes will be accounted in a companion publication involving

more detailed kinetics with axial variations as well as a more detailed deactivation

model. Shown in Fig.5.8b is the variation of wall temperature as a function of feed

(inlet) temperature. Except for the transition period (light-off region), the steady

state wall temperature for both aged and green catalysts almost overlaps. This

happens because the heat generated is proportional to the reactant conversion for

a given reaction. The theoretical adiabatic temperature rise (∆Tadb) with 1.5% re-

ductant is 1420C and the model predicts the temperature rise of 140.40C which is

consistent as around only 98% conversion was observed. The adiabatic tempera-

ture rise is defined as the maximum temperature rise observed for an exothermic

reaction for 100% conversion of reactant to product under adiabatic condition and

is given by

∆Tadb =(−∆H) Xi

Mi Cp, (5.16)

where, ∆H is the heat of reaction, Xi is the mole fraction of reductant (or the

limiting reagent), Mi is the molecular weight of reductant and Cp is the specific

heat of the gas.

5.5 Summary

The model presented and validated here is the simplest non-trivial one that re-

tains all the qualitative features of the TWC. We have demonstrated here that this

simplest model retains high fidelity and is computationally efficient for real-time im-

plementation. The model was also validated for cold start on FTP cycles and good

performance was observed. The present model provides a very efficient method

to control TWC performance based on estimated FOS to minimize the emission

breakthrough and flexibility to switch between partial and full-volume control.

125

400 450 500 550 600 6500

0.2

0.4

0.6

0.8

1

Tf,in (K)

Con

vers

ion

400 450 500 550 600 650 700 7500

0.2

0.4

0.6

0.8

1

Tf,in (K)

Con

vers

ion

Shi as a function of T

No washcoat di ffusionAsymptotic Sh

i

Shi=∞

Shi=Sh

i,∞=2.65

Shi from Eq 13

(b)

(a)

Figure 5.9: Impact of washcoat diffusion on conversion in a TWC: Bifucation plot for

1.5% reductant feed under stoichiometric operation at (a) u=1m/s and (b) u=10m/s

126

A second contribution of this work is the development of a simple aging model

for catalyst activity as well as oxygen storage capacity for the TWC. Specifically, our

model uses a single dimensionless parameter to monitor and update the catalyst

activity. This parameter can be used to identify the green and aged catalysts and

also to tune the control algorithm to achieve the desired emissions performance.

127

Chapter 6 Spatial-temporal Dynamics in a Three-way

Catalytic Converter

6.1 Introduction

In a recent publication (Kumar et al., 2012), a low-dimensional model of the

three-way catalytic converter (TWC), appropriate for real-time fueling control and

TWC diagnostics in automotive applications was proposed. A simplified chemistry

and the axial averaging was used to meet computational requirement of on-board

real time processing. In this work we extend the model to include spatial variation

and discuss the validity of an averaged model. We have also shown the validity of

internal mass transfer approximation by comparing with the detailed solution.

6.2 Kinetic Model

A similar kinetic model as discussed in earlier work (Kumar et al, 2012) is used

in this work. The parameters used are shown in Table 6.1. The constant used in

simulation are showed in Table 6.2.

6.3 Model 1: Low-dimensional Model

A low-dimensional model is derived from a detailed two dimensional model,

by simplifying the mass transfer along the transverse direction due to diffusion by

using internal and external mass transfer concept. In an earlier work by Joshi

et al. (2009) a similar model was used and verified with the detailed COMSOL

Table 6.1: Kinetic parameters for a Pd/Rh based TWC

sl.no. Reaction Ai (unit) Ei ( kJmol.K

)1 A+ 0.5O2 −→ AO 1.5× 1020 mol m−3s−1K 1052 Ce2O3 + 0.5O2 −→ Ce2O4 4.95× 1010 s−1 803 A+ Ce2O4 −→ Ce2O3 + AO 3.0× 107 s−1 75Absorption constant: Ka1 = Aa1 exp(−Ea/RT )Aa1 = 65.5 Ea = −7.99( kJ

mol K)

TOSCgreen = 200 molm3of washcoat

128


Constants Value



εw 0.41τ 8Sh∞ 3.2Nu∞ 3.2Shi,∞ 2.65Λ 0.58

model. The model was observed to validate well with the detailed model. However,

in that model for multiple reaction case a constant asymptotic Sherwood number

was used to compute internal mass transfer coefficient and also the ceria kinetic

which includes gas-solid reaction was not considered. Here, we present a much

general approach and later verify our model with both detailed and experimental

results. The symbols used have same definition as in Kumar et al., 2012 and are

not described here.

The species balance in the fluid phase (for gas phase species) is given by

∂Xfm

∂t= −〈u〉 ∂Xfm

∂x− kmoRΩ

(Xfm − 〈Xwc〉) . (6.1)


εw∂ 〈Xwc〉∂t

=1

CTotalνT r +

kmoδc

(Xfm − 〈Xwc〉) . (6.2)

129


k−1mo = k−1

me + k−1mi, (6.3)



ρfCpf∂Tf∂t


RΩ

(Tf − Ts) , (6.4)


δwρwCpw∂Ts∂t

= δwkw∂2Ts∂x2

+ h (Tf − Ts) + δcrT (−∆H) . (6.5)

Here, δc is the washcoat thickness and δw represents the effective wall thickness

(defined as sum δs + δc, where δs is the half-thickness of wall) , ρw and Cpw are the

effective density and specific heat capacity, respectively, defined as δwρwCpw =

δcρcCpc + δsρsCps and δwkw = δckc + δskswhere the subscript s and c represent

the support and catalyst washcoat, respectively. To quantify the oxygen storage on

ceria, we define the fractional oxidation state (FOS), θ of ceria as,

θ =[Ce2O4]

[Ce2O4] + [Ce2O3], (6.6)

∂θ

∂t=

1


or for the simplified kinetics used in this work,

∂θ

∂t=

1

2TOSC(r2 − r3) , (6.8)

where r2 and r3 are reaction rates for oxidation and reduction of ceria respectively.

130

The gradients in the transverse direction are accounted by the use of internal

and external mass transfer coefficients, computed using the Sherwood number

(Sh) correlations. The external mass transfer coefficient matrix kme ∈ RN×N is

defined by

kme =Df Sh

4RΩ

. (6.9)

Here, Sh is a diagonal matrix given by, Sh =Sh I, where I ∈ RN×N is the identity

matrix. We use the position dependent Sherwood number Sh, defined for the fully

developed flow with constant flux boundary condition (Gundlapally et al. 2011)

Sh = Sh∞ +0.272(P

z)

1 + 0.083(Pz

)23

,

where Sh∞ = 3.2 for rounded square channel. Similarly, the internal mass transfer

coefficient is defined as

kmi =DsShiδc

. (6.10)

The internal Sherwood number matrix, Shi ∈ RN×N , is evaluated as a function of

Thiele matrix (Φ) as follows (Balakotaiah, 2008)

Shi = Shi,∞ + (I + ΛΦ)−1ΛΦ2. (6.11)

We define the effective rate constant as ki,eff =(− 1CTotal

Ri(X)Xi

)X=Xfm

in which case,

Φ2 becomes a diagonal matrix with the diagonal terms defined as

Φ2ii =

δ2c

Ds,i

ki,eff , (6.12)

More detail about the derivation of Φ has been discussed in Kumar et al., (2012).

