Mathematical Modeling of Transport Phenomena in Karthik ... · Karthik Somasundaram, Erik Birgersson, Kenneth Teo Hua Yeong, and Arun Sadashiv Mu-jumdar, Development of a Mathematical

Mathematical Modeling of Transport Phenomena inElectrochemical Energy Storage Systems

Karthik SomasundaramM.Sc.(NUS)

A Thesis SubmittedFor the Degree of Doctor of PhilosophyDepartment of Mechanical EngineeringNational University of Singapore

2012

DECLARATION

I hereby declare that the thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been

used in the thesis.

This thesis has also not been submitted for any degree in any university previously.

___________________

Karthik Somasundaram

17 October 2012

ii

Acknowledgements

I express my sincere gratitude to my advisors Prof. Arun S. Mujumdar and Dr. Erik Birgersson for

their warm encouragement and thoughtful guidance. Prof. Mujumdar with his immense knowledge

and vast experience provided expert guidance on framing and carrying out my research work. It is

a great honour to be mentored by such a great scientist. On the whole, I see him as a wonderful

teacher and an expert mentor to learn the di¤erent aspects of academic as well as industrial research.

I am greatly indebted to Dr. Erik Birgersson for his patient and diligent guidance. He has

helped me to transform ideas into fruitful research with his brilliant analyzing skills. I thank him

for teaching me the art of writing a manuscript and the approach to use scaling analysis for getting

physical insights of a process. Working with Dr. Erik has been a great learning experience for me.

I am grateful to my colleagues and friends for the fruitful discussions, meaningful suggestions

and advices, and also to the academic, technical and secretarial sta¤ of the National University of

Singapore for their continuous support and encouragement. I have learnt many interesting things

during discussion with my colleagues, friends and �nal year undergraduate project students.

I dedicate this thesis to my parents and sisters for their love, endless support and encouragement.

The �nancial support of the National University of Singapore (NUS) is gratefully acknowledged.

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Lithium-Ion Batteries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Electrochemical Capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Objectives and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Literature Review 9

2.1 Mathematical Modeling of batteries . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Mathematical Modeling of Electrochemical Capacitors . . . . . . . . . . . . . . . . 13

3 Mathematical Formulation 16

3.1 Lithium-Ion Batteries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.3 Governing equations (Macroscale) . . . . . . . . . . . . . . . . . . . . . . . 18

3.1.4 Governing equations (Microscale) . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.5 Boundary and initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.6 Constitutive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Electrochemical Capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Analysis of Electrochemical Capacitor Model 29

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32


iii

iv

4.2.2 Governing equations (Microscale) . . . . . . . . . . . . . . . . . . . . . . . . 35


4.2.4 Constitutive relations and parameters . . . . . . . . . . . . . . . . . . . . . 37

4.3 Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.4 Calibration and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.5 Analysis (Microscale) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.6 Analysis (Macroscale) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.6.1 Current collectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.6.2 Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.6.3 Separator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.6.4 Macroscopic time-scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.7 Veri�cation of reduced models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.8 Computational cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5 Analysis of Li-ion Battery Model 60

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.2 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62




5.3 Analysis (Macroscale) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.3.1 Current collectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.3.2 Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.3.3 Separator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.3.4 Macroscopic time-scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3.5 Thermal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6 Automated Model Generation of a Lithium-Ion Bipolar Battery Module 92

v

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92


6.2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96



6.3 Model reformulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.4 Automated Model Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.5 Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.6 Calibration and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.7 Veri�cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.8 Computational cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7 Thermal-ElectrochemicalModel for Passive ThermalManagement of a Spiral-wound Lithium-

Ion Battery 114

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114


7.3 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.3.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120



7.4 Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.5.1 Discharge and power curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.5.2 Edge and geometry e¤ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.5.3 Heat generation and thermal behavior . . . . . . . . . . . . . . . . . . . . . 127

7.5.4 Passive thermal management . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

vi

8 Conclusions and Outlook 140

8.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

8.2 Recommendations for future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Summary

This research work focuses on formulating models for electrochemical energy storage devices, speci�-

cally for Lithium-ion batteries and electrochemical capacitors, for which reduced model formulations

are proposed to lower the computational cost as part of fundamental research; secure guidelines for

design of thermal management systems for battery packs and supercapacitor packs used in high

power applications like electric vehicles, as part of applied research.

The similarities or the common features among these devices are that all these devices consist of

two electrodes in contact with an electrolyte solution; the energy-providing processes take place at

the phase boundary of the electrode/electrolyte interface; electron and ion transport are separated.

Hence by exploiting the similarities of these systems, this work provides a framework for the

rigorous mathematical model development that captures the essential underlying physics of these

electrochemical systems.

In general, the mathematical model for the transport inside these devices accounts for the

transient conservation of species, charge and energy in the solid and the liquid phases developed

based on porous electrode theory. These models are reduced in terms of dimensions as well as

physics by capturing only the leading order phenomena through the use of scaling analysis. The

reduced models secured are compared with the full model without reductions and found to have

good agreement. The highlight of the reduced models in terms of computational cost is presented.

A coupled electrochemical-thermal model is presented for a commercially available spiral-wound

cylindrical (applicable to prismatic cell as well) Li-ion battery. A technique known as automated

model generation is developed to automatically build models for battery packs and supercapacitor

stacks, that could help in various parametric and perturbation studies to optimize the design of

these packs.

As part of the applied research, the models formulated initially are used to evaluate thermal

management systems for Li-ion battery packs, as they are associated with a lot of thermal issues. A

passive thermal management system using a phase change material is evaluated for a commercially

available spiral-wound cylindrical Li-ion battery using a coupled electrochemical-thermal model

developed as part of the fundamental research and its extension to battery packs is presented.

Preface

This thesis presents the study on the modeling of transport phenomena in electrochemical en-

ergy systems The following publications are based on research carried out for this doctoral thesis.

E-book chapter:

1. S.Karthik, E. Birgersson and Arun S. Mujumdar, Modeling of Li-ion Battery, book chapter

in: Mathematical Modeling of Industrial Transport Processes, edited by P. Xu, Z. Wu and A.

S. Mujumdar, TPR group, Singapore, 2009. (ISBN: 978-981-08-6269-5).

Journal:

1. Karthik Somasundaram, Erik Birgersson and Arun Sadashiv Mujumdar, Analysis of a Model

for an Electrochemical Capacitor, Journal of The Electrochemical Society, 158 (11), A1220-

A1230, 2011.

2. Karthik Somasundaram, Erik Birgersson and Arun Sadashiv Mujumdar, Thermal-electrochemical

model for passive thermal management of a spiral-wound lithium-ion battery, Journal of Power

Sources, 203, 84-96, 2012.

3. Karthik Somasundaram, Erik Birgersson and Arun Sadashiv Mujumdar, Model for a Bipolar

Li-ion Battery Module: Automated Model Generation, Validation and Veri�cation, Applied

Mathematics and Computation, In Press, Corrected Proof.

4. Karthik Somasundaram, Erik Birgersson and Arun Sadashiv Mujumdar, Analysis of a Model

for a Li-ion Battery, Manuscript in preparation.

Conferences:

1. Karthik Somasundaram, Erik Birgersson and Arun Sadashiv Mujumdar, A Coupled Thermal-

Electrochemical Reduced Model of a Lithium-Ion Battery, International Conference on Applied

Energy, Singapore, p. 294-303, April 21-23, 2010.

2. Karthik Somasundaram, Erik Birgersson and Arun Sadashiv Mujumdar, Automated Code

Generation for a Lithium-Ion Battery Stack Model, International Conference on Applied En-

ergy, Singapore, p. 304-312, April 21-23, 2010.

3. Karthik Somasundaram, Erik Birgersson, Kenneth Teo Hua Yeong, and Arun Sadashiv Mu-

jumdar, Development of a Mathematical Model for Spiral-wound Li-ion Batteries, Interna-

tional Conference on Materials for Applied Technologies 2011, Singapore, June 26 - July 1,

2011.

4. Karthik Somasundaram, Erik Birgersson, Kenneth Teo Hua Yeong, and Arun Sadashiv Mu-

jumdar, Passive Thermal-management of a Li-ion Battery Cell: a Computational Study,

International Conference on Materials for Applied Technologies 2011, Singapore, June 26 -

July 1, 2011.

List of Tables

4.1 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Scales and nondimensional numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3 Computational cost for the full and reduced set of governing equations . . . . . . . 55

5.1 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.2 Electrolyte, and outer can properties . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.3 Constants in expressions (Ref. [1, 2]) . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.4 Comparison of scales and numerical results, nondimensional numbers . . . . . . . . 86

5.5 Nondimensional numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.1 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.2 Constants in Eqs. (Refs. [1, 2, 3, 4]) . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.3 Change of entropy with state of charge for positive electrode (Ref. [5]) . . . . . . . 102

7.1 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.2 PCM, electrolyte, and outer can properties . . . . . . . . . . . . . . . . . . . . . . 125

x

List of Figures

1.1 Simpli�ed Ragone plot of the energy storage domains for various electrochemical

energy systems. [6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Schematic showing the operating principle of a Li-ion cell. . . . . . . . . . . . . . . 3

3.1 Schematic of a single Li-ion cell showing (a) various functional layers on the macroscale,

and (b) di¤usion of lithium in the active material in the electrodes on the microscale. 17

4.1 Schematic of an electrochemical capacitor cell. . . . . . . . . . . . . . . . . . . . . 30

4.2 Section A-A of a single cell showing (a) various functional layers on the macroscale,

and (b) di¤usion of proton in RuO2 in the electrodes on the microscale. . . . . . . 31

4.3 Cell voltage vs time for galvanostatic charge and discharge at 5 mA: Full model,

2D+3D (line) and experimental data (symbols) from Zheng et al. (Ref. [7]) . . . . 40

4.4 Concentration pro�le of proton in RuO2 in the negative (blue) and positive electrode

(red) at di¤erent times during charge at a current of 5 mA: Full model, 2D+3D

(symbols) and 2D+2D model counterpart (lines) predictions. . . . . . . . . . . . . 43

4.5 Concentration pro�le of proton in RuO2 in the negative (blue) and positive (red)

electrode at di¤erent times during charge at a current of 5000 mA : Full model,

2D+3D (symbols) and 2D+2D model counterpart (lines) predictions. . . . . . . . . 44

4.6 Liquid phase (lines) and solid phase (lines with symbols) potential pro�le in the elec-

trodes and separator at various times predicted by reduced model, 1D+1D during

galvanostatic charging at various currents. . . . . . . . . . . . . . . . . . . . . . . . 49

4.7 Variation of surface concentration during charge at a current 5000 mA in the negative

and positive electrodes. Full model, 2D+3D (symbols) and reduced model, 1D+1D

(lines), and 1D+2D (dotted lines) predictions. . . . . . . . . . . . . . . . . . . . . . 52

xi

4.8 Concentration pro�le of proton in the electrolyte at the end of discharge at var-

ious currents: Full model, 2D+3D (symbols) and reduced model, 1D+1D (lines)

predictions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.9 Cell voltage during galvanostatic charge and discharge at various currents: Full

model, 2D+3D (symbols) and reduced model, 1D+1D (lines) predictions. . . . . . 54

5.1 Schematic of (a) a 18650 Li-ion battery, (b) an axisymmetric representation of the

spiral-wound battery showing the various functional layers, and (c) a layer of the

jelly roll comprising a single cell with the roman numerals indicating the interfaces

of the di¤erent layers and the boundaries. . . . . . . . . . . . . . . . . . . . . . . . 61

5.2 Schematic of (a) a Li-ion cell showing the various functional layers on the macroscale,

and (b) lithium di¤usion in the active material in the electrodes in the microscale. 63

5.3 Electrolyte concentration at various times during discharge at di¤erent C-rates. . . 74

5.4 Solid phase potential at the innermost layer near the centre core at various times

during discharge at di¤erent C-rates. . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.5 Liquid phase potential at the innermost layer near the centre core at various times

during discharge at di¤erent C-rates. . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.6 Local state of charge of the electrodes during discharge : numerical results (contin-

uous) and scales (dotted) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.7 Ecell vs time at various discharge rates. . . . . . . . . . . . . . . . . . . . . . . . . 80

5.8 Total, irreversible, reversible, and ohmic heat generation from the battery during

discharge at 1 C-rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.9 Reversible heat generated during discharge at 1 C-rate: numerical results (continu-

ous) and scales (dotted). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.10 Average temperature of the battery vs time during discharge at various rates (h = 5

W m�2 K�1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.1 Schematic of (a) a bipolar lithium-ion battery module, (b) the various functional

layers on the macroscale, and (c) di¤usion of lithium in the active material of the

electrodes in the microscale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.2 Comparison of cell voltage predicted by the reformulated model (line) and experi-

mental results (symbol) at a current density of 15 A m�2. . . . . . . . . . . . . . . 104

6.3 Cell voltage predicted by the full model (symbols) and reformulated model (lines)

at various discharge rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.4 Comparison of results predicted by the full model (symbols) and the reformulated

model (lines) at various times during discharge. . . . . . . . . . . . . . . . . . . . . 106

6.5 Global veri�cation: Comparison of results for a 10-cell module predicted by the

manually implemented full model (symbols) and the reformulated model (lines) at

various discharge current densities. . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.6 Local veri�cation: Comparison of results predicted by the full model (symbols) and

the reformulated model (lines) at various times during discharge. . . . . . . . . . . 107

6.7 Computational cost comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.1 Schematic of (a) a 18650 Li-ion battery, (b) cross-section of the 18650 battery show-

ing the spiral-wound jelly roll, (c) cross section of the battery with PCM surrounding

it, (d) various functional layers in the jelly roll with the roman numerals indicating

the interfaces of these layers at the inner end of the spiral, (e) outer end of the

spiral with the interfaces of various layers shown by the roman numerals, (f) modi-

�ed computational domain, see numerics, (g) agglomerate structure in the negative

electrode (*- positive electrode also exhibits similar structure) and, (h) di¤usion of

lithium in active material in the electrodes on the microscale. . . . . . . . . . . . . 116

7.2 Comparison of cell performance with (symbols) and without PCM (lines) for gal-

vanostatic discharge at various C-rates. . . . . . . . . . . . . . . . . . . . . . . . . 126

7.3 Local distribution of the following dependent variables at t = 1800 s and t = 3600

s during discharge at 1 C-rate: SOC of positive electrode (a, b), SOC of negative

electrode (c, d), lithium ion concentration in the electrolyte (e, f), and liquid phase

potential (g, h). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7.4 Time history of (a) the average battery temperature and (b) the temperature dif-

ference between the battery core and the surface during discharge at various rates

with (dotted) and without PCM (continuous). . . . . . . . . . . . . . . . . . . . . . 129

7.5 Time history of heat generation by various sources (a,b), total heat generation and

heat generation in various layers (c,d) during discharge at 1 C and 5 C-rates with

(dotted) and without PCM (continuous). . . . . . . . . . . . . . . . . . . . . . . . 130

7.6 Local distribution of temperature at t = 320 s and t = 640 s: without PCM (a, b),

with PCM (c, d) during discharge at 5 C-rate. . . . . . . . . . . . . . . . . . . . . . 132

7.7 Liquid fraction of PCM vs time during discharge at 5 C-rate. . . . . . . . . . . . . 134

Chapter 1

Introduction

1.1 Background

The worldwide increase in demand for energy has put ever-increasing pressure on identifying and

implementing ways to save energy. Energy storage therefore has a pivotal role to play in the e¤ort

to combine a future, sustainable energy supply. Reliable and a¤ordable electricity storage is a

prerequisite for optimizing the integration of renewable energy systems. Electrochemical energy

storage systems have been used for long years for this purpose of electric energy storage. Systems

for electrochemical energy storage include batteries and electrochemical capacitors.

The power and energy capabilities of an energy storage/conversion system is represented in a

Ragone plot as shown in Fig.1.1. Batteries are capable of storing higher energy than the electro-

chemical capacitors as can be inferred from Fig.1.1. Electrochemical capacitors are considered as

high power systems as compared to batteries. Batteries convert chemical energy to electrical energy

via redox reactions at the anode and cathode. In electrochemical capacitors the energy delivering

process is the formation and release of electric double layers at the electrode/electrolyte interface.

In case of pseudocapacitors, energy release also takes place additionally by redox reactions at the

anode and cathode. Both these systems di¤er in their way of energy conversion but have some

electrochemical similarities [6]�energy-providing process occurs at the phase boundary of the elec-

trode/electrolyte interface and the ion and electron transport are separated. This work focusses on

lithium-ion battery and electrochemical capacitor with pseudocapacitance in particular.

1

1.1. Background 2

Figure 1.1: Simpli�ed Ragone plot of the energy storage domains for various electrochemicalenergy systems. [6]

1.1.1 Lithium-Ion Batteries

Among existing rechargeable batteries, lithium-ion batteries are more promising since they have

higher energy density, higher cell voltage (� 4 V) and longer charge retention or shelf life (upto 5 or

10 years) when compared with the conventional aqueous technologies [8]. The consumer electronics

market nowadays is dominated by Li-ion batteries owing to their high energy and power densities,

compactness and light weight compared to the sealed lead acid batteries. A Li-ion cell is made up

of a porous insertion positive (pe) and negative electrode (ne) coated on current collector foils (cc)

with a separator (sp) in between them and a liquid electrolyte �lling the entire cell as shown in

Fig. 1.2. The positive electrode is typically a metal oxide (LiCoO2, LiMn2O4 etc.) mixed with a

binder (polyvinylidene �ouride - PVDF) and a conducting �ller additive (carbon black) adhered

to a current collector of aluminum foil. The negative electrode is typically a graphitic carbon

(mesocarbon microbeads - MCMB) coated on a current collector of copper foil with the binder and

the additive. The separator is a microporous polyethylene or polypropylene �lm. The electrolyte

solution commonly comprises of a lithium salt dissolved in a mixture of organic solvents (example

LiPF6 in ethylene carbonate/dimethyl carbonate). During cycling, the lithium ions are exchanged

1.1. Background 3

between the positive and the negative electrodes. During discharging, the Li-ions deintercalate

from the active material particles in the negative electrode and enter the electrolyte whereas in the

positive electrode, the Li-ions from the electrolyte intercalate into the active material particle as

depicted in Fig. 1.2. The reverse reaction happens during charging. Because of this mechanism,

these batteries are also called as rocking chair batteries. The electrons �ow through the external

circuit owing to the redox reaction taking place in the electrodes. The cell reactions are as shown

below, forward reaction represents charging and the reverse reaction happens during discharging.

Figure 1.2: Schematic showing the operating principle of a Li-ion cell.

In the positive electrode

LiMO2 Li1�xMO2 + xLi+ + xe�

In the negative electrode,

C + xLi+ + xe� LixC

Overall reaction is

LiMO2 + C LixC + Li1�xMO2

1.1. Background 4

where M indicates the metal (for example, Co) used for the positive electrode active material.

Some of the advantages that lithium ion batteries o¤er when compared to the relative battery

types are higher energy density lower self discharge rate (2% to 8% per month), longer cycle life

(greater than 1000 cycles) and a broad temperature range of operation, enabling their use in a wide

variety of applications and no memory e¤ect [8]. The �exibility in design is also an advantage of

Li-ion batteries. They are available in various from factors like cylindrical, prismatic, coin cells,

pouch cells etc. Of these, the cylindrical and prismatic types are of spiral-wound con�guration

whereas layered prismatic batteries are also available.

There are certain disadvantages like ageing e¤ects on service life, higher internal resistance,

capacity loss or thermal runaway when overcharged, and, need for protection circuits for safety.

Li-ion batteries can be dangerous if not handled properly. In fact, there have been several cases of

Li-ion batteries going into thermal runaway in laptop applications leading to recalls by Dell, Apple,

IBM, and other manufacturers. [9, 10] The sensitivity of lithium-ion cells to overcharge can result in

chemical decomposition of positive electrode materials and the electrolyte and/or in the deposition

of metallic lithium at the negative electrode which damage the cell and can result in hazardous

conditions, including gassing and release of �ammable electrolyte solvent vapors, if the cell safety

seal is breached as a result of excessive gas pressure. Hence lithium-ion batteries require accurate

voltage control for every cell, unlike Nickel Metal Hydride and other aqueous electrolyte batteries

that can tolerate signi�cant amounts and rates of overcharge. Accurate and reliable control of cell

voltage and temperature is thus critical requirements for achieving long life and adequate safety

of lithium-ion batteries for all uses, but especially so for automotive applications, which demand a

very long battery life and high levels of safety.

Much research in this �eld concentrates on developing and characterizing stable and safe ma-

terials for electrodes and electrolytes. Another research aspect focusses on analyzing the electro-

chemical and thermal behavior of Li-ion batteries from cell level to pack level through the use of

mathematical modeling. The electrochemical and thermal behavior depends on the design of the

batteries like its shape and size, the kinetic and tranport properties of the materials used for making

the batteries. Experimental study of the performance of the batteries and its dependence on the

design and material properies would be a highly time consuming task and extremely expensive.

Mathematical modeling could help in reducing the number of experiments and most importantly

1.1. Background 5

would provide an understanding of the system and the factors that a¤ect the performance of these

systems by providing an insight into the physical process taking place in these systems. Each of the

above mentioned disadvantages could be addressed through theoretical analysis using mathemati-

cal modeling supported by minimum experiments. Hence, the objective of the thesis is to develop

tools for analyzing and optimizing the design of these Li-ion batteries.

1.1.2 Electrochemical Capacitors

Electrochemical capacitors �ll in the gap between the conventional capacitors and batteries as

shown in the Ragone plot in Fig. 1.1. This fact is responsible for considerable research interest in

electrochemical capacitors. They are also called as supercapacitors or ultracapacitors. They are

divided into two types, namely, electric double-layer capacitors and pseudocapacitors. The charge

storage in electric double-layer capacitor is through the electrochemical double-layer formed by the

accumulation of the positive and negative ionic charges at the electrode/electrolyte interface, quite

analogous to the double-layer in a conventional electrostatic capacitor and there are no faradaic

reactions. In the case of faradaic reactions, charge transfer occurs across the electrode/electrolyte

interface leading to a change in the oxidation state of the electroactive species that participate

in this charge transfer. For non-faradaic processes, there are no electrochemical reactions and is

thus electrostatic in nature. For the pseudocapacitors, there is additional charge storage mechanism

through the faradaic reaction similar to batteries. The capacitance of the electrochemical capacitors

is much higher than that of the conventional electrostatic capacitors because of the extremely high

surface area provided by the porous electrodes.

The construction of the supercapacitor is similar to that of a battery. It consists of two porous

electrodes coated on current collector foils and the electrodes are separated by the separator which

is an ionic conductor. The electrodes and the separator are �ooded with the electrolyte. For a

symmetric capacitor, both the electrodes are made up of the same material. The various electrode

materials used for making electrochemical capacitors are carbon based materials, metallic oxides

and polymeric materials. The carbon based materials store charges predominantly through the

double-layer mechanism whereas the transition metal oxides like RuO2 exhibit pseudocapacitance

through faradaic reactions. The electrolytes for electrochemical capacitors may be aqueous or

organic electrolytes. The decomposition voltage of the electrolyte decides the voltage at which the

1.2. Objectives and outline 6

capacitor can operate. The decomposition voltage of the aqueous electrolytes is lesser than that for

the organic counterparts and hence the voltage of a cell employing aqueous electrolyte is lesser than

that of the organic electrolytes [11]. During charging, current �ows into the positive electrode from

the positive current collector. Since electrons cannot �ow through the separator, the current has

to be transferred to the electrolyte phase through ions. This is achieved by double-layer charging

for an electric double-layer capacitor. Positive charges accumulate on the solid electrode particles

near the interface region with the electrolyte. In order to preserve electroneutrality, negative ionic

charges accumulate on the electrolyte side of the interfacial region, thus forming the double-layer.

This leads to an increase in the potential di¤erence across the electrode/electrolyte interface. The

process is reversed on the negative electrode. The charge stored within the double-layer is released

during discharge leading to the relaxation of the potential. The electrochemistry of the double-layer

is discussed in detail by Bard and Faulkner [12].

Electrochemical capacitors are mainly used for high power applications like secondary power

source in electric vehicles, providing power during peak requirements like acceleration. They could

also be used in load-levelling applications in power generating stations that su¤er from peak power

surges. They are also employed as power boost in some applications like power electronics as

well as to store energy through regenerative braking in automobiles. Honda FCX [13] combines

a fuel cell with the ultracapacitor for powerful performance as well as to store energy produced

during braking. The currrent research interest in electrochemical capacitors focus on developing low

cost electrode materials, ways to minimize self-discharge and optimizing electrolytes. This thesis

focusses on mathematical modeling of electrochemical capacitors and developing tools for analyzing

their behavior from cell level to pack level. As mentioned before, the electrochemical similarities

between the batteries and the electrochemical capacitors are utilized in deriving the mathematical

models. The presence of double-layer capacitance distinguishes the model for an electrochemical

capacitor from that of a battery.

1.2 Objectives and outline

The research aims to develop tools for analyzing the design of electrochemical energy storage

systems from cell to stack/pack level of lithium-ion batteries and electrochemical capacitors. These


tools can well be utilized to optimize the design of these systems as well. In order to develop

these tools, a basic understanding of the physicochemical process taking place inside these systems

is required. In this context, mathematical modeling and computational analysis have come to

play an important role in elucidating basic mechanisms, such as the electrochemical and thermal

behavior of the energy storage systems. These mechanisms invoke series of intrinsically coupled

physicochemical processes which are taking place simultaneously during operation of these systems.

Moreover, mathematical modeling can further save time and money as numerical experiments can

be carried out at a signi�cantly lower cost as compared to practical experiments.

The outline of the thesis is as follows. First, a review of the existing mathematical models

for predicting the electrochemical and thermal behavior of Li-ion batteries and electrochemical

capacitors as well as utilizing the results for designing these systems is presented in Chapter 2.

