KUMAR-

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KUMAR-
© Copyright by Pankaj Kumar 2012
All Rights Reserved
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 advisors 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 system 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 ordinary 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 predicts the exhaust composition of regulated gases (total unburned HC’s, CO, and
N Ox ) 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 oxidation 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
List of Figures
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 (Encyclopedia Britannica inc., 2007) . . . . . . . . . . . . . . . .
Figure 1.2
Schematic of inner and outer control loop in partial volume
catalyst
Figure 1.3
4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Schematic of inner and outer control loop in full volume catalyst . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
Figure 1.4
FTP -75 cycle . . . . . . . . . . . . . . . . . . . . . . . . . .
10
Figure 2.1
Intake manifold pressure variation with throttle position as a
function of time. . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 2.2
41
In-cylinder temporal variation of (a) Total unburned hydrocarbon concentration, (b) Unburned oxygen concentration,
(c) Pressure and (d) Temperature with time . . . . . . . . . .
Figure 2.3
42
Temporal variation of in-cylinder (a) CO2 concentration, (b)
N Ox concentration, (c) CO concentration and (d) Hydrogen
concentration . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 2.4
In-cylinder variation of (a) Pressure and (b) Temperature
during a complete cycle after a periodic state is attained . .
Figure 2.5
43
45
Variation of exhaust gas concentrations with time (a) Unburned hydrocarbon, (b) Exhaust oxygen, (c) Exhaust N Ox
and (d) Exhaust CO . . . . . . . . . . . . . . . . . . . . . . .
Figure 2.6
Figure 2.7
46
Normalized reaction rate as a function of temperature over
a cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
Hydrocarbon conversion with crank angle for =1 . . . . . .
49
xiv
Figure 2.8
Effect of change in air/fuel ratio on peak temperature and
pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 2.9
50
Variation of regulated exhaust gases with air fuel ratio (a)
Predicted from low-dimensional model and (b) Experimentally observed [12] . . . . . . . . . . . . . . . . . . . . . . . .
51
Figure 2.10 (a) Impact of blending on emissions and (b) In-cylinder temperature and pressure under constant air/fuel ratio of =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) N Ox formation rate . . . . . . . . . . .
54
Figure 2.12 Impact of blending on (a) Emissions and (b) In-cylinder temperature and pressure at constant flowrate ( goes leaner) .
56
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 . . . . . . . . . . . . . . . . . . . . . . . . . .
57
Figure 2.14 Impact of engine speed on (a) N Ox and HC emission as
predicted by model, (b) In-cylinder peak temperature and
pressure, (c) Experimentally reported N Ox with change in
speed (Celik, 2008) and (d) Experimentally reported HC and
N Ox 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 . . .
xv
60
Figure 2.16 Impact of change in crevice volume on (a) Hydrocarbon emission, (b) In-cylinder temperature, (c) CO emission and (d)
N Ox emission for
mix;1
=0 and
mix;2
=0 . . . . . . . . . . .
62
Figure 2.17 Impact of change in inlet temperature on (a) Hydrocarbon
emission, (b) N Ox 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 . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 4.6
96
Comparision of model predicted and experimental CO conversion for lean to rich step change at a constant vehicle
speed of 30 mph
Figure 4.7
. . . . . . . . . . . . . . . . . . . . . . . .
97
Comparision of model predicted and experimental HC conversion for lean to rich step change at a constant vehicle
speed of 30 mph
Figure 4.8
. . . . . . . . . . . . . . . . . . . . . . . .
97
Comparision of model predicted and experimental NO conversion for lean to rich step change at a constant vehicle
speed of 30 mph
Figure 4.9
. . . . . . . . . . . . . . . . . . . . . . . .
98
Fractional oxidation state of the catalyst during lean to rich
step change . . . . . . . . . . . . . . . . . . . . . . . . . . .
xvi
99
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
Figure 4.12 comparision of model predicted vs experimentally observed
NO emission at constant vehicle speed of 60 mph . . . . . . 101
Figure 4.13 comparision of model predicted vs experimentally observed
HC emission at constant vehicle speed of 60 mph . . . . . . 102
Figure 4.14 comparision of model predicted vs experimentally observed
CO2 emission at constant vehicle speed of 60 mph . . . . . 103
Figure 4.15 comparision of model predicted vs experimentally observed
O2 emission at constant vehicle speed of 60 mph . . . . . . 104
Figure 5.1
Operating condition: Feed gas A/F ( ) 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
Figure 5.6
. . . . . . . . . . . . . . . . . . 119
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% reductant in feed under stoichiometric operation . . . . . . . . 124
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
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 temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Figure 6.9
Bifurcation plot for uniform activity at u=1 m/s for 1.5% reductant 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 loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 constant space velocity (45868 hr 1 ) and constant feed temperature (T=550K) . . . . . . . . . . . . . . . . . . . . . . . . . 156
Figure 6.17 Effect of change in cell density in ceramic substrate for constant space velocity (45868 hr 1 ) and constant feed temperature (T=550K) . . . . . . . . . . . . . . . . . . . . . . . . . 157
Figure 6.18 Effect of change in cell density in metallic substrate for constant space velocity (45868 hr 1 ) and constant feed temperature (T=550K) . . . . . . . . . . . . . . . . . . . . . . . . . 158
Figure 6.19 Effect of change in cell density in metallic substrate for constant space velocity (45868 hr 1 ) and constant feed temperature (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 catalyst 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 5.1 Numerical constants and parameters used in TWC simulation
108
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 6.2 Numerical constants and parameters used in TWC simulation
129
Table 6.3 Nominal properties of standard and thin walled Cordierite substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Table 6.4 Nominal properties of standard and thin-wall metallic substrate 154
Table 6.5 Physical properties of washcoat, ceramic and metallic substrate 155
xx
Nomenclature
Part I
a
crevice flow parameter
Ath
throttle area (m2 )
B
cylinder bore (m)
hCi
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))
Hjin
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)
Qfinuel
inlet volumetric fuel flowrate ( m3 =s)
Qair
in
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 (m3 P a)/(K mol))
b
R
ratio of connecting rod length to crank radius
R(C)
reaction rate at concentration C (mol/(m3 s))
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
ratio Air/ fuel actual to that at stoichiometry
crank angle (rad)
slope of road (rad)
angular speed (rad/s)
ij
stoichiometric coefficient of ith reaction and jth component
density of in-cylinder gaseous mixture (kg/m3 )
a
density of air (kg/m3 )
heat capacity ratio
Stefan -Boltzman constant W= (m2 K 4 )
emissivity
e
effective efficiency
m
power transmission efficiency
xxiii
Part 2
Symbols
Definition
3
s 1)
K
1
A
pre-exponential factor (mol m
a
pore radius (m)
Cp
specific heat capacity (J kg
D
diffusivity (m2 s 1 )
E
activation energy (J mol 1 )
h
heat transfer coefficient (W m 2 K
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
R
gas constant (J mol 1 K)
R
hydraulic radius of monolith channel (m)
t
time (s)
T
temperature (K)
T OSC
total oxygen storage capacity (mol m 3 )
hui
feed gas velocity (m s 1 )
X
mole fraction
Xf m
cup-mixing mole fraction in fluid phase
hXwc i
volume averaged mole fraction in washcoat
Xs
mole fraction at gas-solid interface
xxiv
1
3
)
1
)
s 1)
Greek symbols
fractional oxidation state of catalyst
w
void fraction (porosity) of washcoat
tortuosity
normalized air/fuel ratio
matrix of stoichiometric coefficient
density (kg/m3 )
w
effective wall thickness (m)
c
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 diagnostics and feedback control requirements to drastically minimize consumed fuel (recent regulations dictate a 54.5 mile per gallon fuel economy by 2025) and keep
harmful emissions to practically zero. Automotive engines are complex electromechanical 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 spatiotemporal scales. The two systems, engine and exhaust after-treatment, are highly
interdependent based on coupled operational constraints and interconnected dynamics with conflicting objectives. Therefore, diagnostics and feedback control
of the overall system are defined and constrained by the complexity of the coupled 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 optimization 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 conservation 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 relatively harmless CO2 , H2 O 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 formation 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 temperature 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-C7 H16 ) has a value of zero octane number and isooctane (C8 H18 ) 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 ratios 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 unthrottled 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 arrangement. 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 mechanical 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 washcoat 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 voltage (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 control
Environmental regulations limit the amount of carbon monoxide (CO), hydrocarbons (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 consumption 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 total 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)
Tier 1
50,0000
0.25
3.4
100,000
0.31
4.2
TLEV
50,000
0.125
3.4
120,000
0.156
4.2
LEV
50,000
0.075
3.4
120,000
0.09
4.2
ULEV
50,000
0.04
1.7
120,000
0.055
2.1
SULEV
120,000
0.01
1
PZEV
150,000
0.01
1.0
ZEV
-0-0-0-
NOx (g/miles)
0.4
0.6
0.4
0.6
0.05
0.07
0.05
0.07
0.02
0.02
-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 automobile 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 consideration. 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 ( ) = 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 coordinates 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/ ) model of turbulence and axis symmetric two dimensional 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 modification are still commonly used. Dinler and Yucel (2010) also used a similar kturbulence 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 (Heywood, 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 phenomenological 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) dynamically in time. As such they are computationally less expensive and are often used
for control application. Hendricks and Sorenson (1990) described MVM model consisting 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 different 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 involving 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 assumption 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, comprising 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 optimization 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 presented 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 Modeling
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 ordinary 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 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 capture 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 discussed.
2.1
Introduction
The plethora of information that a combustion model provides can help understand the complex sub-processes occurring in an internal combustion engine and
especially the various interdependencies between these processes. The combustion 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 models, 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 ignition 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 thermodynamic 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). However, 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 multidimensional 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 using 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 system with time dependent control volume (function of crank angle). The instantaneous volume of the cylinder V (t) as a function of crank angle is given by (Heywood
et. al., 1988),
1
V (t)
= 1 + (rc
Vc
2
b + (1
1) R
b2
R
(cos (t)))
1=2
sin2 (t)
;
(2.1)
b is the ratio of
where, Vc is the clearance volume, rc is the compression ratio, R
connecting rod length to crank radius and (t) is the crank angle at any time t:
Differentiating Eq. 2.1 w.r.t. time we get,
0
dV
B1
= Vc @ (rc
dt
2
where,
=
d
dt
2
6
1) 4sin +
sin cos
b2
R
sin
2
31
7C
1=2 5A
is the angular speed. In the present work,
;
(2.2)
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 stoichiometric 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 effect, 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 Bhattacharya 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 compartment 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 hCi. 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 dimensional 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 coarsegrain 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 single differential equation and an algebraic equation relating the two concentration
20
modes (the cup-mixing concentration, Cm , and the volume averaged concentration, hCi). 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 hCi
1
in
Cm
+
R
(hCi)
=
dt0
Cm
where
hCi =
1
in
t0mix;2 Cm
Cm ;
t0mix;1 Cm ;
is the residence time, t0mix;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 t0mix;2 captures the effect
of nonuniform feeding of the reactants. When both mixing times are zero, Cm = hCi
and the above model reduces to the classical ideal CSTR model. It was shown by
Bhattacharya et al. that when 0 <
tmix;2
1, the above two-mode model has the
same qualitative features as the full CDR equation. In general, the mixing times
t0mix;1 and t0mix;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
t0mix;1 =
|m
{z }1
+
|m
{z }3
+
micromixing
t0mix;2 =
m is
;
(2.3)
| M{z }4
;
(2.4)
macromixing
micromixing
where
| M{z }2
macromixing
the characteristic local scale mixing (also called micromixing) time
present within the tank (which depends on the molecular properties such as species
diffusivities), while
M
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
i;
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
m and
M,
both micro and macro mixing contributions may be important for liquid phase reactions 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 respectively. While case 2 corresponds to the batch operation condition. Model
development is first discussed for a general case and then special cases are discussed.