131

The initial and boundary conditions are given by

Xfm,j(x) = X0fm,j(x) @t = 0, (6.13)

〈Xwc,j(x)〉 =⟨X inwc,j(x)

⟩@t = 0, (6.14)

Tf (x) = T 0f (x) @t = 0, (6.15)

Ts(x) = T 0s (x) @t = 0, (6.16)

Xfm,j(t) = X infm,j(t) @x = 0, (6.17)

Tf (x, t) = T inf (t) @x = 0, (6.18)

∂Ts∂x

= 0 @x = 0, (6.19)

∂Ts∂x

= 0 @x = L. (6.20)

6.3.1 Discretized Model

Eqs. 6.1-6.8 are solved by discretizing using finite difference method with up-

winding. For interior points we have,

132

dXfm(i, j)

∂t= −〈u〉 Xfm(i, j)−Xfm(i− 1, j)

4x − kmo(i, j)

RΩ

(Xfm(i, j)− 〈Xwc〉 (i, j)) ,

(6.21)

εwd 〈Xwc〉dt

=1

CTotalνT r(i, j) +

kmo(i)

δc(Xfm(i, j)− 〈Xwc〉 (i, j)) , (6.22)

ρfCpfdTfdt

= −〈u〉 ρfCpfTf (i)− Tf (i− 1)

4x − h

RΩ

(Tf (i)− Ts(i)) , (6.23)

δwρwCpwdTsdt

= δwkwTs(i+ 1)− 2Ts(i) + Ts(i− 1)

4x2(6.24)

+h(i) (Tf (i)− Ts(i)) + δc

Nr∑k=1

rk(i) (−∆Hk(i)) , (6.25)

dθ(i)

dt=

1

2TOSC(r2(i)− r3(i)) . (6.26)

To get discretized model for the boundary condition, integrate Eq. 6.5 from x = 0

to 4x2,

δwρwCpw∂Ts∂t

4x2

= δwkw

((∂Ts∂x

)x=4x

2

−(∂Ts∂x

)x=0

)(6.27)

+h(1) (Tf − Ts)4x2

+4x2δc

Nr∑i=1

ri (−∆Hi) . (6.28)

133

Using boundary condition Eq.6.19 and central difference for(∂Ts∂x

)x=4x

2

gives

δwρwCpwdTs(1)

dt= 2δwkw

(Ts(2)− Ts(1)

4x2

)(6.29)

+h(1) (Tf (1)− Ts(1)) + δc

Nr∑i=1

ri(1) (−∆Hi(1)) . (6.30)

Similarly integrating from x = L − 4x2

to x = L and using boundary condition

Eq.6.20 gives

δwρwCpwdTs(N)

dt= δwkw

(−2

Ts(N)− Ts(N − 1)

4x2

)(6.31)

+h(N) (Tf (N)− Ts(N)) + δc

Nr∑i=1

ri(N) (−∆Hi(N)) .(6.32)

Eqs. 6.21 and 6.23-are solved for discretization points i=2 toN, with boundary

conditions Xfm,j(1) = Xinfm,j(t) and Tf (1) = T inf (t). Eq 6.22 is solved for i=1 to N.

Eq 6.24 is solved for i=2 to N-1. While for i=1 and N, Eq 6.29 and 6.31 are used

respectively. Eq 6.26 is solved for i=1 to N. Thus the total number of variables

equal to NC(2N − 1) + (3N − 1), where NC is number of gaseous component and

N is the axial grid points.

6.3.2 Experimental Validation

The model was validated using the experimental results collected during vari-

ous drive cycles. The detail about the experimental setup and operating conditions

can be found in Kumar et al. 2012. Shown in Fig. 6.1 and 6.2 are the instantaneous

oxidant and reductant emissions, respectively, for an idle vehicle speed. A delib-

erate lean-rich experiments were performed. The dash-dotted (blue) curve rep-

resents the feed gas composition while the solid (black) and dashed (red) curves

represent the model estimated and measured emissions at TWC exit, respectively.

134

900 950 1000 1050 1100 1150 1200 1250 13000

5000

10000

15000

oxid

ant p

pm

feedexit estimatedexit measured

Figure 6.1: Experimental validation for oxidant emission at idle vehicle speed with

an aged catalyst

The overall trends are accurately predicted. A good match is observed for re-

ductant emission, while small error was observed for oxidant emission particularly

during fast lean-rich oscillatory steps. This could be due to various mechanism.

Like, the assumed total ceria capacity may be higher than actual, in which case

the model will predict longer breakthrough delay. Or, the assumed kinetic form for

ceria oxidation may not be correctly represented by linear mass action form.

It is interesting to observe that the step change response in oxidant (Fig 6.1)

is much sharper as that compared to reductant (Fig 6.2) implying the ceria reduc-

tion to be slower compared to oxidation at the operating conditions of experiment,

which is around 650K mean feed temperature and idle vehicle speed. Another

possible reason for such a behavior is that by switching from lean to rich feed, the

feed temperature reduces, reducing the observed reaction rate while step change

from rich to lean increases the feed temperature. The model performance is also

135

900 950 1000 1050 1100 1150 1200 1250 13000

0.5

1

1.5

2

2.5

3

3.5

4x 104

redu

ctan

t ppm


Figure 6.2: Experimental validation for reductant emission with an aged catalyst at

idle vehicle speed.

compared at other operating conditions and good match was observed.

Shown in Fig.6.3 and 6.4 are the model validation over an FTP cycle. Only the

first 300s starting with cold start is shown because once the catalyst lights-off the

conversion becomes high the model matches properly. Comparison over a FTP cy-

cle showed the biggest improvement as compared to the averaged model proposed

in Kumar et al, (2012). With an averaged model the, estimated and measured val-

ues starts to match after roughly 30s while with spatial variation included, the cold

start emission is predicted accurately. Another advantage with a spatial model as

compared to detailed model is with respect to the degree of generalization. Even

for different catalyst load, the model gives reasonably accurate prediction by just

updating the storage capacity or the catalyst activity.

It is interesting to observe that at time t=0s, the feed to TWC has very high

oxidant concentration, implying a leaner feed. This observation cannot be veri-

136

0 50 100 150 200 250 3000

1

2

3

4

5x 104

conc

entr

atio

n pp

m


Figure 6.3: Oxidant emission for first 300s of FTP (ac=0.3)

0 50 100 150 200 250 3000

1

2

3

4

5

6x 104

conc

entr

atio

n pp

m


Figure 6.4: reductant emission for first 300s of FTP (ac=0.3)

137

fied from measured A/F as for the first few seconds the sensors (UEGO) are not

warmed up thus they do not give meaningful information. However A/F ratio can be

computed from the measured emission using Spindt or Brettscheinder equation.