The mathematical models are then presented for both the energy systems in Chapter 3. The

computational cost for solving the formulated models depends upon the complexity of the models

i.e., the level of detail and resolution depends on the treatment of electrochemical and thermal

phenomena. In addition to this, the geometry investigated is also a factor in determining the

computational cost. Hence, in order to simulate these systems at a reasonable computational cost

without compromising the accuracy, some type of analysis has to be carried out to derive reduced

models. Scaling analysis is an e¤ective tool to obtain model reductions and this is illustrated for the

model of an electrochemical capacitor in Chapter 4 and a Li-ion battery in Chapter 5. Once faster

and e¢ cient single cell models are obtained, then these can be utilized to build models for a battery

pack or an electrochemical capacitor stack as mathematical modeling that aims to elucidate and

resolve the salient features that can be found in a typical pack/stack is highly challenging: �rst,

we have to capture the three-dimensional nature and the multiple length and time scales from

the functional layers and groups in the cell to the module to the pack level; second, we need to

consider the highly coupled, non-linear behavior of the transport phenomena together with the

relevant electrochemistry and local e¤ects such as heat generation; and third, we also have to

account for the intrinsic transient nature of a battery/electrochemical capacitor and degradation

over time. Further, generally, when a mathematical model is solved numerically, a commercial

software is employed, in which the geometry/design, governing equations, boundary conditions,

constitutive relations and operating conditions are implemented � this is often a tedious, time


consuming task, requiring a specialist that knows the particular software well. This numerical

complexity is addressed with automated code generation to facilitate simulations and automate the

procedure of drawing the geometry, meshing, implementing the mathematical equations, solving

and postprocessing. This approach will allow for a large reduction in time, since the time spent

on manually setting up and solving a complex battery pack as well as any human errors in doing

so are removed. In addition, and perhaps most importantly, the automated script will allow for

completely automated multi-objective optimization, statistical modeling, perturbation studies and

wide-ranging parameter studies for various applications, such as battery and thermal management,

design, and overall optimization. The details of the automated model generation are discussed in

Chapter 6.

The application of the models derived above are demonstrated in Chapter 7 wherein a passive

thermal management system is proposed and analyzed for a commercially available spiral-wound

cylindrical Li-ion battery based on the coupled electrochemical-thermal model developed for a

Li-ion battery. Finally, the thesis ends with conclusions and outlook for future work in Chapter 8.

The contributions from this thesis are the development of reduced models based on model refor-

mulation and scaling analysis, a tool for studying battery packs/ electrochemical capacitor stacks

and the application of the detailed model to a commercially available spiral-wound 18650 battery.

Through model reformulation, a reduction in the dependent variable is achieved on microscale and

hence, an improvement in the computational cost. Through scaling analysis, criteria are developed

to justify reduction in dimensionality of the models as well as to obtain scales to estimate the in�u-

ence of the design variables on the performance of the system. These scales can be used as a rule

of thumb in the design of these storage systems. The automated model generation developed using

the interface between Comsol Multiphysics and Matlab is demonstrated in this thesis for studying

a bipolar Li-ion battery stack. This tool can be modi�ed to incorporate any kind of geometries and

materials. Finally, the developed models are applied to study the design of a spiral-wound battery

and the results are used to design a passive thermal management system for it.

Chapter 2

Literature Review

2.1 Mathematical Modeling of batteries

The review presented here complements the work of Thomas et al. [14] and Bandhauer et al. [15]

for the modeling e¤orts in Li-ion batteries.

The basic modeling framework consists of porous electrode theory, concentrated solution theory,

Ohm�s law, kinetic relationships, and charge and material balances [16]. Porous electrode theory

treats the porous electrode as a superposition of active material, electrolyte, and �ller, with each

phase having its own volume fraction [17]. The material balances are averaged about a volume small

with respect to the overall dimensions of the electrode but large with respect to the pore dimensions.

This allows one to treat electrochemical reaction as a homogeneous term, without having to worry

about the exact shape of the electrode�electrolyte interface. Concentrated solution theory provides

the relationship between driving forces (such as gradients in chemical potential) and mass �ux [18].

The �ux equation is then used in a standard material balance to account for the transient change of

concentration due to mass �ux and reaction. A charge balance is also needed to keep track of how

much current has passed from the electrode into the electrolyte. Ohm�s law describes the potential

drop across the electrode and also in the electrolyte. In the electrolyte, Ohm�s law is modi�ed to

include the di¤usion potential. Finally, the Butler�Volmer equation generally is used to relate the

rate of electrochemical reaction to the di¤erence in potential between the electrode and solution,

using a rate constant (exchange current density) that depends on the composition of the electrode

and the electrolyte.

9

2.1. Mathematical Modeling of batteries 10

Doyle et al. [19] presented a model for a full cell comprising lithium anode, soild polymer elec-

trolyte and insertion composite cathode using concentrated solution theory. Galvanostatic charge

and discharge of the full cell was simulated using the one dimensional model and the optimization

of system parameters was discussed. This model was further extended by Fuller et al. [20] for dual

insertion electrodes with liquid electrolyte and validated the simulation results with experimental

data. Later Doyle et al. [1] presented a generic one-dimensional model for lithium as well as

lithium-ion batteries and which became the basis for further developments. The model described

the potential distribution in the solid and the electrolyte phases by the porous electrode theory

developed by Newman [17], material balance in the electrolyte phase by the concentrated solution

theory and material balance in the solid phase by the Fick�s law of di¤usion in spherical coordinates.

The model was used to explore di¤erent system designs to achieve higher speci�c energy. All the

above said models describe the transport of species and charge in the solid and electrolyte phases

under isothermal conditions.

Bernardi et al. [21] presented a general energy balance for battery systems. They accounted

for heat generation occurring due to electrochemical reactions, phase changes, mixing e¤ects and

ohmic resistance in the energy balance. Later, Pals and Newman [22] included this energy balance

and extended the electrochemical model by Doyle et al. [19] to predict cell temperature. They

adopted an average heat generation method where the entire battery is assumed to be at uniform

temperature. Song and Evans [23]and Gu et al. [24] used local heat generation terms instead of

the average heat generation method. The coupling between the electrochemical and thermal model

is achieved through temperature dependent transport properties and heat generation terms.

Based on the above said electrochemical-thermal models, attempts were made to study various

aspects of Li-ion batteries �e¤ect of side reactions on battery performance and aging [25, 26, 27,

28, 29, 30, 31, 32], analysis of thermal behavior [2, 3, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43,

44, 45, 46, 47, 48], design of thermal management systems for battery systems [49, 50, 51, 52] �

from cell level to battery pack level. The complexity of the models depends on the level of detail

and accuracy required and the computing power available. This chapter presents a review of the

existing literature that are relevant to the work carried out in the thesis.

Hatchard et al. [53] developed a one-dimensional model to study the response of batteries to

oven heating, an abuse test and got reasonable agreements between their model calculations and


experimental results. They predicted the critical size for a cylindrical as well as a prismatic battery

at which the thermal runaway would occur.

The side reactions were taken into account by Botte et al. [3] in the energy balance for insertion

battery systems in order to see the e¤ect on the thermal behavior of the cell. Srinivasan et al. [2]

showed the two-dimensional e¤ects in a cell with large aspect ratio using the models and analyzed

the di¤erent sources of heat generation in the Li-ion cell. Kandler et al. [39] studied the behavior of

an electric vehicle Li-ion battery pack through a one-dimensional electrochemical, lumped thermal

model under high pulse power operations and found that the solid state di¤usion of Lithium is the

limiting mechanism in these conditions. Also, they observed that the inner regions of the active

material particles in the electrodes remain unutilized under high rate pulse discharges.

Li-ion batteries are commercially available in various form factors like cylindrical or prismatic

cells, coin cells, pouch cells etc. Of these, the cylindrical and prismatic cells are of spiral wound

type. The e¤ect of the various geometries requires the use of two- or three-dimensional models to

study the thermal behavior. All of the above models dealt with a monolithic structure of a Li-ion

battery and are mostly one-dimensional. Gomadam et al. [37] established criteria for assessing

the signi�cance of the spiral heat conduction and found that it is negligible for the case of Li-ion

batteries. The spiral wound con�guration of Li-ion batteries was studied for their thermal behavior

using a two-dimensional thermal model developed by Chen et al. [41]. It properly represented

the con�guration of the hollow core, the spiral, the contact layer, and the case in a battery to

avoid deviations due to improper approximation of the spiral geometry. The simulation results

showed that the maximum temperature does not occur exactly at the centre as there is no heat

generation in the hollow core but it is in the circular region near the hollow core and enhancing

the surface emissivity of the outer can is an e¤ective strategy for thermal management. Zhang [47]

studied the contribution of various heat sources for an 18650 cell using a thermal model developed

with one-dimensional electrochemical equations coupled with a cell-lumped energy equation under

galvanostatic discharge and found that the ohmic heat is the largest heat generation source that

contributes to 54% of the total heat generation on an average. This fact was further con�rmed

by Jeon et al. [48] through their modeling e¤orts validated with experimental data. They found

that the ohmic heat is the highest contributor at higher discharge rates whereas at lower discharge

rates, it is the heat generated due to entropy changes.


Al-Hallaj et al. [49] studied a novel passive thermal management system using a phase change

material (PCM) for a Li-ion battery pack using a one-dimensional thermal model with lumped

parameters. They suggested that this type of system should be e¤ective for batteries under cold

ambient conditions and in space applications. Chen et al. [38] developed a three-dimensional

model to examine the thermal behavior of the layered structure of the cell stacks in a battery

pack. The model predicted the temperature distribution in a battery pack under various cooling

conditions. Under forced convection, the temperature of the entire battery pack is reduced but there

is nonuniform temperature distribution as compared to natural convection that a¤ects the battery

performance. Radiation of heat from the outer surface of the battery was found to be an important

process of heat dissipation, especially under natural convection. Kizilel et al. [52] demonstrated

the advantages of using a passive thermal management system using a PCM over conventional

air cooling system through the thermal modeling of a battery module with heat generation data

obtained from experiments. They showed that the usage of PCM results in uniform temperature

distribution under normal and stressed operating conditions as well it prevents the propagation of

thermal runaway in a battery pack.

In addition to the use of the developed models for analyzing the performance of batteries and

design of thermal management systems, attempts were being carried out to improve the computa-

tional e¢ ciency for solving the models [4, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63]. Battery modeling

involves multi-scales. In the macro-scale, it includes the mass transport in the liquid phase, elec-

tronic conduction in the solid phase and ionic conduction in the liquid phase; in the micro-scale,

it includes the di¤usion of lithium in the active material of the porous electrodes. Solving the

di¤usion equation in a pseudo-dimension along with the macro-scale governing equations is com-

putationally expensive and hence methods like Duhammel�s superposition method [1] , di¤usion

length approach [54], and polynomial approximations [55] were employed. Subramanian et al. [4]

reformulated the mathematical model to reduce the computational time to simulate a cell. Various

numerical techniques such as orthogonal collocation [60, 63] or proper orthogonal decomposition

[59] were used by researchers to improve the computational e¢ ciency further.

Apart from the above mentioned models developed based on the mechanistic approach, there are

other models like equivalent circuit models [64, 65, 66, 67] that are �t to experimental data under

various conditions. These models are extremely simple and are faster to compute but they have

2.2. Mathematical Modeling of Electrochemical Capacitors 13

their own disadvantages. The models are only as good as the experimental data they are trained

to, and thereby do not provide the ability to extrapolate beyond the range of this data. In addition,

changes in design of the cell do not permit the use of the same models, and the task of building

prototype cells, collecting data and training the model has to be repeated. More importantly, as

these models are empirical in nature, they provide little, if any, insight into the working of the cell.

2.2 Mathematical Modeling of Electrochemical Capacitors

A detailed review of the history of the development of electrochemical capacitors and their applica-

tions can be obtained from the works of Conway [68, 69, 70], Burke [71] ,Kotz et al. [11], Simon et

al. [72] and Kurzweil [73]. A concise review on the modeling e¤orts of electrochemical capacitors

from cell level to stack level is provided below.

Posey and Morozumi [74] studied the potentiostatic and galvanostatic transients which occur

during charging of the double layer in porous and tubular electrodes of �nite length through a one

dimensional model. Faradaic reactions are not considered in the model. Newman and Tiedemann

[17] developed a generic model describing the theory of the operation and behavior of porous elec-

trodes by which almost almost any system can be treated. In particular, equations were developed

to provide the bases for examining the behavior of speci�c systems such as primary and secondary

batteries, adsorption and double-layer charging, and �ow-through electrochemical reactors. John-

son and Newman [75] developed a model for a porous electrode to analyze desalting processes in

terms of ionic adsorption on porous carbon. Pillay and Newman [76] analyzed the e¤ects of side

reactions on the performance of electrochemical double-layer capacitors through a one-dimensional

model developed based on the macrohomogenous theory of porous electrodes reviewed by Newman

and Tiedemann [17]. They studied the in�uence of oxygen and hydrogen evolution on the perfor-

mance of a typical system with carbon electrodes and sulfuric acid electrolyte. Using Tafel kinetics,

the authors conclude that the two side reactions result in considerable e¢ ciency losses during early

cycles even when operating the cell within the thermodynamic stability window of the electrolyte

and concluded that the consideration of the side reactions for a good design of the cell is essential.

Srinivasan and Weidener [77] carried out performance studies of electrochemical capacitors under

operating conditions of constant current and electrochemical impedance spectroscopy by solving the


mathematical model analytically. The e¤ect of cell design parameters on energy and power density

of the cell was established as well as heat generation was estimated through the one-dimensional

model.

Carbon materials have been widely used as electrodes for electrochemical capacitors. Also, metal

oxides like Ruthenium dioxide [7, 70, 73] gained popularity as electrode materials due to the presence

of faradaic reactions resulting in pseudocapacitance in addition to the double-layer capacitance .

Modeling e¤orts were then carried out on pseudocapacitors. Lin et al. [78] developed a model for

an electrochemical capacitor that considered both double-layer and faradaic processes. The one-

dimensional model they developed for a RuO2 supercapacitor was used to study its behavior under

galvanostatic charge and discharge conditions. Through the model, they concluded that smaller

particle sizes resulted in improved performance and the faradaic process increased the volumetric

energy density of the capacitor corresponding to a range of power density. Further extension of

this model [79] was carried out to study the e¤ects of composition variations, particle packing,

and concentration polarization. The model investigated the e¤ects of varying carbon type, carbon

mass/volume fraction, and discharge current density on the performance of RuO2/C electrochemical

capacitors. Among various types of activated carbons, those with large micropore surface areas and

low meso- and macropore surface areas are preferred because they give high double-layer capacitance

and favor e¢ cient packing of RuO2 nanoparticles, thus maximizing faradaic pseudocapacitance.

The model for RuO2 supercapacitor was further extended by Kim et al. [80] by considering the

proton di¤usion in RuO2 particle, thus distinguishing the amorphous and crystalline natures of the

which was not previously considered. A pseudo two-dimensional model was presented by following

the procedure of Doyle et al.[1] and the e¤ect of particle size, porosity and volume fraction of

C/RuO2 on the performance of the cell was studied.

In tandem with the model development, solution methodologies for the models are also in-

vestigated. Subramanian et al. [81] presented analytic solutions for the porous electrode with

double-layer charging and a faradaic reaction.A dynamic model was developed to replace the an-

alytic solution and the e¤ects of various design parameters on the performance of the cell was

studied. These e¤orts are attempted to improve the computational e¢ ciency while solving the

models. Verbrugge et al. [82] demonstrated the application of the model to determine if the reac-

tion distribution is su¢ ciently uniform to access all regions of the electrode and con�rmed that at


higher discharge rates, there is non-uniform charge distribution and lower electrode utilization.

As electrochemical capacitors are used for high power applications, thermal management is

one of the main issues for safer design of these systems. The heat generation in ultracapacitors

was studied by Schi¤er et al. [83] through experiments and it was found that the heat generation

comprises of the irreversible ohmic heating and the reversible heating due to the change of entropy

during charging and discharging. Guillemet et al.[84] analyzed the thermal behavior of an ultraca-

pacitor system under steady state through various modeling approaches depending upon the level

of accuracy required. The model predictions indicated the need for a thermal management system

to avoid detrimental e¤ects due to overheating of the ultracapacitor pack.

Apart from the detailed transport models mentioned above, equivalent electrical circuit models

[85, 86, 87, 88, 89, 90] were also developed to study the electrochemical and thermal behavior of

supercapacitor systems. These models are not as accurate as the detailed transport models but are

much faster to solve and are generally used for dynamic system studies.

In general, mathematical models form an e¤ective tool for analyzing and optimizing the design

of electrochemical capacitors from cell level to stack level. Selection of appropriate model entails

the level of accuracy required and the computational cost involved. One of the objectives of this

thesis is to reduce the computational cost through the use of techniques like scaling analysis that

provides criteria for selection of appropriate model that represent the given electrochemical system

through the scales and nondimensional numbers secured from the analysis.

Chapter 3

Mathematical Formulation

3.1 Lithium-Ion Batteries

3.1.1 Introduction

A schematic representation of a Li-ion cell sandwich is shown in Fig. 3.1. The cell sandwich consists

of �ve layers: porous composite insertion negative (ne) and positive electrode layers (pe) coated on

current collector foils (cc) with a separator (sp) in between them. The porous insertion electrodes

constitutes their respective active material particles �for example, graphite for negative electrode

and LiCoO2 for positive electrode �held together by a PVDF binder and a �ller material. The

negative current collector is usually copper and the positive is aluminum. The separator is a thin

�lm made up of polyethylene and is used to electrically isolate the two electrodes. The entire cell is

�lled with an electrolyte solution (typically LiPF6 salt in a 1:2 liquid mixture of ethylene carbonate

and dimethyl carbonate solvent)

The Li-ions shuttle between the insertion electrodes during charging and discharging. During

discharging, the Li-ions deintercalate from the active material particles in the negative electrode

and enter the solution phase whereas in the positive electrode, the Li-ions from the solution phase

intercalate into the active material particle. This leads to a concentration gradient within the cell

that drives the Li-ions from the negative electrode to the positive electrode as stated before. The

cell voltage reduces as the equilibrium potential of the electrodes are a function of the lithium

concentration on the surface of the active material particles. These various phenomena �mass

16

3.1. Lithium-Ion Batteries 17

Figure 3.1: Schematic of a single Li-ion cell showing (a) various functional layers on themacroscale, and (b) di¤usion of lithium in the active material in the electrodes on the microscale.

transport of lithium in the solid phase in the electrodes, mass transport of lithium ions in the

liquid phase in the electrodes and separator, charge transport in the solid phase in the electrodes

and current collectors and charge transport in the liquid phase in the electrodes and separator �

are to be accounted in the mathematical description of the system. Such a model is termed as the

electrochemical model of the system. The electrochemical model predicts the cell voltage as well as

the potential and concentration distributions within the cell during charging or discharging. Due

to the kinetic, ohmic and mass transfer resistances, heat is generated from the cell that leads to a

temperature distribution within the cell in addition to the concentration and potential distributions.

Hence to have a complete description of the system, the energy transport in the cell should also

be considered which is termed as the thermal model of the system. The detailed derivation of the

system of equations mentioned below could be obtained from Newman et al. [18].

3.1.2 Assumptions

Some of the assumptions made for developing the model are listed below.

� The electrolyte is assumed to be a binary electrolyte and Li+ ions are the only electroactive


species [18, 19].

� Charge transfer across the electrode/electrolyte interface is assumed to be described by Butler-

Volmer type kinetic expression [18, 19].

� Side reactions are neglected.

� Concentration dependent exchange current density is considered to account for the variation

of the current with the solid and solution phase concentrations [18, 19].

� The active material particles in the electrodes are assumed to be spherical.

� Uniform distribution of active materials of the same size in the electrodes.

� Volume changes in the cell due to its operation are neglected and a constant porosity is used

[18, 19].

� The temperature dependent transport and kinetic parameters are described by Arrhenius

type expressions [18, 19].

� Double layer e¤ects are considered negligible [14].

3.1.3 Governing equations (Macroscale)

The transport of ions and electrons in the cell between the electrodes is referred as the transport at

the macroscale, which includes species transport in the liquid/electrolyte phase, electronic charge

conduction in the solid phase and ionic charge conduction in the liquid/electrolyte phase. The

mathematical model comprises conservation of species and charge in the liquid- and the solid-phase

as well as the conservation of energy [18]

r � is = �J (pe, ne, cc), (3.1)

r � il = J (pe, ne, sp), (3.2)

"l@cl@t+r �Nl =

J

F(pe, ne, sp), (3.3)

(�Cp)eff @T

@t+r � q = Q (pe, ne, sp, cc), (3.4)


where the �uxes are given by

is = ��effs r�s; (3.5)

il = ��effl r�l +2RT�effl

F

�1� t0+

�r(ln cl); (3.6)

Nl = �Deffl rcl +ilt0+

F; (3.7)

q = �keffrT: (3.8)

with the source term expressed as

J =

8>>>><>>>>:Aif (ne, pe)

0 (sp, cc)

: (3.9)

In the above equations, is is the solid-phase current density, J is the transfer current per unit

volume, �effs is the e¤ective conductivity of the solid phase, �s is the solid phase potential, il is

the solution phase current density, �effl is the e¤ective electric conductivity of the solution phase,

�l is the solution phase potential, � is the number of moles of ions into which a mole of electrolyte

dissociates, R is the universal gas constant; T is the absolute temperature, F is the Faraday�s

constant, t0+ is the transference number of the cation, "l is the volume fraction of electrolyte in the

electrodes and the separator, and Nl is the molar �ux of the cations; � is the e¤ective density of the

various functional layers, Cp is the e¤ective speci�c heat capacity, q is the conductive heat �ux, Q is

the heat generation per unit volume, Deffl is the e¤ective di¤usion coe¢ cient in the liquid/solution

phase, keff is the e¤ective thermal conductivity, A is the speci�c surface area for the faradaic

reaction per unit volume, and if is the charge transfer current density; the electrolyte is a binary

electrolyte with the concentration of the electrolyte cl de�ned as [18] cl = c+=�+ = c�=��, where

c+ and c� are the concentrations of the cations and anions respectively, and �+ and �� represent

the number of cations and anions produced by the dissociation of one mole of electrolyte. (The

governing equations are applicable in the layers mentioned inside parentheses.)


3.1.4 Governing equations (Microscale)

At the microscale, lithium di¤uses into the spherical active material particles (Fig. 3.1b), whence

the conservation of lithium inside the active material can be stated as

@cs@t

=1

r2@

@r

�r2Ds

@cs@r

�; (3.10)

where cs(x; y; r; t) is the concentration of protons in the active material particle of the electrode;

t represents the time, Ds is the di¤usion coe¢ cient of lithium in the active material, and r is the

radial coordinate inside an agglomerate. Here the di¤usion length approach [54] or a polynomial

approximation [55] approach is employed, such that the governing equations in the microscale are

reduced to

Dsls(csurfs � cavgs ) = � if

F; (3.11)

dcavgsdt

= � 3ifFR

; (3.12)

csurfs and cavgs are the volume-averaged surface and average concentrations of Li-ion in the active

material, if is the charge transfer current density and R is the radius of the active material in

the electrodes. The approximation mentioned above is obtained from the 2-parameter model or

the parabolic pro�le approximation from Subramanian et al. [55, 56]. At higher discharge rates, it

might not give accurate predictions of the surface concentration and hence higher order polynomials

like the 3-parameter or the 4-parameter models [55, 56] are employed.

3.1.5 Boundary and initial conditions

The generic boundary conditions are provided in this chapter. Boundary conditions speci�c to the

geometries and operating conditions are provided in individual chapters.

At the left and right sides of the current collectors and the top and bottom walls (see Fig. 3.1

for placement of roman numerals):

is � ex = 0;q � ex = h(T � Ta) (I); is � ey = il � ey = Nl � ey = 0;q � ey = h(T � Ta) (V,VII); (3.13)


At the current collector/electrode interface:

is � exjII+ = is � exjII� ; q � exjII+ = q � exjII� ; il � ex = Nl � ex = 0 (II) (3.14)

At the electrode/separator interface:

is � ex = 0; il � exjIII+ = il � exjIII� ; Nl � exjIII+ = Nl � exjIII� ; q � exjIII+ = q � exjIII� (III) (3.15)

At the top side of the negative current collector, the solid phase potential is set to be zero:

�s = 0 (IV) (3.16)

At the top side of the positive current collector, the current density is prescribed:

is � ey = �iapp (VI) (3.17)

In Eqs. 3.13 - 3.17, ex denotes the unit vector along the x-direction, ey denotes the unit vector

along y-direction, h is the heat transfer coe¢ cient, Tamb is the ambient temperature, and iapp is the

applied current density. The boundary conditions for the heat transfer can be varied by changing

the value of the heat transfer coe¢ cient to simulate a wide range of operating scenarios. The two

extreme cases of an insulated cell as well as a cell with in�nite heat transfer could be simulated by

setting h = 0 and h!1 respectively.

The boundary conditions for the solid-state di¤usion (microscale) are as follows:

Ds@cs@r

= 0 at r = 0 (3.18)

�Ds@cs@r

=ifFat r = R (3.19)


At t = 0,

cs = c0s; cl = c

0l ; T = T0 (3.20)

�s =

8>>>><>>>>:0 (ne)

�0s (pe)

(3.21)

�l = �0l (ne,pe,sp) (3.22)

3.1.6 Constitutive relations

The constitutive relations used in general are provided in this section. Relations speci�c to certain

cases are provided in the respective chapters. The source term, J , is expressed as

J =

8>>>><>>>>:Aif (ne, pe)

0 (sp, cc)

; (3.23a)

where A is the speci�c surface area for the faradaic reaction per unit volume; the local charge

transfer current density is given by the Butler-Volmer equation for electrode kinetics

if = i0

�exp

��a�F

RT

�� exp

��c�FRT

��: (3.24)

Here, i0 is the exchange current density, �a and �c are the anodic and cathodic transfer coe¢ cients

and � is the overpotential. The speci�c interfacial area is related to the particle radius, volume

void fraction of �ller �f , polymer matrix �p, and, solution phase "l [1] as

A = 3(1� "l � "f � "p)R

: (3.25)

The exchange current density is given by

i0 = Fk0

qcl(cmaxs � csurfs )csurfs ; (3.26)


k0 is the reaction rate constant and cmaxs is the maximum lithium concentration in the electrodes.