[Note: Bold letters represent vectors/matrices while scalers are written in normal font]. In a matrix form we can write mass balance for N number of interacting
perfectly mixed cells as,
d
(VR (t)C(t)) = Qin (t)Cin (t)
dt
where,8
>
< Vi
VR =
>
: 0
9
>
,i=j =
;
; i 6= j >
Q=
8
>
<
>
:
Qe (t)C(t) + Q(t)C(t)
n
j=1;i
Qcji
22
Qcij
9
>
; i=j =
; i 6= j
>
;
VR (t)R(C);
Qin =
8
>
< Qin
i
>
: 0
(2.5)
9
>
,i=j =
;
; i 6= j >
and
8
>
< Qex
i
Qe =
>
: 0
9
>
,i=j =
;
;
; i 6= j >
where, VR RN
N
and Vi is the volume of ith cell, C = [C1 C2 :::::]T RN
1
where
Ci is the spatially averaged concentration of each cell. Q, Qin and Qe RN
N
, Qcij is
the circulation flowrate from cell i to cell j, Qin
and Qex
i
i are the inlet and exit flow
from ith cell, respectively. The reaction rate vector R(C) RN 1 . The total volume
P
of the reactor is given by, VR = Vi , and the fractional volume are defined as i =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)
Q(t)C(t) =
. Rearranging Eq. 2.5 we get,
d
(VR (t) C(t)) + VR (t) R(C(t)) + Qe (t)C(t)
dt
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,
(2.7)
Q(t)y0 = 0;
with y0 = [1 1 1 1 :::::]T . Similarly adjoint eigenvector is given by
v0T Q(t) = 0 ;
with v0T = [1 1 1 1 :::::] : Let
= 1/jjQ(t)jj, then for the limit
(2.8)
! 0 (i.e. very fast
circulation flowrate), from Eqs. 2.6 and 2.7 we get C = hCi y0 i.e. when circulation
flow rate is very high, all the cells are in perfect communication and the concentration is uniform, given by hCi : For small but finite ( << 1), there exist a deviation
23
from equilibrium state, the concentration then is given by
C = hCi y0 + C0 ;
(2.9)
where,
C0 = w1 +
2
(2.10)
w2 + :::
Lets define the inner product as
yT x;
hx; yi
where x
RN
1
and y
RN
1
RN
and
N
(2.11)
is the volume fractions. The volume
averaged concentration is defined as
Pn
Vi Ci
:
C = Pi=1
n
i=1 Vi
v0T
hCi = hC; v0 i
(2.12)
Taking the inner product of Eq. 2.9 with v0 and using result from Eq. 2.12 gives
0
the solution to C ; v0 as
D
E
0
(2.13)
C ; v0 = 0:
Multiplying Eq. 2.6 by v0T on LHS and using Eq. 2.8 we get
0 = v0T
d
(VR (t) C(t)) + VR (t) R(C(t)) + Qe (t)C(t)
dt
Qin (t)Cin (t) : (2.14)
Now simplifying,
Term 1: VR (t)v0T C(t) = VR (t) hCi y0 ;
Term 2: v0T
R(C) = v0T
0
R( hCi y0 )+ R ( hCi y0 )C0
0
= R(hCi)+R (hCi) hC0 ; v0 i + O ( 2 ) = R(hCi)+O ( 2 ) ;
Term 3: (v0 )T Qe (t)C(t) = qe (t)Cm (t);
TQ C
v0
e
PN
ex
i=1 Qi
where, cup mixing concentration Cm
24
=
PN
ex
i=1 Qi Ci
P
N
ex ;
Q
i=1 i
Term 4: (v0 )T Qin (t)Cin (t) = qin (t)Cm;in (t);
where, Cm;in
Also qin (t) =
vT Qin C
P0N
in
i=1 Qi
X
=
PN
in
i=1 Qi Ci;in
PN
;
in
i=1 Qi
Qin
i and qe (t) =
X
is the total cumulative inlet and exit
Qex
i
flowrates respectively. Substituting above simplification in Eq. 2.14 we get
d
(VR hCi) + VR R( hCi )
dt
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 jjQ(t)jj >> 1, i.e. very
fast circulation, the concentration will be uniform and equal to hCi. Thus for zeroth
order, the model equation reduces to
d
(VR hCi) + VR R( hCi )
dt
qin (t)Cm;in (t) + qe (t) hCi = 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 hCi : 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,
0
QC =
d
(VR hCi )y0 + VR R(hCi )y0 + Qe (t) hCi y0
dt
Qin (t)Cin (t):
(2.17)
Now if we assume that although the total volume is a function of time, volume
fractions
are constant, i.e., each cell is varying at constant rate. Then, we can
rewrite Eq. 2.17 as
0
QC =
d
(VR hCi ) + VR R(hCi ) ( y0 ) +Qe (t) hCi y0
dt
Substituting Eq. 2.16 into Eq. 2.18 we get
25
Qin (t)Cin (t):
(2.18)
0
QC = [qin (t)Cm;in (t)
qe (t) hCi )] ( y0 ) +Qe (t) hCi y0
Qin (t)Cin (t):
(2.19)
Rearranging Eq. 2.19 we get,
0
C = inv(Q) qin (t) y0
Qin (t)Cin (t)
Cm;in (t)
Cm;in (t)
inv(Q) [qe (t)
Qe (t)] hCi 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 hCi can be related by
0
vT Qe C hCi v0T Qe y0 + vT0 Qe C
Cm = 0
=
;
qe
qe
(2.21)
0
v T Qe C
Cm = hCi + 0
qe
(2.22)
Substituting the result obtained in Eq. 2.20 after regularization we get
v0T Qe
Qin (t)Cin (t)
inv(Q) qin (t) y0
Cm;in (t) (2.23)
qe
Cm;in (t)
v0T Qe
inv(Q) [qe (t)
Qe (t)] hCi y0 ;
qe
Cm = hCi +
Cm = hCi +
where
mix;1 ;
mix;2
mix;2 Cm;in
mix;1 Cm ;
are the dimensionless mixing time given by
26
(2.24)
mix;2
=
v0T Qe
qe
mix;1
=
inv(Q) qin (t) y0
Qin (t)Cin (t)
;
Cm;in (t)
(2.25)
Qe (t)] y0 :
(2.26)
and
v0T Qe
qe
inv(Q) [qe (t)
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
mix;2
mix;1
=
= 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
mix;1
=
mix;2
= 0:
3. Exhaust stroke: Only exhaust valve is open. As there is no inflow
but
mix;1
mix;2
=0
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(hCj i)
1
=
Fjin
dt
V
Fj +
NR
X
ij Ri (hCi)V
i=1
dV
hCj i
dt
Fj;cr
;
(2.27)
"
NR
X
d(Ccr;j )
1
=
Fj;cr +
dt
Vcr
i=1
ij Ri (Ccr )Vcr
in
hCj i = tmix;2 Cm;j
tmix;1 Cm;j ,
(2.29)
a)Ccrj );
(2.30)
Cm;j
Fj;cr = Qcr (aCm;j
(1
#
#
;
(2.28)
where the suffix i and j stands for the reaction number and the gaseous component, respectively. Here, NR is the total number of reactions, Cm;j and hCj i is the
flow (velocity) weighted concentration and volume averaged concentration of the
j-th component, respectively. Fjin and Fj are the molar flow rates in and out of the
cylinder, respectively.
ij
gives the stoichiometric coefficient defined in standard
notation as negative for reactant and positive for products. Ri (hCi) 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 concentration 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 hCj i ; 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
in
. As discussed in the earlier section
by the inlet cup mixing concentration Cm;j
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 qualitative 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 hCi. 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
p
jP
Qcr = Qcr;0
(2.31)
Pcrevice j;
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
X
Cm;j
j
!
and
Pcrevice =
Nc
X
Ccr;j
j
(2.32)
RT
!
(2.33)
RTwall
In the above expressions R=8.314 J/K mol, is the universal gas constant and N c is
the total number of gaseous components. The total inlet volumetric flow rate is the
f uel
cumulative sum of the air (Qair
in ) and fuel flow rates (Qin ). The air flow rate can
be computed using the first principles based air path dynamic model (Franchek et.
al., 2006) as follows
Qair
in =
vol
Vd
:
2 2
Here, Vd is the engine displacement (Vd = (rc
(2.34)
1)Vc ). The volumetric efficiency
vol
of the induction process is given by
vol
=E
k
1
Pamb
Pman
rc
+
(rc
where E is an experimentally determined quantity,
1)
,
(2.35)
is specific heat ratio, Pamb and
Pman are the ambient and inlet manifold pressures, respectively. The inlet concentration 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 increase in blending (for example, xe ; the mole fraction of ethanol in the fuel) as well
as for an increase in desired
air=f uel
( (air=f
). Thus for a given air flow, the required
uel)
s
fuel flow rate can be computed as
Qfinuel =
1
(10:6
7:6xe )
Qair
in .
(2.36)
[The numerical factors 10.6 and 7.6 follow from the stoichiometry of fuel combustion 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
requirement.
The exit volumetric flow rate is modeled as the flow through an orifice and is
expressed as
Q = Cd Ath
p
2 (P
(2.37)
Pout ) = ;
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
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
X
Fjin Hjin
j=1
Nc
X
Fj Hj =
j=1
@ 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, Hjin 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
b=
E
m
X
(2.39)
Nj Ej ;
j=1
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
b Neglecting changes in K.E and P.E we have,
given by E.
b =
E
Nc
X
Nj Ej
j=1
Nc
X
Nj (Hj
=
P Vj )
j=1
Nc
X
=
Nj (Hj )
j=1
=
Nc
X
Nj (Hj )
P
Nc
X
Nj Vj
j=1
P VR ;
(2.40)
j=1
where Hj , Vj are the molar enthalpy and molar volume respectively. Differentiating the above equation w.r.t. time, we get
32
c
b X
dE
Nj
=
dt
j=1
N
where NT ot =
Nc
P
dHj
dt
+
Nc
X
Hj
dNj
dt
j=1
d
(NT ot RT ) ;
dt
(2.41)
Nj :
j=1
Also from the mass balance equation we have
dNj
= Fjin
dt
Fj +
NR
X
(2.42)
ij Ri (hCi)VR ;
i=1
substituting Eq. 2.42 in Eq. 2.41 we get,
c
b X
dE
=
Nj Cpj
dt
j=1
N
Nc
X
dT
dt
dT
Nj R
dt
j=1
Nc
Nc
X
dNj X
RT
+ Hj
dt
j=1
j=1
Fjin
Fj +
NR
X
ij Ri (hCi)VR
i=1
(2.43)
substituting above result in energy balance equation, we get
Nc
X
Nc
X
Fjin Hjin
WS +
:
Qheat
Fj Hj =
Nj Cpj
j=1
j=1
j=1
Nc
X
Nc
X
+ Hj
Nc
X
dT
dt
Fjin
dT
Nj R
dt
j=1
Fj +
j=1
NR
X
RT
Nc
X
dNj
j=1
dt
!