λ =

[CO2] + [CO]2

+ [O2] + [NO]2

+

[[Hcv ]

43.5

3.5+[CO][CO2]

− Ocv2

]([CO2] + [CO])(

1 + [Hcv ]4− Ocv

2

)([CO2] + [CO] + Cfactor[HC])

, (6.33)

where

λ =normalized A/F ratio,

[XX]= gas concentration in % volume,

Hcv =atomic ratio of oxygen to carbon in the fuel,

Cfactor =number of carbon atom in each of the HC molecules being measured,

A similar expression can be derived in terms of the reduced species,

λ =[AO] + 2[O2]

[A] + [AO]. (6.34)

Shown in Fig 6.5 is the λ at the TWC inlet as measure by Eq.6.34. While

shown in Fig 6.3.2 is the commanded λ. From Fig.6.34 and 6.5, it can be concluded

that although the commanded λ is rich in the cold start while the observed λ

at TWC inlet is leaner. This behavior may be observed because not all the fuel

injected vaporizes and some sticks to the valve as ’puddle’ which will reduce the

fuel content in TWC feed. Also, connecting lines will have some air at time t=0,

which can also dilute the fuel mixture entering TWC. This phenomena, may actually

be advantageous with respect to emission. As before light-off, if the feed has lower

fuel content, it will reduce the cumulative CO and HC emission from tailpipe. In

the later section, we discuss the effect of various design parameters on light-off

temperature and emissions.

138

0 50 100 150 200 250 300 350 4000

0.5

1

1.5

com

man

ded

λ

Time s

thresholdful

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

30

35

Time s

λ

fulthreshold

Figure 6.5: Observed lambda as computed using chemical composition

139

6.4 Model 2: Validation with Detailed Model

In this model we solve the detailed diffusion reaction equation for the washcoat.

The solid temperature is assumed to be relatively uniform along the washcoat thick-

ness and an averaged value is used.

The species balance equation in the fluid phase (for gas phase species) is given

by

∂Xfm

∂t= −〈u〉 ∂Xfm

∂x− kmeRΩ

(Xfm −Xs) . (6.35)

Here kme is the external mass transfer coefficient as defined by Eq.6.9 and Xs

is the concentration at the fluid-solid (washcoat) interface.


εw∂Xwc

∂t=

1

CTotalνT r + De

∂2Xwc

∂y2. (6.36)

The energy balance in the fluid phase is given by

ρfCpf∂Tf∂t


RΩ

(Tf − Ts) , (6.37)


δwρwCpw∂Ts∂t

= δwKw∂2Ts∂x2

+ h (Tf − Ts) +

δc∫0

rT (Xs, Ts)dy

(−∆H) . (6.38)

Ceria balance

∂θ

∂t=

1

2TOSC(r2 − r3) , (6.39)

with the boundary conditions:

Xfm,j(x) = X0fm,j(x) @t = 0, (6.40)

140

Tf (x) = T 0f (x) @t = 0, (6.41)

Ts(x) = T 0s (x) @t = 0, (6.42)

Xfm,j(t) = X infm,j(t) @x = 0, (6.43)

Tf (x, t) = T inf (t) @x = 0, (6.44)

∂Ts∂x

= 0 @x = 0, (6.45)

∂Ts∂x

= 0 @x = L, (6.46)

Xwc = Xs @y = 0, (6.47)

∂Xwc,j

∂y= 0 @y = δc, (6.48)

where Xs is defined as

kme (Xfm −Xs) = −De

(∂Xwc

∂y

)y=0

. (6.49)

6.4.1 Discretized model

Discretization Eq. 6.38 for the interior points (k = 2 to M − 1),

εwdXwc(i, k)

dt=

1

CTotalνT r(i, k) + De

Xwc(i, k + 1)− 2Xwc(i, k) + Xwc(i, k − 1)

∆y2.

(6.50)

To get 2nd order accuracy, we integrate eq. 6.38 from y = 0 to ∆y2

and substitute

the boundary conditions Eq. 6.47 and 6.49

εwdXwc(i, 1)

dt

∆y

2=

∆y

2

1

CTotalνT r(i,1) +

(De

∂Xwc

∂y

)y= ∆y2

y=0

, (6.51)

141

εwdXwc(i, 1)

dt=

1

CTotalνT r(i,1) + 2De

Xwc(i, 2)−Xwc(i, 1)

∆y2(6.52)

+2

∆ykme (Xfm(i)−Xwc(i, 1)) , (6.53)

where,

Xwc(i, 1) = Xs.

Similarly for the boundary condition at y = δc, integrating eq. 6.38 from y =

δc − ∆y2

to ∆y2

and substituting the boundary conditions Eq. 6.48, we get

εwdXwc(i,M)

dt=

1

CTotalνT r(i,M)− 2De

Xwc(i,M)−Xwc(i,M − 1)

∆y2. (6.54)

6.4.2 Case 1: Single reaction

Shown in Fig 6.6 is the comparison of the low-dimensional model with the de-

tailed model for a case of a single reaction. The solid (red) curve represents the

conversion as predicted by detailed solution. The dash-dot (green) curve, , dotted

(black) curve and dashed (blue) curve represents the prediction with low-dimension

model for different approximations of internal mass transfer case. The asymptotic

case implies a internal mass transfer coefficient (kmi) computed using the constant

internal Sherwood number of 2.6, while kmi = ∞ will imply no washcoat diffusion

limitation. This is the case which is achieved if kmo = kme in Eqs. 6.1 and 6.2 and is

the one most commonly used in literature where washcoat diffusion is not consid-

ered. The dashed (blue) curve represents the case where internal mass transfer

coefficient is computed using the method proposed in Eq 6.12 and was found to

be the most accurate representation of the detailed model. The rate kinetic used

is shown below

142

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

time s

conv

ersi

on

Detailedasymptoticno diffusion limitationinternal mass transfer

Figure 6.6: Model comparision of internal mass transfer concept with detailed

model for a single reaction

CO + 0.5O2 −→ CO2 (6.55)

rate =1018 exp(−90000

RT)XCOXO2

G, (6.56)

where,

G = T

(1 + 65.5 exp(

7990

RT)XCO

)2

(6.57)

Shown in Fig.6.7 is the temperature transient for the same simulation as in

Fig.6.6. The temperature gradient is not much affected and all the different model

gives almost the same temperature profile. The constants used in simulation are

tabulated below.

143

0 50 100 150 200300

350

400

450

500

550

600

650

700

time s

tem

pera

ture

Detailedasymptoticno diffusion limitationinternal mass transfer

Figure 6.7: Model comparision of internal mass transfer concept with detailed

model for a single reaction

u 1 m/s

X inf,CO 1.5%

X inf,O2

0.75%

T 0f 300 K

T 0s 300 K

T inf 550

6.4.3 Case 2 Multiple reaction including ceria kinetics

For the case including ceria kinetics, the approximation used in computation of

Thiele modulus using Eq.6.12 introduces an additional error as compared to case

involving all reactions between gas phase species only. While for all other species,

a gas phase concentration is used, for ceria a solid phase concentration is used.