The overpotential is de�ned as

� = �s � �l � Ueffref; i; i = ne, pe (3.27)

in which U effref is the e¤ective open circuit potential of the electrode with respect to the solid lithium

electrode. The internal heat generation rate Q [2] is de�ned as

Q = J�+JT@Uref; i@T

+�effs (r�s)2+�effl (r�l)2+

2RT�efflF

(1�t0+)r(ln cl)�r�l; i = ne, pe (3.28)

The �rst term on the right-hand-side (RHS) in the heat generation, Eq. 3.28, captures the ir-

reversible heat generation arising due to the reaction that is responsible for the deviation of the

potential from the equilibrium potential; the second term is the reversible heat due to the changes

in entropy; the third term quanti�es the ohmic heating in the solid phase; and the last two terms

re�ect the ohmic heating in the solution phase. �ne and �pe are the state of charge of the negative

and positive electrodes respectively, and are de�ned as

�ne = �pe =csurfs

cmaxs

: (3.29)

The overall state of charge of the battery here refers to the local state of charge of the electrode

that limits the battery capacity. The electrode capacity is de�ned as

Ci = wi(1� "l � "f � "p)�iCth; i = ne,pe, (3.30)

where wi represents the thickness of the electrodes and Cth is the theoretical capacity of the electrode

material. The total battery capacity will be the minimum of the two electrode capacities.

The e¤ective (i.e., temperature-dependent) open-circuit potential of an electrode is approxi-

mated by a �rst order Taylor-series expansion around a reference temperature Tref [91]:

U effref; i = Uref; i + (T � Tref )@Uref; i@T

; i = ne, pe (3.31)


The e¤ective properties are de�ned as

�effs = �s(1� "l � "f � "p); (3.32)

�effl = �l" l ; (3.33)

keffi = ki(1� "l) + kl"l; i = ne, pe, sp (3.34)

(�Cp)effi = (�Cp)i (1� "l) + (�Cp)l "l; i = ne, pe, sp (3.35)

Deffl = Dl" l ; (3.36)

where is the Bruggemann constant.

The electrolyte conductivity is expressed as a function of concentration cl [1]:

�l =

4Xi=0

aicil; (3.37)

where ai are constants.

The physical properties �di¤usion coe¢ cients (both solid and liquid) and ionic conductivity

(liquid) �are dependent on temperature, the function of which is typically written in the form of

an Arrhenius expression [2, 91]:

�(T ) = �(Tref ) exp

�Ea;�R

�1

Tref� 1

T

��(3.38)

where �(T ) is a placeholder for a temperature-dependent property, Tref is a reference temperature,

and Ea;� is the activation energy.

The di¤usion length, ls, for the spherical electrode particles is estimated as [54]

ls =R

5: (3.39)

The initial values of the solid phase and the liquid phase potential are given by

�0s = Uref; pe(�0pe)� Uref; ne(�0ne); (3.40)

�0l = �Uref; ne(�0ne); (3.41)

3.2. Electrochemical Capacitors 25

where �0pe and �0ne denote the initial state of charge (SOC) of the positive and negative electrodes

respectively. The solid-phase potential di¤erence between the positive current collector and the

negative current collector is de�ned as the cell voltage, Ecell:

Ecell = �sjII � �sjV : (3.42)

The current density at which the battery becomes completely discharged until the cut-o¤voltage

in 1 hour is taken as the 1 C-rate. (Complete discharge here implies the local SOC in the positive

electrode reaches one.) The parameters used for simulation are provided in the respective chapters.

3.2 Electrochemical Capacitors

As mentioned before, the similarities present between the electrochemical capacitors and batteries

greatly simplify the development of a model for an electrochemical capacitor. The construction of

the supercapacitor is the same as that of a battery consisting of two porous electrodes coated on

current collector foils with a separator in between. The model considered here accounts for both

the double-layer and faradaic reaction mechanisms of charge storage but is an isothermal model

unlike the Li-ion battery. By exploiting the similarities, the model for an electrochemical capacitor

is obtained by adding the double-layer charge storage mechanism to the model presented above for

a Li-ion battery. This requires a modi�cation of equations for the conservation of charge as well as

the species. The transfer current per unit volume J will now have two terms, one representing the

faradaic reaction and the other representing the double-layer charging [18, 92]. Except the source

term J , all other equations remain the same as that for a Li-ion battery. Therefore, in order to

avoid repetition, the governing equations, boundary conditions and constitutive relations for the

electrochemical capacitor were provided in the next chapter. Another reason for providing the

model equations of the electrochemical capacitor in the next chapter is to have a continuity for the

reader to ensure better understanding of the model analysis being carried out and avoid confusion

between the battery and capacitor models.


Nomenclature

ai constants in electrolyte conductivity expression

A speci�c surface area for the faradaic reaction per unit volume, m�1

Ci electrode capacity, Ah m�2

Cth theoretical capacity of electrode material, mAh g�1

Cp speci�c heat capacity, J kg�1 K�1

cl electrolyte concentration, mol m�3

cs concentration of lithium in active material in the electrodes, mol m�3

cavgs average concentration of lithium in the active materials, mol m�3

csurfs surface concentration of lithium in active materials, mol m�3

Dl di¤usion coe¢ cient of electrolyte, m2 s�1

Ds di¤usion coe¢ cient of lithium in the active material in the electrodes, m2 s�1

Ea;� activation energy for a variable �, kJ mol�1

ex; ey coordinate vectors

F Faraday�s constant, 96487 C mol�1

h height of the battery, m

h heat transfer coe¢ cient, W m�2 K�1

iapp applied current density, A m�2

i0 exchange current density, A m�2

il liquid phase current density, A m�2

is solid phase current density, A m�2

if faradaic transfer current density, A m�2


J local charge transfer current per unit volume, A m�3

k thermal conductivity, W m�1 K�1

k0 reaction rate constant, mol m�2s�1�mol m�3

�1:5ls di¤usion length, m

Nl species (lithium ion) �ux, mol m�2 s�1

Q volumetric heat generation, W m�3

q conductive heat �ux, W m�2

R gas constant, J mol�1 K�1

R radius of active material, m

r radial coordinate

t time, s

t0+ transference number of cation

T temperature, K

Ta; T0 ambient and initial temperature, K

Tref reference temperature, 298:15 K

Uref open circuit potential of the electrode, V

wi thickness of the layer i, m

Greek

�a anodic transfer coe¢ cient

�c cathodic transfer coe¢ cient

"l volume fraction of the eletrolyte in the electrodes and separator

"f volume fraction of the conductive �ller additive in the electrodes


"p volume fraction of the polymer in the electrodes

� overpotential, V

�+; �� number of cations and anions into which a mole of electrolyte dissociates

� density, kg m�3

�l ionic conductivity of electrolyte, S m�1

�s electric conductivity of solid matrix, S m�1

�l liquid phase potential, V

�s solid phase potential, V

� local state of charge of the electrodes

�(T ) placeholder for a temperature dependent property

Bruggeman constant (= 1:5)

Subscripts

cc current collector

ne negative electrode

pe positive electrode

sp separator

l liquid/electrolyte

Superscripts

0 initial values

eff e¤ective values

max maximum values

Chapter 4

Analysis of Electrochemical Capacitor

Model

4.1 Introduction

Electrochemical capacitors �which store charges through electrostatic charging of a double layer

as well as through a Faradaic reaction �not only have a higher energy density than conventional

electrostatic capacitors but also possess higher power density than batteries in general. Due to

these advantages, electrochemical capacitors are being considered for energy storage in a range of

applications: e.g., in electrical and hybrid vehicles [70, 71].

Mathematical modeling and numerical simulations of electrochemical energy systems like bat-

teries and electrochemical capacitors plays a vital role in their design and estimation of performance;

however, deriving and solving these models numerically is a challenging task for two main reasons:

First, a model that aims to capture the physical and chemical phenomena has to consider conserva-

tion of charge, species and energy together with relevant constitutive relations for the double-layer

charging and Faradaic reaction; and second, it has to do so for all functional layers and groups

within the cell at varying length scales (macroscale and microscale), as illustrated in Fig. 4.1 and

4.2.

Thus far, a number of mathematical models starting from models that describe the transport

inside a single cell to equivalent circuit models have been developed. The transport models account

for the charge and ion transport taking place in a cell, most of which consider electric double-layer

29

4.1. Introduction 30

Figure 4.1: Schematic of an electrochemical capacitor cell.

capacitors that do not have a Faradaic reaction [74, 76, 77, 82, 93]. In contrast to the conventional

electrochemical capacitors that store charges in double-layers electrostatically, pseudocapacitors

store charges faradaically as well through charge transfer between electrode and electrolyte. The

faradaic process allows pseudocapacitors to achieve higher capacitances than the conventional ones.

The electrochemical capacitor with Faradaic reaction is modeled by only a few [78, 79, 80]. Lin

et al. developed a electrochemical capacitor model with a carbon/RuO2 composite electrode [78]

that includes the faradaic reaction of Ruthenium dioxide (RuO2), but did not consider the proton

di¤usion in RuO2. Kim and Popov included and studied the proton di¤usion in RuO2 [80]. The

ionic transport in nanoporous carbon was explored by Zuleta with an agglomerate hypothesis [94],

which was applied by Malmberg [95] to model an alkaline electrochemical capacitor electrode to

study high current operation; Malmberg also validated his model for the case of a single electrode

and a half cell. All of the these models are one-dimensional (1D) on the macroscale, based on

postulated 1D transport and are not validated with experimental results for a single electrochemical

capacitor cell. The other class of models, the equivalent electrical circuit models [75, 86, 87, 89] are


able to predict the overall performance of electrochemical capacitors, but fail to do so on a local,

mechanistic level. In addition, the physical properties of the materials of the capacitors are often

not considered by these models [87] �hence their predictions are not as accurate as the transport

models.

Figure 4.2: Section A-A of a single cell showing (a) various functional layers on the macroscale,and (b) di¤usion of proton in RuO2 in the electrodes on the microscale.

To extend the previous work on modeling of electrochemical capacitors, a transient, isothermal

model comprising conservation of charges and species at the macro- and micro-scale (see Fig. 4.1

and 4.2) is considered that also accounts for double-layer charging and a Faradaic reaction for

electrodes made of RuO2 that exhibit pseudocapacitance. In short, the transport of ions and

electrons in the cell between the electrodes is referred as the transport at the macroscale, which

includes species transport in the liquid/electrolyte phase, electronic charge conduction in the solid

phase and ionic charge conduction in the liquid/electrolyte phase; and the di¤usion of ions in

the active material present in the electrodes is referred to as the transport at the microscale,

4.2. Mathematical Formulation 32

which includes di¤usion of protons in the active material (RuO2) of the porous electrodes. After

calibration and validation with experimental data obtained from Zheng et al., [7] scaling arguments

are employed to (i) secure the scales and nondimensional numbers that characterize the cell, (ii)

show in what limits a 1D model can be employed, (iii) secure limits when the microscale governing

equation can be simpli�ed, and (iv) demonstrate how model reductions result in a signi�cant

improvement in the computational cost without compromising the accuracy in certain limits �the

latter is key for deriving models that not only capture the detailed, mechanistic behavior of a

single cell but an entire stack. Finally, conclusions are drawn and extensions are highlighted �e.g.,

inclusion of conservation of energy and heat generation �to the validated mathematical framework

presented here.

4.2 Mathematical Formulation

A model for a three-dimensional (3D) electrochemical capacitor cell is considered that consists of

two porous electrodes (ne and pe for the negative and positive electrode respectively) with an ion-

conductive separator (sp) sandwiched between them and current collectors (cc) at both ends, as

shown in Fig. 4.1. The two electrodes are made of RuO2 with a solution of H2SO4 electrolyte �lling

the pores of the electrode and separator, similar to the conditions in the experiments by Zheng et al.

[7]. This cell exhibits pseudocapacitance that depends on the charge storage mechanism as well as

the structure of the material. The two di¤erent mechanisms that account for charge storage through

pseudocapacitance in metal oxides are adsorption/desorption and insertion/deinsertion of a species

in the active material; [72, 96] for RuO2, the main charge-storage mechanism is the protonation

of the oxide inside the agglomerates at the microscopic level. [7] An amorphous structure of

hydrated RuO2 �which is the one considered here �has a higher speci�c capacitance than the

rutile crystalline structure, since Ru ions inside the agglomerates also participate in the Faradaic

reaction in the amorphous structure as opposed to the crystalline structure where the reaction

is mainly con�ned to the surface.[7, 97, 98, 99]The Faradaic reaction can be described with the

following reaction mechanism [100] (see Fig. 4.2):

H0:8+�RuO2 H0:8RuO2+�H++�e�


In order to ensure a tractable analysis, the discussion is limited to an isothermal model and outline

extensions to account for thermal e¤ects in the Conclusions. The model itself is based on the

porous-electrode theory developed by Newman and Tiedemann [17, 18] and embodies the following

main assumptions:

1. Physical properties like the di¤usion coe¢ cient, transference number are not dependent on

electrolyte concentration;

2. The exchange current density is not concentration dependent;

3. Isotropic material properties;

4. The double layer capacitance per unit area is a constant;

5. Uniform distribution of active materials of the same size in the electrodes;

6. The active material is assumed to be spherical; i.e., only the radial direction is considered at

the microscale;

7. Charge storage through insertion/deinsertion inside the agglomerates is considered at the

microscale and adsorption/desorption assumed negligible for the amorphous RuO2 considered

here ;

8. Temperature e¤ects and side reactions are assumed negligible.

The assumption of isothermal conditions, #8, coupled with zero-�ux condition that can be

invoked on the left and right sides (marked "B" in Fig. 4.1) of the cell along with the porous

nature of the electrodes and separator and solid nature of the current collectors allow for a model

reduction from 3D (x; y; z) to two dimensions (x; y) since there are then no variations in the

dependent variables in the z-direction. One can thus e¤ectively start with a 2D mathematical

representation of a 3D cell. Galvanostatic charge and discharge is considered.


The mathematical model accounts for transient conservation of charge and species (cations) in two

phases: the solution phase and the solid phase. The governing equations at the macroscale are as


follows [79, 80, 92]:

r � is = �J (ne,pe,cc) (4.1)

r � il = J (ne,pe,sp) (4.2)

�+"@cl@t+r �N+ =

J

F(ne,pe,sp) (4.3)

Here, the �uxes are de�ned as

is = ��sr�s (4.4)

il = ��effl r�l ��RT�effl

F

�s+n�+

+t0+z+�+

�r(ln cl) (4.5)

N+ = ��+Deffl rcl +ilt0+

z+F(4.6)

and the source term is expressed as

J =

8>>>><>>>>:�AdCd

z+

�dq+dq

�@@t (�s � �l)�Af

s+n if (ne, pe)

0 (sp, cc)

(4.7)

In the above equations, is is the solid phase current density, J is the transfer current per unit

volume, �effs is the e¤ective conductivity of the solid phase, �s is the solid phase potential, il

is the solution phase current density, �effl is the e¤ective electric conductivity of the solution

phase, �l is the solution phase potential, � is the number of moles of ions into which a mole of

electrolyte dissociates, R is the universal gas constant; T is the absolute temperature, F is the

Faraday�s constant, s+ is the stoichiometric coe¢ cient of the cations in the electrode reaction, n

is the number of electrons transferred in the reaction, z+ is the charge number of the cation, t0+ is

the transference number of the cation, " is porosity of the electrodes and the separator, N+ is the

molar �ux of the cations; Deffl is the e¤ective di¤usion coe¢ cient in the liquid/solution phase, Ad

is the speci�c interfacial area for the double-layer capacitance per unit volume, Af is the speci�c

surface area for faradaic reaction/pseudocapacitance per unit volume, q+ and q are the charge of

the cations and the total charge stored in the interfacial region (dq+=dq = �1, as only cations are


adsorbed [14] ), Cd is the double layer capacitance per unit area and if is the Faradaic transfer

current density for the RuO2 redox reaction. The governing equations are applicable in the layers

mentioned inside brackets.

The electrolyte is a binary electrolyte with the concentration of the electrolyte cl de�ned as [18]

cl =c+�+

=c��

(4.8)

where c+ and c� are the concentrations of the cations and anions respectively; and �+ and ��

represent the number of cations and anions produced by the dissociation of one mole of electrolyte.

4.2.2 Governing equations (Microscale)

At the microscale, protons di¤use into RuO2 particles (see Fig. 4.2b) whence the conservation of

protons inside the active material can be stated as

@cs@t

=1

r2@

@r

�r2Ds

@cs@r

�(4.9)

where cs(x; y; r; t) is the concentration of protons in a RuO2 particle; t represents the time, Ds is

the di¤usion coe¢ cient of protons in RuO2, and r is the radial coordinate. In the microscale, the

Ru ions involved in the faradaic reaction are only from the surface in case of crystalline structure

of RuO2, whereas for amorphous material, in addition to the surface, Ru ions from the bulk of

the material also participates in the reaction, thus accounting for higher charge storage. However,

since the focus of this study is mainly on the macroscale related to the design and optimization of

a cell, the structure of the materials are not analyzed in depth, but the physical properties used in

the simulation taken from ref. 7 correspond to an amorphous structure. [7]

This transport equation 4.9 in the radial direction has to be solved in the entire electrode and

thus adds complexity to the overall model, because the proton concentration at the surface of the

spherical active material particle is coupled to the concentration and the �ux at the macroscale

for the charge and material transport in the electrolyte through the boundary condition at the

solid electrode/electrolyte interface. Therefore, for a 2D model at the macroscale that embodies

features at the microscale, the microscale model has to be solved in three dimensions numerically

with one representing the direction of di¤usion of proton (radial coordinate r) and the other two


representing the macroscale coordinates (x,y). This model is thus referred to as a 2D+3D model.


At the left and right sides of the current collectors and the top and bottom walls (see Fig. 4.2 for

placement of roman numerals):

is � ex = 0 (I); is � ey = 0; il � ey = 0;Nl � ey = 0 (V,VII) (4.10)

At the current collector/electrode interface:

is � exjII+ = is � exjII� ; il � ex = Nl � ex = 0 (II) (4.11)

At the electrode/separator interface:

is � ex = 0; il � exjIII+ = il � exjIII� ; Nl � exjIII+ = Nl � exjIII� (III) (4.12)

At the top side of the negative current collector, the solid phase potential is set to be zero:

�s = 0 (IV) (4.13)

At the top side of the positive current collector, the current density is prescribed:

is � ey = �iapp (VI) (4.14)

In Eqs. 4.10 - 4.14, ex denotes the unit vector along the x-direction, ey denotes the unit vector

along y-direction, and iapp is the applied current density. A constant current is applied until the

capacitor is charged to the cell voltage (1 V in our case) and then immediately discharged at the

same current until the cell voltage reaches zero.


The boundary conditions for the solid-state di¤usion (microscale) are as follows:

Ds@cs@r

= 0 at r = 0 (4.15)

�Ds@cs@r

=ifFat r = R (4.16)

At t = 0,

cs = c0s; cl = c

0l (4.17)

�s =

8>>>><>>>>:0 (ne)

�0s (pe)

(4.18)

�l = 0 (ne,pe,sp) (4.19)

4.2.4 Constitutive relations and parameters

The transfer current density is given by Butler-Volmer kinetics:

if = i0

�exp

��a�F

RT

�� exp

��c�FRT

��(4.20)

where i0 is the exchange current density, �a and �c are the anodic and cathodic transfer coe¢ cients

respectively, and � is the overpotential, which is de�ned as

� = �s � �l � U (4.21)

Here, U is the equilibrium potential of the RuO2 electrode (vs. saturated calomel electrode [SCE]),

de�ned as [80]

U = 2U0

�1:3� MRuO2

�RuO2csurfs

�(4.22)

where U0 is the initial equilibrium voltage before charging, MRuO2 is the molecular mass of RuO2;

�RuO2 is the density of RuO2; and csurfs is the surface concentration of protons in RuO2; i.e.,

csurfs = cs(r = R).

The speci�c surface area per unit volume for the double-layer capacitance (Ad) and the Faradaic

4.3. Numerics 38

reaction (Af ) are given by [28]

Ad = Af =3(1� ")R

(4.23)

The e¤ective ionic conductivity and di¤usion coe¢ cient in the porous layers are expressed with the

common Bruggeman correction factor as [80, 92]

�effl = �l"1:5;Deffl = Dl"

0:5 (4.24)

The parameters are summarized in Table 4.1.

4.3 Numerics

The commercial �nite-element solver, COMSOL Multiphysics 3.5a [103], was employed to solve

the full model (2D+3D) and the reduced model (1D+1D and 1D+2D) that will be secured in the

analysis later. In short, quadratic elements were implemented for all dependent variables; the direct

solver UMFPACK was chosen as linear solver with a relative convergence tolerance of 10�6; and

solutions for all models were tested for mesh independence.

The full model is simulated in COMSOL by coupling two distinct geometries: The �rst captures

the 2D geometry representing the total cell with the two electrodes and a separator in between

them as well as the current collectors at both the ends, where the macroscale governing equations

are solved; and the second geometry captures the solid-phase di¤usion of protons in RuO2 in the

electrodes which is numerically represented by a 3D geometry with x and y -axes representing the

coordinates of the electrode as in the macroscale and the z-axis representing the active material

radius, r.

Numerically, the di¤usion in the 3D numerical geometry is modeled as anisotropic with a non-

zero di¤usion coe¢ cient in the z-direction and zero in the x- and y-direction: The surface concen-

tration of protons at r = R in the 3D micro-scale geometry is coupled to the concentration and

�ux in the 2D macro-scale geometry for charge and material transport in the electrolyte.

Charge and discharge currents are applied at the respective boundaries with a smoothed Heav-

iside function for the full and the reduced models.

All computations were carried out on a PC with a dual-core processor 2.33 GHz and a total of

4.3. Numerics 39

Table 4.1: Parameters

Parameter Value Units Reference

Cd 4:75 F m�2 calibrated

c0l 5:3� 103 mol m�3 [7]

c0s;ne;pe 2:2� 104 mol m�3 calculated from ref. [79]

Dl 2:9� 10�9 m2 s�1 [101]

Ds 10�15 m2 s�1 [79]

h 1:3� 10�2 m [7]

i0 0:1 A m�2 [79]

wcc; wne; wsp; wpe (10; 580; 50; 580)� 10�6 m [7]

MRuO2 0:13 kg mol�1 -

R 20� 10�9 m [7]

s+; n � (0 � � � 1) - -

T 298:15 K -

t+0 0:74 - [101]

U0 0:97 V [7]

�a; �c 0:5; 0:5 - assumed

"sp; "ne;pe 0:7; 0:05 - [102]

�; �+; �� 3; 2; 1 - -

�RuO2 2:3� 103 kg m�3 [79]

�l 76 S m�1 [101]

�s;ne; �s;pe; �s;cc (3:1; 3:1; 810)� 104 S m�1 [7]

�0s;ne;pe 0; 0 V -

4.4. Calibration and Validation 40

4 GB random access memory (RAM). The real execution times (wall-clock time) and peak memory

usage were estimated from the graphical user interface of COMSOL with all unnecessary processes

stopped to ensure accurate times.

4.4 Calibration and Validation

The nonlinear, multi-scale and coupled nature of the the governing equations as well as the high

number of parameters result in a complex model, which requires some form of calibration with a

training set and validation with a test set. Here, the experimental charge and discharge curves from

Zheng et al. [7] at a current of 5 mA are employed: One experimental point at the charge curve

is chosen for the training set (black symbol in Fig. 4.3) to heuristically calibrate the di¤erential

capacitance, Cd; and the remaining points for charge as well as discharge provide the test set for

validation. The matching conditions between the simulation and the experiment were the various

physical properties of the electrode materials other than Cd, the operaitnf conditions i.e., the

discharge and charging rate, and the geometric dimensions of the cell as mentioned in Table 4.1.

Overall, good agreement is found, as can be inferred from Fig. 4.3.

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

t / hour

Ece

ll / V

Figure 4.3: Cell voltage vs time for galvanostatic charge and discharge at 5 mA: Full model,2D+3D (line) and experimental data (symbols) from Zheng et al. (Ref. [7])

4.5. Analysis (Microscale) 41

4.5 Analysis (Microscale)

The model equations mentioned above were taken from the available literature and from this section

onwards, analysis was carried out on those models on the micro and macroscales that contributes

to a signi�cant part of this thesis. The di¤usion of protons into RuO2 occurs at a time scale

determined by the radius of the agglomerate and the di¤usion coe¢ cient:

tmicro �R2

Ds: (4.25)

At the macroscale, there is another time scale that corresponds to the charge /discharge time,

which will be secured later in the macroscale analysis. Taking the ratio of these two time scales

yields the nondimensional number

�1 �tmicrotmacro

; (4.26)

based on which the solid phase di¤usion can be analyzed.

First, let us consider the case where �1 � 1. In this limit, the solid-phase di¤usion reduces to a

quasi-steady-state problem, which can be solved analytically since the micro-scale di¤usion is much

faster than the macroscopic time for charge/discharge of the capacitor; thus, Eq. 4.9 becomes

0 =1

r2@

@r

�r2Ds

@cs@r

�(4.27)

Integrating and substituting the boundary condition given by Eq. 4.16 results in

cs = �C1r+ C2 (4.28)

such that

cs = constant (4.29)

since the constant C1 has to be equal to zero in order to bound the solution in the sphere when

r ! 0 m.