;
ij Ri (hCi)VR (2.44)
i=1
which simplifies to
:
Qheat
WS +
Nc
X
Fjin
Hjin
Hj
j=1
NR
X
Ri (hCi)VR (4HR;i )T + RT
i=1
=
Nc
X
j=1
where (4HR;i )T =
PNc
j=1
ij Hj .
Nc
X
dNj
j=1
Nj Cpj
dT
dt
Nc
X
j=1
Nj R
dt
dT
; (2.45)
dt
Assuming that the shaft work is equal to the work
33
!
;
done by the piston (P V_ R ) we get;
dT
1
=
PNc
dt
VR j=1 hCj i Cpj
3
PNc in
in
_
Hj 7
6 Qspark Qcoolant P VR + j=1 Fj Hj
4
PNR
PNc dNj 5 :
R
+
R
(hCi)V
(
4H
)
+
RT
i
R
R;i
i=1
j=1 dt
T
2
:
:
(2.46)
The above equation gives the temporal variation of temperature inside the combustion 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
P
j
hCj i V
+R T
Cp j
X d(hCj i V )
dt
R
+
"
! Qspark
NR
X
i
Qcoolant
PV +
Nc
X
Fjin Hjin
H
j
j=1
Ri (hCi) V ( 4HR;iT ) + Qcr (1
a)
Nc
X
Ccrj Hjcr
j=1
In the above equation, Qspark is the rate of energy added by the spark and is modeled 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 constant 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
(2.47)
H
j
#
:
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 onedimensional 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,
q_g = hc;g (T
T4
Tw;g ) +
4
Tw;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,
is the emissivity and
10 8 W= (m2 K 4 ) : Heat transfer within the
is the Stefan -Boltzman constant 5:67
cylinder wall is given as,
q_w =
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,
q C = hc;c (Tw;c
Tc ) ;
(2.50)
where hc;c is coolant heat transfer coefficient and Tc is coolant temperature. Assuming pseudo steady state and neglecting radiation, the above three equations
can be combined to give the wall heat flux as
q=
(T
1
hc;g
+
35
Tc )
l
k
+
1
hc;g
:
(2.51)
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
0:2 0:8
p T
0:55
w0:8 :
(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 = C1 Sp + C2
Vd Tr
(p
pr Vr
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 assuming that the cylinder pressure equilibrates with the inlet pressure at bottom
dead center (V = rc Vc ):
pm =
rc Vc
V
pin
(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 hydrocarbons 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
no
1
2
3f
3b
4f
4b
5f
5b
6
7
Table 2.2: Global kinetics for propane and ethanol combustion
reaction
orders of reaction
ref
0:25
1:5
17
C8 H18 + 2 O2 ! 3CO + 4H 2 O
[C8 H18 ] [O2 ]
Westbrook et al., 1981
0:1
1:85
(CH 2 )n +nO2 ! nCO + nH 2 O [(CH 2 )n ] [O2 ]
(modified)
0:5
:25
1
CO+ 2 O2 ! CO2
[CO] [H2 O] [O2 ]
Westbrook et al., 1981
1
CO2 ! CO+ 2 O2
[CO2 ]
Westbrook et al., 1981
CO + H 2 O ! CO2 +H 2
[CO] [H2 O]
Jones et al., 1988
CO2 +H 2 ! CO + H 2 O
[CO2 ] [H2 ]
N2 +O2 ! 2N O
[O2 ]0:5 [N2 ]
Heywood et al., 1988
2
0:5
2N O ! N2 +O2
[NO] [O2 ]
1
H2+ 2 O2 ! H2 O
[H2 ] [O2 ]0:5
Marinov et. al., 1995
C2 H5 OH + 2O2 ! 2CO + 3H 2 O [C2 H5 OH]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 twolump 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 temperature 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 combustion 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 system 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 . Reaction 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
Ea
11
1
5.7 10
0
30000
2
1.2 1011
30000
14
3f
3.98*10
0
40000
8
3b
5 10
0
40000
4f
2:75 1012
0
20000
4b
5f
6 1016
0:5 137281
5b
6
1:8 1013
0
34500
12
7
1:8 10
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 dynamics. 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 manifold 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 ordinary 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 (C8 H18 , (CH 2 )n ; O2 , CO2 , H2 O,
38
N2 , CO, NO, H2 , C2 H5 OH) balances each for the crevice and combustion cylinder 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 initialized 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 variables. Since our goal here is to validate the model by comparing the model predictions with available experimental data and examine some key trends (and not an
exhaustive parametric study), we have fixed the values of some of these parameters 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 concentrations, the temperature and pressure inside the combustion chamber under
stoichiometric ( = 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 (Figure 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 cm
Clearance length
1:27 cm
rpm
1500
Compression ratio, rc
9
^
Crank length/Rod, R
4
Cylinder wall thickness, l
1 cm
Coolant thermal conductivity,hc;c 750 W=m2 K
Wall thermal conductivity, k
54 W=mK
Ambient temperature
298 K
Ambient pressure, Pamb
1 atm
Exhaust manifold pressure, Pout 1.1 atm
Inlet manifold pressure, Pman
0.8 atm
Coolant temperature, Tc
373 K
Cd
0.1
E
0:6
1:33
tmix;1
0:2
tmix;2
0
a corresponding peak in temperature and pressure caused by the heat released
by the highly exothermic combustion reactions. These sharp gradients in temperature, 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, especially 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 4 crank angle rotation. At
= 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
1
0.9
Intake manifold pressure (atm)
0.8
0.7
I n cre a s in g
th ro tt le a n g le
0.6
0.5
0.4
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Tim e ( s )
0.7
0.8
0.9
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
= 1800 , when the piston reaches
the bottom dead center (BDC) the compression stroke begins. The compression work done by the piston leads to an increase in the in-cylinder temperature
41
1
5
4
2
x 10
2.5
x 10
(b)
2
1.5
oxygen (ppm)
Unburned hydrocarbon (ppm)
(a)
1
1.5
1
0.5
0.5
0
0
0.2
0.4
0.6
Time (s)
0.8
0
0
1
25
0.2
0.4
0.6
Time (s)
0.8
1
3500
(c)
(d)
3000
20
Temperature (K)
pressure (atm)
2500
15
10
2000
1500
1000
5
500
0
0
0.2
0.4
0.6
Time (s)
0.8
1
0
0
0.2
0.4
0.6
Time (s)
0.8
Figure 2.2: In-cylinder temporal variation of (a) Total unburned hydrocarbon concentration, (b) Unburned oxygen concentration, (c) Pressure and (d) Temperature
with time
42
1
4
14
x 10
8000
7000
12
NO conc (ppm)
8
2
CO (ppm)
(b)
6000
10
6
4
5000
4000
3000
2000
(a)
2
1000
0
0
0.2
0.4
0.6
Time (s)
0.8
0
0
1
0.2
0.4
0.6
Time (s)
0.8
1
4
3.5
x 10
1600
3
1400
(c)
1200
Hydrogen (ppm)
CO (ppm)
2.5
2
1.5
1
1000
800
600
400
0.5
0
0
(d)
200
0.2
0.4
0.6
Time (s)
0.8
1
0
0
0.2
0.4
0.6
Time (s)
0.8
Figure 2.3: Temporal variation of in-cylinder (a) CO2 concentration, (b) N Ox concentration, (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 propagation introducing a delay to obtain peak temperature. This phenomenon can be
43
1
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
= 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 predicted 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 12%, 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
25
(a)
Pressure (atm)
20
15
10
5
0
0
100
200
300
400
500
Crank angle (deg)
600
700
2500
(b)
Temperature(K)
2000
1500
1000
500
0
0
100
200
300
400
500
Crank angle (deg)
600
700
Figure 2.4: In-cylinder variation of (a) Pressure and (b) Temperature during a complete cycle after a periodic state is attained
45
410
400
3000
(a)
O conc (ppm)
380
370
2
HC conc (ppm)
390
360
2500
2000
350
1500
340
330
0
0.2
0.4
0.6
Time (s)
0.8
1
0
6000
(b)
0.2
0.4
0.6
Time (s)
(c)
(d)
2500
NO conc (ppm)
CO conc (ppm)
1
3000
5000
4000
3000
2000
1000
0
0.8
2000
1500
1000
0.2
0.4
0.6
Time (s)
0.8
1
500
0
0.2
0.4
0.6
Time (s)
0.8
Figure 2.5: Variation of exhaust gas concentrations with time (a) Unburned hydrocarbon, (b) Exhaust oxygen, (c) Exhaust N Ox 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
1
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 conversion 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 complex combustion mechanism for predicting the regulated gas emissions. Unlike the
Wiebe function based model where burn duration and ignition delay are the predefined 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 ratio ( ) on the exhaust gas composition. Shown in Figure 2.8 is the variation of
peak temperature and pressure with change in : 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
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
Normalized reaction rate
1
HC oxidation
CO oxidation
N oxidation
0.8
2
0.6
0.4
0.2
0
1000
Normalized reaction rate
1
0.8
1500
2000
Temperature (K)
2500
slow burn HC
WGS
H oxidation
2
0.6
0.4
0.2
0
1000
1500
2000
Temperature (K)
2500
Figure 2.6: Normalized reaction rate as a function of temperature over a cycle
48
1
conversion
0.8
0.6
fast burn HC
slow burn HC
Total HC
0.4
0.2
0
300
320
340
360
380
400
420
crank angle (degree)
440
460
480
Figure 2.7: Hydrocarbon conversion with crank angle for =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 manipulated to change
in :
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 . This is observed
because the NOx formation is a strong function of temperature and oxygen concentration (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
22.5
2500
22
2450
21.5
2400
21
λ=1
Rich
Le an
2350
2300
0.8
Pressure (atm)
Temperature (K)
2550
20.5
0.85
0.9
0.95
1
λ
1.05
1.1
1.15
1.2
20
1.25
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 increase in oxygen concentration and the peak for NO concentration is observed at a
slightly leaner condition, around
= 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
6000
1200
λ=1
(a)
CO
1000
4000
800
3000
600
2000
400
1000
200
0
0.85
0.9
Rich
0.95
1
1.05
Normalized air to fuel ratio
1.1
Lean
1.15
x
HC
NO ppm
mole fraction ppm
5000
0
1.2
8000
8
x
mole fraction ppm (HC and NO )
(b)
6000
X
6
CO
4000
2000
0
0.7
4
2
HC
0.8
0.9
1
1.1
Normalized air to fuel ratio
1.2
Figure 2.9: Variation of regulated exhaust gases with air fuel ratio (a) Predicted
from low-dimensional model and (b) Experimentally observed [12]
51
0
1.3
CO %
NO
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 ethanolgasoline 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. Therefore, 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 qualitative 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 formation 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
1000
2500
NO x
NO x
2400
CO
600
400
2300
2200
HC
200
0
0
CO ppm
mole fraction ppm
800
2100
0.1
0.2
0.3
0.4
0.5
0.6
Ethanol volume fraction
0.7
0.8
0.9
2000
1
2485
21.6
2480
2475
0
21.5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Ethanol volume fraction
0.8
0.9
21.4
1
Figure 2.10: (a) Impact of blending on emissions and (b) In-cylinder temperature
and pressure under constant air/fuel ratio of =1
53
Pressure (atm)
Temperature (K)
(b)
C H oxid.