Shown in Fig. 6.4.3 is the comparison of low-dimensional model prediction with de-

tailed model for the kinetics shown in Table 6.1. Even with the simplifications used

144

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

time s

conv

ersi

on Detailedinternal mass transferno diffusion limitationasymptotic

in computing internal mass transfer coefficient, it gives the best representation of

the detailed model as compared to having no washcoat diffusion limitation assump-

tion or using an asymptotic value. The computational time for low-dimensional

model was roughly 30-40 times faster as compared to than detailed model.

Shown in Fig.6.4.3 is the temperature profile for different model assumption.

For the case involving gas phase reaction only, the temperature curve for all model

overlapped (Fig.6.7), while with ceria kinetic included the low-dimensional model

shows faster temperature rise as compared to detailed model.

6.5 Effect of design parameters on catalyst light-off and con-

version efficiency

For a square channel with rounded corners, we examine the effect of various

parameter on light-off and emission using the low-dimensional model as described

in model 1. Light-off curve for base case of uniform activity in a ceramic substrate

catalyst with 1.5% reductant is shown in Fig 6.9. The light-off occurs at around 480

145

0 20 40 60 80 100 120 140300

350

400

450

500

550

600

650

700

time s

tem

pera

ture

DsKmiKmeSh

∞

K. Shown in Fig. 6.8 is the solid steady state temperature profile along the length

of the catalyst for different inlet feed temperature. Before light-off the steady-state

temperature is uniform as feed temperature. Near light-off temperature, the end of

the catalyst becomes warmer as compared to front, while as the feed temperature

increases, the temperature front moves towards the entrance and almost a uniform

temperature is achieved.

6.5.1 Effect of change in washcoat thickness

Shown in figure 6.10 is the effect of change in washcoat thickness for an uniform

catalyst loading with ac=1 for 1.5% reductant concentration at stoichiometry. The

solid (black) curve represent the base case of 30 µm washcoat thickness while

the dash (red) curve and dash-dot (blue) curve represent washcoat thickness of

20 and 40 µm, respectively. The feed gas speed was assumed constant at 1m/s

(space velocity 45868 hr−1). Changing the washcoat thickness, changes the total

catalyst loading. Higher the catalyst loading, the lower is the light-off temperature.

146

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08350

400

450

500

550

600

650

700

750

800

length m

solid

tem

pera

ture

Tf,in=499 K

Tf,in=350 K

Tf,in=443 K

Tf,in=488 K

Tf,in=511 K

Tf,in=564 K

Tf,in=650 K

Figure 6.8: Steady state axial temperature for different inlet feed temperature

However, because of the diffusion limitations, increasing the washcoat thick-

ness may not lead to further increase in the transient time after a critical washcoat

thickness value as is seen in Fig6.11. Also, increasing the washcoat thickness

for the given CPSI, reduces the channel open area hydraullic radius. As stated in

earlier publication (Pankaj et al 2012), for the case of a single reaction, the relative

values of RΩ

Dfand δc

Dsat 700K are 2.04 and 120.6, respectively

1

kmo=

δcDsShi

+4RΩ

DfSh∞. (6.58)

Thus, the system is internal mass transfer limited. Now, by changing δc, we also

reduce RΩ,making RΩ

Dfeven smaller compared to δc

Dsmaking the system even more

diffusion limited in which case the entire washcoat thickness is not utilized and

increasing the thickness does not improves conversion.

147

350 400 450 500 550 600 650 7000

0.2

0.4

0.6

0.8

1

conv

ersi

on

Tf,in KFigure 6.9: Bifurcation plot for uniform activity at u=1 m/s for 1.5% reductant at

stoichiometry

6.5.2 Non-uniform catalyst activity

Keeping the total amount of catalyst constant, the catalyst density is varied

along the length of the channel. A higher loading at the inlet decreases the light-off

time and would reduce the cold start emission. The parameter ac is the catalyst

loading, which for the case of uniform distribution will be equal to 1 as given in case

1. A continuous profile can be used for ac to get optimal conversion, however such

an arrangement is difficult to implement, so we compare the case of two brick in

series with higher loading in front brick. The total length is also kept constant for

comparison. The front brick is assumed to be 1/5 of the total length.

∫ x=L

x=0

acdx = constant,

148

440 450 460 470 480 490 500 510 5200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

conv

ersi

on

Tf,in K

δwc =30

δwc =20

δwc =40

Figure 6.10: Effect of change in washcoat thickness on catalyst light-off

ac =

b for 0<x< L10

(10− b)/9 for x> L10

. (6.59)

For the formulation shown in Eq. 6.59, the case with b=1 represents uniform

distribution, while b=10 will represent 100% of the catalyst is deposited in the first

brick of length L/10 while the remaining brick has no catalyst loading.

The steady state concentration is not influenced by the catalyst distribution.

However, it does influences the transient time for light-off as seen in Fig. 6.14.

Shown in Fig 6.13 is the axial temperature profile with 50% of catalyst loading on

first 10% of the catalyst length. Closer to the light-off temperature a discontinuity

in a profile can be seen, because of the non-uniform heating caused by different

149

0 50 100 150 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Conv

ersi

on

Time s

δwc =30

δwc =20

δwc =40

δwc =50

Figure 6.11: Effect of change in washcoat thickness on exit conversion efficiency

transient

0 50 100 150 200300

350

400

450

500

550

600

650

Time s

Exit

solid

tem

pera

ture

K

δwc =30

δwc =20

δwc =40

δwc =50

Figure 6.12: Effect of change in washcoat thickness on exit temperature transient

150

350 400 450 500 550 600 650 7000

0.2

0.4

0.6

0.8

1

conv

ersi

on

Tf,in K

uniformnonuniform

reactivity. At very high temperature the conversion reaches 100% and the entire

catalyst achieves a uniform temperature with slightly lower temperature at the front

caused by cooling of the reactor with the incoming feed.

It may be noted that increasing the loading for the front of catalyst reduces the

light off-time and hence can help in reducing the cold start emission. However,

increasing the loading in front catalyst by redistributing total catalyst content may

lead to lower steady state conversion. Shown in fig 6.14 is the transient response

observed starting from cold start with constant feed inlet temperature of 550K and

feed velocity u=1m/s with 1.5% reductant at stoichiometry. The dotted (blue) curve

represents the conversion obtained for the base case with uniform activity. While

the dash (red) and dash-dot (black) and solid (green) curves represent the con-

version obtained for non-uniform loading with 40, 80 and 95% catalyst distributed

in first 10% of the length, respectively. It can be seen that a=9.5 gives the fastest

151

0 0.02 0.04 0.06 0.08350

400

450

500

550

600

650

700

750

800

length m

tem

pera

ture

T=438 K

T=484 KT=495 K

T=569 K

T=500 K

T=350 K

T=650 K

Figure 6.13: Steady state temperature profile for non-uniform catalyst loading

light-off, however it shows lower steady state conversion as compared to uniform

distribution. At a higher feed inlet temperature the difference observed in the light-

off time will reduce as is shown in Fig 6.15 where the inlet feed temperature was

taken as 650K while keeping other parameters constant.