The fact that cs is constant at leading-order at the microscale by integrating Eq. 4.9 over the

4.5. Analysis (Microscale) 42

sphere results in

RZ0

r2@cs@tdr =

RZ0

@

@r

�r2Ds

@cs@r

�dr (4.30)

R3

3

@cs@t

= R2 Ds@cs@r

��r=R

(4.31)

From the boundary condition given by Eq. 4.16, and from Eq. 4.31, the following equation is

obtained@cs@t

= � 3ifRF

(4.32)

The microscale coordinate r is thus eliminated from the system of governing equations and the

overall problem is now reduced from 2D+3D to 2D+2D by replacing Eq. 4.9 with Eq. 4.32: i.e.,

2D on the macroscale (x; y) and 2D on the microscale (x; y) due to coupling between the two scales.

When �1 & 1, the di¤usion equation at the microscale has to be solved numerically or by

introducing an approximation of, e.g., a polynomial for the solution of cs in analogy with the

treatment of Li-ion batteries [24, 55, 56] to further reduce the dimensionality.

In order to verify the two limits for �1, the concentration pro�le of the protons in RuO2

predicted by the 2D+3D model is compared with the 2D+2D counterpart in both the positive and

the negative electrodes at two di¤erent currents of 5 mA (macroscopic time scale around 104 s) and

5000 mA (macroscopic time scale around 10�1 s), as illustrated in Fig. 4.4 and 4.5 respectively.

Clearly, at 5 mA for which �1 � 1, the concentration does not depend on the radius and remains

constant throughout the active material as indicated by the straight lines in Fig. 4.4 at various

times during charging; at 5000 mA, however, the concentration varies with respect to the radius as

shown in Fig. 4.5 since �1 > 1: The nondimensional number, �1; thus decides the model reduction

from 2D+3D to 2D+2D. In this context, the scale for the macroscopic time scale secured in the

next section together with the time scale for the microscale, given by Eq. 4.25, predicted a value of

�1 � 10�5 and �1 � 1 for 5 and 5000 mA, which agree well with the numerically computed values

from the 2D+3D model.

4.6. Analysis (Macroscale) 43

0 1 2 3 4

x 108

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

x 104

r / m

cs /

mol

m

3

2700 s

0 s

5500 s

8200 s

11000 s

2700 s

5500 s

8200 s

11000 s

Figure 4.4: Concentration pro�le of proton in RuO2 in the negative (blue) and positive electrode(red) at di¤erent times during charge at a current of 5 mA: Full model, 2D+3D (symbols) and2D+2D model counterpart (lines) predictions.

4.6 Analysis (Macroscale)

The microscale analysis so far has resulted in the nondimensional number, �1; which presents

the limit at which the overall model can be reduced from a 2D+3D a 2D+2D. The analysis is

proceeded with the aim to not only �nd the macroscopic time scale and other relevant scales and

nondimensional numbers but also the limit at which can further be reduced the model to 1D at

leading-order at the macroscale.

4.6.1 Current collectors

In the current collectors, the magnitude of the potential drop in the x and y directions can be

estimated from Eq.4.4 by an order-of-magnitude estimate as

��sx �[isx]wcc�s

; ��sy �[isy]h

�s(4.33)

where ��sx and ��sy represent the scales of the potential drop in the x- and y-directions respec-

tively; and [isx] and [isy] are the corresponding scales for the respective current densities.


0 1 2 3 4

x 108

2

2.05

2.1

2.15

2.2

2.25

2.3

2.35

2.4

2.45

2.5x 10

4

r / m

cs /

mol

m

3

0.075 s

0.05 s

0.025 s

0.025 s

0.05 s

0.075 s

0.1 s

0.1 s

Figure 4.5: Concentration pro�le of proton in RuO2 in the negative (blue) and positive (red)electrode at di¤erent times during charge at a current of 5000 mA : Full model, 2D+3D (symbols)and 2D+2D model counterpart (lines) predictions.

The current density scales can be obtained from the boundary condition for �s and the overall

conservation of charge: A current density of iapp is applied at the top of the positive current collector

(VI, Eq. 4.14), whence the current scale in the y-direction is

[isy] � iapp (4.34)

Now, in order to ensure overall conservation of charge,

[isx] � iappwcc=h (4.35)

Combining the scales for the potential drops and current densities, it is found that

��sx �iappw

2cc

�sh(4.36)

��sy �iapph

�s(4.37)

Further, taking the ratio of the potential drops in the x and y-direction in the current collector


yields the nondimensional number, �2, de�ned by

��sx��sy

� w2cch2

� �2 (4.38)

Typically, �2 � 1 since h � wj (j= ne, sp, pe); i.e., the potential drop in the x- is much smaller

than the drop in the y-direction in the current collector.

In order to safely reduce the model to a 1D counterpart at the macroscale, the ohmic drop in

the current collectors should be much smaller than the cell potential: i.e., ��sy � Ecell where

Ecell = �sjVI � �sjIV (4.39)

This criteria can be conveniently be captured by de�ning a nondimensional number, �3; as the

ratio of the potential drop in the current collector in y-direction to the cell voltage, given by

��sy �iapph

�s� Ecell (4.40)

such that,

�3 �iapph

�sEcell(4.41)

If �3 � 1; one can thus safely work with a 1D model at the macroscale, which can be expressed as

@

@x

��effs

@�s@x

�= �J (ne,pe,cc) (4.42)

@

@x

��effl

@�l@x

��RT�effl

F

�s+n�+

+t0+z+�+

�@

@x(ln cl)

!= J (ne,pe,sp) (4.43)

�+"@cl@t+@

@x

��+Deffl

@cl@x

+ilt0+

z+F

�=J

F(ne,pe,sp) (4.44)

for which the same interface conditions at the current collector/electrode and electrode/separator

interfaces II and III respectively are retained, whereas the boundary conditions at the current

collectors are modi�ed as follows. At the negative current collector:

�s = 0 (4.45)


At the positive current collector:

��effs@�s@x

= �i0app = �iappwcch

(4.46)

Since the area of current �ow di¤ers from the 1D to the 2D macroscopic formulation, the current

density is adjusted in order to have the same current �ow through the cell; e.g., the value of iapp

for 5 mA current is 39:37 A m�2: The overall reduced models are now 1D+2D and 1D+1D, where

the former considers cs(x; r; t) and the latter cs(x; t):

Note that the scaling arguments are valid except for region of in�uence near the top of the

cell where y � wcc; this constraint does not pose a problem for typical electrochemical capacitors

unless one is speci�cally only interested in that small region of in�uence. Also, from now on in the

analysis, the x- and y-components will not be distinguished for simplicity.

The postulate #3 earlier can easily be relaxed for the scales above; typically, however, the ratio

between in- and through-plane conductivities is not large enough to o¤set the ratio between the

height and width, whence the scales presented here should still be valid.

4.6.2 Electrodes

In the electrodes, the total current is the sum of the ionic current (il) and electron current (is),

whereas in the separator the entire current is carried by the ions. For the electrodes, the scale for

total current density at any x-location of the cell should be equal to the applied current density

i0app in order to satisfy overall conservation of charge; hence,

[is] + [il] � i0app (4.47)

where the ionic current consists of migration current and the di¤usion current as indicated by Eq.

4.5. The scale for the ionic current is taken to be the maximum of the two terms in Eq. 4.5:

[il] � max �effl

��lwne;pe

;�RT�effl

F

�s+n�+

+t0+z+�+

�1

[cl]

�clwne;pe

!(4.48)


At this stage, ��l; [cl];and �cl are unknown. Therefore it is assumed that

�effl��lwne;pe

��RT�effl

F

�s+n�+

+t0+z+�+

�1

[cl]

�clwne;pe

(4.49)

which is justi�ed by comparison with the numerical solution (2D+3D; not shown here).

The scale for the ionic current can thus be written as

[il] � �effl��lwne;pe

(4.50)

For the electron current, a similar expression can be found from Eq. 4.4, such that the sum of the

two currents can be expressed as

� �effs��swne;pe

� �effl��lwne;pe

� i0app (4.51)

At the electrode/separator interface (III), the total current is carried by ions, whereas at the

current-collector/electrode interface (II), the total current is carried by electrons. Therefore, both

the current densities are bounded between a minimum of zero current density to a maximum of

i0app in the electrodes, which in turn implies that

��s � i0app

wne;pe

�effs(4.52)

��l � i0app

wne;pe

�effl(4.53)

The ratio of the potential drop in the solid phase and the liquid phase yields the nondimensional

number �4:

�4 ��s��l

��effl

�effs(4.54)

In general, the solid-phase conductivity is signi�cantly larger than the liquid phase conductivity in

an electrochemical capacitor, whence �4 � 1. For a current of 5 mA (i0app = 39:37 A/m

2) the scales

for the potential drop in the solid and liquid phase are 8 � 10�7 V and 2 � 10�2 V respectively,

which agree well with the numerical counterparts from the 2D+3D solution of around 4� 10�7 V

and 1:3 � 10�2 V; similarly, at a current of 5000 mA (i0app = 39:37 � 103 A/m2), the scales are


7� 10�4 V and 5� 10�1 V as compared to the numerical counterparts of around 7� 10�4 V and

5� 10�1 V for the potential drop in the solid and liquid phase respectively.

Now, since the electrodes are identical in terms of composition and the potential drop in the

solid phase is signi�cantly lower than that in the liquid phase (see Fig. 4.6a and b), the potential

drop in the liquid phase in each of the electrodes during charge/discharge can at most only amount

to around half of the total cell voltage. This implies that there is a limit to the potential drop in

the liquid phase in the electrodes; it also implies that the length over which the liquid potential

drop occurs may no longer be the entire width of the electrode but rather a width decided by the

applied current density. The latter corresponds to the fact that at high charge/discharge rates, the

reaction happens much faster near the electrode/separator interface (III) and the remaining part

of the electrode is not utilized due to the small time scale. In order to estimate the length scale

over which the liquid potential drop takes place at rapid charge/discharge, by introducing

��l �Ecell2

(4.55)

into Eq. 4.53:Ecell2

� i0app[wne;pe]

�effl(4.56)

from which the length scale can be found as

[wne;pe] ��effl Ecell2i0app

(4.57)

where [wne;pe] is introduced to distinguish it from the �xed length of the electrodes, wne;pe. The

decrease in length over which the liquid potential drop occurs in the electrodes is illustrated in Fig.

4.6b, where the liquid potential drop is concentrated near the electrode/separator interface.

In summary, the characteristic length scale over which the liquid-potential drop occurs can be

de�ned as

[wne;pe] � min �effl Ecell2i0app

; wne;pe

!(4.58)

This scale can be used as a design parameter in optimizing the thickness of the electrodes for an

electrochemical capacitor designed to operate under certain current ranges. From the numerical

point of view, the scale can be employed to adjust the mesh density to capture the potential drop


0 0.2 0.4 0.6 0.8 1 1.2

x 103

0

0.2

0.4

0.6

0.8

1

x / m

φ l ,φ s /

V

t increasing

(a) 5 mA: 2700 s, 5500 s, 8200 s, 11000 s

0 0.2 0.4 0.6 0.8 1 1.2

x 103

0

0.2

0.4

0.6

0.8

1

x / m

φ l ,φ s /

V

t increasing

(b) 5000 mA: 0:02 s, 0:05 s, 0:07 s, 0:10 s

Figure 4.6: Liquid phase (lines) and solid phase (lines with symbols) potential pro�le in theelectrodes and separator at various times predicted by reduced model, 1D+1D duringgalvanostatic charging at various currents.

to ensure that it is fully resolved.

In the scales for the remaining variables, the length scale in the electrodes will be determined

by Eq. 4.58. Finally, a nondimensional number, �5 is de�ned, which represents the ratio of the

two length scales:

�5 �wne;pe[wne;pe]

=2i

0appwne;pe

�effl Ecell(4.59)

�5 is thus the deciding factor for determining the macroscopic length scale in the electrodes for the

potential drop in the liquid phase.

4.6.3 Separator

For the separator �where the total current is carried by ions �the scale for potential drop across

the separator is determined by scaling with migration (which is the dominating mechanism) in Eq.

4.5; i.e.,

il � �effl��lwsp

� i0app (4.60)


whence

��l � i0app

wsp

�effl

As one example, for i0app = 39:37 A/m

2and �effl = 45 S/m, a potential drop of nearly 4� 10�5 V

is predicted from the scale that matches well with the simulation results (2D+3D), for which the

liquid potential drop amounts to around 5� 10�5 V.

4.6.4 Macroscopic time-scale

The macroscopic time-scale can be secured from Eq. 4.2 and 4.7 in an electrode:

r � il = �AdCdz+

�dq+dq

�@

@t(�s � �l)�Af

s+nif (4.61)

By balancing the left-hand side with the �rst term on the right-hand side, which corresponds to

the double layer phenomenon, it is found that

i0app

[w]� AdCd

�dq+dq

�Ecell[tmacro]

where the di¤erence between the solid and liquid phase potentials is bounded as: �s��l . Ecell:The

macroscopic time-scale can thus be obtained as

[tmacro] � [w]AdCd�dq+dq

�Ecelli0app

(4.62)

Finally, inserting the length scale, [wne;pe] ; from Eq. 4.58,

[tmacro] �

8>>>><>>>>:wne;peAdCd

�dq+dq

�Ecelli0app; �5 � 1

�effl AdCd2

�dq+dq

�E2celli20app

; �5 > 1

(4.63)

As the current density increases, the time for charging or discharging decreases: This fact is

brought out clearly by the macroscopic time-scale obtained above. The time scale for 5 mA current

amounts to 10:4 � 103 s, which agrees well with the simulation result of 10:9 � 103 s (�5 � 1; see

Fig. 4.3); similarly, for 5000 mA current, the time scale amounts to 0:2 s and the simulation result

4.7. Veri�cation of reduced models 51

Table 4.2: Scales and nondimensional numbers

5 mA 5000 mA

Current collectors

��sxiappw2cc�sh

5� 10�11 V 5� 10�8 V

��syiapph�s

8� 10�5 V 8� 10�2 V

Electrodes

[w] min

��effl Ecell

2i0app; wne;pe

�6� 10�4 m 1� 10�5 m

��s i0app

[w]

�effs8� 10�7 V 7� 10�4 V

��l i0app

[w]

�effl

2� 10�2 V 5� 10�1 V

Separator

��l i0app

wsp

�effl

4:4� 10�5 V 4:4� 10�2 V

Time-scales

[tmicro]R2

Ds4� 10�1 s 4� 10�1 s

[tmacro] [w]AdCd Ecelli0app11� 103 s 2� 10�1 s

Nondimensional numbers

�1R2i

0app

[w]AdCdDsEcell 3� 10�5 2

�2w2cch2

6� 10�7 6� 10�7

�3iapph�sEcell

8� 10�5 10�1

�4�effl

�effs3� 10�5 3� 10�5

�52i0appwne;pe

�effl Ecell5� 10�2 5� 102

to 0:1 s (�5 > 1; see Fig. 4.9b).

The scales and nondimensional numbers are summarized in Table 4.2 together with the predicted

scales for low and high currents.

4.7 Veri�cation of reduced models

Some additional form of local and global veri�cation is needed for the reduced model predictions �

both 1D+1D and 1D+2D �in order to ensure their accuracy.

Starting with the local level, the surface concentration of protons in the active material in both

the positive and negative electrodes is compared during charge at the two currents of 5 mA and

4.7. Veri�cation of reduced models 52

0 2000 4000 6000 8000 10000 120001.6

1.8

2

2.2

2.4

2.6

2.8x 10

4

t / s

cssu

rf /

mol

m

3

cs,nesurf

cs,pesurf

(a) Full model, 2D+3D (negative electrode -5: a,O: b, +: c; positive electrode - 4: d, �: e, �: f)and reduced model, 1D+1D (lines), and 1D+2D

(dotted lines) predictions at 5 mA.

0 0.05 0.1 0.15 0.2 0.252

2.05

2.1

2.15

2.2

2.25

2.3

2.35

2.4

2.45x 10

4

t / s

cssu

rf /

mol

m

3

cs,pesurf

cs,nesurf

(b) Full model, 2D+3D (negative electrode -5:a, O: b, +: c; positive electrode - 4: d, �: e, �:f) and reduced model, 1D+1D (lines), and

1D+2D (dotted lines) predictions at 5000 mA.

Figure 4.7: Variation of surface concentration during charge at a current 5000 mA in thenegative and positive electrodes. Full model, 2D+3D (symbols) and reduced model, 1D+1D(lines), and 1D+2D (dotted lines) predictions.

5000 mA, which represent slow and fast charge. The predictions agree well with the full model as

shown in Fig. 4.7a for a current of 5 mA where the nondimensional number �1 � 1 and �3 � 1;

for a current of 5000 mA, however, the predictions from the 1D+1D are not good as can be inferred

from Fig. 4.7b as �1 > 1. Therefore, the solid-state di¤usion has to be solved numerically along

the radial coordinate to ensure more accurate results; i.e. with the 1D+2D model. Note that the

surface concentrations are taken at the separator/electrode interface for the reduced models (III)

and for the full model, the values are taken at points marked as a,b,c,d,e and f in Fig. 4.2 to also

study the 2D to 1D reduction. Clearly, there is no di¤erence between the various points a-f for low

currents, but at the higher current, the local values start to deviate, which indicates that it is no

longer �safe�to solve a 1D model; the latter fact is mirrored by �3 � 10�1 at 5000 mA, which is

thus no longer negligible at leading order (typically taken as 10�2 or less).

The model predictions are also compared in terms of the concentration of the electrolyte at the

4.8. Computational cost 53

0 0.2 0.4 0.6 0.8 1 1.2

x 103

4800

4900

5000

5100

5200

5300

5400

5500

5600

5700

5800

x / m

c l / m

ol m

3

iapp

decreasing

(a) 5 mA, 50 mA, and 100 mA

0 0.2 0.4 0.6 0.8 1 1.2

x 103

4600

4800

5000

5200

5400

5600

5800

6000

x / m

c l / m

ol m

3

iapp

decreasing

(b) 500 mA, 1000 mA, and 5000 mA

Figure 4.8: Concentration pro�le of proton in the electrolyte at the end of discharge at variouscurrents: Full model, 2D+3D (symbols) and reduced model, 1D+1D (lines) predictions.

end of charge at various current densities as shown in Figs. 4.8a and b. The reduced model results

agree well with that of the full model at various current densities. Note that for the full model, the

concentration pro�le is obtained for the center (0 � x � w; y = h=2) of the cell.

Finally, for the global veri�cation, the galvanostatic charge and discharge of the electrochemical

capacitor cell is simulated at various current densities with the variation of cell voltage, de�ned in

Eq. 4.39, over time presented in Fig. 4.9a and b. Overall, the reduced model predictions agree

well with the full model at currents until 1000 mA; at higher currents, as expected, the agreement

is not good due to the aforementioned reasons as well as due to the fact that the potential drop

in the current collectors (both the positive and negative electrode) is not negligible as it reaches

nearly 0:15 V (from Eq. 4.37) �amounting to around 15% of the total cell voltage.

4.8 Computational cost

Even after having a well established, reliable model, a high computational cost �memory re-

quirement and solution time �might still prohibit the wide use of the model in designing and/or


0 500 1000 15000

0.2

0.4

0.6

0.8

1

t / s

Ece

ll /

V

50 mA100 mA

(a) 50 mA and 100 mA

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

t / s

Ece

ll /

V

1000mA5000mA

(b) 1000 mA and 5000 mA

Figure 4.9: Cell voltage during galvanostatic charge and discharge at various currents: Fullmodel, 2D+3D (symbols) and reduced model, 1D+1D (lines) predictions.

optimizing systems or in carrying out wide-ranging parameter studies. Therefore, after establishing

a reliable model, the focus should be placed on ensuring that the computational cost is kept to

a minimum. In the latter context, it is noted that a stack model is typically highly simpli�ed

as pointed out earlier due to the often prohibitive computational cost associated with solving a

mechanistic model for tens or even hundreds of cells in a stack. Thus, in order to simulate the

behavior of a stack that consists of hundreds of cells, a computationally �inexpensive�model for a

single cell is needed that can be employed as a building block to construct the overall stack model

�which is what the reduced models presented here, aim for.

The improvement in computational cost is illustrated for the computational time, memory and

degrees of freedom (DoF) for the reduced model �1D+1D model for 5 mA and 1D+2D model for

5000 mA �in Table 4.3 for a (i) single cell and cells with a mesh density that emulates a stack of

(ii) ten and (iii) one hundred cells. Note that the mesh for 5000 mA is �ner in order to ensure

accurate results for sharp gradients near the interfaces and that this case is not solved for (iii)

due to the associated computational cost. Overall, the DoF for the reduced models is around one

order less than that required for the full model in all the three cases. There is also a signi�cant

4.9. Conclusion 55

Table 4.3: Computational cost for the full and reduced set of governing equations

5 mA 5000 mA

Full set Reduced set Full set Reduced set

(2D+3Dmodel)

(1D+1Dmodel)

(2D+3Dmodel)

(1D+2Dmodel)

Case (i) DoF 4� 103 5� 102 10� 103 4� 103

(1-cell mesh) Time (s) 61 7 200 18

Memory (GB) 0:3 0:17 0:5 0:25

Case (ii) DoF 4� 104 6� 103 105 4� 104

(10-cell mesh) Time (s) 2� 103 31 3000 250

Memory (GB) 1 0:2 2 0:4

Case (iii) DoF 4� 105 6� 104 106 4� 105

(100-cell mesh) Time (s) � 280 � 3600

Memory (GB) � 0:3 � 2

reduction in the computational time required for the reduced set of equations: For case (ii), the

time required to solve the reduced set is 2 orders of magnitude less than the full set.

4.9 Conclusion

In this work, reduced models for an electrochemical capacitor is presented, analyzed with scaling

arguments, calibrated and validated with experimental data. The �delity of the reduced models are

established by comparison with the full model: overall, good agreement is obtained. Besides pro-

viding rigorous arguments for model reductions, the secured scales capture the key characteristics

and the nondimensional numbers identify the conditions which have to be satis�ed for reductions

in dimensionality at the micro- as well as macro-scopic scale.

The reduction in the computational cost is highlighted through the reduction in the DoF,

solution time and the memory requirement for three cases: a single cell, a 10-cell and a 100-cell

stack. The reduced models can be employed as a basic building block for a stack model through an

automated procedure similar to the stack model by Ly et al. for a fuel cell [104]. In addition, and

perhaps most importantly from a thermal-management point of view, the models can be extended

4.9. Conclusion 56

to account for the equation of change for energy and heat generation. The latter would feature

ohmic heating and heating from the Faradaic reaction, provided the functional relation of the open-

circuit potential of the electrodes with temperature is known to determine the entropic e¤ect. The

scaling arguments should lead to additional model reductions and identi�cation of key scales and

nondimensional numbers for the thermal aspects of an electrochemical capacitor.

Furthermore, the proton transport in the case of nanodots of hydrous ruthenium oxide [98, 99,

105] as well as thin �lms made up of these nanodots can be analyzed within the mathematical

framework and the related scales be secured.

As a �nal note, the model can be extended to account for the additional charge storage mecha-

nism of the adsorption/desorption process that occurs in other metal oxides like MnO2 in addition

to the insertion/deinsertion mechanism modeled here.

4.9. Conclusion 57

Nomenclature

Ad speci�c interfacial area for double-layer capacitance per unit volume, m�1

Af speci�c interfacial area for pseudocapacitance (RuO2) per unit volume, m�1


c0l initial electrolyte concentration, mol m�3

c0s initial concentration of proton in RuO2, mol m�3

cs concentration of proton in RuO2, mol m�3

csurfs surface concentration of protons on RuO2, mol m�3


Ds di¤usion coe¢ cient of proton in RuO2, m2 s�1

ex; ey; ez coordinate vectors


h height of the electrochemical capacitor cell, m








MRuO2 molecular mass of RuO2, kg mol�1

Nl species (proton) �ux, mol m�2 s�1

4.9. Conclusion 58


R radius of active material (RuO2), m

t time, s


T Temperature, K

U equilibrium potential of RuO2 vs. SCE, V

U0 initial equilibrium potential of RuO2 before charging, V

Greek



" porosity of the electrodes and separator

� overpotential

� number of moles of ions into which a mole of electrolyte dissociates


�RuO2 density of RuO2, kg m�3


�s electronic conductivity of solid matrix, S m�1



�0s initial solid phase potential, V

�1;�2;�3;�4;�5 non-dimensional numbers

4.9. Conclusion 59

Subscripts




sp separator

Chapter 5

Analysis of Li-ion Battery Model

5.1 Introduction

Mathematical modeling and numerical simulations of electrochemical energy systems like batteries

and electrochemical capacitors play a vital role in their design and estimation of performance.

Various mathematical models exist in literature to predict the electrochemical and thermal behavior

of a Li-ion battery [1, 2, 3, 4, 19, 21, 23, 24, 40, 41, 42, 47, 48, 53, 106, 107, 108, 109]. These models

investigate various designs of the Li-ion battery such as coin, pouch, cylindrical and prismatic cells.

Of these designs, the coin cells and pouch cells have a rectilinear geometry whereas the cylindrical

and prismatic cells are mostly of spiral-wound geometry. These geometries as well as the treatment

of the electrochemical and thermal phenomena determines the model complexity. In essence, these

models typically consider the transient equations of change for species, charge and energy together

with relevant boundary conditions and constitutive relations.