8
1
(a )
8
C H oxid.
2
8
CO oxi d.
8
C H oxid.
2
2
N oxid
0 .8
2
Normalized reaction rate
Normalized reaction rate
(b)
C H oxid.
1
2
C H O H oxid
2
5
0 .6
0 .4
0 .2
0 .8
CO oxid.
N oxid
2
0 .6
0 .4
0 .2
0
0
1000
1500
2000
T em pera ture (K )
1000
2500
1500
2000
Tem pera ture (K )
2500
5
10
x 10
50
E0
E50
9
E0
E5 0
(c)
8
40
7
NO reaction rate
(d)
5
4
3
x
CO oxidation rate
6
2
1
30
20
10
0
0
340
350
360
370
380
390
400
cra n k a n g le(deg )
410
420
430
340
350
360
370
380
C ra nk a ng l e (deg )
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) N Ox formation rate
54
390
400
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 ). 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
1400
3000
(a)
2500
1000
2000
800
1500
600
1000
400
200
0
0
0.1
500
CO
HC
NO
0.2
0.3
x
0.4
0.5
0.6
Ethanol volume fraction
0.7
0.8
0.9
3000
0
1
22
(b)
P
20
T
2000
18
1500
16
1000
0
0.1
0.2
0.3
0.4
0.5
0.6
Ethanol volume fraction
0.7
0.8
0.9
Pressure (atm)
Temperature (K)
2500
14
1
Figure 2.12: Impact of blending on (a) Emissions and (b) In-cylinder temperature
and pressure at constant flowrate ( goes leaner)
56
CO ppm
mole fraction ppm
1200
4500
(b)
concentration (ppm)
4000
NO x
3500
HC
3000
2500
2000
1500
1000
250
300
350
400
450
500
imep, Kpa
550
600
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
650
700
2000
2700
650
32
(b)
(a )
2600
500
28
2500
2400
24
Pressure (atm)
450
Temperature (K)
1000
HC ppm
550
NO
x
ppm
1500
350
2300
0
1000
1500
2000
2500
3000
rpm (rev /m in)
3500
2200
1000
250
4000
3500
1500
2000
2500
3000
rpm (rev/min)
20
4000
3500
3000
(c)
mole fraction ppm
(d)
2500
x
NO ppm
3000
2000
2500
NO
x
HC
2000
1500
1000
1200
1400
1600
1800
2000
Speed rev/min
2200
2400
1500
1000
1200
1400
1600
Speed rev/min
1800
Figure 2.14: Impact of engine speed on (a) N Ox and HC emission as predicted by
model, (b) In-cylinder peak temperature and pressure, (c) Experimentally reported
N Ox with change in speed (Celik, 2008) and (d) Experimentally reported HC and
N Ox with change in engine speed (Heywood, 1988)
2.3.4
Sensitivity of the model
i
Sensitivity ( @X
) is defined as how a desired output (Xi : peak temperature ,
@pj
peak pressure, exit concentration of hydrocarbon, CO and NOx ) varies with system parameters (pj ) like dimensionless mixing time (
mix;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
2000
Sensitivity to mixing time
In the base case model considered in this work there is no valve overlap. Thus,
mix;2 =0,
for all of the stages in the engine cycle and
mix;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
time
mix;1 ;
mix;1
is a positive parameter. Increasing the mixing
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
mix;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 combustible leftover gases act like a diluent (or inert), increasing the specific heat of
the system thereby reducing the temperature inside the cylinder. Since NO formation is very sensitive to temperature change, the concentration of NOx drops as
the mixing time increases. The HC conversion is a weak function of
mix;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
mix;1
on emissions, as well as, peak temperature and
pressure.
It may be noted that in the limit
mix;1
! 0 the model reduces to a classical
one-mode ideal combustion chamber model with Cm = hCi. Though this onemode 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
2400
2800
CO
2500
NO x
1600
2000
(a)
1200
1500
CO ppm
mole fraction ppm
2000
800
0
0
1000
HC
400
0.05
0.1
0.15
τmix1
0.2
0.25
500
0.35
0.3
2700
25
P
max
(b)
2500
T
20
Pressure (atm)
Temperature (K)
2600
max
15
2400
2300
0
0.05
0.1
0.15
τ
0.2
0.25
0.3
10
0.35
mix1
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
mix;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 energy rate accordingly leads to an earlier start of ignition as compared to the slow
61
460
2660
(a )
450
(b)
2650
440
Temperature (K)
HC ppm
430
420
410
2640
2630
400
2620
390
380
0
0 .5
1
1 .5
2
2 .5
3
3 .5
4
4 .5
crev ice v o lu m e (% o f clea ra n ce v o lu m e)
2610
0
5
2350
1
2
10
2400
2300
2350
(c )
(d)
2300
2200
2250
ppm
2250
x
2150
2200
NO
CO ppm
3
4
5
6
7
8
9
crev ice v o lu m e (% o f clea ra n ce v o lu m e)
2100
2150
2050
2100
2000
2050
1950
0
0 .5
1
1 .5
2
2 .5
3
3 .5
4
4 .5
crev ice v o lu m e (% o f clea ra n ce v o lu m e)
5
2000
0
0 .5
1
1 .5
2
2 .5
3
3 .5
4
4 .5
crev ice v o lu m e (% o f clea ra n ce v o lu m e)
Figure 2.16: Impact of change in crevice volume on (a) Hydrocarbon emission, (b)
In-cylinder temperature, (c) CO emission and (d) N Ox emission for mix;1 =0 and
mix;2 =0
62
5
spark, and the unburned hydrocarbon decreases and NOx emission increases. A
3.5 times shorter ignition duration resulted in around 10% reduction in hydrocarbon 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 emission, temperature and pressure trend could be predicted quite accurately using
the low dimensional model developed here; the hydrocarbon emission was underdetermined. 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 hydrocarbon 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 component 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 incylinder temperature and pressure. The in-cylinder temperature increases, as expected, 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 cylinder 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 hydrocarbon emission increases by around 3% for 10% increase in feed temperature
(compared to ambient conditions) and the NOx emission shows around 4% increase. 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 aftertreatment 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 ignition (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
1010
360
(b)
1000
990
355
NO concentration (ppm)
350
980
970
960
x
Unburned hydrocarbon (ppm)
(a)
345
950
940
340
930
290
335
290
300
310
320
T K
330
340
300
310
320
T K
330
340
350
in
350
in
2494
22
(c)
Pressure (atm)
Temperature (K)
(d)
21.5
2492
2490
2488
21
20.5
20
2486
19.5
2484
290
300
310
320
T K
330
340
350
19
290
300
310
320
T K
330
340
in
in
Figure 2.17: Impact of change in inlet temperature on (a) Hydrocarbon emission,
(b) N Ox 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
350
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 requires 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 control 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),
I
d (t)
= Te (t)
dt
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 literature (Saerens et. al., 2009) as
:
mf Qv
Te (t) = e
;
2
where Qv is the heating value of gasoline and
e
(2.56)
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,
Fv vv = 2
Tl
m;
(2.57)
where Fv is the force acting on the wheel of the vehicle and vv is the vehicle velocity
and
m
is the power transmission efficiency,
vv = Rw
2
;
iD iG
(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 = Mv
v2
dvv
+ SCx a v + fr Mv g cos + Mv g sin ;
dt
2
(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,
cient and
a
is the density of air, fr is the friction coeffi-
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 manifold. 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 composition after combustion. This will give us ; 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 fundamentals 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 verified 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 formation 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 ( ), while the NOx exhibits a maxima occurring for slightly leaner mixture 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; Assumptions: (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 discussed above, the quantitative predictions of the model can be improved by ex68
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 Ignition
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 homogeneous 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(hCj i)
1
=
Fjin
dt
V
Fj +
NR
X
ij Ri (hCi)V
i=1
hCj i
dV
dt
;
(3.1)
"
NR
X
d(Ccr;j )
1
=
Fj;cr +
dt
Vcr
i=1
ij Ri (Ccr )Vcr
in
hCj i = tmix;2 Cm;j
tmix;1 Cm;j ,
(3.3)
a)Ccrj ):
(3.4)
Cm;j
Fj;cr = Qcr (aCm;j
(1
#
Fj;cr
#
(3.2)
;
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
P
j
hCj i V
+R T
Cp j
X d(hCj i V )
dt
R
+
!