6.5.3 Effect of cell density

Cell density have a strong impact on the catalyst light-off. Generally, a high cell

density is used for close-coupled catalyst while a lower cell density is used for un-

der body reactor. Increasing the cell density, generally, reduces the wall thickness

and also reduces the hydraulic diameter of the open flow area. This reduces the

heat capacity of the catalyst leading to faster light-off. Shown in Table 6.3 is the

properties of different CPSI cordierite substrate (Heck and Farrauto, 2002). Sim-

ilar specification for metallic substrate are shown in Table 6.4 (Heck and farrauto,

152

0 10 20 30 40 50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time s

Con

vers

ion

a=1a=4a=8a=9.5

Figure 6.14: Effect of change in loading profile on conversion transient at constant

feed temperature of T=550K

153

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

time s

conv

ersi

on

a=4a=2a=1

Figure 6.15: Effect of change in loading profile on conversion transient at constant

feed temperature of T=650K

Table 6.3: Nominal properties of standard and thin walled Cordierite substrate

Cell density(cell/in2) 400 600 900 1200

wall thickness (mili in) 6.5 4 2.5 2.5

Hydraulic diameter (mm) 1.1 0.94 0.78 0.67

Heat capacity (J/K l) 352 270 209 240

2002). The metallic substrate differs from cordierite, mainly because of their lower

specific heat capacity, higher density and lower wall thickness. The physical prop-

erty of washcoat (Santos and Costa 2008), ceramic and metallic substrate (Heck

and farrauto, 2002) are listed in Table 6.5.

Table 6.4: Nominal properties of standard and thin-wall metallic substrate

Cell density(cell/in2) 400 500 500 600 600

wall thickness (mili in) 2 1.5 2 1.5 2

Hydraulic diameter (mm) 0.98 0.89 0.88 0.85 0.84

Heat capacity (J/K l) 408 371 445 408 482

154

Table 6.5: Physical properties of washcoat, ceramic and metallic substrate

property Washcoat Cordierite substrate Metallic substrate

Specific heat capacity (J/kg K) 950 891 515

Thermal conductivity (W/m K) 1 1 13

density (kg/m3) 2790 1630 7200

Shown in Fig.6.16 and 6.17 are the effect of change in cell density for a ceramic

substrate. For each case the wall thickness and hydraulic radius is updated using

the Table 6.3 and 6.4. The feed gas speed is assumed constant at 1 m/s (constant

space velocity) and uniform activity of ac=1 is used with the kinetics shown in Table

6.1. The feed inlet temperature was also assumed constant at 550K. The feed

concentration is taken as 1.5% reductant at stoichiometry. The effective specific

heat and density was also updated using relation δwρwCpw = δcρcCpc + δsρsCps

and δwKw = δcKc + δsKs, the values for ρc, Cpc, ρs and Cps are shown in Table

6.5. The simulation shown are for a single channel with same washcoat thickness,

this implies the catalyst in each channel is constant for different cases however

this implies higher CPSI catalyst will have overall higher catalyst loading for same

volume.

Shown in Fig. 6.16 and 6.17 are the effect of change in cell density with

cordierite substrate on light off duration. It can be seen that increasing the cell

density decreases the light-off time, however the CPSI of 1200 did not perform

better then 900. This is because the wall thickness stayed same in both case and

the hydraulic radius was reduced, thus the volumetric flowrate of gas has reduced

for constant space velocity.

Shown in Fig. 6.18 and 6.19 are the effect of change in cell density with a

metallic substrate. From the results on metallic substrate, it can be concluded that

CPSI of 500 and 1.5 wall thickness, performs best due to lower wall thickness

and higher flow area. A comparison of metallic and ceramic substrate is shown

in Fig: 6.20. The solid (red) curve represents a metallic substrate with 500CPSI

155

0 20 40 60 80 100 120 1400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

Con

vers

ion

CPSI=900CPSI=400CPSI=600CPSI=1200

Figure 6.16: Effect of change in cell density in ceramic substrate for constant space

velocity (45868 hr−1) and constant feed temperature (T=550K)

and 1.5 milli inch wall thickness, while dotted (blue) and dashed-dot (black) curve

represents the ceramic substrate with 600 and 900 CPSI, respectively. A ceramic

substrate catalyst with 900 CPSI gives best performance.

However, if the catalyst loading is also changed to keep the total mass of cat-

alyst constant for different CPSI arrangement, the metallic substrate gives faster

light off due to its lower specific heat as shown in Fig.6.22,

156

0 50 100 150 200300

350

400

450

500

550

600

650

700

Time (s)

Exit

tem

pera

ture

K

CPSI=900CPSI=400CPSI=600CPSI=1200

Figure 6.17: Effect of change in cell density in ceramic substrate for constant space


6.6 Conclusions

The spatial-temporal dynamics of TWC was studied. The model is validated

with an experimental result. Comparing with the averaged model the major im-

provement in performance was observed in predicting the light-off behavior. The

averaged model take longer for conversion to start because the entire catalyst

needs to be brought to the catalyst ignition temperature while in spatial model a

localized high temperature, like front of catalyst (for front end ignition) will lead to

finite conversion observed at the exit of the catalyst. With respect to FTP cy-

cle, the averaged model starts agreeing with the experiments from around 40 sec.

while the detailed model gives good agreement from the start. The operating con-

dition used in close coupled TWC leads to front end ignition and after the catalyst

157

0 50 100 150 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

Conv

ersi

onCPSI=400 δ=2CPSI=500 δ=1.5CPSI=500 δ=2CPSI=600 δ=1.5

Figure 6.18: Effect of change in cell density in metallic substrate for constant space


lights-off the temperature front were not sharp. In contrast, the catalyst oxidation

state (FOS) showed a very sharp front propagation. We also validate the internal

mass transfer concept approximation with the detailed model for the case of single

reaction as well as multiple reactions involving ceria kinetics.