In view of reducing the complexities and to obtain model reductions, similar to the analysis

carried out for an electrochemical capacitor in the previous Chapter, an analysis for a Li-ion battery

is carried out in this Chapter. A transient, electrochemical-thermal model comprising conservation

of charges, species and energy at the macroscale and conservation of species in the micro-scale

(see Fig. 5.1 and 5.2) is considered in this study. In short, the transport of ions and electrons

and energy in the cell between the electrodes is referred as the transport at the macroscale, which

includes species transport in the liquid/electrolyte phase, electronic charge conduction in the solid

phase, ionic charge conduction in the liquid/electrolyte phase and heat conduction in the solid

60


Figure 5.1: Schematic of (a) a 18650 Li-ion battery, (b) an axisymmetric representation of thespiral-wound battery showing the various functional layers, and (c) a layer of the jelly rollcomprising a single cell with the roman numerals indicating the interfaces of the di¤erent layersand the boundaries.

phase as well as the electrolyte phase; and the di¤usion of ions in the active material present in

the electrodes is referred to as the transport at the microscale, which includes di¤usion of lithium

in the active material of the porous electrodes. Scaling arguments are employed to (i) secure the

scales and nondimensional numbers that characterize the cell both thermally and electrochemically,

(ii) obtain criteria for reduction to a 1D model, and (iii) establish the dependence of the various

parameters on the thermal and electrochemical behavior of the cell that helps in the design of the

batteries.



A commercially available spiral-wound cylindrical Li-ion 18650 battery (Fig. 5.1a), for which an

axisymmetric two-dimensional cross-section of the battery is resolved, as illustrated in Fig. 5.1b,

is considered in this study where the functional layers �positive electrode (pe), negative electrode

(ne), current collector (cc) and separator (sp) � are wound up in the form of a jelly roll. The

dimensions of the various functional layers are taken from a Sony cell [110] with the number of

wounds determined to be 16 based on the diameter of the battery and the thickness of the wounds.

Generally, the porous electrodes consist of active material, conductive �ller additive, binder

and liquid electrolyte (el). In this study, the active material in the negative electrode is LixC6 and

in the positive electrode it is LiyMn2O4. The salt is LiPF6 in a nonaqueous 1:2 liquid mixture

of ethylene carbonate and dimethyl carbonate. The electrochemical reactions that occur at the

electrode/electrolyte interface during charge and discharge are then

LixC6Disch arg e�

Ch arg eC6 + xLi+ + xe�;

Liy�xMn2O4 + xLi+ + xe�Disch arg e�

Ch arg eLiyMn2O4;

where x is the stoichiometric coe¢ cient or the number of moles of lithium present in the graphite

structure, C6, and y is the number of moles of lithium in the spinel structure of manganese dioxide

Mn2O4; Li+ is the lithium ion.

As depicted in Fig.5.2, there are two main scales involved in the modeling of a Li-ion cell: the

macro- and the micro-scale. In short, the transport of ions and electrons in the cell between the

electrodes is referred as the transport at the macroscale, which includes species transport in the

liquid electrolyte, electronic charge conduction in the solid phase and ionic charge conduction in

the liquid electrolyte; and the di¤usion of ions in the active material present in the electrodes is

referred to as transport at the microscale, which includes di¤usion of lithium in the active material

of the porous electrodes.

The model is based on the porous-electrode theory developed by Newman and Tiedemann

[17, 18] and embodies the following main assumptions:


Figure 5.2: Schematic of (a) a Li-ion cell showing the various functional layers on the macroscale,and (b) lithium di¤usion in the active material in the electrodes in the microscale.

1. The electrolyte is assumed to be a binary electrolyte and Li+ ions are the only electroactive

species [1];

2. Uniform distribution of active materials of the same size in the electrodes;

3. The active material is assumed to be spherical; i.e., only the radial direction needs to be

considered at the microscale;

4. Side reactions are assumed negligible.

5. Volume changes in the cell due to its operation are neglected and a constant porosity is used

[1].

6. The temperature dependent transport and kinetic parameters are described by Arrhenius

type expressions.


7. Double layer e¤ects are considered negligible as no rapid pulse charge or discharge cycles are

studied [14, 111].

8. The battery is considered to have a continuous tab design [14] (Refer Fig.5.1b).

9. Edge e¤ects because of the spiral are not considered in the analysis [112]. This is accomplished

by assuming no reaction in the innermost and the outermost electrode layers.


The governing equations are as provided in Chapter 3. The equations applicable for the current

collecting tab (tb) are the same as that applicable for the current collector and conservation of

energy is solved in the outer casing/can (ca) as well.


At the interface I between the current collector/electrode or the current collecting tab/current

collector (see Fig. 5.1c and 5.1c for placement of roman numerals), continuity of energy �ux and

solid-phase current is speci�ed; insulation is speci�ed for the ionic �ux and current:

n � isjI+ = n � isjI� ; n � qjI+ = n � qjI� ;n � il = n �Nl = 0 (I). (5.1)

At the electrode/separator interfaces, continuity of energy �ux and ionic �ux as well as ionic

current is de�ned and since there is no �ow of electrons across the interface, insulation for solid

phase current is de�ned.

n � is = 0; n � iljII+ = n � iljII� ; n � qjII+ = n � qjII� n �NljII+ = n �NljII� (II). (5.2)

At the interface III between the electrode/electrolyte (see Fig. 5.1b and 5.1c for placement of roman

numerals), continuity for the energy �ux as well as the ionic �ux of lithium ions (ionic current) is

speci�ed, whereas insulation is speci�ed for the solid-phase current:

n � is = 0, n � iljIII+ = n � iljIII� ; n �NljIII+ = n �NljIII� (III).; (5.3)


At the interface IV between the separator/electrolyte (see Fig. 5.1b and 5.1c for placement of

roman numerals), continuity for the energy �ux as well as the ionic �ux of lithium ions (ionic

current) is speci�ed:

n � iljIV+ = n � iljIV� ; n �NljIV+ = n �NljIV� (IV). (5.4)

At the current collector/electrolyte and current collecting tab/electrolyte interfaces, insulation is

de�ned for the solid phase current and continuity for the energy �ux:

n � is = 0, n � qjV+ = n � qjV� (V). (5.5)

At the electrolyte/can interface, there is continuity of energy �ux and no �ow of ions:

n � qjVI+ = n � qjVI� ;n � il = n �Nl = 0 (VI). (5.6)

The current density is prescribed at the top part of the outer can:

n � is = �iapp (VII): (5.7)

The current is collected from the bottom of the outer can or otherwise this end is grounded:

�s = 0 (VIII). (5.8)

At the outer surface of the can, Newton�s law of cooling is speci�ed :

n � q = h(T � Ta) (VII. VIII, IX). (5.9)

In Eqs. 5.1 - 5.9, n denotes the unit normal vector for a given boundary or interface, and iapp is

the applied current density. The battery is discharged under galvanostatic conditions at various

current densities.


At t = 0,

csurfs = cavgs = c0s; cl = c0l ; T = T0 (5.10)

�s =

8>>>><>>>>:0 (ne,ncc)

�0s (pe,pcc)

(5.11)

�l = �0l (ne,pe,sp) (5.12)


The generic relations are given in Chapter 3 and speci�c relations are provided in this section. The

entropic heat as a function of state of charge for the negative electrode is expressed as

@Uref; ne@T

=n1 exp(n2�ne + n3)

n4 + n5 exp(n6�ne + n7)+ n8�ne + n9�

2ne + n10; (5.13)

and for the positive electrode, it is written as

@Uref; pe@T

= p1 + p2�pe + p3�2pe + p4�

3pe + p5 exp(p6�pe) + p7 exp

��pe + p8

p9

�2+ p10 sin(p11�pe)

+ p12 sin(p13�pe + p14) + p15 sin(p16�pe + p17), (5.14)

in which ni and pi [2] are constants obtained from curve �tting for experimental data; The open-

circuit potential for the positive electrode de�ned as [1]

Uref; pe = P1 +P2 tanh (P3�pe +P4) +P5

1

(P6 � �pe)P7+P8

!

+P9 exp�P10�

8pe

�+P11 exp (P12 (�pe +P13)) ; (5.15)

and for the negative electrode as

Uref; ne = N1 +N2 exp (N3�ne) +N4 exp (N5�ne) ; (5.16)

here, Ni and Piare constants (refer Table 5.3) obtained by curve-�tting with experimental data.


The temperature dependence of the di¤usion coe¢ cients (both solid and liquid) and ionic con-

ductivity are expressed as [2, 91]

Ds(T ) = Ds(Tref ) exp

�Ea;DsR

�1

Tref� 1

T

��; (5.17)

Dl(T ) = Dl(Tref ) exp

�Ea;DlR

�1

Tref� 1

T

��; (5.18)

�l(T ) = �l(Tref ) exp

�Ea;�lR

�1

Tref� 1

T

��; (5.19)

where Tref is a reference temperature, and Ea is the activation energy. The solid-phase potential

di¤erence between the top side of the positive current collector (VI) and the negative current

collector (IV) is de�ned as the cell voltage:

Ecell = �sjVI � �sjIV : (5.20)

The length of the spiral is de�ned as

L = �N (D + d)2

; (5.21)

where N is the total number of wounds, D is the diameter at the end of the spiral (taken to be

the diameter of the battery) and d is the diameter at the start of the spiral (taken to be the core

diameter). The volume of each layer is then calculated as

Vi = 2wihL; i = ne, pe, sp. (5.22)

The number 2 in the above equation is because the electrode layers are coated on both sides of the

current collector and the presence of two separator sheets whereas for the current collectors, the

volume need not be multiplied by that factor of 2:The parameters for these constitutive relations

are summarized in Table 5.1, 5.2 and 5.3. As the model equations and parameters were taken from

a previously calibrated and validated model [1] , calibration and validation are not carried out here.



Parameter Unit cc (-) ne sp pe cc (+) Reference

c0l mol m�3 - 2� 103 - [1]

Cp J kg�1K�1 3:8� 102 7:0� 102 7:0� 102 7:0� 102 8:7� 102 [2]

c0s mol m�3 - 1:5� 104 - 3:9� 103 - [1]

cmaxs mol m�3 - 2:6� 104 - 2:3� 104 - [1]

Dl m2 s�1 - 7:5� 10�11 - [1]

Ds m2 s�1 - 3:9�10�14 - 1:0�10�13 - [1]

Ea;Dl kJ mol�1 - 10 - [2]

Ea;Ds kJ mol�1 - 4 - 20 - [2]

Ea;�l kJ mol�1 - 20 - [2]

h m 60� 10�3 -

h W m�2 K�1 5 -

iapp(1C) A m�2 2:3� 105 -

k W m�1 K�1 3:8� 102 0:05� 102 0:01� 102 0:05� 102 2:0� 102 [2]

k0 mol2:5m�6:5s�1 - 2� 10�11 - 2� 10�11 - -

Rb m 9� 10�3 -

R m - 12:5�10�6 - 8:5� 10�6 - [1]

Ta; Tref K 298:15 [2]

wi m 18� 10�6 88� 10�6 25� 10�6 80� 10�6 25� 10�6 [110]

�a; �c - - 0:5 - 0:5 - [1]

"p - - 0:14 - 0:19 - [1]

"l - - 0:36 0:72 0:44 - [1]

"f - - 0:03 - 0:07 - [1]

�0i - - 0:56 - 0:17 - [1]

� kg m�3 9:0� 103 1:9� 103 1:2� 103 4:1� 103 2:7� 103 [1], [2]

�s S m�1 6:0� 107 1� 102 - 3:8 3:8� 107 [2]


Table 5.2: Electrolyte, and outer can properties

Parameter Unit Electrolyte[34] Outer can

cp J (kg K)�1 700 475

k W (mK)�1 1 44:5

� kg m�3 1200 7850

Table 5.3: Constants in expressions (Ref. [1, 2])

Const. Unit Value Const. Unit Value Const. Unit Value

n1 mV K�1 344:1347 p3 mVK�1 �26:0645 N3 - �3:0

n2 - �32:9633 p4 mVK�1 12:7660 N4 V 10

n3 - 8:3167 p5 mVK�1 4:3127 N5 - �2000

n4 - 1 p6 - 0:5715 P1 V 4:1983

n5 - 749:0756 p7 mVK�1 �0:1842 P2 V 0:0565

n6 - �34:7909 p8 - �0:5169 P3 - �14:5546

n7 - 8:8871 p9 - 0:0462 P4 - 8:6094

n8 mV K�1 �0:8520 p10 mVK�1 1:2816 P5 V �0:0275

n9 mV K�1 0:3622 p11 - �4:9916 P6 - 0:9984

n10 mV K�1 0:2698 p12 mVK�1 �0:0904 P7 - 0:4924

a0 S m�1 1:0793� 10�2 p13 - �20:9669 P8 - �1:9011

a1 S m2 mol�1 6:7461� 10�4 p14 - �12:5788 P9 V �0:1571

a2 S m5 mol�2 �5:2245�10�7 p15 mVK�1 0:0313 P10 - �0:0474

a3 S m8 mol�3 1:3605� 10�10 p16 - 31:7663 P11 V 0:8102

a4 S m11 mol�4 �1:172� 10�14 p17 - �22:4295 P12 - �40

p1 mV K�1 �4:1453 N1 V �0:16 P13 - �0:1339

p2 mV K�1 8:1471 N2 V 1:32 -


5.3 Analysis (Macroscale)

5.3.1 Current collectors

The curvature e¤ects are considered negligible as the thickness of a unit cell comprising the �ve

layers as shown in Fig. 5.2 is very smaller than the radius, Rb;of the 18650 battery [113]. Hence, the

analysis for the electrochemical model can be carried out in a cartesian coordinate system rather

than cylindrical coordinates. The analysis of the electrochemical model is quite similar to that of

the electrochemical capacitor.

The value of the applied current density iapp changes from the radial coordinate (refer Fig. 5.1a

& b) to the cartesian coordinate (refer Fig. 5.2) as there is a change in the area. In the analysis

carried out below, the value of iapp corresponds to the cartesian coordinate system. The potential

drop in the current collector around the total wound of length, L is given by ��s;L � iappL=�s

which can be neglected when ��s;L � Ecell. The ratio of this potential drop to the overall cell

voltage provides a nondimensional number �1 given by

�1 �iappL�sEcell

(5.23)

For a given battery, �1 is decided by the charge/discharge rate. The magnitude of the potential

drop in the x and y directions in the current collector can be estimated from Eq.3.5 by an order-

of-magnitude estimate as

��sx �[isx]wcc�s

; ��sy �[isy]h

�s(5.24)

where ��sx and ��sy represent the scales of the potential drop in the x- and y-directions respec-

tively; and [isx] and [isy] are the corresponding scales for the respective current densities.

The current density scales, are obtained from the boundary condition for �s and the overall

conservation of charge: A current density of iapp is applied at the top of the positive current collector

(VI, Eq. 3.17), and hence the current scale in the y-direction is

[isy] � iapp (5.25)


Now, in order to ensure overall conservation of charge,

[isx] � iappwcc=h (5.26)

Therefore, the scale for potential drops become

��sx �iappw

2cc

�sh(5.27)

��sy �iapph

�s(5.28)

The ratio of the potential drops in the x and y-direction in the current collector yields the nondi-

mensional number, �2, de�ned by��sx��sy

� w2cch2

� �2 (5.29)

Typically, �2 � 1 since h � wj (j= ne, sp, pe); i.e., the potential drop in the x-direction is much

smaller than the drop in the y-direction in the current collector.

In order to safely reduce the electrochemical model to a 1D counterpart at the macroscale,

the ohmic drop in the current collectors should be much smaller than the cell potential: i.e.,

��sy � Ecell This criteria can be conveniently captured by de�ning a nondimensional number, �3;

as the ratio of the potential drop in the current collector in y-direction to the cell voltage, given by

��sy �iapph

�s� Ecell (5.30)

such that

�3 �iapph

�sEcell(5.31)

If �3 � 1; a 1D electrochemical model is su¢ cient at the macroscale, which can be expressed as

@

@x

��effs

@�s@x

�= �J (ne,pe,cc) (5.32)

@

@x

��effl

@�l@x

+2RT�effl

F

�1� t0+

� @@x(ln cl)

!= J (ne,pe,sp) (5.33)

"@cl@t+@

@x

��Deffl

@cl@x

+ilt0+

F

�=J

F(ne,pe,sp) (5.34)


for which the same interface conditions at the current collector/electrode and electrode/separator

interfaces II and III respectively are retained, whereas the boundary conditions at the current

collectors are modi�ed as follows. At the negative current collector:

�s = 0 (5.35)

At the positive current collector:

��effs@�s@x

= �i0app = �iappwcch

(5.36)

Since the area of current �ow di¤ers from the 1D to the 2D macroscopic formulation, the current

density is adjusted in order to have the same current �ow through the cell; e.g., the value of i0app

for 750 mA current is 12:5 A m�2:

Note that the scaling arguments are valid except for region of in�uence near the top of the cell

where y � wcc; this constraint does not pose a problem for typical Li-ion batteries unless one is

speci�cally only interested in that small region of in�uence. Also, from now on in the analysis, the

x- and y-components will not be distinguished for simplicity.

5.3.2 Electrodes

In the electrodes, the total current is the sum of the ionic current (il) and electron current (is),

whereas in the separator the entire current is carried by the ions. For the electrodes, the scale for

total current density at any x-location of the cell should be equal to the applied current density

i0app in order to satisfy overall conservation of charge; hence

[is] + [il] � i0app (5.37)

where the ionic current consists of migration current and the di¤usion current as indicated by Eq.

3.6. The scale for the ionic current is taken to be the maximum of the two terms in Eq. 3.6:

[il] � max �effl

��lwne;pe

;2RT�effl

F

�1� t0+

� 1[cl]

�clwne;pe

!(5.38)


At this stage, ��l; [cl]; T and �cl are unknown. The scale for the electrolyte concentration and

temperature is taken to be the initial values c0l and T0: From Eq. 3.1 and 3.7, the change in

electrolyte concentration in the electrodes can be estimated. Inserting Eq. 3.7 in 3.1 results in

"l@cl@t+r �

�Deffl rcl

�=J

F(1� t0+): (5.39)

By taking a constant di¤usivity, the above equation scales as follows

"l�cl[tdi¤]

+Deffl

�clw2ne;pe

� [J ]

F(1� t0+): (5.40)

By balancing the transient term with the di¤usion term in the above equation, the time scale for

electrolyte di¤usion is estimated to be

[tdi¤] �"lw

2ne;pe

Deffl

; (5.41)

the time scale for di¤usion is then calculated to be around 30 s which is very much lesser as

compared to the discharge time scale that will be estimated later in the analysis and hence the

di¤usion term can be balanced with the term on the RHS in Eq. 5.40 to secure �cl

�cl �[J ]w2ne;pe

FDeffl

(1� t0+): (5.42)

The scale for the transfer current per unit volume is obtained from Eq. 3.9 and is given by

[J ] � A [if ] (5.43)

where [if ] represents the scale for the transfer current density that could be estimated from the

solid phase conservation of charge equation Eq. 3.1 as follows

r � is = �J (5.44)

�iswne;pe

� A [if ] (5.45)


2 2.1 2.2

x 103

1800

1850

1900

1950

2000

2050

2100

2150

2200

2250

r / m

cl /

mol

m3

0 s

pe sp ne

50 s

3000 s

1000 s

(a) 1 C-rate

2 2.1 2.2

x 103

1200

1400

1600

1800

2000

2200

2400

2600

2800

r / m

cl /

mol

m3

250 s

0 s

500 s700 s

pe sp ne

(b) 5 C-rate

Figure 5.3: Electrolyte concentration at various times during discharge at di¤erent C-rates.

The solid phase current is varies between from zero in the electrode/separator interface to i0app in

the electrode/current collector boundary and hence �is � i0app which yields

[if ] �i0app

Awne;pe: (5.46)

Substituting Eq. 5.46 in Eq. 5.43 results in

[J ] �i0app

wne;pe; (5.47)

and by inserting this scale in Eq. 5.42 provides

�cl �i0appwne;pe

FDeffl

(1� t0+): (5.48)

The above estimate for the concentration variation agrees well with the simulation results as shown

in Table 5.4. The numerical results in Fig. 5.3 shows the electrolyte concentration pro�le in the

innermost layer of the battery at various times during discharge at 1 C and 5 C-rates.

By assuming that

�effl��lwne;pe

�2RT�effl

F

�1� t0+

� 1[cl]

�clwne;pe

; (5.49)


which is justi�ed by comparison with the numerical solution, the scale for the ionic current can be

written as

[il] � �effl��lwne;pe

(5.50)

For the electron current, a similar expression can be found from Eq. 3.5, such that the sum of the

two currents can be expressed as

� �effs��swne;pe

� �effl��lwne;pe

� i0app (5.51)

At the electrode/separator interface (III), the total current is carried by ions, whereas at the

current-collector/electrode interface (II), the total current is carried by electrons. Therefore, both

the current densities are bounded between a minimum of zero current density to a maximum of

i0app in the electrodes, which in turn implies that

��s � i0app

wne;pe

�effs(5.52)

��l � i0app

wne;pe

�effl(5.53)

The ratio of the potential drop in the solid phase and the liquid phase yields the nondimensional

number �4:

�4 ��s��l

��effl

�effs(5.54)

In general, the solid-phase conductivity is signi�cantly larger than the liquid phase conductivity in

a Li-ion battery, whence �4 � 1. For a current of 750 mA (i0app = 12:5 A/m

2) the scales for the

potential drop in the solid and liquid phase in the positive electrode are 8� 10�4 V and 6� 10�3

V and in the negative electrode are 2� 10�5 V and 9� 10�3 V respectively, which agree well with

the numerical counterparts from the solution of around 4� 10�4 V and 5� 10�3 V in the positive

electrode and 8�10�5 V and 1�10�2 in the negative electrode respectively; similarly, at a current

of 3750 mA (i0app = 62:5 A/m

2), the scales are 4� 10�3 V and 3� 10�2 V in the positive electrode

4� 10�5 V and 6� 10�2 V in the negative electrode as compared to the numerical counterparts of

around 2� 10�3 V and 2� 10�2 V in the positive and 5� 10�5 V and 3� 10�2 V in the negative

electrodes for the potential drop in the solid and liquid phase respectively. The initial liquid phase


2 2.1 2.2

x 103

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

r / m

φ s / V

sp nepe0.1 s

3000 s

3600 s

1000 s

(a) 1 C-rate

2 2.1 2.2

x 103

0

0.5

1

1.5

2

2.5

3

3.5

4

r / m

φ s / V

700 s

460 s

230 s0.1 s pe sp ne

(b) 5 C-rate

Figure 5.4: Solid phase potential at the innermost layer near the centre core at various timesduring discharge at di¤erent C-rates.

conductivity in the electrodes is utilized to calculate the above scales. The solid and liquid phase

potentials at various time during discharge for the above mentioned currents are shown in Figs.5.4

and 5.5 from which a rough estimate for the potential drops can be obtained. The scales provided

here agree well with the potential drops in the innermost as well as the outermost layer of the

battery. The numerical results provided above and Figs.5.4 and 5.5 corresponds to the innermost

layer in the battery.

The overpotential scale in the two electrodes is determined from the Butler Volmer Kinetics

given by Eq. 3.24. The negative electrode acts as the anode during discharging and cathode

during charging and vice versa. For the negative electrode acting as anode under discharging, the

overpotential scale is obtained by balancing the �rst term in the RHS with the transfer current

density to obtain

[�ne] � ln�[if ]

[i0]

�R [T ]

�aF; (5.55)

and for the positive electrode acting as cathode during discharging, the second term in RHS is

considered and hence ��pe�� ln

�[i0]

[if ]

�R [T ]

�cF; (5.56)


2 2.1 2.2

x 103

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

r / m

φ l / V

pe sp ne

3600 s

3000 s

1000 s

0.1 s

(a) 1 C-rate

2 2.1 2.2

x 103

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

r / m

φ l / V

nesppe

700 s

460 s

230 s

0.1 s

(b) 5 C-rate

Figure 5.5: Liquid phase potential at the innermost layer near the centre core at various timesduring discharge at di¤erent C-rates.

The scale for the exchange current density is calculated from Eq. 3.26

[i0] � Fk0r[cl] (c

maxs;i �

hcsurfs;i

i)hcsurfs;i

i; i = ne, pe (5.57)

The surface concentration of lithium ion varies from the initial value to the maximum concentration

and hence the range of the exchange current density can be obtained by assuming [cl] � c0l .

Therefore by substituting the values in the above Eq. results in,

[i0;ne] � 0:70� 1:1; 0:10 � �ne � 0:56;

[i0;pe] � 0:74� 0:14; 0:17 � �pe � 0:99;

the scale for i0 is then taken as the average value and hence

[i0;ne] � 0:90 (5.58)

[i0;pe] � 0:44 (5.59)

Substituting the scales [i0] and [if ] in Eqs. 5.55 and 5.56 and taking the initial temperature as


the scale for temperature leads to

[�ne] � ln

i0app

0:90Awne

!RT0�aF

; and (5.60)

��pe�� ln

�0:44Awpei0app

�RT0�cF

: (5.61)

The scale for the overpotential in the negative electrode during discharge was found to be

9 � 10�3 V for a discharge rate of 1C and 9 � 10�2 at 5C-rate respectively and similarly for the

positive electrode, the scale for the overpotential at discharge rates of 1C and 5C are �0:06 V and

�0:14 V respectively that matches well with the numerical counterparts as shown in Table 5.4.

The local state of charge of the electrodes can be determined by estimating the surface concen-

tration csurfs using Eq. 3.11 and 3.12. Scaling the two eqns. provides

hcsurfs

i� [cavgs ]� [if ]

F

lsDs; (5.62)

[cavgs ] � cavgs;0 �3 [if ] t

FR; (5.63)

where t represents the time from the start of discharge. The local state of charge �pe/ne can be

estimated fromhcsurfs

iusing Eq. 3.29. The prediction of the scales matches well with the numerical

results as shown in Fig. 5.6.