"
NR
X
i
Qcoolant
PV +
Nc
X
Fjin Hjin
H
(3.5)
j
j=1
Ri (hCi) V ( 4HR;iT ) + Qcr (1
71
a)
Nc
X
j=1
Ccrj Hjcr
H
j
#
:
Pressure (atm)
30
20
10
0
0
100
200
300
400
500
Crank angle (de g)
600
700
100
200
300
400
500
Crank angle (de g)
600
700
Temperature(K)
3000
2500
2000
1500
1000
0
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
1
hc;g
+
72
Tc )
l
k
+
1
hc;g
:
(3.6)
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 balance 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 assumes 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 constant energy source, which was assumed to add energy uniformly. While in a real
73
4
x 10
8000
NO conc (ppm)
CO
2
(ppm)
10
5
0
0
1
2
Tim e (s)
6000
4000
2000
0
0
3
1
2
Tim e (s)
3
1
2
Tim e (s)
3
4
x 10
Hydrogen (ppm)
CO (ppm)
3
2
1
0
0
1
2
Tim e (s)
3
1000
500
0
0
Figure 3.2: In-cylinder emissions for a HCCI engine
74
8000
CO conc (ppm)
HC conc (ppm)
30
20
10
0
-10
-20
1
2
Time (s)
4000
2000
0
3
1
2
Time (s)
3
1
2
Time (s)
3
14
H O conc (%)
3000
2500
2000
13.95
13.9
2
NO conc (ppm)
6000
1500
1000
0
1
2
Time (s)
3
13.85
13.8
0
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 Converter 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 approximating 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 reductants. 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) system. 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
H2 O while reducing NOx to N2 . The air-fuel mixture entering the TWC is often
78
Table 4.1: LEV II Emission standards
under 8500 lbs, g/mi [CEPA, 2011]
Category 50,000 miles/ 5 years
NMOG CO NOx PM
LEV
0.075
3.4 0.05 ULEV
0.040
1.7 0.05 SULEV
-
for passenger cars and light duty vehicles
120,000 miles / 11 years
HCHO NMOG CO NOx PM
0.015 0.09
4.2 0.07 0.01
0.118 0.055
2.1 0.07 0.01
0.01
1.0 0.02 0.01
HCHO
0.018
0.011
0.004
quantified using the normalized air to fuel ratio (A/F), defined as
=
Thus,
(A=F )actual
:
(A=F )stoichiometry
> 1 corresponds to a (fuel) lean operation while
< 1 corresponds to a
rich operation. It is well known that there exists a narrow zone around stoichiometry
( = 1) where the TWC efficiency is simultaneously maximum for all the major pollutants (Heywood, 1988; Heck et al., 2009). Thus, gasoline engines are normally
controlled to operate around stoichiometry. However, in real world operating conditions, 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 storage, 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 oxygen 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 oxygen sensor with sharp transition around stoichiometry, UEGO can be used to measure 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 volume 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 fueling 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 consideration 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 measurement, 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 emissions 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 feasible. 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 system, 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 empirically 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 appropriate 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 respect to oxygen storage and is yet computationally efficient enough for implementation 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 using 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 converter and the physical phenomena occurring over a single channel. The TWC can
be modeled as a three-dimensional system involving convection-diffusion and reaction 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 represented 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 axially 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
d
@X
fm
=
@t
hui
d
@X
fm
@x
kmo d
Xf m
R
\
hX
wc i :
(4.1)
The species balance in the washcoat (for gas phase species) is
\
1
@ hX
wc i
=
w
@t
CT otal
T
b
r+
kmo
c
d
X
fm
\
hX
wc i :
(4.2)
The energy balance in the fluid phase is
f Cpf
cf
@T
=
@t
hui
f Cpf
cf
@T
@x
h
R
cf
T
Tbs ;
(4.3)
and the energy balance for the washcoat is
w w Cpw
@ Tbs
=
@t
w kw
@ 2 Tbs
cf
+h T
@x2
with the boundary conditions given by
83
Tbs + cb
rT (
H) :
(4.4)
0
\
X
f m;j (x) = Xf m;j (x)
@t = 0
in
hX\
wc;j (x)i = Xwc;j (x)
@t = 0
(4.5)
(4.6)
0
\
T
f (x) = Tf (x)
@t = 0
(4.7)
0
T[
s (x) = Ts (x)
@t = 0
(4.8a)
Xf m;j (t) = Xfinm;j (t)
Tf (x; t) = Tfin (t)
@x = 0
@x = 0
(4.9)
(4.10)
@Ts
=0
@x
@x = 0
(4.11)
@Ts
=0
@x
@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
Xf m
1
=
L
ZL
0
d
X
f m dx:
(4.13)
Similar defination were used for other variables as hXwc i ; Tf and Ts : We asd
sume X
f m (L) = Xf m ; i.e the exit concentration is assumed to be same as the
84
concentration within the reactor, a continuous stirred tank reactor (CSTR) assumption. 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
dXf m
=
dt
hui
Xf m
L
kmo
(Xf m
R
Xin
f m (t)
hXwc i) :
(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
w
1
d hXwc i
=
dt
CT otal
T
r+
kmo
(4.15)
hXwc i) :
(Xf m
c
The overall mass transfer coefficient matrix (kmo ) is given by
kmo1 = kme1 + kmi1 ;
(4.16)
where kme and kmi are the external and internal mass transfer coefficient matrices.
The averaged energy balance in the fluid phase is
f Cpf
dTf
=
dt
hui
f Cpf
L
Tf
h
(Tf
R
Tfin (t)
Ts ) ;
(4.17)
and the average energy balance for the washcoat is
w
dTs
= h (Tf
w Cpw
dt
Ts ) +
c
Nr
X
ri (
Hi ) :
(4.18)
i
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,
c
is the washcoat thickness and
(defined as sum
s
+ c ; where
s
w
represents the effective wall thickness
is the half-thickness of wall) ,
w
and Cpw are the
effective density and specific heat capacity, respectively, defined as
c c Cpc
+
s s Cps ,
w w Cpw
=
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),
of ceria as,
=
[Ce2 O4 ]
:
[Ce2 O4 ] + [Ce2 O3 ]
(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 Ce2 O3 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 is proportional to the rate of change
of Ce2 O4 : Thus,
1
d
=
(rstore
dt
2T OSC
rrelease ) ;
(4.20)
where rstore and rrelease are the rate of formation and concumption of Ce2 O4 , respectively.
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
N
transfer to the washcoat. Here, the column vectors, Xf m and Xin
f m (t) 2 R ; repre-
sent the exit and inlet mole fractions of the species in the fluid phase, respectively.
The column vector, r 2 RN r where each element ri 8 i 2 [1; N r]; represents the
rate of the ith reaction. The parameters N and N r represent the total number
86
of gaseous species and reactions, respectively. The stoichiometric matrix,
RN r
N
; is a matrix of stoichiometric numbers with rows representing the reaction
index while the columns represents species index. The average feed gas velocity,
hui ; is computed using the measured air mass and known A/F ratio (or ). The
total concentration (CT otal ) is computed at the channel inlet using the ideal gas law
CT otal =
P
:
RTfin (t)
(4.21)
Here, P represents the total gas pressure, assumed constant as one atm. By expressing Eq. 5.1 and 5.2 in mole fractions, we inherently assume that CT otal 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 experimentally. 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 ,
T otal 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 2 RN
N
is defined
by
kme =
Df Sh
:
4R
Here, Sh is a diagonal matrix given by, Sh =Sh1 I, where I 2 RN
(4.22)
N
is the identity
matrix. The asymptotic value of Sh1 depends on the flow geometry as well as
the kinetics. Here, we use a constant value corresponding to the fast reaction
87
inlet
exit
Mole fraction
0.14
0.135
0.13
0.125
1800
1850
1900
1950
2000
Time (s)
2050
2100
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 2 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 9 Tf1:68
[in m2 s 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 =
Ds Shi
c
88
:
(4.23)
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 =
w
97a
r
Ts
;
Mi
(4.24)
where Mi is the molecular mass of the ith species. Here,
ity,
w
is the washcoat poros-
is the tortuosity; a is the mean pore size and Ds is in m2 s 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 2 RN
N
; is evaluated as a function of
Thiele matrix ( ) as follows (Balakotaiah, 2008)
Shi = Shi;1 + (I +
)
1
2
(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 constants kef f by the Jacobian of the rate vector evaluated at the washcoat-gas interfacial conditions]. For the case of a square channel with a rounded square flow
area, the internal asymptotic Sherwood matrix is given by, Shi;1 = Shi;1 I;where
Shi;1 = 2:65 and the constant
2 RN
N
= 0:58 (Joshi et al., 2009). The Thiele matrix,
2
, is defined as
2
=
2
1
c (Ds )
d (R(X))
CT otal
dX
1
=
2
1
c (Ds ) (
(4.26)
J):
X=XS
The Jacobian, J= CT1otal dR(X)
is the derivative of the rate vector w.r.t concentration,
dX
evaluated at gas-washcoat interfacial concentrations. Here, R(X) =
89
T
r(X); i.e.
R(X) 2 RN and represents the overall reaction rate for each species. For nonlinear kinetics with multiple species, the Jacobian might become a non-diagonal
matrix. This happens because of the coupling between the species due to reactions. 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
(kme Xf m + kmi hXwc i) :
(4.27)
For computational simplification, J can be evaluated at bulk (fluid phase) conditions. 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., k k >> 1); Shi approaches
. The above procedure for calculating kmi is valid only if the matrix
2
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,
2
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. However, 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
=
c
Ds Shi
90
+
4R
:
Df Sh1
(4.28)
Since Sh1 =4 is approximately unity but Shi can have values above two, the importance of internal and external mass transfer depends on the relative values of
and
c
Ds
R
Df
. 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;ef f =
2
Ri (X)
1
CT otal Xi
in which case,
X=Xf m
becomes a diagonal matrix with the diagonal terms defined as
2
ii
=
2
c
Ds;i
(4.29)
ki;ef f ;
where Ri (X) represent the net rate of formation of the ith species (thus, for reactants
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 (N u) correlation, as follows
h=
N u kf
:
4R
(4.30)
An asymptotic value of N u=N u1 = 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:
Xf m;j = Xf0m;j
@t=0
hXwc;j i = Xf0m;j
@t=0
Tf0
@t=0
Ts = Ts0
@t=0
Tf =
91
j 2 [1; ::; N ]
(4.31)
Table 4.2: Numerical constants and parameters used in TWC simulation
Constants Value
a
10 10 9 m
R
181 10 6 m
30 10 6 m
c
2 s
63:5 10 6 m
kf
0:0386 W m 1 K 1
Cpf
1068 Jkg 1 K
Cpw
1000 Jkg 1 K
2000 kg m 3
w
0:41
w
8
Sh1
3:2
N u1
3:2
Shi;1
2:65
0:58
sl.no. Reaction
H(J=mol)
1
CO + 0:5O2 ! CO2
-2.83e5
2
H2 + 0:5O2 ! H2 O
-2.42e5
3
C3 H6 + 4:5O2 ! 3CO2 + 3H2 O
-1.92e6
4
N O + CO ! CO2 + 0:5N2
-3.73e5
5
N O + H2 ! H2 O + N2
-3.32e5
6
CO + H2 O
CO2 + H2
-4.1e4
7
C3 H6 + 3H2 O
3CO + 6H2
3.74e5
8
CO + Ce2 O4 ! Ce2 O3 + CO2
-1.83e5
1
1
1
9
C
H
+
Ce
O
!
Ce
O
+
CO
+
H
O
-1.14e5
2 4
2 3
2
9 3 6
3
3 2
10
Ce2 O3 + 0:5O2 ! Ce2 O4
-1e5
11
Ce2 O3 + N O ! Ce2 O4 + 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 composition, 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 becomes 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 LevenbergMarquardt method is applied. The advantage of using a GA method is it’s a heuristic 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 .
errorT otal =
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 = RM SE (
Here,
pred
and
act
pred
actual ):
(4.33)
are the predicted and actual conversions, respectively. The
conversion is computed using the relation
pred
pred
actual
Xin
Xexit
=
;
actual
Xin
(4.34)
actual
actual
Xin
Xexit
:
actual
Xin
(4.35)
and
act
=
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
actual
actual
and
is the measured concentration at the TWC inlet, while Xexit
Here, Xin
pred
Xexit
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 conversion 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
f itnessi = 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 problems.