158

0 50 100 150 200300

350

400

450

500

550

600

650

700

Time (s)

Exit

tem

pera

ture

K

CPSI=400 δ=2CPSI=500 δ=1.5CPSI=500 δ=2CPSI=600 δ=1.5

Figure 6.19: Effect of change in cell density in metallic substrate for constant space


159

0 20 40 60 80 100 120 1400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

Conv

ersi

on

M : CPSI=500 δ=1.5C: CPSI=600 δ=4C: CPSI=900 δ=2.5

Figure 6.20: Comparision of ceramic and metallic substrate for constant feed tem-

perature of 550 K and constant space velocity and composition

160

0 50 100 150 200300

350

400

450

500

550

600

650

700

Time (s)

Exit

tem

pera

ture

K


Figure 6.21: Comparision of ceramic and metallic substrate for constant feed tem-

perature of 550 K and constant space velocity and composition

161

0 50 100 150 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time s

conv

ersi

on


Figure 6.22: Comparision of metallic and ceramic substrate for same catalyst load-

ing

162

Chapter 7 Conclusions and Recommendations for Fu-

ture Work

7.1 In-cylinder combustion modeling

7.1.1 Summary and conclusions

The internal combustion engine cylinder is modeled as an open system i.e a

chemical reactor exchanging mass (air and fuel) and energy (spark and piston

work) with the surrounding. An averaged model in terms of bulk properties is de-

veloped to specify the species balance. The model is developed by first consider-

ing the combustion cylinder to be comprised of N smaller compartments, and the

species balance is written for each compartment and then LS reduction method is

applied to achieve a low dimensional model in two modes, namely volume aver-

aged and flow averaged concentration related through dimensionless mixing times.

These dimensionless mixing times incorporates the non-ideality of finite mixing

time between the reactants. In the limit, both the mixing times approaches zero,

the two concentrations becomes equal implying single uniform concentration within

the reactor (perfect mixing). It has also been shown (Kumar et at, 2010) that in ab-

sence of mixing time model predicts slightly higher temperature and consequently

higher NOx emissions. The model incorporates the importance of crevice as well

which are known to be one of the major reasons for unburned hydrocarbon emis-

sions. Crevice is modeled as a small isolated compartment exchanging mass and

energy continuously with the reactor. The flow to the crevice is guided by the pres-

sure difference between crevice zone and the cylinder. When the pressure inside

the crevice is higher than in-cylinder flow is out of crevice, while otherwise the flow

is into the crevice.

A first law of thermodynamic for an open system is used to derive an energy

balance equation in terms of averaged cylinder temperature over the total reactor

163

volume that includes the contribution from heat exchange with the coolant or wall of

the reactor, piston work, heat generated by reaction, flow work and spark energy.

Spark being modeled as an energy source enables us to study the effect of spark

timing and intensity on combustion. In the present work for simplicity, gasoline has

been assumed to be comprised of 80% fast burning (FHC) and 20% slow burning

hydrocarbon (SHC) (two-lump model). The fast burning component has been rep-

resented as an iso-octane, as it exhibits property similar to gasoline while the slow

burning was represented by (CH2)2, to increase the carbon to hydrogen ratio closer

to real gasoline where C:H ratio is typically of the order of 1:1.875. Global reac-

tion kinetics available in literature has been used to simulate the kinetic behavior.

The low-dimensional model developed involves total of 10 different species and

consist of mass balance for each species in crevice and cylinder and an energy

balance for the total of 21 ODE’s . The model was then verified for different oper-

ating conditions and was observed to agree qualitatively with the results reported

in the literature. The basic finding can be summarized as:

1. Out of all the regulated emission NOx formation is most sensitive to peak

temperature and occurs at very high temperature (above 1800 K).

2. CO and hydrocarbon emission decreases with increase in air-fuel ratio (λ)

while the NOx exhibits a maxima occurring at slightly leaner mixture.

3. The peak temperature occurs for slightly rich condition.

4. Ethanol blending decreases CO and hydrocarbon emissions while NOx emis-

sion may be higher or lower depending on the mode of operation.

5. Reducing the crevice volume can reduce the unburned hydrocarbon emis-

sions.

6. Advancing the spark timing will lead to increase in NOx emission.

164

7. Although the model was developed for SI engine, it was observed that the

model assumptions are more justified for HCCI engine. The major problem

with HCCI engine wide commercialization is in its difficulty in controlling the

ignition timing. The ignition in any system will be function of temperature and

fuel composition. Thus, with a predictive model as the one proposed in this

work that utilizes the detailed kinetics, the ignition timing can be predicted and

with current technology like variable valve and variable compression ratio, the

igniting can be controlled for proper functioning of HCCI engine.

7.1.2 Recommendations for future work

The low-dimensional combustion model can be extended for the case of HCCI,

GDI or variable valve engines. Some of the steps to improve the model accuracy

and applicability are outlined below,

1. The current version has modeled gasoline as 2-lump but to characterize

gasoline properly, we need to model gasoline comprising of more lumps, like

with 5 lump model (involving species from straight chain aliphatic, branched

alkanes, cyclic and aromatic compounds). Also as the engine emissions are

very sensitive to the combustion kinetics. Thus, a more detailed kinetic study

is required to determine the important reaction steps and gaseous interme-

diates in gasoline combustion. This extension is expected to greatly increase

the model predictive accuracy.

2. The present model assumes constant engine speed thus a major improve-

ment to the current model can be obtained by integrating it with the torque

balance model. The torque balance will relate the engine speed as a func-

tion of mass air flow and engine load and would increase the applicability of

model.

165

3. In the current work, the inlet manifold pressure was assumed constant, how-

ever with variable speed the manifold pressure will change and needs to be

integrated with the model.

4. The current model assumes a single averaged temperature throughout the

reactor, because of the point source nature of spark there exist large temper-

ature gradients. Hence, the current model can be extended to include spatial

variation so as to correctly capture the flame front propagation. This will also

improve the model prediction of pressure delay after spark ignition.

5. The low-dimensional combustion model can be used to design a robust inner

loop controller. Given the throttle angle position, the model can compute

emissions and the normalized air fuel ratio (λ), which can be used to compute

the optimal fueling profile for the desired operation.

7.2 Three-way catalytic converter modeling

7.2.1 Summary and Conclusions

The main contribution of this work is the development and validation of a funda-

mentals based reduced order model that is useful for TWC control and diagnostics.

In developing such a model, we have used three main approximations.First, we

have simplified the problem of multi-component diffusion and reaction in the wash-

coat and approximated the transverse gradients in the gas phase and washcoat

by using multiple concentration modes and overall mass transfer coefficients. The

external mass transfer coefficient is computed using the Sherwood number corre-

lation, while the internal mass trasfer coefficient is evaluated as a fucntion of Thiele

modulus. Various approximations for computing the Thiele modulus for multiple re-

action case is discussed. This approximation is validated by comparing with the

detailed model solution. For a single reaction case involving gas phase species

only, the internal mass transfer coeefiecient model was found to overlap exactly

166

with the detailed model solution. For the case of multiple reaction involving solid

species balance, some difference was observed. However, internal mass transfer

concept model was found to be best representative as compared with an asymp-

totic value or no washcoat diffusion model.

Second, we have simplified the complex catalytic chemistry by lumping all the

oxidants and reductants. A detailed kinetic model was studied and it was observed

that for predicting the fractional oxidation state (FOS) of the catalyst, it is not

important to track different effluent (HC, CO, H2, H2O, CO2, N2, NH3, O2) separately

and a good estimete of the FOS and TOSC can be obtained by tracking the net

behavior of oxidant and reductant.