5.3.3 Separator

For the separator �where the total current is carried by ions �the scale for potential drop across

the separator is found by scaling with migration (which is the dominating mechanism) in Eq. 3.6;

i.e.,

il � �effl��lwsp

� i0app (5.64)

whence

��l � i0app

wsp

�effl

As one example, for i0app = 12:5 A/m2and �effl = 0:1 S/m, a potential drop of nearly 3 � 10�3

V is predicted from the scale that matches well with the simulation results, for which the liquid

potential drop amounts to around 2� 10�3 V.


0 500 1000 1500 2000 2500 3000 35000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t / s

θ i

θpe

θne

(a) 1 C-rate

0 100 200 300 400 500 600 7000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t / s

θ i

θp e

θn e

(b) 5 C-rate

Figure 5.6: Local state of charge of the electrodes during discharge : numerical results(continuous) and scales (dotted)

5.3.4 Macroscopic time-scale

The macroscopic time-scale can be secured from Eq. 3.12 in an electrode:

[tmacro] =�cavgs FR

3 [if ](5.65)

Substituting the scale for if from Eq. 5.46 leads to

[tmacro] =�cavgs FRAwne=pe

3i0app(5.66)

As the current density increases, the time for charging or discharging decreases: This fact is brought

out clearly by the macroscopic time-scale obtained above. The time scale for 700 mA current

amounts to 3:9 � 103 s, which agrees well with the simulation result of 3:6 � 103 s; similarly, for

3500 mA current, the time scale amounts to 7:8 � 102 s and the simulation result to 7:1 � 102 s.

The variation of cell voltage with time is shown in Fig.5.7 that shows the time of discharge for

various C-rates and the scales agree well with the numerical results.

The scales are summarized in Table 5.4 together with the numerical counterparts and the

nondimensional numbers are summarized in Table 5.5 for low and high currents.


0 500 1000 1500 2000 2500 3000 3500

2.8

3

3.2

3.4

3.6

3.8

4

t / s

Ece

ll / V 1C

2C

5C

Figure 5.7: Ecell vs time at various discharge rates.

5.3.5 Thermal Analysis

The electrochemical analysis was carried out for a unit cell of the spiral-wound battery whereas

the thermal analysis has to be carried out for the entire battery. This is because the electrochem-

ical phenomenon is governed locally and the thermal behavior is determined by the overall heat

generation from the di¤erent layers and the heat removal rate from the battery surface. Hence,

simpli�cation to cartesian coordinate is not possible and therefore cylindrical coordinates are em-

ployed. The main objective of this thermal analysis is to provide an estimate of the heat generated

in the battery at various C-rates. Expressing Eq. 3.1 in cylindrical coordinates gives

(�Cp)b@T

@t=krr

@

@r

�r@T

@r

�+ kz

@2T

@z2+Q; (5.67)

subjected to the following boundary and initial conditions

@T

@r= 0; at r = 0 (5.68)

kr@T

@r= h (T � Ta) ; at r = Rb; (5.69)

kz@T

@z= h (T � Ta) ; at z = 0;H; (5.70)


where Rb and H denotes the radius and height of the battery respectively. The battery has

anisotropic thermal conductivity in the radial and axial direction and is given by [41]

kr =

PiwiP

i

wiki

; i = ne, pe,sp,cc, (5.71)

kz =

PiwikiPiwi; i = ne, pe,sp,cc, (5.72)

where kr and kz represent the thermal conductivity in the radial and axial direction respectively.

The overall e¤ective thermal capacity of the battery, (�Cp)b ; is de�ned as

(�Cp)b =

Piwi�iCpiPiwi

; i = ne, pe,sp,cc. (5.73)

The anisotropic thermal conductivity and the overall thermal capacity of the battery is taken to

be that of the jelly roll comprising the functional layers and the e¤ect of the electrolyte present in

the core of the battery as well as the outer casing are considered negligible in this analysis. Scaling

Eq.5.67 results in

(�Cp)b�T

�t� kr�Tr

R2;kz�TzH2

; [Q]; (5.74)

[Q] is the scale for the heat generation term that will be determined from Eq. 3.28.

From Eq. 5.74, the criteria for ignoring the axial conduction was obtained by the taking the

ratio between the �rst and second terms on the right hand side and is given by

kzR2

krH2= �5 � 1 (5.75)

From Eq. 3.28, the scale for heat generation can be written as

[Q] � max X

i

[Ji] [�i] ;Xi

[Ji] [T ]

�@Uref; i@T

�;

Xi

�effl;i

��l;iwi

�2+2RT�effl

F(1� t0+)

1

[cl]

�cl��lw2ne;pe

+ �effs;i

��s;iwi

�2!!i = ne, pe, cc. (5.76)

All the terms in the RHS of Eq. 5.76 depends on the charge/discharge rate (C-rate). The �rst


term on the RHS represents the scale for the irreversible heat generated. This is applicable only

in the electrodes where the electrochemical reaction takes place. The second term indicates the

scale for the reversible heat generated due to entropy changes and as the previous term, this is

also valid only in the electrodes. The entropy change is related to the change of the equilibrium

potential of the electrode with temperature and this term may be positive or negative leading to

heat generation or heat absorption respectively. The third term represents the ohmic heating in

the liquid phase. The ohmic heating in the solid phase is relatively lower than that in the liquid

phase as owing to the higher conductivity in the solid phase than in the liquid phase.

The variation of the open circuit potential with temperature in the reversible heat scale is

obtained by inserting the state of charge computed using the surface concentration estimated from

Eq. 5.62 into Eqs. 5.13 and 5.14 for negative and positive electrodes respectively. Substituting the

di¤erent scales obtained above in Eq. 5.76 results in

[Q] � i0appmax

0@Xi

[�i]

[wi];Xi

[T ]

[wi]

�@Uref; i@T

�;Xi

i0apph�effl;i

i + i0apph�effs;i

i + 2RTi0app

F 2Deffl c0l(1� t0+)2

1A ;i = ne; pe; cc: (5.77)

The above scales are to be multiplied by the respective volumes of heat generation �the volume

of the electrodes for the �rst two scales and the volume of the entire battery for the third scale for

ohmic heating �to calculate the total heat generated by these respective terms. The summation

of all the three terms will provide the overall heat generated by the battery. At 1 C-rate, the

scales predicted 6� 10�2 W and 1:5� 10�2 W for the irreversible and ohmic heating respectively,

which matches well with the numerical results as shown in Fig. 5.8. The comparison of the scales

and the numerical results for reversible heating in the positive and negative electrodes at 1 C-rate

is provided in Fig. 5.9 good agreement is found. Overall, the scales determined above are able

to predict the heat generation from the battery at a reasonable accuracy. Table 5.4 provides a

comparison between the results predicted by the scales and the numerical computation.

There are three possible time scales, the observation time [to], the characteristic time scale

for heat conduction [tcond] and the time scale characterizing the heat loss to the ambient [text].

The time scale for the heat generation is same as that of the observation time and hence not


0 500 1000 1500 2000 2500 3000 35000.1

0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

t / s

Hea

t gen

erat

ion

/ W

Qrev

Qtotal

Qohmic

Qirrev

Figure 5.8: Total, irreversible, reversible, and ohmic heat generation from the battery duringdischarge at 1 C-rate.

considered separately. The observation time is nothing but the macroscopic discharge time scale,

[tmacro];obtained above. The time scale for heat conduction is obtained by balancing the transient

term with the radial heat conduction term in Eq. 5.74 to yield

[tcond] �(�Cp)bkr

R2 � R2

�(5.78)

where � is the thermal di¤usivity of the battery. The decrease in the internal energy of the battery

depends upon the heat loss from the battery and the corresponding time scale is obtained by

balancing the transient term with the heat loss at the boundary described by Newton�s law of

cooling

(�Cp)b

ZV

@T

@tdV �

ZA

h (T � Ta) dA (5.79)

(�Cp)b�T

[text]V � h�TA (5.80)

[text] �(�Cp)b V

hA(5.81)

The ratio of the external time scale for heat loss to the di¤usion time scale provides a nondimensional

number

�5 �[tcond]

[text]� hA

(�Cp)b V

(�Cp)bR2

kr� hR

kr(5.82)


0 500 1000 1500 2000 2500 3000 35000.06

0.04

0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

t / s

Qre

v / W

Qrev,ne

Qrev,pe

Figure 5.9: Reversible heat generated during discharge at 1 C-rate: numerical results (continuous)and scales (dotted).

�5 is the Biot number, Bi, which is the ratio between the heat convection to the ambient and the

heat conduction within the battery. If Bi � 0:1, the battery can be treated as a lumped system

as the conduction within the battery is so high than the convection to the ambient that results in

a negligible temperature gradient within the battery. Under such cases, the conduction within the

battery can safely be neglected leading to an overall energy balance as follows

ZV

(�Cp)b@T

@tdV =

ZV

QdV �ZA

h (T � Ta) dA (5.83)

(�Cp)b@T

@tV = QV � h (T � Ta)A; (5.84)

@T

@t+

hA

(�Cp)b VT =

Q

(�Cp)b+

hA

(�Cp)b VTa (5.85)

where V is the total volume of the battery and A is the surface area for heat transfer. Eq. 5.85 is

a �rst order ordinary di¤erential equation subjected to the initial condition, T = T0 at t = 0;that

can be solved to obtain the variation of temperature during charge or discharge. An average

value for the overall heat generation from the battery is assumed to treat Q as constant in the

above equation. On the other hand, for Bi � 0:1 corresponding to higher h in case of forced

convection liquid cooling systems, the conduction term cannot be considered negligible as there

will be temperature gradient within the battery and hence the above solution becomes invalid and

numerical simulations are to be carried out. The increase in the average temperature of the battery

5.4. Conclusion 85

under various discharge rates is shown in Fig. 5.10. As the discharge rate increases, the average

0 500 1000 1500 2000 2500 3000 3500300

305

310

315

320

325

330

335

340

345

350

355

t / s

Tav

g / K

1C

2C

5C

Figure 5.10: Average temperature of the battery vs time during discharge at various rates (h = 5W m�2 K�1).

temperature of the battery rises faster and also reaches temperature more than the safe limits

that shows the need for a thermal management system. The practical application of the model

developed is demonstrated through the design a passive thermal management system using a phase

change material for a Li-ion battery in Chapter 6.

5.4 Conclusion

In this work, the model for a spiral wound cylindrical 18650 Li-ion battery is presented and analyzed

with scaling arguments. The scales obtained from the analysis provided estimates of the various

variables in the model. The estimated scales are veri�ed by comparison with the numerical results:

overall, good agreement is obtained. Besides providing rigorous arguments for model reductions,

the secured scales captured the key characteristics and the nondimensional numbers identi�ed the

conditions which have to be satis�ed for reductions in dimensionality.

The analysis of the electrochemical model provided criteria to reduce the model from two-

dimensions to one-dimension and also provided an insight into the e¤ect of the design adjustable

parameters such as porosity, width of the functional layers, radius of the active material particle in

the electrodes and the physical properties of the materials on the performance of the battery. The

thermal analysis mainly provided an estimate of the various sources of heat generation in the battery

5.4. Conclusion 86

Table 5.4: Comparison of scales and numerical results, nondimensional numbers

750 mA (1 C-rate) 3750 mA (5 C-rate)

scale simulated scale simulated

Current collectors

��sxiappw2cc�sh

;V 5�10�11 5� 10�11 2�10�10 2:5� 10�10

��syiapph�s; V 3� 10�5 2� 10�5 2� 10�4 1:5� 10�4

Negative electrode

��s i0app

wne�effs

, V 2� 10�5 4� 10�5 10�4 6� 10�5

��l i0app

wne�effl

, V 9� 10�3 10�2 5� 10�2 3� 10�2

�cli0appwne;pe

FDeffl

(1� t0+), mol m�3 2� 102 [0:4� 2]� 102 9� 102 [1� 9]� 102

[�ne] ln

�i0app

0:90Awne

�RT0�aF

, V 2� 10�2 [2� 3]� 10�2 10�1 [8� 10]� 10�2

Qirrev i0app

[�ne]wne

Vne, W 1� 10�2 [2� 3]� 10�2 4� 10�1 [3� 6]� 10�1

Qohmic

�i0app

�effl;ne

+i0app

�effs;ne+

2RTi0app

F 2Deffl c0l

(1� t0+)2�Vne,

W

9� 10�3 [7� 13]� 10�3 2� 10�1 [2� 3]� 10�1

Positive electrode

��s i0app

wp e

�effs, V 8� 10�4 4� 10�4 4� 10�3 2� 10�3

��l i0app

wp e

�effl

, V 6� 10�3 5� 10�3 3� 10�2 2� 10�2

�cli0appwne;pe

FDeffl

(1� t0+), mol m�3 2� 102 [0:3� 1:5]�102 7� 102 [0:7� 6]� 102��pe�

ln

�0:44Awpei0app

�RT0�cF

, V 6� 10�2 [4� 14]� 10�2 10�1 [12� 22]�10�2

Qirrev i0app[�pe]wp e

Vpe; W 5� 10�2 [2� 09]� 10�2 5� 10�1 [4� 8]� 10�1

Qohmic

�i0app

�effl;p e

+i0app

�effs;p e+

2RTi0app

F 2Deffl c0l

(1� t0+)2�Vpe;

W

9� 10�3 [4� 9]� 10�3 2� 10�1 [1� 2]� 10�1

Separator

��l i0app

wsp

�effl

, V 3� 10�3 2� 10�3 2� 10�2 1:2� 10�2

Qohmic

�i0app

�effl;sp

+2RTi

0app

F 2Deffl c0l

(1� t0+)2�Vsp, W 2� 10�3 [2� 4]� 10�3 6� 10�2 [5� 8]� 10�2

Time-scales

[tmacro]�cavgs FRAwne=pe

3i0app, s 4� 103 3:6� 103 8� 102 7� 102

[tdi� ]"lw

2ne;pe

Deffl

, s 130 (pe), 170 (ne)

[tcond]R2

� ,�� = kr

(�Cp)b

�, s 35

[text](�Cp)bVhA (h = 5 Wm�2K), s 1:2� 103

5.4. Conclusion 87

Table 5.5: Nondimensional numbers

750 mA (1 C-rate) 3750 mA (5 C-rate)

�1iappL�sEcell

2� 10�3 10�2

�2w2cch2

2� 10�7

�3iapph�sEcell

2� 10�5 10�4

�4�effl

�effs7� 10�2 (pe), 1:5� 10�3 (ne)

�5hRkr

1:5� 10�2

under various discharge rates. Further, it established the dependence of the heat generation on

the various design parameters and physical properties of the materials. While designing a battery,

these scales can be utilized to estimate the e¤ects of the various design adjustable parameters as

well as the e¤ect of di¤erent materials used for making those batteries.

Further, the models can be reduced in dimensions based on the criteria provided by the nondi-

mensional numbers that will lead to an improvement in the computational cost. This reduced

models could then be used for studying large battery modules and packs. The numerical procedure

could be automated to implement the geometry and the mathematical formulation as demonstrated

in Chapter 6 that can be used for wide-ranging parameter studies.

The scaling analysis provided criteria for model reduction, scales to estimate the in�uence of

the design variables on the performance of the system. The scales provided here can be used as a

rule of thumb and an optimization program could be used to optimize these design parameters for

the given operating conditions. These scales and criteria derived in this chapter contribute to the

fundamental knowledge in this �eld.

5.4. Conclusion 88

Nomenclature

A speci�c surface area for the faradaic reaction per unit volume, m2/m3

Bi Biot number

Cp speci�c heat capacity, J kg�1 K�1


cs concentration of lithium in active material in the electrodes, mol m�3

cavgs average concentration of lithium in the active materials, mol m�3

csurfs surface concentration of lithium in active materials, mol m�3

D diameter at the outer end of the spiral, m

d diameter at the inner end of the spiral, m

Dl di¤usion coe¢ cient of electrolyte, m�2 s�1

Ds di¤usion coe¢ cient of lithium in the active material in the electrodes, m�2 s�1

Ea activation energy for a variable, kJ mol�1

ex; ey coordinate vectors



h heat transfer coe¢ cient, Wm�2 K�1






5.4. Conclusion 89


k thermal conductivity, W m�1 K�1

kr radial thermal conductivity, W m�1 K�1

kz axial thermal conductivity, W m�1 K�1

k0 reaction rate constant, mol m�2s�1�mol m�3

�1:5L total length of the spiral, m

ls di¤usion length, m

N no. of wounds in the jelly roll (= 16)


n normal vector

ni; pi constants in the entropic heat term for negative and positive electrodes

Ni;Pi constants in the open circuit potential for negative and positive electrodes




R radius of active material, m

r radial coordinate

tcond time scale for heat conduction, s

t time, s

tdi¤ time scale for electrolyte di¤usion, s

text time scale for heat transfer to external atmosphere, s

tmacro macroscopic or discharge time scale, s

5.4. Conclusion 90


T Temperature, K

Ta; T0 ambient and initial temperature, K

Tref reference temperature, 298:15 K

Uref open circuit potential of the electrode, V

V volume of the various layers in the jelly roll, m3


Greek



"l volume fraction of the eletrolyte in the electrodes and separator

"f volume fraction of the conductive �ller additive in the electrodes

"p volume fraction of the polymer in the electrodes

� overpotential, V


� density, kg m�3


�s electronic conductivity of solid matrix, S m�1



� local state of charge of the electrodes

Bruggeman constant (= 1:5)

5.4. Conclusion 91

�1;�2;�3;�4;�5 nondimensional numbers

Subscripts




sp separator

l liquid/electrolyte

Superscripts

0 initial values

eff e¤ective values

max maximum value

Chapter 6

Automated Model Generation of a

Lithium-Ion Bipolar Battery Module

6.1 Introduction

The main characteristics of a Li-ion battery � high energy density, no memory e¤ect, and slow

self-discharge �have made it ideal for energy storage in consumer electronics. The Li-ion battery

is also a potentially attractive candidate for hybrid and electric vehicles as well as for large-scale

energy storage for intermittent power generation and smart grids. Common to all these already

established as well as currently developed applications for Li-ion batteries is that there are as of yet

no mature and standardized designs, thermal and battery management strategies. There is thus

a need for tools that can provide cost-e¤ective solutions for design and management of a battery

system.

In general, a battery pack is a multi-module system containing a number of battery modules,

each of which can comprise a large number of battery cells; the cells, in turn, are typically in the

form of coin cells, cylindrical or prismatic cells. In addition, supervisory control and management

circuits are integrated into the pack to ensure optimal performance by minimizing nonuniformities

among cells and modules. Based on the application, a Li-ion battery pack can comprise in the

range of two to several thousands of single cells; e.g., the Tesla electric car, which is powered by

a battery pack with 6800 Li-ion cells [114]. Therefore, di¤erent length scales can be found in a

battery pack (see Fig. 6.1): agglomerate structures in the electrode layer that are typically O(10�6

92


m); functional layers that are around O(10�5m) in thickness; and the module and pack itself with

a typical length scale of O(10�1 � 1 m). Overall, the design from the agglomerate to the cell to

the system level plays an important role in the achieved energy and power density as well as in the

performance and safety, both of which are interrelated.

Figure 6.1: Schematic of (a) a bipolar lithium-ion battery module, (b) the various functionallayers on the macroscale, and (c) di¤usion of lithium in the active material of the electrodes inthe microscale

Mathematical modeling that aims to elucidate and resolve the salient features that can be found

in a typical battery pack is highly challenging: �rst, the three-dimensional nature and the multiple

length and time scales from the functional layers and groups in the cell to the module to the

pack level have to be captured; second, the highly coupled, non-linear behavior of the transport

phenomena together with the relevant electrochemistry and local e¤ects such as heat generation

have to be accounted for; and third, the intrinsic transient nature of a battery and degradation over


time must be considered. It is thus not surprising that the majority of detailed, mechanistic models

for Li-ion batteries have focused on the cell level so far [1, 2, 3, 4, 19, 21, 24, 30, 31, 39, 40, 43, 53,

62, 109, 115, 116, 117, 118, 119]; see Refs. [14, 15] for reviews of these. At the battery module or

pack level, modeling studies are generally simpli�ed with a loss in the level of detail and resolution

of the salient features of the electrochemical and transport phenomena [23, 39, 51, 52, 120]. The

main reasons behind the simpli�cations can be found in the prohibitive computational cost and

complexity that is incurred once one resolves the physicochemical phenomena at a module or pack

level with tens or even hundreds of single cells.

In order to reduce the complexity and computational cost of battery systems comprising more

than one cell, the commonly employed coupled electrochemical-thermal mathematical formulation

for a Li-ion battery cell [1, 2] is reformulated at the microscale to reduce the number of depen-

dent variables. The reformulated model is veri�ed and the reduction in computational cost is

demonstrated for the reformulated single-cell model for a Li-ion battery module designed in bipolar

con�guration, which is illustrated in Fig. 6.1, followed by exploring automated model generation

in the context of battery modules based on numerical building blocks introduced for fuel cell stacks

[104].

Here, the bipolar con�guration [121, 122], which strives to minimize ohmic losses between

adjacent cells and to provide a more uniform current and potential distribution over the active

surface area in each cell, serves as a good stepping stone due to its relatively simple design compared

to battery modules comprising a large number of spiral-wound single-cells. Further, the automation

of the model brings with it a number of bene�ts: a reduction in computational time of several

orders of magnitude and complete removal of human errors after veri�cation, as the entire pre-

processing � drawing of geometry, meshing, and introduction of governing equations, boundary

and initial conditions as well as constitutive relations at multiple length scales �for the numerical

implementation is automated; and an easy-to-use automated numerical framework, which allows

for cost-e¤ective multi-objective optimization of key features, such as the battery design, operating

conditions, and management strategies.



The Li-ion battery modeled in this study consists of a negative electrode (petroleum coke), a

positive electrode (cobalt oxide), and a separator (Celgard 2400) in between them as illustrated in

Fig. 6.1. The electrodes and the separator in the battery are �lled with an electrolyte solution of

LiClO4 salt in 1:2 ethylene carbonate:diethyl carbonate solvent.

The reactions that occur in the two electrodes during charge and discharge are (forward reaction

representing discharge) as follows:

LixC6�LiC6+xLi++xe�

LiCoO2+xLi++xe��LixCoO2

As depicted in Fig.6.1, there are two main scales involved in the modeling of a Li-ion cell: the

macro- and the micro-scale. In short, the transport of ions and electrons in the cell between the

electrodes is referred as the transport at the macroscale, which includes species transport in the

liquid electrolyte, electronic charge conduction in the solid phase and ionic charge conduction in

the liquid electrolyte; and the di¤usion of ions in the active material present in the electrodes is

referred to as transport at the microscale, which includes di¤usion of lithium in the active material

of the porous electrodes.

The battery module is constructed by placing a bipolar plate between the cells as depicted in

Fig. 6.1a. For simplicity, the properties and thickness of the bipolar plate is assumed to be that of

the positive current collector.

The bipolar, symmetric nature of the battery module allows for a reduction in dimensionality

from three (x; y; z) to two dimensions (x; y) with the postulate that the heat �ux in the z-direction is

negligible compared to the heat �uxes in the x- and y-directions, which essentially implies insulation

at the boundaries in the z-direction. This postulate can easily be relaxed; for our purposes, however,

it serves as a good stepping stone to explore and verify the model reformulation as well as automated

model generation at the module level. The latter can be extended in a straight-forward manner to

also encompass the pack level.


6.2.1 Governing equations

The governing equations in the macro and microscale are as provided in chapter 3. The equations

applicable for the bipolar plate are the same as that applicable for the current collector.


At the left and right sides of the current collectors and the top and bottom walls (see Fig. 6.1a for

placement of roman numerals),

is�ex = 0;q�ex = h (T � Tamb) (I); is�ey = il�ey = Nl�ey = 0 (V,VII); q�ey = h (T � Tamb) (V;VII):

(6.1)

At the current collector/electrode and the bipolar plate/electrode interface,

is � exjII+ = is � exjII� ; q � exjII+ = q � exjII� ; il � ex = Nl � ex = 0 (II). (6.2)

At the electrode/separator interface,

is � ex = 0; il � exjIII+ = il � exjIII� ; q � exjII+ = q � exjII� Nl � exjIII+ = Nl � exjIII� (III). (6.3)

At the top side of the negative current collector, the solid phase potential is set to be zero and

convective heat transfer to the ambient:

�s = 0;q � ey = h (T � Tamb) (IV). (6.4)

At the top side of the positive current collector, the current density is prescribed and convective

heat transfer to the ambient:

is � ey = �iapp;q � ey = h (T � Tamb) (VI): (6.5)

In Eqs. 6.1 - 6.5, ex and ey denote the unit vectors in the x- and y- directions respectively, Tamb

is the ambient temperature, and iapp is the applied current density.


At t = 0,

cs = cavgs = c0s; cl = c

0l ; (6.6)

�s =

8>>>><>>>>:(j � 1)�0s (ne)

j�0s (pe)

j = 1; 2; :::n; (6.7)

�l = j�0l (ne,pe,sp), j = 1; 2; :::n; (6.8)

T = T0 (ne,pe,sp,bp,cc), (6.9)

where j is an index denoting the cell number in the bipolar module built with n number of cells.


The generic constitutive relations are as mentioned in chapter 3. The system speci�c relations are

provided here.

The entropic heat as a function of state of charge for the negative electrode is expressed as

@Uref; ne@T

=n1 exp(n2�ne + n3)

n4 + n5 exp(n6�ne + n7)+ n8�ne + n9�

2ne + n10; (6.10)

in which ni [2] are constants obtained from curve �tting for experimental data and �ne represent the

state of charge of the negative electrode. For the positive electrode, the change in entropy with the

state of charge is obtained from experiments conducted by Reynier et al. [5]. The change in entropy

(�S) is related to the change in the open circuit potential of the electrode with temperature by

[123]@Uref@T

=�S

aF(6.11)

where a is the number of electrons involved in the reaction (here, a is 1).