4.3.1
Experimental Set-up
The experimental data was collected using a 3.5L-4V-V6 2008 model year Mercury. For our experiment, we sampled the right bank only. Table 4.4 lists the properties 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 percent 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 =1
95
80
3
60
2
40
1
20
Vehicle speed mph
λ in
4
0
0
580 1000 1500 2000 2500 3000 3500 4000 4500 5000
time s
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 experimentally 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 observed. 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 followed 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
1
calc
expm
CO conversion
0.8
0.6
Lean
Rich
0.4
0.2
0
1600
1620
1640
time(s)
1660
1680
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
1
calc
expm
0.95
HC conversion
0.9
0.85
0.8
0.75
0.7
0.65
1600
1620
1640
time(s)
1660
1680
Figure 4.7: Comparision of model predicted and experimental HC conversion for
lean to rich step change at a constant vehicle speed of 30 mph
97
1.2
1
NO conversion
0.8
calc
expm
0.6
0.4
0.2
0
-0.2
1600
1620
1640
time(s)
1660
1680
Figure 4.8: Comparision of model predicted and experimental NO conversion for
lean to rich step change at a constant vehicle speed of 30 mph
and requires high temperature for its formation, which is unlikely in TWC environment. 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 reduced 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 conditions 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
fractional oxygen content
1
0.8
0.6
0.4
0.2
0
1630
1632
1634
1636
time s
1638
1640
1642
Figure 4.9: Fractional oxidation state of the catalyst during lean to rich step change
530
brick
fe e dgas
0
Temperature C
520
510
500
490
480
1600
1620
1640
Time (s)
1660
1680
Figure 4.10: Catalyst wall (brick) and feedgas temperature for a lean to rich step
change experiment
99
14000
calc(midbe d)
meas(midbe d)
meas(fe e dgas)
CO concentration ppm
12000
10000
8000
6000
4000
2000
0
2550
2600
2650
2700
time
Figure 4.11: comparision of model predicted vs experimentally observed CO emission 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 using 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
1400
NO concentration ppm
1200
1000
800
calc(midbe d)
meas(midbe d)
meas(fe e dgas)
600
400
200
0
2550
2600
2650
Time (s)
2700
Figure 4.12: comparision of model predicted vs experimentally observed NO emission at constant vehicle speed of 60 mph
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
600
HC concentration ppm
500
400
300
calc(midbe d)
meas(midbe d)
meas(fe e dgas)
200
100
0
2550
2600
2650
Time (s)
2700
Figure 4.13: comparision of model predicted vs experimentally observed HC emission at constant vehicle speed of 60 mph
102
5
1.38
x 10
calc(midbe d)
meas(midbe d)
meas(fe e dgas)
1.34
1.32
1.3
1.28
2
CO concentration ppm
1.36
1.26
1.24
1.22
2550
2600
2650
Time (s)
2700
Figure 4.14: comparision of model predicted vs experimentally observed CO2
emission at constant vehicle speed of 60 mph
103
14000
2
O concentration ppm
12000
10000
8000
calc(midbe d)
meas(midbe d)
meas(fe e dgas)
6000
4000
2000
0
2550
2600
2650
Time (s)
2700
Figure 4.15: comparision of model predicted vs experimentally observed O2 emission at constant vehicle speed of 60 mph
104
Chapter 5 Low-dimensional Three-way Catalytic Converter 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 reduced order model consists of seven ordinary differential equations and captures
the essential features of a TWC providing estimates of the oxidant and reductant emissions, fractional oxidation state (FOS), and total oxygen storage capacity
(TOSC). The model performance is tested and validated using data on actual vehicle 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 capture 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 efficiency 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
dXf m
=
dt
hui
Xf m
L
Xin
f m (t)
105
kmo
(Xf m
R
hXwc i) :
(5.1)
The species balance in the washcoat (for gas phase species) is
w
1
d hXwc i
=
dt
CT otal
T
r+
kmo
(5.2)
hXwc i) :
(Xf m
c
The overall mass transfer coefficient matrix (kmo ) is given by
kmo1 = kme1 + kmi1 ;
(5.3)
where kme and kmi are the external and internal mass transfer coefficient matrices.
The external mass transfer coefficient matrix kme 2 RN
N
is defined by
Df Sh
:
4R
kme =
(5.4)
Here, Sh is a diagonal matrix given by, Sh =Sh1 I, where I 2 RN
N
is the identity
matrix. The asymptotic value of Sh1 with a numerical value of 3.2 corresponding
to rounded square channel is used. The internal Sherwood number matrix, Shi
2 RN
N
; is evaluated as a function of Thiele matrix ( ) as follows (Balakotaiah,
2008)
Shi = Shi;1 + (I +
)
1
2
An effective rate constant for each reactant species as ki;ef f =
in which case,
2
(5.5)
:
Ri (X)
1
CT otal Xi
;
X=Xf m
is a diagonal matrix with the diagonal terms defined as
2
ii
=
2
c
Ds;i
(5.6)
ki;ef f :
The energy balance in the fluid phase is
f Cpf
dTf
=
dt
hui
f Cpf
L
Tf
106
Tfin (t)
h
(Tf
R
Ts ) ;
(5.7)
and the energy balance for the washcoat is
w
dTs
= h (Tf
w Cpw
dt
Ts ) +
c
Nr
X
ri (
Hi ) :
(5.8)
i
The oxygen storage on ceria, we define the fractional oxidation state (FOS),
of
ceria as,
d
1
=
(rstore
dt
2T OSC
rrelease ) ;
(5.9)
or
d
1
=
(r2
dt
2T OSC
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:
Xf m;j = Xf0m;j
@t=0
hXwc;j i = Xf0m;j
@t=0
Tf0
@t=0
Ts = Ts0
@t=0
Tf =
j 2 [1; ::; N ]
(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. 5.15.2) 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 approaches 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
Table 5.1: Numerical constants and parameters used in TWC simulation
Constants Value
a
10 10 9 m
R
181 10 6 m
30 10 6 m
c
2 s
63:5 10 6 m
kf
0:0386 W m 1 K 1
Cpf
1068 Jkg 1 K
Cpw
1000 Jkg 1 K
2000 kg m 3
w
0:41
w
8
Sh1
3:2
N u1
3:2
Shi;1
2:65
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 reduced.
We combine all the chemical species into three different groups. We define the
net reducing agent ‘A’ as
3
y
[A] = (2 + )[CHy ] + [CO] + [H2 ] + [N H3 ];
2
2
(5.12)
where [CHy ] represents the general representation of hydrocarbon present in gasoline fuel. For the fuel used in this work, y=1.865. The net oxidizing group is defined
as
1
[Ox ] = [O2 ] + [N O]:
2
108
(5.13)
From here on, unless specified otherwise, O2 will be used to represent total oxidants. The oxidation products are defined as
[AO] = [CO2 ] + [H2 O]:
(5.14)
The constant coefficients appearing in Eqs.5.12-5.14 come from the stoichiometric number that is required for the complete combustion of the individual reactant
to the final products (CO2 , H2 O 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, Auckenthaler’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 reactions 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 H2 O and CO2 can oxidize ceria as shown by Möller et al. (2009). However, this occurs only at very high temperature (>800K) and large change in the
concentration of H2 O and CO2 and absence of other reductants. For the conditions
considered in this work the variations in the H2 O and CO2 concentrations are small
and hence we have neglected the oxidation of ceria by these species. Further, the
109
sl.no. Reaction
Table 5.2: Global reaction kinetics
Reaction rate ( mmol
3 :s )
A + 12 O2 ! AO
Ce2 O3 + 12 O2 ! Ce2 O4
A + Ce2 O4 ! Ce2 O3 + AO
1
2
3
kJ
- H ( mol
)
E
A1 exp( RT1 )XO2 XA
c T (1+K X )2
s
a1 A
E2
2
O2
RT
E3
r1 = a
r2 =ac A exp(
)X (1
) T OSCgreen
r3 =ac A3 exp( RT )XA T OSCgreen
283
100
183
Table 5.3: Kinetic parameters for a Pd/Rh based TWC with specifications shown in
Table 5
kJ
)
sl.no. Reaction
Ai
(unit)
Ei ( mol:K
20
3
1
1
A + 0:5O2 ! AO
1:5 10
mol m s K 105
2
Ce2 O3 + 0:5O2 ! Ce2 O4
4:95 1010 s 1
80
7
1
3
A + Ce2 O4 ! Ce2 O3 + AO 3:0 10
s
75
Absorption constant: Ka1 = Aa1 exp( Ea =RT )
Aa1 = 65:5
Ea = 7:99( molkJK )
TOSCgreen = 200 m3 of mol
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 reduction, 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 expression 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) =
110
T
r(X).
Ordering the species as A; O2 ; AO; Ce2 O3 and Ce2 O4 ; we have
2
T
1
6
6 1
6
6 2
6
=6
6 1
6
6
6 0
4
0
0
1
2
0
1
1
3
1
7
7
07
7
7
17
7
7
7
17
5
1
T
and
r = r1 r2 r3
[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
T
in Eq.