Third, we have simplified the axial variations in temperature and concentration

by using averaging over the axial length scale. This reduces the computational time

significantly. However, this approximation also introduces a slight error, particularly

during the light-of period. However, once the catalayst is ignited the model agrees

well with the experimentaly observed value.

A fourth contribution of this work is the development of a simple aging model for

catalyst activity as well as oxygen storage capacity for the TWC. Specifically, our

model uses a single dimensionless parameter to monitor and update the catalyst

activity. This parameter can be used to identify the green and aged catalysts and

also to tune the control algorithm to achieve the desired emissions performance.

In conclusion, the model presented and validated here is the simplest non-trivial

one that retains all the qualitative features of the TWC. We have demonstrated

here that this simplest model retains high fidelity and is computationally efficient

for real-time implementation. The present model provides a very efficient method

to control TWC performance based on estimated FOS to minimize the emission

breakthrough and flexibility to switch between partial and full-volume control.

167

7.2.2 Recommendations for future work

The model presented and validated here is the simplest non-trivial one that

retains all the qualitative features of the TWC, to improve the model accuracy,

1. More detailed kinetic study can be performed to determine the interaction of

Ce with precious metal. Also the reversibility of the ceria reaction with CO

and oxygen will be worth investigating.

2. It was observed that prolong rich phase in TWC can lead to ammonia for-

mation. The kinetic model presented in this work did not involve ammonia

reactions and can be updated to predict ammonia formation. Such a model

can be used in series with SCR as well for the urealess NOx reduction.

3. The averaged model works well with warmed up catalyst however, during cold

start the model is not as accurate. The model error in cold start conditions

can be reduced by replacing the total length of the TWC by the ignited length.

For front end ignition, the ignited length may be estimated from the work of

Ramanathan et al. (2004).

4. The TWC model is experimentally validated, once the combustion model is

validated as well a good extension of the work will be to combine engine with

the TWC model and optimize the behavior of an integrated system to obtain

optimal fueling profile for high gas mileage and low emissions.

168

References

Auckenthaler, T., Onder, C., Geering, H., Frauhammer, J., 2004. Modeling of a

Three-Way Catalytic Converter with Respect to Fast Transients of λ-Sensor Rele-

vant Exhaust Gas Components. Industrial & Engineering Chemistry Research, 43,

4780–4788.

Auckenthaler, T.S., 2005. Modelling and Control of Three-Way Catalytic Con-

verters. PhD Thesis, Swiss Federal Institute of technology Zurich.

Balakotaiah, V., 2008. On the relationship between Aris and Sherwood num-

bers and friction and effectiveness factors. Chemical Engineering Science, 63,5802–

5812.

Baker, D.R., Verbrugge, M.W., 2004. Mathematical analysis of potentiometric

oxygen sensors for combustion-gas streams. AIChE Journal, 40 (9), 1498-1514.

Bayraktar, H., Durgun, O., 2003. Mathematical modelling of spark-ignition en-

gine cycles. Energy sources. 25:651-666.

Bayraktar, H. 2005. Experimental and theoretical investigation of using gasoline–

ethanol blends in spark-ignition engines. Renewable Energy. 30:1733–1747.

Bird, R.B., Stewart, W.E., Lightfoot, E.N., 2002. Transport Phenomena. John

wiley & Sons, Inc, New York ISBN 9971-51-420-6.

Bhatia, D., Clayton, R.D., Harold, M., Balakotaiah, V., 2009. A global kinetic

model for NOx storage and reduction on Pt/BaO/Al2O3 monolithic catalysts. Catal-

ysis Today, 147, S250-S256.

Bhatia, D., Harold, M.P., Balakotaiah, V., 2009. Kinetic and bifurcation analysis

of the cooxidation of CO and H2 in catalytic monolith reactors. Chemical Engineer-

ing Science, 64, 1544-1558.

Bhattacharya, M., Harold, M.P., Balakotaiah, V., 2004. Mass-transfer coefficient

in washcoated monoliths. AIChE Journal. 50, 2939-2955.

169

Bhattacharya, M., Harold, M.P., Balakotaiah, V., 2004. Low-dimensional models

for homogeneous stirred tank reactors. Chem Engg Sci., 59, 5587-5596.

Blumberg, P.N., George, A.L., Rodney, J.T., 1975. Phenomenological models

for reciprocating internal combustion engines. Prog Energy Comb Sci., 5, 123-167.

Brandt, E.P., Wang, Y., Grizzle, J.W., 1997. A simplified three-way catalyst

model for use in on-board SI engine control and diagnostics. In Proceedings of the

ASME Dynamic Systems and Control, 653-659.

Brailsford, A.D., Yussouff, M., Logothetis, E.M., 1997. A first-principles model

of the zirconia oxygen sensor. Sensors and Actuators B, 44 (1-3), 321-326.

Carpenter M.H., Ramos, JI., 1985. Mathematical modelling of spark-ignition

engines. Applied Mathematical Modelling. 9, 40-52.

Celik MB. Experimental determination of suitable ethanol–gasoline blend rate at

high compression ratio for gasoline engine. Applied Thermal Engg. 2008;28:396–

404.

CEPA (California Environmental Protection Agency), 2011. Low-Emission vehi-

cle regulations and test procedures. Retrieved from http://www.arb.ca.gov/msprog/levprog/test_proc.htm

Chatterjee, D., Deutschmann, O., Warnatz, J., 2002. Detailed surface reaction

mechanism in a three-way catalyst. Faraday Discuss., 119, 371-384.

Curran HJ, Gaffuri P, Pitz WJ, Westbrook CK. Comprehensive modeling study

of iso-octane oxidation. Combust Flame. 2002;129:253–280.

Dinler, N., Yucel N., 2010. Multidimensional calculation of combustion in an

idealized homogeneous charge engine. Thermal science, 14 ,1001-1012.

Dubien, C., Schweich, D., Mabilon, G., Martin, B., Prigent, M., 1998. Three-

way catalytic converter modelling: fast- and slow-oxidizing hydrocarbons, inhibit-

ing species, and steam-reforming reaction. Chemical Engineering Science 53 (3),

471-481).

170

Fiengo, G., Cook, J.A., Grizzle, J.W., 2002. Fore-aft oxygen storage control.

American Control Conference, 2, 1401- 1406.

Franchek MA, Mohrfeld J , Osburn A. Transient fueling controller identification

for spark ignition engines.Journal of dynamic systems measurement and control-

Transaction of the ASME. 2006;128:499-509.

Goldberg, D.E., 1998. Genetic Algorithms in Search, Optimization and Machine

Learning. Addision- Wesley: Reading, MA.

GM press release, 2007. GM takes new combustion technology out of lab and

on to the road. Retreived from http://www.gm-press.com/yu/site/article.php?aid=2216.