The open-circuit potential for the positive electrode is de�ned as [4]

Uref; pe =

5Pi=0Pi�

2ipe

5Pi=0pi�

2ipe

; (6.12)


where �pe represent the state of charge of the positive electrode, and for the negative electrode as

[1]

Uref; ne = N1 +N2 exp (N3�ne) +N4 exp (N5�ne) ; (6.13)

here, Ni and Piare constants obtained by curve-�tting with experimental data. The transference

number and electrolyte di¤usivity are given by [3]

t0+ =z+D0+

z+D0+ � z�D0�; (6.14)

Dl =D0+D0� (z+ � z�)z+D0+ � z�D0�

; (6.15)

where z+ and z� represent the charge of the anion and cation respectively (here, z+ = 1and

z� = �1) and D0+ and D0� are the temperature dependent di¤usion coe¢ cients of anion and

cation in the solvent. The electrolyte conductivity at the reference temperature is written as [3]

�ljTref = a1 + " l ca2l

ha3 exp

�a4 (a5cl + a6)

2 + a7cl + a8

�i(6.16)

where is the Bruggemann constant. The temperature dependence of the di¤usion coe¢ cients

(both solid and liquid) and ionic conductivity are expressed as [2, 3, 91]

For the positive electrode: Ds(T ) = DsjTref exphEa;DsR

�1

Tref� 1

T

�iFor the negative electrode: Ds(T ) = C1 exp

�Ea;DsRT

�Electrolyte di¤usivity: D0+ = C2 exp

�Ea;D0+RT

�D0� = C3 exp

�Ea;D0�RT

�Electrolyte conductivity: �l(T ) = C4 �ljTref exp

�Ea;�lRT

�where Tref is a reference temperature, Ci are constants and Ea is the activation energy.

The e¤ective conductivities and di¤usivity are approximated as

�effs = �s"; (6.17)

Deffl = Dl" : (6.18)


The solid-phase potential di¤erence between the top side of the positive current collector (VI)

and the negative current collector (IV) is de�ned as the module voltage:

Emodule = �sjVI � �sjIV : (6.19)

The parameters for these constitutive relations are summarized in Table 6.1, 6.2 and 6.3.



Parameter Unit cc (-) ne sp pe cc (+)� Reference

c0l mol m�3 - 1� 103 - [4]

Cp J kg�1 K�1 3:8� 102 1:4� 103 2:0� 103 1:3� 103 8:7� 102 [38]

c0s mol m�3 - 2:0� 104 3:2� 104 - calibrated

cmaxs mol m�3 - 2:6� 104 5:1� 104 - [1, 4]

DljTref m2 s�1 - 2:6� 10�10 - [3]

DsjTref m2 s�1 - 3:9� 10�14 1:0� 10�13 - [3]

Ea;D0+ kJ mol�1 - 10 - [3]

Ea;D0� kJ mol�1 - 38 - [3]

Ea;Ds kJ mol�1 - 4:1 - 20y - [2, 3]

Ea;�l kJ mol�1 - 18 - [3]

h m 10�1 [121]

h W m�2 K�1 5 -

iapp A m�2 1:8� 103�� calculated,[121]

k W m�1 K�1 3:8� 102 1:0 0:3 1:6 2:0� 102 [38]

R m - 10�5 - 10�5 - [3]

Tamb; Tref K 298:15 -

k0 - - 2� 10�11 - 2� 10�11 - -

wi m 800� 10�6 88� 10�6y 25� 10�6y 80� 10�6y 800� 10�6 [4, 121]

�a; �c - - 0:5 - 0:5 - [1]

"p - - 0:14 - 0:19 - [1]

"l - - 0:485 0:41y 0:385 - [4]

"f - - 0:033 - 0:025 - [1]

� kg m�3 9:0� 103 1:9� 103 103 2:3� 103 2:7� 103 [38]

�s S m�1 6:0� 107 102 - 102 3:8� 107 [2, 4]

� - properties of bipolar plate are assumed to be that of the positive current collector exceptwbp which is taken as 10�5 m from ref. [121], ** - the current is applied at boundary VIand hence adjusted for the change in area as described in y - assumed

6.3. Model reformulation 101

Table 6.2: Constants in Eqs. (Refs. [1, 2, 3, 4])

Constant Unit Value Constant Unit Value Constant Unit Value

n1 mV K�1 344:1347 P3 V 342:909 N5 - �2000

n2 - �32:9633 P4 V �462:471 a1 S m�1 0:0001

n3 - 8:3167 P5 V 433:434 a2 - 0:855

n4 - 1 p0 - �1 a3 S m�1 0:00179

n5 - 749:0756 p1 - 18:933 a4 - �0:08

n6 - �34:7909 p2 - �79:532 a5 - 0:00083

n7 - 8:8871 p3 - 37:311 a6 - 0:6616

n8 mV K�1 �0:8520 p4 - �73:083 a7 - 0:0010733

n9 mV K�1 0:3622 p5 - 95:96 a8 - 0:855

n10 mV K�1 0:2698 N1 V �0:16 C1 m2 s�1 2� 10�13

P0 V �4:656 N2 V 1:32 C2 m2 s�1 10�8

P1 V 88:669 N3 �3:0 C3 m2 s�1 26� 10�4

P2 V �401:119 N4 V 10 C4 S m�1 1500

6.3 Model reformulation

At this stage, a total of six dependent variables need be solved for the macroscale and microscale

governing equations: �s; �l; cl; csurfs ; cavgs ; and T . Out of these, the underlying equations are di¤er-

ential equations for all dependent variables except for csurfs , which is given by an implicit algebraic

equation (Eq. 3.11). The latter suggests that a reformulation should be possible in order to remove

csurfs as a dependent variable. This is indeed the case if Eq. 3.11 and 3.12 are combined to give

csurfs = cavgs +lsDs

dcavgsdt

R

3; (6.20)

in which csurfs (cavgs ; dcavgs =dt). It is thus su¢ cient to solve only for cavgs at the microscale instead of

as previously for both cavgs and csurfs , because every csurfs can be replaced by Eq. 6.20; however,

in doing so, several of the constitutive relations �e.g., if ; i0; �n and �p �now become nonlinear

functions of the time derivative of a dependent variable, which could have implications on the

convergence behavior and numerical discretization of the mathematical formulation. This reformu-

6.4. Automated Model Generation 102

Table 6.3: Change of entropy with state of charge for positive electrode (Ref. [5])

�pe �S, J mol�1 K�1 �pe �S, J mol�1 K�1

0:490123 �23:8843 0:664198 �27:6033

0:498765 �30:0826 0:718519 �33:5537

0:507407 �38:0165 0:77037 �43:9669

0:516049 �38:2645 0:823457 �56:1157

0:524691 �19:1736 0:849383 �55:124

0:534568 2:64463 0:875309 �58:0992

0:540741 12:314 0:902469 �58:843

0:550617 15:7851 0:928395 �60:8264

0:577778 �17:438 0:954321 �61:0744

0:585185 �16:6942 0:981481 �43:719

0:61358 �20:6612 0:997531 �45:2066

lation will be justi�ed a posteriori by comparing the solutions of the local and global behavior of

a single cell and module from the full set of equations with the reformulated counterpart.

6.4 Automated Model Generation

An automated model generation demonstrated for a fuel cell stack [104] is introduced here for a

battery module. In short, the procedure automates the steps that are necessary to obtain numer-

ical solutions to the mathematical model: viz., drawing the geometry, meshing, implementing the

mathematical equations, solving and post-processing. This approach thus allows for a large reduc-

tion in time, since the time spent on manually setting up and solving a complex battery module

is shortened considerably. In addition, and perhaps most importantly, the automated script will

allow for completely automated studies and solutions, which opens up avenues for, e.g., automated

multi-objective optimization of not only operating conditions but also of the design and layout of

a battery module or pack.

6.5. Numerics 103

6.5 Numerics

The commercial �nite-element solver, Comsol Multiphysics 3.5a (see Ref. 103 for details), which

allows for the solution of generic di¤erential equations, was employed to implement the full model

and the reformulated counterpart. The former comprises six dependent variables (�s; �l; cl; csurfs ; cavgs ;

and T ) and the latter �ve (�s; �l; cl; cavgs ; and T ); the governing equations were implemented with

the general form of the PDE mode for all dependent variables except temperature, T , which was

implemented with the convection-and-conduction heat transfer mode.

The automated model generation was carried out in the commercial general-programming-

environment Matlab 2008 [124] by exploiting the bidirectional interface between Comsol Mul-

tiphysics and Matlab. In essence, the automated model generation is based on a Matlab script

that manipulates a COMSOL-associated structure, known as a Fem structure, which contains the

numerical formulation of the entire model.

The discharge current was applied with a smoothed Heaviside function in which the current

is ramped up from zero to its actual value in a time period that is much smaller than the overall

time scale to ensure converged solutions whilst not impacting the solution. In addition, to obtain a

robust numerical formulation, a relational operator was introduced for the exchange current density

to ensure that it does not become negative:

i0 = Fk0

rclc

surfs

h�cmaxs � csurfs

��cmaxs > csurfs

�+ c�

i; (6.21)

where c� = 10�12 mol m�3 is a negligible concentration chosen so that it does not a¤ect the solution

at leading order; i.e., c� � cmaxs � csurfs .

The direct solver UMFPACK was chosen as linear solver with a convergence tolerance of 10�4.

For a single cell, which e¤ectively becomes a building block for the battery module, the full model

was resolved with 4:6� 102 linear elements amounting to 9:1� 103 degrees of freedom (DoF); the

reformulated counterpart had the same number of linear elements, albeit with 8:1 � 103 DoF due

to model reformulation. The number of elements were set after a mesh-independence study.

The computations were carried out on a workstation with two quad-core processors (3.2 GHz)

and a total of 64 GB RAM. The wall-clock time and peak memory usage were estimated from

6.6. Calibration and Validation 104

COMSOL�s graphical user-interface and Matlab�s tic and toc commands with all unnecessary

processes stopped to secure reasonably accurate times.

6.6 Calibration and Validation

The nonlinear, multi-scale and coupled nature of the the governing equations as well as the high

number of parameters result in a complex model, which requires some form of calibration. As shown

in Fig. 6.2, the initial concentration of lithium in the active material in the electrodes is calibrated

with the �rst data point from an experimental discharge curve [121] for a bipolar single cell at a

current density of 1:5 mA cm�2 and subsequently validated with the remaining data points; overall,

good agreement was found. The discharge capacity of the cell, which is limited by the carbon in the

anode, was found to be 255 mAh g�1 experimentally [121]. The matching conditions between the

simulation and the experiments were the design adjustable parameters, the operating conditions

(discharge current) and the physical properties of the electrode, separator and current collector as

provided in Table 6.1.

0 50 100 150 200 2502.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4

4.2

Capacity / mAh g 1

Ece

ll / V

Figure 6.2: Comparison of cell voltage predicted by the reformulated model (line) andexperimental results (symbol) at a current density of 15 A m�2.

6.7. Veri�cation 105

6.7 Veri�cation

The reformulated model requires veri�cation with the full model due to the introduction of time

derivatives in the constitutive relations to ensure that the numerical solver captures these. Fur-

thermore, the automated procedure has to be veri�ed as there are a large number of functional

layers in the module and associated equations.

The single cell reformulation is veri�ed on a global and local level as depicted in Figs. 6.3

and 6.4 respectively. Overall, good agreement is achieved for various discharge current densities

ranging from 1:5 mA cm�2 to 7:5 mA cm�2 (the local behavior for 3:0 mA cm�2 and 7:5 mA cm�2

discharge is not shown), which suggests that the numerical solver is handling the reformulated model

adequately. The behavior of the single cell, which will later be employed as a building block to

construct a battery module, is as expected with smooth discharge curves and a rapid drop towards

the end of the discharge, as can be seen for the 1:5 mA cm�2 discharge in Fig. 6.3. Similarly,

the local liquid concentration exhibits an increase in the negative electrode and a corresponding

depletion at the positive electrode.

0 1000 2000 3000 4000 5000

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4

4.2

t / s

Ece

ll / V

15 A m 275 A m 2 30 A m 2

Figure 6.3: Cell voltage predicted by the full model (symbols) and reformulated model (lines) atvarious discharge rates.

At the module level for ten cells, the reformulated model and the automated model generation

are veri�ed with a manually implemented counterpart for the full set of equations, as shown in Fig.

6.5 for the global behavior and in Fig. 6.6 for the local behavior. In particular, Fig. 6.5b shows


8 9 10

x 104

900

920

940

960

980

1000

1020

1040

1060

1080

x / m

cl /

mol

m3

5300 s

10 s

50 s

0 s

a) Concentration of lithium ion in the electrolyteinside the porous electrodes and separator in a

single cell�y = 4:9� 10�2 m, iapp = 15 A m�2

�.

8 9 10

x 104

1.4

1.2

1

0.8

0.6

0.4

0.2

0

x / m

φ l / V

875 s1750 s

2625 s

3500 s

4375 s

5250 s

0 s

b) Liquid phase potential in the electrolyteinside the porous electrodes and separator in asingle cell

�y = 4:9� 10�2 m, iapp = 15 A m�2

�.

Figure 6.4: Comparison of results predicted by the full model (symbols) and the reformulatedmodel (lines) at various times during discharge.

that as the discharge proceeds, the heat generated by the cells increases the temperature rapidly of

the module as there is heat transfer only from the top of the module with the other sides insulated.

This type of boundary condition and thermal envelope can be found in large battery modules for,

e.g., electric vehicles [2].

Clearly, the rapid temperature increase during discharge indicates the need for a thermal man-

agement system in order to ensure that the module operates within a safe temperature range.

In this context, modeling studies could aid in the design of a tailored thermal management sys-

tem wherein the automated procedure with the reformulated model could signi�cantly reduce the

complexity and computational cost, as highlighted in the following section.

6.8 Computational cost

Even after having derived and implemented a well-established, reliable model, a high computational

cost, i.e. memory requirement and solution time, might still prohibit its wide use. Therefore, after

con�rming the reliability, the focus should be placed on ensuring that the computational cost is


0 1000 2000 3000 4000 5000

24

26

28

30

32

34

36

38

40

42

t / s

Mod

ule

volta

ge /

V

15 A m 275 A m 2 30 A m 2

a) Module voltage vs. time

0 1000 2000 3000 4000 5000295

300

305

310

315

320

325

330

t / s

Tav

g / K

75 A m 2

30 A m 2

15 A m 2

b) Average module temperature vs. time

Figure 6.5: Global veri�cation: Comparison of results for a 10-cell module predicted by themanually implemented full model (symbols) and the reformulated model (lines) at variousdischarge current densities.

0.5 1 1.5 2 2.5 3 3.5 4

x 103

900

920

940

960

980

1000

1020

1040

1060

1080

x / m

cl /

mol

m3

a) Concentration of lithium ion (cl) in theelectrolyte inside the porous electrodes and

separator�y = 4:9� 10�2 m, iapp = 15 A m�2

�.

1 1.5 2 2.5 3 3.5

x 103

0

5

10

15

20

x / m

φ l / V

b) Liquid phase potential in the porouselectrodes and separator�

y = 4:9� 10�2 m, iapp = 15 A m�2�

Figure 6.6: Local veri�cation: Comparison of results predicted by the full model (symbols) andthe reformulated model (lines) at various times during discharge.

6.9. Conclusion 108

kept to a minimum �this is especially the case for a battery module or pack.

Starting with the computational cost in terms of the time to set up the model in the numerical

software for a single cell up to a battery module comprising 50 single cells (see Fig. 6.7a), the

automated model generation for the reformulated model requires around 1, 3, and 11 s for a single

cell, a 10-cell and a 50-cell module, respectively. This time should be contrasted with the time it

takes to set up the mathematical model manually: e.g., the aforementioned manually-constructed

battery module comprising ten cells took around half an hour, which is due to the fact that each

single cell comprises 5 functional layers, 12 boundaries, and 4 interfaces, for all of which the

governing equations have to be set properly. A 10-cell module would thus comprise 50 functional

layers and 160 boundary/interface conditions. Furthermore, there is a decrease in setup time for

the reformulated model as compared to the original formulation as the number of cells increases,

which originates from having to implement one dependent variable less.

The solution time, i.e. the time to solve the model once properly pre-processed in the numerical

software, is reduced signi�cantly for the reformulated model, as can be inferred from Fig. 6.7a: The

di¤erence in solution time is around 40 %. The reduction in solution time can be partly related to

the computational cost in terms of DoF and the memory requirements, which are illustrated in Fig.

6.7b. Here several features are apparent: �rst is that the DoF increase linearly with the number

of cells, which is expected since more numerical building blocks are simply stacked with increasing

module size; and second is that the reformulated version requires less DoF and thus less memory.

In this case, a direct solver has been employed, which explains the overall memory cost on the order

of gigabytes. For larger battery modules, one would at one stage need to switch to iterative solvers

to reduce the memory requirements, albeit at the loss of robustness. Another alternative would

be to implement numerical techniques such as orthogonal collocation [60, 63] or proper orthogonal

decomposition [59] in C or Fortran to improve the computational e¢ ciency further.

6.9 Conclusion

A reformulated mathematical model for a Li-ion battery module model is presented, veri�ed with

the original formulation, and solved numerically through automated model generation.

The advantage of the reformulation is highlighted through the reduction of the computational

6.9. Conclusion 109

a) Time required for for setting up (full - H,reduced - O) and solving (full - �, reformulated -

�) the automated module model.

b) Memory requirement (full - H, reduced - O)and DoF (full - �, reformulated - �) for the

automated module model.

Figure 6.7: Computational cost comparison.

cost in terms of DoF, computational time and memory requirement for simulating a battery module;

e.g., solving a 50-cell battery module takes around 1:5 � 103 s for the reformulated model as

compared to 2:6 � 103 s for the original model �a reduction in time near to half. As the number

of cells in the module increase to hundreds or even thousands, even larger gains can be expected.

The automated procedure carries out all the steps necessary in solving the mathematical model

numerically �from drawing the geometry, meshing, implementing the equations, solving and post-

processing of the results. Once veri�ed, the numerical framework thus lends itself well to e¢ cient,

fast automated studies for optimization, design, and wide-ranging parameter studies; e.g., since

the procedure allows for automated drawing of the geometry, design and layout, one could envision

a multi-objective optimization of not only operating conditions but the actual design itself or the

number of cells in the module. Other types of studies that could be carried out include but are not

limited to statistical analysis, thermal runaway studies of an entire battery module, and detailed

management-strategy and design studies.

The methodology can be extended to include other types of cell designs, such as spiral-wound

batteries, as well as additional physical phenomena, such as �uid dynamics for cooling purposes.

6.9. Conclusion 110

Nomenclature

ai constants in electrolyte conductivity expression (units given in Table 6.2)

A speci�c interfacial area per unit volume, m�1

a number of electrons in the reaction, -


Cp e¤ective speci�c heat capacity, J kg�1 K�1

cs concentration of lithium in the active material, mol m�3

cavgs average concentration of lithium in the active material, mol m�3

cmaxs maximum concentration of lithium in the active material, mol m�3

csurfs surface concentration of lithium in the active material in the electrodes, mol m�3

D0+ di¤usion coe¢ cient of cation in solvent, m2 s�1

D0� di¤usion coe¢ cient of anion in solvent, m2 s�1


Ds di¤usion coe¢ cient of lithium in the active material, m2 s�1

Ea activation energy for a variable, kJ mol�1

ex; ey; ez coordinate vectors


h height of the battery cell / module, m

h convective heat transfer coe¢ cient, W m�2 K�1




6.9. Conclusion 111



j index for cell number in the module, -


k e¤ective thermal conductivity, W m�1 K�1

k0 reaction rate constant, -




n number of cells in the module

ni constants in the entropic heat term (negative electrode; see Table 6.3 for units)

Ni constants in the open circuit potential (negative electrode; see Table 6.2 for units)

pi, Pi constants in the open circuit potential (positive electrode; see Table 6.2 for units)

Ci constants, see Table 6.2 for units




R radius of active material in the electrodes, m

t time, s


T temperature, K

Uref; i open circuit potential of the electrode i, V

6.9. Conclusion 112

x; y; z coordinates

z+ charge number of cation, -

z� charge number of anion, -

Greek



"l volume fraction of electrolyte

"f volume fraction of conductive �ller additive

"p volume fraction of polymer phase

� overpotential

�ne state of charge of negative electrode

�pe state of charge of positive electrode


� e¤ective density, kg m�3


�s electronic conductivity of solid matrix, S.m�1



�S change in entropy, J mol�1 K�1

Subscripts

amb ambient

bp bipolar plate

6.9. Conclusion 113




ref reference value

sp separator

Superscripts

0 initial value

eff e¤ective value

Chapter 7

Thermal-Electrochemical Model for

Passive Thermal Management of a

Spiral-wound Lithium-Ion Battery

7.1 Introduction

Recent years have seen several recalls of commercial Li-ion batteries due to overheating. The heat

originates mainly from the electrochemical reactions that occur during charge/discharge of the bat-

tery as well as Joule heating �if this heat is not dissipated properly, it can lead to overheating of the

battery, and, in a worst-case scenario, thermal runaway. The latter typically occurs in conjunction

with �abuse�conditions, high power draw, and/or manufacturing defects. Generally, some form of

protection is thus needed to prevent overheating, either by means of electronic control circuits that

monitor charge/discharge rates and/or temperature, or by means of a properly designed thermal

management system. Furthermore, for the purpose of operating the battery within safe operat-

ing temperature limits, either one of two following thermal management strategies are typically

introduced: active systems with air/liquid cooling [51] or passive systems with, for example, phase-

change materials (PCMs)[49, 50, 51, 52, 120, 125]; see Bandhauer et al. [15] for a detailed review

of these. The main advantages of passive cooling with PCM viz-a-viz active cooling are a simpli�ed

design, absence of parasitic power consumption, smaller temperature gradients with air as coolant

114


under normal and stressed operating conditions [51], and that propagation of energy from cell to

cell arising from thermal runaway inside a battery pack can be reduced [52] .

Various mechanistic mathematical models have been developed in order to predict the transient

electrochemical and thermal behavior of a Li-ion cell in a rectilinear geometry [1, 2, 3, 4, 19, 21,

23, 24, 40, 109] and a spiral-wound geometry [41, 42, 47, 48, 53, 106, 107, 108]. In essence, these

models typically consider the transient equations of change for species, charge and energy together

with relevant boundary conditions and constitutive relations. The level of detail and resolution

depends on the treatment of electrochemical and thermal phenomena: The most detailed models

[2, 3, 23, 24] solve the governing equation in the form of partial di¤erential equations (PDEs), thus

resolving the local transport phenomena, electrochemistry and heat generation. Loss of detail is

often incurred when one of the governing equations is simpli�ed to an ordinary di¤erential equation

(ODE) or a phenomenological expression, thus only truly accounting for the global behavior of the

battery cell and, to a lesser extent, the local behavior; for example, when only the equation of

change for energy is solved locally together with an expression for heat generation from some

form of approximative relation [40, 41, 48, 53, 106, 107] of the electrochemical reactions or from

experimental measurements [51, 52]; or when the equations of change describing the electrochemical

phenomena on a local level are coupled with some form of lumped-parameter model for the thermal

part on the global level [39, 47]. The general trend here is that detailed local models have been

employed for rectilinear geometries and simpli�ed counterparts for spiral-wound geometries �the

latter is typically signi�cantly more expensive to solve from the numerical point-of-view as can be

inferred from intricate geometrical features in Fig. 7.1.

In view of the lack of detailed, local resolution for modeling and simulation of coupled electro-

chemistry, transport phenomena and heat generation in spiral-wound geometries, the aim of this

work is twofold: First, to develop a coupled thermal-electrochemical model for a cylindrical spiral-

wound lithium-ion battery without compromising local resolution, which can easily be applied to

spiral-wound prismatic cells as well; and second, to apply the derived model to investigate the

design and operation of a passive thermal management system based on PCM. In short, the model

considers transient conservation of charges, species and energy; it couples the electrochemical and

thermal behavior through the heat generation arising from reversible, irreversible and ohmic heating

as well as through the temperature-dependent transport and electrochemical . The PCM, in turn,


Figure 7.1: Schematic of (a) a 18650 Li-ion battery, (b) cross-section of the 18650 battery showingthe spiral-wound jelly roll, (c) cross section of the battery with PCM surrounding it, (d) variousfunctional layers in the jelly roll with the roman numerals indicating the interfaces of these layersat the inner end of the spiral, (e) outer end of the spiral with the interfaces of various layersshown by the roman numerals, (f) modi�ed computational domain, see numerics, (g) agglomeratestructure in the negative electrode (*- positive electrode also exhibits similar structure) and, (h)di¤usion of lithium in active material in the electrodes on the microscale.


is wrapped around the battery and solved for in terms of conservation of energy. The results are

discussed with emphasis on transient behavior and temperature distribution in the various layers

of the spiral-wound battery under galvanostatic discharge at various rates.


A commercially available spiral-wound cylindrical Li-ion 18650 battery (Fig. 7.1a) is considered

in this study, for which a two-dimensional cross-section of the battery is resolved, as illustrated in

Fig. 7.1b, where the functional layers �positive electrode (pe), negative electrode (ne), current

collector (cc) and separator (sp) �are wound up in the form of a jelly roll. The dimensions of the

various functional layers are taken from a Sony cell [110] with the number of wounds determined

to be 15 based on the diameter of the battery and the thickness of the wounds.

Generally, the porous electrodes consist of active material, conductive �ller additive, binder

and liquid electrolyte (el). In this study, the active material in the negative electrode is LixC6 and

in the positive electrode it is LiyMn2O4. The salt is LiPF6 in a nonaqueous 1:2 liquid mixture

of ethylene carbonate and dimethyl carbonate. The electrochemical reactions that occur at the

electrode/electrolyte interface during charge and discharge are then

LixC6Disch arg e�

Ch arg eC6 + xLi+ + xe�;

Liy�xMn2O4 + xLi+ + xe�Disch arg e�

Ch arg eLiyMn2O4;

where x is the stoichiometric coe¢ cient or the number of moles of lithium present in the graphite

structure, C6, and y is the number of moles of lithium in the spinel structure of manganese dioxide

Mn2O4; Li+ is the lithium ion.