5.2]. For example, the net rate of A production is -( r1 +r3 ): Similarly, the net rate
of formation of Ce2 O4 is r2
r3 , which is used to calculate the fractional oxidation
state (FOS) of ceria ( ); 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
T OSC = ac T OSCgreen ;
(5.15)
where ac is the normalized activity. For a green or a fresh catalyst ac = 1 and it
reduces as the catalyst ages. T OSCgreen is the maximum storage capacity for a
green catalyst. For a given catalyst age, it is assumed that TOSC remains constant. 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 enthalpy 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 Ce2 O4 to Ce2 O3 ) 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 oxidation. 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 T OSC 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 performed 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 sensor, 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 comparison of model predicted and experimentally observed total oxidant and reductant 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 predicts 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
1.1
770
1.08
765
in
1.04
755
1.02
750
1
745
0.98
740
0.96
735
(K)
760
λ
T
λ
1.06
in
T
0.94
850
900
950
730
1000 1050 1100 1150 1200 1250 1300 1350
Time (s)
Figure 5.1: Operating condition: Feed gas A/F ( ) 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 reduced 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 ( ) while the solid (blue) curve
gives the FOS of the catalyst. From t=875s to 910s, as the feed is lean, the catalyst 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
10000
5000
2
O concentration (ppm)
(a)
15000
reductant concentration (ppm)
0
3
900
x 10
1000
1100
Time(s)
1200
1300
4
calc(midbed)
meas(midbed)
meas(feedgas)
(b)
2.5
2
1.5
1
0.5
0
900
1000
1100
Time(s)
1200
1300
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
1
1.1
1.08
0.8
1.06
0.6
λ
FOS
1.04
1.02
0.4
FOS
λ
0.2
1
0.98
0
900
1000
1100
Tim e (s)
1200
1300
0.96
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 reactivity 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 experimentally 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 approximately 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 expected, 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
calc(midbed)
meas(midbed)
(a)
meas(feedgas)
10000
5000
2
O concentration (ppm)
15000
0
1800
1850
1900
1950
Time (s)
2000
2050
2100
reductant concentration (ppm)
4
3
x 10
(b)
2.5
2
1.5
1
0.5
0
1800
1850
1900
1950
Time (s)
2000
2050
2100
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
O2 concentration (ppm)
15000
(a)
10000
5000
reductant concentration (ppm)
0
900
4
x 10
1000
1100
Time(s)
4
1200
1300
cal c(m idbe d)
m e as(m idbe d)
(b)
m e as(fe e dgas)
3
2
1
0
900
1000
1100
Time(s)
1200
1300
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
kJ
sl.no. Reaction
Ai
(units)
Ei ( M ol:K
) TOSC( m3 of mol
washc
1
A + 0:5O2 ! AO
2:34 1019 mol m 3 s 1 K 90
2
Ce2 O3 + 0:5O2 ! Ce2 O4
1:16 1011 s 1
80
10
7
1
3
A + Ce2 O4 ! Ce2 O3 + AO 3:10 10
s
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 precious 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 predicted 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 measured 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 concentration 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 catalyst 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 behavior 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 results 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 temperature, respectively, as a function of inlet feed temperature for a stoichiometric (i.e.,
121
4
2
O concentration (ppm)
7
x 10
(a)
6
5
cal c(m idbe d)
m e as(m idbe d)
m e as(fe e dgas)
4
3
2
1
0
0
50
100
150
200
250
300
reductant concentration (ppm)
Time(s)
8
x 10
4
(b)
6
4
2
0
0
50
100
150
Time(s)
200
250
300
Figure 5.6: Comparision of (a) oxidant and (b) reductant emissions with threshold
catalyst over a FTP cycle
122
lambda
FOS
1
1.05
λ
FOS
1.025
0.5
1
0.975
0
0
200
400
600
800
Time (s)
1000
1200
0.95
1400
Figure 5.7: Change in FOS over bag one and two of a FTP cycle
= 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 diffusional resistance and thus the observed activation energy, as is shown by Joshi et
123
1
Conversion
0.8
(a)
0.6
0.4
green
aged
0.2
0
400
450
T
500
(K)
550
600
f,in
750
700
650
T
S
(b)
600
550
green
aged
500
450
400
400
450
500
T (K)
550
600
f,in
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% reductant is 1420 C and the model predicts the temperature rise of 140.40 C which is
consistent as around only 98% conversion was observed. The adiabatic temperature 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 =
where,
(
H) Xi
;
Mi Cp
(5.16)
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 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 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
1
Conversion
0.8
S h as a fu nction of T
i
0.6
No wash coat di ffusion
Asym ptoti c Sh
0.4
i
0.2
(a)
0
400
450
500
550
T
f,in
600
650
(K)
1
S h =∞
i
Conversion
0.8
S h =S h =2.65
0.6
i
S h from Eq 13
i,∞
i
0.4
0.2
(b)
0
400
450
500
550
T
f,in
600
(K)
650
700
750
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)
1
A + 0:5O2 ! AO
1:5 1020
mol m 3 s 1 K
2
Ce2 O3 + 0:5O2 ! Ce2 O4
4:95 1010 s 1
3
A + Ce2 O4 ! Ce2 O3 + AO 3:0 107
s 1
Absorption constant: Ka1 = Aa1 exp( Ea =RT )
Aa1 = 65:5
Ea = 7:99( molkJK )
TOSCgreen = 200 m3 of mol
washcoat
128
kJ
Ei ( mol:K
)
105
80
75
Table 6.2: Numerical constants and parameters used in TWC simulation
Constants Value
a
10 10 9 m
R
181 10 6 m
30 10 6 m
c
2 s
63:5 10 6 m
kf
0:0386 W m 1 K 1
Cpf
1068 Jkg 1 K
Cpw
1000 Jkg 1 K
2000 kg m 3
w
0:41
w
8
Sh1
3:2
N u1
3:2
Shi;1
2:65
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
@Xf m
=
@t
hui
@Xf m
@x
kmo
(Xf m
R
hXwc i) :
(6.1)
The species balance in the washcoat (for gas phase species) is
w
@ hXwc i
1
=
@t
CT otal
T
r+
kmo
c
129
(Xf m
hXwc i) :
(6.2)
The overall mass transfer coefficient matrix (kmo ) is given by
kmo1 = kme1 + kmi1 ;
(6.3)
where kme and kmi are the external and internal mass transfer coefficient matrices.
The energy balance in the fluid phase is
f Cpf
@Tf
=
@t
hui
f Cpf
@Tf
@x
h
(Tf
R
(6.4)
Ts ) ;
and the energy balance for the washcoat is
w
Here,
c
w Cpw
@Ts
=
@t
w kw
@ 2 Ts
+ h (Tf
@x2
is the washcoat thickness and
(defined as sum
s
+ c ; where
s
w
Ts ) +
T
cr
(
H) :
represents the effective wall thickness
is the half-thickness of wall) ,
w
and Cpw are the
effective density and specific heat capacity, respectively, defined as
c c Cpc
+
s s Cps
and
w kw
=
c kc
+
(6.5)
s ks where
w w Cpw
=
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),
=
of ceria as,
[Ce2 O4 ]
;
[Ce2 O4 ] + [Ce2 O3 ]
@
1
=
(rstore
@t
2T OSC
rrelease ) ;
(6.6)
(6.7)
or for the simplified kinetics used in this work,
@
1
=
(r2
@t
2T OSC
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 2 RN
N
is
defined by
kme =
Df Sh
:
4R
(6.9)
Here, Sh is a diagonal matrix given by, Sh =Sh I, where I 2 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 = Sh1 +
0:272( Pz )
2
1 + 0:083( Pz ) 3
;
where Sh1 = 3:2 for rounded square channel. Similarly, the internal mass transfer
coefficient is defined as
kmi =
Ds Shi
(6.10)
:
c
The internal Sherwood number matrix, Shi 2 RN
N
; is evaluated as a function of
Thiele matrix ( ) as follows (Balakotaiah, 2008)
Shi = Shi;1 + (I +
)
We define the effective rate constant as ki;ef f =
2
1
2
(6.11)
:
Ri (X)
1
CT otal Xi
in which case,
X=Xf m
becomes a diagonal matrix with the diagonal terms defined as
2
ii
More detail about the derivation of
=
2
c
Ds;i
ki;ef f ;
(6.12)
has been discussed in Kumar et al., (2012).
131
The initial and boundary conditions are given by
Xf m;j (x) = Xf0m;j (x)
@t = 0;
in
(x)
hXwc;j (x)i = Xwc;j
(6.14)
Tf (x) = Tf0 (x)
@t = 0;
(6.15)
Ts (x) = Ts0 (x)
@t = 0;
(6.16)
Xf m;j (t) = Xfinm;j (t)
Tf (x; t) = Tfin (t)
6.3.1
@t = 0;
(6.13)
@x = 0;
@x = 0;
(6.17)
(6.18)
@Ts
=0
@x
@x = 0;
(6.19)
@Ts
=0
@x
@x = L:
(6.20)
Discretized Model
Eqs. 6.1-6.8 are solved by discretizing using finite difference method with upwinding. For interior points we have,
132
dXf m (i; j)
=
@t
w
hui
Xf m (i; j)
d hXwc i
1
=
dt
CT otal
f Cpf
dTf
=
dt
w w Cpw
hui
T
Xf m (i
4x
r(i; j) +
kmo (i; j)
(Xf m (i; j)
R
Tf (i)
w kw
hXwc i (i; j)) ;
(6.21)
kmo (i)
hXwc i (i; j)) ;
(Xf m (i; j)
c
f Cpf
dTs
=
dt
1; j)
Tf (i
4x
Ts (i + 1)
+h(i) (Tf (i)
1)
h
(Tf (i)
R
Ts (i)) ;
2Ts (i) + Ts (i 1)
4x2
Nr
X
Ts (i)) + c
rk (i) (
(6.22)
(6.23)
(6.24)
Hk (i)) ;
(6.25)
k=1
1
d (i)
=
(r2 (i)
dt
2T OSC
(6.26)
r3 (i)) :
To get discretized model for the boundary condition, integrate Eq. 6.5 from x = 0
to
4x
;
2
w
@Ts 4x
=
w Cpw
@t 2
w kw
@Ts
@x
+h(1) (Tf
133
x= 4x
2
@Ts
@x
4x 4x
Ts )
+
2
2
c
x=0
Nr
X
i=1
!
ri (
(6.27)
Hi ) :
(6.28)
Using boundary condition Eq.6.19 and central difference for
w w Cpw
dTs (1)
= 2
dt
Ts (2) Ts (1)
4x2
w kw
+h(1) (Tf (1)
Ts (1)) +
@Ts
@x x= 4x
2
gives
(6.29)
c
Nr
X
ri (1) (
Hi (1)) : (6.30)
i=1
4x
2
Similarly integrating from x = L
to x = L and using boundary condition
Eq.6.20 gives
w w Cpw
dTs (N )
=
dt
w kw
2
Ts (N )
+h(N ) (Tf (N )
Ts (N
4x2
Ts (N )) +
1)
c
(6.31)
Nr
X
ri (N ) (
Hi (N ))(6.32)
:
i=1
Eqs. 6.21 and 6.23-are solved for discretization points i=2 toN, with boundary
in
conditions Xf m;j (1) = Xin
f m;j (t) and Tf (1) = Tf (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 N C(2N
1) + (3N
1); where N C 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 various 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 deliberate lean-rich experiments were performed. The dash-dotted (blue) curve represents the feed gas composition while the solid (black) and dashed (red) curves
represent the model estimated and measured emissions at TWC exit, respectively.
134
feed
exit estim ated
exit m easured
oxidant ppm
15000
10000
5000
0
900
950
1000
1050
1100
1150
1200
1250
1300
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 reductant 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 reduction 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
4
x 10
4
feed
exit estim ated
exit m easured
3.5
reductant ppm
3
2.5
2
1.5
1
0.5
0
900
950
1000
1050
1100
1150
1200
1250
1300
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 cycle showed the biggest improvement as compared to the averaged model proposed
in Kumar et al, (2012). With an averaged model the, estimated and measured values 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
5
x 10
4
feed
exit estim ated
exit m easured
concentration ppm
4
3
2
1
0
0
50
100
150
200
250
300
Figure 6.3: Oxidant emission for first 300s of FTP (ac=0.3)
6
x 10
4
feed
exit estim ated
exit m easured
concentration ppm
5
4
3
2
1
0
0
50
100
150
200
250
Figure 6.4: reductant emission for first 300s of FTP (ac=0.3)
137
300
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.
[CO2 ] +
[CO]
2
+ [O2 ] +
[N O]
2
+
[Hcv ]
3:5
4 3:5+ [CO]
[CO ]
Ocv
2
([CO2 ] + [CO])
2
=
1+
[Hcv ]
4
Ocv
2
;
(6.33)
([CO2 ] + [CO] + Cfactor [HC])
where
=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,
=
Shown in Fig 6.5 is the
[AO] + 2[O2 ]
:
[A] + [AO]
(6.34)
at the TWC inlet as measure by Eq.6.34. While
shown in Fig 6.3.2 is the commanded : From Fig.6.34 and 6.5, it can be concluded
that although the commanded
is rich in the cold start while the observed
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
1.5
commanded λ
threshold
ful
1
0.5
0
0
50
100
150
200
Tim e s
250
300
350
400
35
30
ful
threshold
25
λ
20
15
10
5
0
0
50
100
150
200
Tim e s
250
300
350
400
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 thickness and an averaged value is used.
The species balance equation in the fluid phase (for gas phase species) is given
by
@Xf m
=
@t
hui
@Xf m
@x
kme
(Xf m
R
(6.35)
Xs ) :
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.