Griffin, M. D., Anderson, J. D., Jr., Jones, E., 1979. Computational Fluid Dy-

namics Applied to Three-Dimensional Nonreacting Inviscid Flows in an Internal

Combustion Engine. J.Fluids Eng, 101(3), 367.

Gupta, N., Balakotaiah, V., 2001. Heat and mass transfer coefficients in cat-

alytic monoliths. Chemical Engineering Science, 56,4771-4786.

Heck, R.M., Farrauto, R.K., Gulati S.T., 2009. Catalytic air pollution control:

commercial technology. John wiley & Sons, Inc, New Jersey ISBN 978-0-470-

27503-0.

Hendricks, E. and Sorenson, S., 1990. Mean Value Modelling of Spark Ignition

Engines. SAE Technical Paper 900616.

Heywood, J.B., 1988. Internal Combustion Engine Fundamentals. McGraw-

Hill, Inc., New York, ISBN 0-07-028637-X.

Joshi, S., Harold, M., Balakotaiah, V., 2009. Low-dimensional models for real

time simulations of catalytic monoliths. AIChE Journal 55, 1771-1783.

Jones WP, Lindstedt RP. Global reaction scheme for hydrocarbon combustion.

Combust Flame. 1988;73:233-249.

Joshi, S.Y., Harold, M.P., Balakotaiah, V., 2009. On the use of internal mass

transfer coefficients in modeling of diffusion and reaction in catalytic monoliths.

171

Chemical Engineering Science. 64(23), 4976-4991.

Joshi, S.Y., Ren, Y., Harold, M.P., Balakotaiah, V., 2010. Determination of kinet-

ics and controlling regimes for H2 oxidation on Pt/Al2O3 monolithic catalyst using

high space velocity experiments. Applied catalysis B: Environmental. 102, 481-

495.

Koc M, Sekmen Y, Topgul T, Yucesu HS. The effects of ethanol–unleaded gaso-

line blends on engine performance and exhaust emissions in a spark-ignition en-

gine. Renewable Energy. 2009;34:2101-2106.

Kaspar, J., Fornasiero, P., Graziani, M., 1999. Use of CeO2-based oxides in the

three-way catalysis. Catalysis Today, 50, 285-298.

Konstantas, G., Stamatelos, A.M., 2007. Modelling three-way catalytic convert-

ers: An effort to predict the effect of precious metal loading. Proceedings of the

Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering,

221 (3), 355-373.

Kumar, P., Gupta, A., Jayaraman, V.K., Kulkarni, B.D., 2008. Aligning Time

Series with Genetically Tuned Dynamic Time Warping Algorithm. Advances in

Metaheuristics for Hard Optimization 251-261.

Marinov NM, Westbrook CK, Pitz WJ. Detailed and global chemical kinetics

model for hydrogen. 8th international symposium on transport properties, 1995.

Makki, I., Gopichandra, S., Kerns, J., Smith, S., 2005 Jun 14. Engine con-

trol and catalyst monitoring with downstream exhaust gas sensors. United States

patent US 6,904,751.

Möller, R., Votsmeier, M., Onder, C., Guzzella, L., Gieshoff, J.. 2009. Is oxygen

storage in three-way catalysts an equilirium controlled process? Appl Catal. 91:

30–38.

Montreuil, C. N., Williams, S. C., Adamczyk, A. A., 1992. Modeling Current

Generation Catalytic Converters: Laboratory Experiments and Kinetic Parameter

172

OptimizationSteady State Kinetics; Society of Automotive Engineers, Warrendale,

PA, SAE Paper 920096.

Muske, K.R., Jones, P., 2004. Estimating the oxygen storage level of a three-

way automotive catalyst. American Control Conference, 2004. Proceedings of the

2004 , 5, 4060-4065.

Najafi G, Ghobadian B, Tavakoli T, Buttsworth DR, Yusaf TF, Faizollahnejad M.

Performance and exhaust emissions of a gasoline engine with ethanol blended

gasoline fuels using artificial neural network. Applied Energy. 2009;86:630–639.

Oh, S.H., Cavendish, J.C., 1982. Transient of monolith catalytic converters:

Response to step changes in feed stream temperature as related to controlling

automobile emission. Ind. Eng. Chem. Prod. Res. Dev. 21 (1), 29–37.

Pontikakis, G.N., Stamatelos, A.M., 2004. Identification of catalytic converter

kinetic model using a genetic algorithm approach. Proceedings of the Institution of

Mechanical Engineers Part D Journal of Automobile Engineering. 218 (12), 1455-

1472.

Pontikakis, G.N., Konstantas G S, Stamatelos A M., 2004. Three-way catalytic

converter modeling as a modern engineering design tool. Journal of Engineering

for Gas Turbines and Power. 126 (4), 906-924.

Ramanathan, K., West, D.H., Balakotaiah, V.,2004. Optimal design of catalytic

converters for minimizing cold-start emissions. Catalysis Today, 98(3), 357-373.

Rao, S.K., Imam, R., Ramanathan, K., Pushpavanam, S, 2009. Sensitivity

Analysis and Kinetic Parameter Estimation in a Three Way Catalytic Converter.

Industrial & Engineering Chemistry Research. 48 (8), 3779-3790.

Riegel, J., Neumann, H., Wiedenmann, H.M., 2002. Exhaust gas sensors for

automotive emission control. Solid state ionics. 152-153, 783-800.

Saerens B, Vandersteen J, Persoons T, Swevers J, Diehl M, Van den Buick E.

Minimization of the fuel consumption of a gasoline engine using dynamic optimiza-

173

tion. Applied Energy. 2009;86:1582-1588.

Santos, H., Costa, M., 2008. Evaluation of the conversion efficiency of ceramic

and metallic three way catalytic converters. Energy Conversion and Management,

49 (2), 291-300.

Siemund, S., Leclerc, J.P., Schweich, D., Prigent, M., Castagna, F., 1996.

Three-way monolithic converter: Simulations versus experiments, Chemical En-

gineering Science, 51 (15), 3709-3720.

Sodré JR. Modelling NOx emissions from spark-ignition engines. Proceedings

of the Institution of Mechanical Engineers, Part D- Journal of Automobile Engg.

2000;214:929-934.

Verhelst, S., Sheppard, C.G.W., 2009. Multi-Zone thermodynamic modelling of

spark-ignition engine combustion-overview. Energy Convers Manage., 50, 1326-

1335.

Voltz, S.E., Morgan, C.R., Liederman, D., Jacob, S. M., 1973. Kinetic study

of carbon monoxide and propylene oxidation on platinum catalysts. Industrial &

Engineering Chemistry Product Research and Development, 12 (4), 294–30.

Westbrook CK, Dryer FL. Simplified reaction mechanisms for the oxidation of

hydrocarbon fuels in flames. Combust Sci Technol. 1981;27:31-43.

Yang, L., Kresnawahjuesa, O., Gorte, R.J., 2000. A calorimetric study of oxygen-

storage in Pd/ceria and Pd/ceria–zirconia catalysts. Catalysis letters. 72 (1-2),

33-37.

174

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