The species balance in the washcoat (for gas phase species) is
1
@Xwc
=
w
@t
CT otal
T
r + De
@ 2 Xwc
:
@y 2
(6.36)
The energy balance in the fluid phase is given by
f Cpf
@Tf
=
@t
hui
f Cpf
@Tf
@x
h
(Tf
R
(6.37)
Ts ) ;
and the energy balance for the washcoat is
w w Cpw
@Ts
=
@t
2
w Kw
@ Ts
+ h (Tf
@x2
0
1
Zc
Ts ) + @ rT (Xs ; Ts )dy A (
H) :
(6.38)
0
Ceria balance
@
1
=
(r2
@t
2T OSC
r3 ) ;
(6.39)
with the boundary conditions:
Xf m;j (x) = Xf0m;j (x)
140
@t = 0;
(6.40)
Tf (x) = Tf0 (x)
@t = 0;
(6.41)
Ts (x) = Ts0 (x)
@t = 0;
(6.42)
Xf m;j (t) = Xfinm;j (t)
(6.43)
@x = 0;
Tf (x; t) = Tfin (t)
(6.44)
@x = 0;
@Ts
=0
@x
@x = 0;
(6.45)
@Ts
=0
@x
@x = L;
(6.46)
Xwc = Xs
@Xwc;j
=0
@y
@y = 0;
(6.47)
@y =
(6.48)
c;
where Xs is defined as
kme (Xf m
6.4.1
Xs ) =
De
@Xwc
@y
Discretized model
Discretization Eq. 6.38 for the interior points (k = 2 to M
w
(6.49)
:
y=0
dXwc (i; k)
1
=
dt
CT otal
T
r(i; k) + De
Xwc (i; k + 1)
1),
2Xwc (i; k) + Xwc (i; k
y2
1)
:
(6.50)
To get 2nd order accuracy, we integrate eq. 6.38 from y = 0 to
y
2
and substitute
the boundary conditions Eq. 6.47 and 6.49
dXwc (i; 1) y
y 1
=
w
dt
2
2 CT otal
T
141
@Xwc
r(i;1) + De
@y
y=
y
2
;
y=0
(6.51)
w
1
Xwc (i; 2) Xwc (i; 1)
dXwc (i; 1)
T
=
r(i;1) + 2De
dt
CT otal
y2
2
+
kme (Xf m (i) Xwc (i; 1)) ;
y
(6.52)
(6.53)
where,
Xwc (i; 1) = Xs :
Similarly for the boundary condition at y =
y
2
c
w
6.4.2
to
y
2
c;
integrating eq. 6.38 from y =
and substituting the boundary conditions Eq. 6.48, we get
1
dXwc (i; M )
=
dt
CT otal
T
r(i;M )
2De
Xwc (i; M )
Xwc (i; M
y2
1)
:
(6.54)
Case 1: Single reaction
Shown in Fig 6.6 is the comparison of the low-dimensional model with the detailed 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 = 1 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 considered. 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
1
conversion
0.8
0.6
Detailed
asym ptotic
no diffus ion lim itation
internal m ass transfer
0.4
0.2
0
0
50
100
tim e s
150
200
Figure 6.6: Model comparision of internal mass transfer concept with detailed
model for a single reaction
(6.55)
CO + 0:5O2 ! CO2
rate =
1018 exp(
90000
)XCO XO2
RT
G
(6.56)
;
where,
G=T
7990
)XCO
1 + 65:5 exp(
RT
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
700
650
temperature
600
550
500
450
Detailed
asym ptotic
no diffusion lim itation
internal m ass transfer
400
350
300
0
50
100
tim e s
150
200
Figure 6.7: Model comparision of internal mass transfer concept with detailed
model for a single reaction
u
1 m/s
in
Xf;CO
1.5%
in
Xf;O
2
0.75%
Tf0
300 K
Ts0
300 K
Tfin
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 detailed model for the kinetics shown in Table 6.1. Even with the simplifications used
144
1
conversion
0.8
0.6
Detailed
internal m ass transfer
no diffusion lim itation
asym ptotic
0.4
0.2
0
0
10
20
30
40
50
tim e s
60
70
80
90
in computing internal mass transfer coefficient, it gives the best representation of
the detailed model as compared to having no washcoat diffusion limitation assumption 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 conversion 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
700
650
Ds
Km i
Km e
Sh
temperature
600
550
∞
500
450
400
350
300
0
20
40
60
80
100
120
140
tim e s
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 m washcoat thickness while
the dash (red) curve and dash-dot (blue) curve represent washcoat thickness of
20 and 40 m; 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
800
T
f,in
=650 K
750
T
solid temperature
700
f,in
=564 K
650
T
f,in
=511 K
600
T
550
f,in
=499 K
T
f,in
=488 K
500
T
450
f,in
=443 K
400
T
350
0
0.01
0.02
f,in
=350 K
0.03
0.04
length m
0.05
0.06
0.07
0.08
Figure 6.8: Steady state axial temperature for different inlet feed temperature
However, because of the diffusion limitations, increasing the washcoat thickness 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
R
Df
and
c
Ds
at 700K are 2.04 and 120.6, respectively
1
kmo
=
c
Ds Shi
+
4R
:
Df Sh1
Thus, the system is internal mass transfer limited. Now, by changing
reduce R ;making
R
Df
even smaller compared to
c
Ds
(6.58)
c;
we also
making 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
1
conversion
0.8
0.6
0.4
0.2
0
350
400
450
500
T
f,in
550
K
600
650
700
Figure 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.
Z
x=L
ac dx = constant,
x=0
148
1
0.9
0.8
conversion
0.7
δ
wc
δ
wc
δ
wc
=30
=20
=40
0.6
0.5
0.4
0.3
0.2
0.1
0
440
450
460
470
480
T K
490
500
510
520
f,in
Figure 6.10: Effect of change in washcoat thickness on catalyst light-off
2
6
ac = 4
b for
(10
L
0<x< 10
L
b)=9 for x> 10
3
7
5:
(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
1
0.9
0.8
Conversion
0.7
0.6
0.5
0.4
δ
wc
δ
wc
δ
wc
δ
wc
=30
=20
=40
=50
0.3
0.2
0.1
0
0
50
100
Time s
150
200
Figure 6.11: Effect of change in washcoat thickness on exit conversion efficiency
transient
650
600
Exit solid temperature K
550
δ
wc
δ
wc
δ
wc
δ
wc
=30
=20
=40
=50
500
450
400
350
300
0
50
100
Time s
150
200
Figure 6.12: Effect of change in washcoat thickness on exit temperature transient
150
1
uniform
non-uniform
conversion
0.8
0.6
0.4
0.2
0
350
400
450
500
T
f,in
550
K
600
650
700
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 conversion 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
800
T=650 K
750
T=569 K
temperature
700
650
T=500 K
600
T=495 K
550
T=484 K
500
T=438 K
450
400
350
0
T=350 K
0.02
0.04
length m
0.06
0.08
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 lightoff 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 under 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). Similar specification for metallic substrate are shown in Table 6.4 (Heck and farrauto,
152
1
0.9
0.8
a=1
a=4
a=8
a=9.5
0.7
Conversion
0.6
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
40
50
Time s
60
70
80
Figure 6.14: Effect of change in loading profile on conversion transient at constant
feed temperature of T=550K
153
90
1
conversion
0.8
0.6
a=4
a=2
a=1
0.4
0.2
0
0
5
10
15
time s
20
25
30
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 property 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
and
w Kw
=
c Kc
+
s Ks ;
the values for
c ; Cpc ;
w w Cpw
s
=
c c Cpc
+
s s Cps
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
1
0.9
0.8
CPSI=900
CPSI=400
CPSI=600
CPSI=1200
Conversion
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
20
40
60
80
Time (s)
100
120
140
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 catalyst 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
700
650
Exit temperature K
600
550
500
450
CPSI=900
CPSI=400
CPSI=600
CPSI=1200
400
350
300
0
50
100
Time (s)
150
200
Figure 6.17: Effect of change in cell density in ceramic substrate for constant space
velocity (45868 hr 1 ) and constant feed temperature (T=550K)
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 improvement 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 cycle, the averaged model starts agreeing with the experiments from around 40 sec.
while the detailed model gives good agreement from the start. The operating condition used in close coupled TWC leads to front end ignition and after the catalyst
157
1
0.9
0.8
CPSI=400
CPSI=500
CPSI=500
CPSI=600
Conversion
0.7
0.6
δ=2
δ=1.5
δ=2
δ=1.5
0.5
0.4
0.3
0.2
0.1
0
0
50
100
Time (s)
150
200
Figure 6.18: Effect of change in cell density in metallic substrate for constant space
velocity (45868 hr 1 ) and constant feed temperature (T=550K)
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
700
Exit temperature K
650
600
CPSI=400
CPSI=500
CPSI=500
CPSI=600
δ=2
δ=1.5
δ=2
δ=1.5
550
500
450
400
350
300
0
50
100
Time (s)
150
200
Figure 6.19: Effect of change in cell density in metallic substrate for constant space
velocity (45868 hr 1 ) and constant feed temperature (T=550K)
159
1
0.9
0.8
Conversion
0.7
M : CPSI=500 δ=1.5
C: CPSI=600 δ=4
C: CPSI=900 δ=2.5
0.6
0.5
0.4
0.3
0.2
0.1
0
0
20
40
60
80
Time (s)
100
120
140
Figure 6.20: Comparision of ceramic and metallic substrate for constant feed temperature of 550 K and constant space velocity and composition
160
700
650
Exit temperature K
600
550
500
M : CPSI=500 δ=1.5
C: CPSI=600 δ=4
C: CPSI=900 δ=2.5
450
400
350
300
0
50
100
Time (s)
150
200
Figure 6.21: Comparision of ceramic and metallic substrate for constant feed temperature of 550 K and constant space velocity and composition
161
1
0.9
0.8
M : CPSI=500 δ=1.5
C: CPSI=600 δ=4
C: CPSI=900 δ=2.5
conversion
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
50
100
time s
150
200
Figure 6.22: Comparision of metallic and ceramic substrate for same catalyst loading
162
Chapter 7 Conclusions and Recommendations for Future Work
7.1
7.1.1
In-cylinder combustion modeling
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 developed to specify the species balance. The model is developed by first considering 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 averaged 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 absence 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 emissions. 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 pressure 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 represented 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 reaction 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 operating 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 ( )
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 emission may be higher or lower depending on the mode of operation.
5. Reducing the crevice volume can reduce the unburned hydrocarbon emissions.
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 intermediates 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 improvement 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 function 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, however 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 temperature 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 ( ), which can be used to compute
the optimal fueling profile for the desired operation.
7.2
7.2.1
Three-way catalytic converter modeling
Summary and Conclusions
The main contribution of this work is the development and validation of a fundamentals 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 washcoat 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 correlation, while the internal mass trasfer coefficient is evaluated as a fucntion of Thiele
modulus. Various approximations for computing the Thiele modulus for multiple reaction 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 asymptotic 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 , H2 O, 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 formation. 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
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