The materials for the positive and negative current collectors and the outer can are aluminium,

copper and stainless steel, respectively.

The PCM, which is coated around the battery (Fig. 7.1c), is taken to be para¢ n wax impreg-

nated in a graphite matrix. Para¢ n wax has a high latent heat but a low thermal conductivity,

whence the graphite matrix is provided to enhance the heat transfer rate between the cell and the

ambient by conduction [51, 126]. Overall, PCMs have the advantage of storing and releasing heat


within a narrow temperature range as latent heat. However, most of the PCMs have a low thermal

conductivity [127] and various attempts have been carried out to improve the conductivity and in

turn, the e¢ ciency of the thermal energy storage [126]. In addition, factors like mechanical strength

and electrical properties will constrain the thickness and type of the PCM that can be employed.

The length scales ranging from the agglomerate level on the order of 10�7 m to the cell level

on the order of 10�1 m are resolved: The transport in the cell on length scales larger than that

of the agglomerates (Fig. 7.1g) is referred to as the transport at the macroscale, which includes

mass transfer in the electrolyte describing the movement of mobile ionic species, material balances,

current �ow and electroneutrality based on concentrated-solution theory [18], electronic charge

conduction in the solid phase and energy transfer in the solid/liquid phases (Fig. 7:1b-f); the

di¤usion of ions in the active material in the electrodes is referred to as the transport at the

microscale, which includes di¤usion of lithium in the active material of the porous electrodes (Fig

7.1g).

The main postulates and features of the model are as follows:

1. Reduction in dimensionality. The 3D battery (Fig. 7.1a) is reduced to a 2D cross-section

(7.1b) through the middle of the battery, which is justi�ed by the following arguments: First,

insulated conditions for the energy �ux are prescribed at the top and bottom surface of the

battery (z-direction in Fig. 7.1a), which is a common assumption [37, 41, 47, 106, 107, 128].

Second, as a �rst approximation, it is assumed that the air temperature is constant around

the battery, whence the temperature di¤erence, �Tz, inside the battery in the axial direction

is zero; i.e., �Tz = 0: Third, potential losses in the axial direction in the current collectors

are negligible at leading order, since the potential drop in the z-direction, ��s;z � iapph=�s �

10�4 V� Ecell � 1 V, for typical operating and material properties: iapp � 106 A m�2 (� 1

C-rate), h � 10�3 m, �s � 107 S m�1, where iapp is the applied current density, h is the

height of the battery, �s is the electrical conductivity and Ecell is the cell voltage. Fourth, the

placement of current collecting tabs on the behavior of the cell is assumed to be negligible.

For the resulting 2D cross-section, the current is prescribed as entering from the innermost

boundary of the current collector (II) and leaving at the outermost current-collector boundary

(V), which is justi�ed by the potential drop in the tangential direction, ��s;t, around the total


wound of length, L, which is negligible at leading order compared to the overall cell voltage;

i.e., ��s;t � iappL=�s � 10�2 V � Ecell � 1 V, for a typical length of a wounded layer, L

� 10�1 m. Note that this condition leads to errors when the C-rate is around 10 or higher,

since iapp � 107 A m�2 (� 10 C-rate), such that ��s;t � iappL=�s � 10�1 V < Ecell � 1 V.

Here, the discussion is limited to C-rates from one to �ve.

2. Natural convection. Natural convection in the type of PCM that is considered here has been

shown to be negligible [129]. Inside the battery, negligible natural convection is postulated

as a �rst approximation by letting gravity act in the z-direction, in which there are no tem-

perature/concentration gradients due to the conditions outlined in #1 above; however, that

the model could be extended to include natural convection [130, 131, 132, 133].

3. Electrochemistry and related phenomena. Side reactions inside the battery and double-layer

capacitance are not considered, which is, again, a common assumption [24, 91, 116].

4. Material properties. Uniform distribution of active materials of uniform size in the electrodes

and PCM is assumed. Further, phase change for the PCM does not occur at a single temper-

ature but rather over a given melting range, which is assumed to be captured reasonably well

with a linear relation between latent heat in the �mushy�region �where a liquid and solid

phases coexist �of the PCM. The emissivity of PCM is taken to be the emissivity of pure

graphite alone.

5. Microscale properties. The active material is assumed to be spherical; i.e., only the radial

direction has to be considered at the microscale.

6. Contact resistance. The contact resistance between the PCM and the battery as well as

between functional layers is assumed to be negligible as a �rst approximation.

The governing equations, boundary conditions and constitutive relations are summarized in

Appendices A-D. The various parameters and constants are summarized in Tables 7.1, and

7.2.



7.3.1 Governing equations

The governing equations for the battery are provided in chapter 3. The governing equation for the

PCM alone is speci�ed here. The model for the PCM comprises only the conservation of energy

[49, 134]:

�@H

@t+r � q = 0 (PCM),

where the heat �ux is given by

q = �krT:

In the above equations, � is the density of the PCM, Cp is the speci�c heat capacity of the PCM,

q is the conductive heat �ux, H is the enthalpy of the PCM, and k is the thermal conductivity of

the PCM.


At the interface I between the electrode/electrolyte or the separator/electrolyte (see Fig. 7.1d and

7.1e for placement of roman numerals), continuity is speci�ed for the energy �ux as well as the

ionic �ux of lithium ions (ionic current), whereas insulation is speci�ed for the solid-phase current:

n � is = 0, n � iljI+ = n � iljI� ; n �NljI+ = n �NljI� (I), (7.1)

The current density is prescribed at the positive current collector in the inner end of the spiral:

n � is = �iapp (II): (7.2)

At the current collector/electrode interfaces, continuity of energy �ux and solid-phase current is

speci�ed; insulation is speci�ed for the ionic �ux and current:

n � isjIII+ = n � isjIII� ; n � qjIII+ = n � qjIII� ;n � il = n �Nl = 0 (III). (7.3)

At the electrode/separator interfaces, continuity of energy �ux and ionic �ux as well as ionic


current is de�ned and since there is no �ow of electrons across the interface, insulation for solid

phase current is de�ned.

n � is = 0; n � iljIV+ = n � iljIV� ; n � qjIV+ = n � qjIV� n �NljIV+ = n �NljIV� (IV). (7.4)

The negative current collector at the outer end of the spiral is grounded:

�s = 0 (V). (7.5)

At the electrolyte/can interface, there is continuity of energy �ux and no �ow of ions:

n � qjVI+ = n � qjVI� ;n � il = n �Nl = 0 (VI). (7.6)

At the outer surface of the can, both convection and radiation is considered :

n � q = h(T � Ta) + ��(T 4 � T 4a ) (VII). (7.7)

When the battery is covered with PCM, there is continuity of energy �ux from the can to the

PCM and on the outer surface of the PCM (VIII), Newton�s law of cooling is speci�ed along with

radiative energy transfer as given by Eq. 7.7. In Eqs. 7.1 - 7.7, n denotes the unit normal vector

for a given boundary or interface, iapp is the applied current density, � is the emissivity, and � is the

Stefan-Boltzmann constant. The battery is discharged under galvanostatic conditions at various

current densities.

At t = 0,

cs = cavgs = c0s; cl = c

0l (7.8)

�s =

8>>>><>>>>:0 (ne)

�0s (pe)

(7.9)

�l = �0l (ne,pe,sp) (7.10)

T = T0 (7.11)



The generic constitutive relations are provided in chapter 3. Some of the system speci�c relations

are provided in this section.

The entropic change as a function of state of charge and the open-circuit potential for the

electrodes and the electrolyte conductivity as a function of concentration are as provided in chapter

4. The phase change for the material considered here takes place over a given melting range, which

is captured the following functional form, H(T ) [134]:

H(T ) =

8>>>>>>>>><>>>>>>>>>:

CpT for T < Ts (solid region),

CpT +T�TsTl�TsL for Ts � T � Tl (mushy region),

CpT + L for T > Tl (liquid region),

(7.12)

where L is the latent heat of the phase change and Ts and Tl are the start and end temperatures

of the phase change respectively.

The average temperature for the battery is de�ned as

hT i = 1

�R2b

ZZ

T (x; y)dA; (7.13)

where Rb is the radius of the battery and is the region of integration representing the entire

battery as shown in Fig. 7.1b. The temperature di¤erential is de�ned as

�T = T jr=0 �1

2�Rb

2�RbI0

Tds; (7.14)

where r is the radial coordinate and the second term on the right hand side represents the average

temperature at the outer surface of the can. The total heat generation and the heat generation in

7.4. Numerics 123

various layers is de�ned as

hQi = hZZ

QdA (7.15)

hQii = hZZ

QidA; i = ne, pe, sp, el, cc (7.16)

where h is the height of the battery considering only the jelly roll without the top cap assembly.

7.4 Numerics

The commercial �nite-element solver, COMSOL Multiphysics 4.1 [103], was employed to solve

the 2D model after importing the geometry from AutoCAD 2011 [135], in which the 2D spiral-

wound battery with/without PCM was created with the helix command by setting the height of

the helix to zero (z-direction). The wounds were drawn one after the other starting from the centre

(x = 0; y = 0) in order to avoid interference of the various layers that occurred when all the wounds

were drawn in a single step. The imported geometry appeared as a curve in COMSOL, whence

it had to be coerced to a solid, followed by splitting the formed solid into separate subdomains

representing the various functional layers. In addition, the narrowing gap of liquid electrolyte

between the jelly roll and the outer can, as illustrated in Fig. 7.1f, was dealt with by treating the

electrolyte present in the vicinity of the region where the jelly roll touches the outer can as part of

the latter.

Linear elements were implemented for all dependent variables: �s; �l; cl; csurfs ; T and cavgs ; the

direct solver UMFPACK was chosen as linear solver with a relative convergence tolerance of 10�3;

and solutions for all models were tested for mesh independence.

Charge and discharge currents, iapp, were applied at the respective boundaries with a smoothed

Heaviside function.

Furthermore, in order to avoid numerical instabilities due to negative values in the current

density when the local state of charge (SOC) approaches unity (i.e., fully charged), a relational

operator was introduced for the exchange current density and local SOC, to ensure that the former

7.4. Numerics 124


Parameter Unit cc (-) ne sp pe cc (+) Reference

c0l mol m�3 - 2� 103 - [1]

cp J kg�1 K�1 3:8� 102 7:0� 102 7:0� 102 7:0� 102 8:7� 102 [2]

c0s mol m�3 - 1:5� 104 - 3:9� 103 - [1]

cmaxs mol m�3 - 2:6� 104 - 2:3� 104 - [1]

Dl m2 s�1 - 7:5� 10�11 - [1]

Ds m2 s�1 - 3:9� 10�14 - 1:0� 10�13 - [1]

Ea;Dl kJ mol�1 - 10 - [2]

Ea;Ds kJ mol�1 - 4 - 20 - [2]

Ea;�l kJ mol�1 - 20 - [2]

h m 60� 10�3 -

h W m�2 K�1 5 -

iapp (1 C) A m�2 4:5� 105 -

k W m�1K�1 3:8� 102 0:05� 102 0:01� 102 0:05� 102 2:0� 102 [2]

Rb m 9� 10�3 -

R m - 12:5� 10�6 - 8:5� 10�6 - [1]

Tamb; Tref K 298:15 [2]

wi m 18� 10�6 88� 10�6 25� 10�6 80� 10�6 25� 10�6 [110]

�a; �c - - 0:5 - 0:5 - [1]

"p - - 0:14 - 0:19 - [1]

"l - - 0:36 0:72 0:44 - [1]

"f - - 0:03 - 0:07 - [1]

�0i - - 0:56 - 0:17 - [1]

� kg m�3 9:0� 103 1:9� 103 1:2� 103 4:1� 103 2:7� 103 [1], [2]

�s S m�1 6:0� 107 1� 102 - 3:8 3:8� 107 [2]

7.5. Results and Discussion 125

Table 7.2: PCM, electrolyte, and outer can properties

Parameter Unit PCM [51] Electrolyte[41] Outer can

cp J (kg K)�1 1980 2055 475

k W (mK)�1 16:6 0:6 44:5

� kg m�3 866 1129:95 7850

L J kg�1 181� 103 - -

Ts K 325 - -

Tl K 328 - -

� - 0:9 (graphite alone) - 0:8

does not become negative or exactly zero:

i0 = Fk0

rclc

surfs

h�cmaxs � csurfs

��cmaxs > csurfs

�+ c�

i; (7.17)

�pe = min

csurfs

cmaxs

; 1

!; (7.18)

here, i0 is the exchange current density, F is the Faraday�s constant, k0 is the reaction rate constant,

cl is the concentration of Li-ions in the electrolyte, csurfs is the concentration of Li-ions on the surface

of the active material, cmaxs is the maximum concentration of Li-ions in the active material, �pe

is the local SOC of the positive electrode, and c� = 10�12 mol m�3 is a negligible concentration

chosen such that it does not a¤ect the solution at leading order; i.e., c� � cmaxs � csurfs .

The relational operators are introduced only for the positive electrode since the latter determines

the capacity of the considered battery; more on this later in the discussion.

All computations were carried out on a workstation with two quad-core processors (3.2 GHz,

with a total of eight processor cores) and a total of 64 GB random access memory (RAM).

7.5 Results and Discussion

The behavior during discharge of a spiral-wound lithium-ion battery without a passive thermal-

management system is �rst discussed, such that only natural convection and radiation provide heat

transfer with the surrounding. (The focus is on the discharge process, since charging exhibits a

similar behavior.) Thereafter, the behavior of the cell with a PCM layer surrounding the battery


0 500 1000 1500 2000 2500 3000 35002.6

2.8

3

3.2

3.4

3.6

3.8

4

4.2

Time / s

Ece

ll / V

1C

2C

5C

(a) Cell voltage vs time with (symbols) andwithout PCM (lines).

0 500 1000 1500 2000 2500 3000 35000

2

4

6

8

10

12

Time / s

Pow

er /

W

1C

2C

5C

(b) Power vs time with (symbols) and withoutPCM (lines).

Figure 7.2: Comparison of cell performance with (symbols) and without PCM (lines) forgalvanostatic discharge at various C-rates.

is discussed.

7.5.1 Discharge and power curves

The global behavior in terms of cell voltage and power during discharge for three di¤erent C-rates

of the battery, is shown in Fig. 7.2a-b. Overall, the battery discharge occurs, as expected, �rst

gradually with decreasing cell voltage and power with respect to time, followed by a sharp drop

towards the end of discharge. From the numerical point-of-view, the latter is re�ected in a slowing

convergence rate and requires the introduction of relational operators as introduced in Numerics

earlier. Further, the discharge curves are more or less the same for a battery with and without

PCM; more on this later in the discussion.

7.5.2 Edge and geometry e¤ects

In a spiral-wound Li-ion cell, the inner and outer ends of the spiral-wound jelly roll � see Fig.

7.1d and e respectively �are exposed to the liquid electrolyte both in the core as well as in the

gap between the jelly roll and the outer can. Furthermore, the innermost layer comprising the

positive electrode is only in contact with the electrolyte in the core and a current collector in


the �rst wound; similar for the negative layer in the outermost wound, which is only in contact

with a current collector and the separator layer. These edge and geometrical aspects from the

spiral-wound geometry can thus be expected to give rise to localized e¤ects and deviations from

the average behavior. This is indeed the case, as can be inferred from Fig. 7.3, for the SOC,

lithium-ion concentration in the electrolyte and liquid-phase potential during discharge at 1 C-rate

halfway through, t = 30 min, and at the end, t = 60 min, for a lithium-ion battery without PCM.

Here, two main features are apparent: First, is the uniformity in the interior of the cell and second,

the expected deviations that occur at the inner and outer edges of the spiral wounds. For example,

the SOC of the positive electrode at the inner edge of the spiral half-way through the discharge is

around 0:2 (Fig. 7.3a), which is close to the initial SOC of 0:17; at the outer edge of the spiral and

in the interior, the SOC is around 0:9 and 0:55 respectively. There is thus a lack of lithium-ions

in the �rst wound, and an excess in the outer. At the end of the discharge, the SOC approaches

1 for the positive electrode except at the inner edge of the spiral where it is around 0:3 ( Fig.

7.3b). The same phenomena are observed for the negative electrode (albeit reversed) as well as for

the lithium-ion concentration and liquid-phase potential. These �ndings, in turn, suggest that the

positive electrode at the inner edge of the spiral and the negative electrode at the outer edge of the

spiral are not completely utilized �hence, these regions can be left uncoated while manufacturing

a battery.

7.5.3 Heat generation and thermal behavior

As alluded to in the introduction, thermal management is key in ensuring not only a safe operation

but also to improve the cycle-life of a lithium-ion battery. In this case, the battery without a

PCM is only cooled through natural convection and radiation, which gives rise to a signi�cant

temperature increase inside the cell during discharge, as shown in Fig. 7.4a. The temperature

increases by around 8 K, 20 K, and 50 K above the ambient temperature for discharge rates of 1 C,

2 C, and 5 C, respectively, with a maximum temperature of 350 K reached at the end of discharge

at the highest C-rate considered here, which indicates the need for a thermal management system.

The radiation from the cell accounts for around 60 % of the total heat transfer to the ambient as

compared to 40 % by natural convection. The temperature di¤erential, �T , between the core and

the outer can of the battery (Fig. 7.4b) is negligible compared to the overall, average temperature


Figure 7.3: Local distribution of the following dependent variables at t = 1800 s and t = 3600 sduring discharge at 1 C-rate: SOC of positive electrode (a, b), SOC of negative electrode (c, d),lithium ion concentration in the electrolyte (e, f), and liquid phase potential (g, h).


0 500 1000 1500 2000 2500 3000 3500295

300

305

310

315

320

325

330

335

340

345

350

Time / s

Ave

rage

tem

pera

ture

/ K

5C

2C

1C

(a)

0 500 1000 1500 2000 2500 3000 3500

0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time / s

∆ T /

K

5C

1C2C

(b)

Figure 7.4: Time history of (a) the average battery temperature and (b) the temperaturedi¤erence between the battery core and the surface during discharge at various rates with(dotted) and without PCM (continuous).

inside the battery; i.e. �T=Tavg � 10�3 � 1; which indicates that the energy transport is mainly

limited by the energy exchange with the ambient.

The rise in temperature originates from the electrochemical reaction and ohmic heating as

charge is passed through the battery. The total heat generation rate and the contribution from

each of the heat sources (see Eq. 3.28) during discharge at 1 C-rate and 5 C-rate are shown in

Fig. 7.5a and b respectively. At a 1 C-rate, the reversible heat generation is higher than the other

sources, amounting for nearly 50 % of the total heat generation, followed by ohmic and irreversible

heating. The reason can be traced back to the functional form of @Uref; i=@T given by Eqs. 5.13

and 5.14, which determines the behavior of the reversible heating with time: At a 1 C-rate, the

reversible heat is negative initially and then it changes sign as the discharge proceeds, whereas

the other heat sources are always positive. The ohmic heat generation starts rising during the

initial period of discharge due to the currents that are passed through the battery and then almost

becomes a constant for the rest of the discharge.

At a 5 C-rate, however, the ohmic heat generation is the highest contributor, amounting for

nearly 50 % of the total heat generation, followed by the irreversible and the reversible heat sources.

The ohmic heat generation becomes increasingly larger due to the increasing current �ow through


0 500 1000 1500 2000 2500 3000 35000.1

0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Time / s

Tot

al a

nd p

arti

al h

eat g

ener

atio

n / W

Total heat generation

Reversible

Ohmic

Irreversible

(a)

0 100 200 300 400 500 6000

0.5

1

1.5

2

2.5

3

3.5

Time / s

Tot

al a

nd p

arti

al h

eat g

ener

atio

n / W Total heat generation

Ohmic

Reversible

Irreversible

(b)

0 500 1000 1500 2000 2500 3000 35000.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Time / s

Tot

al a

nd p

artia

l hea

t gen

erat

ion

/ W

el

ccsp

pe

ne

(c)

0 100 200 300 400 500 600

0

0.5

1

1.5

2

Time / s

Tot

al a

nd p

arti

al h

eat g

ener

atio

n / W

ne

pe

cc

elsp

(d)

Figure 7.5: Time history of heat generation by various sources (a,b), total heat generation andheat generation in various layers (c,d) during discharge at 1 C and 5 C-rates with (dotted) andwithout PCM (continuous).


the battery. However, a drop in the ohmic heat generation is observed after the initial increase

unlike 1 C rate where it remains constant. The increase in battery temperature is mirrored by

a decrease in the electric resistance, whence ohmic heating decreases during the discharge; in

the herein derived model, the electric resistance of the battery is coupled with the temperature-

dependent ionic conductivity and mass di¤usivity in the solid phase. In contrast, the irreversible

heating is almost constant with time with a slight increase towards the end of discharge.

A comparison of the total heat generation (Fig. 7.5a-b) with the power delivered by the battery

(Fig. 7.2b) suggests that nearly 10 % of the battery power is lost as heat at a 1 C-discharge rate;

in comparison, 25 % is lost through heat for the 5 C-rate.

Besides noting the relative magnitude of the various heat sources during discharge, it is also of

interest to see which layer contributes the most to the heat generation (Fig. 7.5c-d): The negative

electrode generates most of the heat, amounting to nearly 60 % of the total heat both at 1 C and

5 C-rates, followed by the positive and then by the remaining layers. The reason for this behavior

can be found in the ohmic and reversible heating, which are higher in the negative electrode than

in the positive electrode due to lower ionic conductivity and the thicker, less porous nature of the

negative electrode �the former is mainly dependent on the material properties and the latter are

design-adjustable . The heat generation in the current collector, electrolyte and separator, on the

other hand, only comprise ohmic heating, whence they remain almost constant throughout the

galvanostatic discharge.

Finally, the local temperature distribution is addressed, as illustrated in Fig. 7.6, half-way and

at the end of discharge for a 1 C-rate. Compared to the other dependent variables discussed earlier,

the temperature does not exhibit any edge e¤ects; instead, the temperature distribution is near-to

axially symmetric (z-axis).

7.5.4 Passive thermal management

Thus far, it has been identi�ed that the limiting heat removal from the cell by natural convection

and radiation is responsible for the overall temperature increase and that the temperature distri-

bution inside the cell is axially symmetric with an overall temperature gradient that is negligible

compared to the average temperature in the battery. One approach to manage the thermal envelope

could be in the form of forced or mixed external convection with air/liquid through active thermal


Figure 7.6: Local distribution of temperature at t = 320 s and t = 640 s: without PCM (a, b),with PCM (c, d) during discharge at 5 C-rate.


management; another approach, and the one pursued here, is through passive thermal management

with a PCM as an additional layer surrounding the battery cell. In essence, the disadvantage of

doing so is that the additional layer e¤ectively adds additional resistance to heat removal from the

battery; the advantage, however, is that doing so increases the thermal capacitance.

Returning to Fig. 7.2, the discharge curves and power characteristics remain una¤ected at

leading order when a PCM layer with thickness of 1 mm is added to the cell �which, at �rst sight,

would suggest that the passive thermal management is ine¤ective and unnecessary. In contrast,

however, the average temperature (Fig. 7.4a) is lowered as compared to the same cell without the

PCM, whereas the average temperature gradient inside the cell (Fig. 7.4b) is slightly higher. The

former can be explained by the increase in the thermal capacitance of the system (battery+PCM)

and the latter by the increased resistance to energy transfer out of the system. Further, the PCM

has not yet reached its melting temperature under the discharge rates of 1 C and 2 C, whence the

advantage of cooling through phase-change is not realized in these two cases. Under a discharge

rate of 5 C, however, the heat generated by the battery increases its temperature to the melting

range of the PCM, such that the battery is around 18 K cooler at the end of discharge as compared

to without PCM �a substantial decrease. Furthermore, at a 5 C-rate, the temperature di¤erential

with PCM reaches a maximum of 1:2 K and then decreases rapidly owing to the drop in heat

generation from the battery and the loss of heat to the ambient through convection and radiation;

it increases again as heat generation starts to rise.

The layer with PCM also a¤ects the heat generated by the battery at the 5 C-rate whereas at

a 1 C-rate, there is no noticeable di¤erence in heat generation, as shown in Fig. 7.5. Overall, the

total heat generation increases by nearly 7 % with PCM as compared to without at a 5 C-rate. This

corresponds to the fact that the presence of PCM keeps the battery temperature lower as compared

to the battery without PCM, which in turn lowers the ionic conductivity and mass di¤usivity, thus

increasing the resistance of the battery and �nally leading to an increase in ohmic heating.

The PCM also maintains the temperature uniformity within the cell unlike forced-convection

cooling [49, 51], for which appreciable temperature di¤erences between the core and the outer

surface of the battery can be established. The temperature distribution inside the battery with the

PCM is shown in Figs. 7.6c-d, in which the temperature rise from t = 320 s to t = 640 s is 16 K

without PCM and 4 K with PCM at a 5 C-rate. The maximum temperature di¤erence between the


core of the battery and the outer can is 0:8 K without PCM and 1 K with PCM and this increase

in gradient is due to the additional resistance for heat transfer to the ambient as mentioned above.

The state of the PCM can readily be identi�ed as solid, liquid or mushy based on the temperature

of the system. When it is in the mushy region, another parameter is needed to exactly de�ne its

state: the liquid fraction. For 1 C and 2 C-rates, since the PCM has not yet reached its melting

temperature range, the liquid fraction is zero whereas at a 5 C rate, the wax present inside the

graphite matrix starts melting, but does not reach a completely liquid state at the end of discharge

as the liquid fraction is 0:7 (Fig. 7.7), indicating that it lies in the two phase or the mushy region.

The volume of the PCM in the layer surrounding the battery should thus be chosen so that it is not

completely liquid at the end of discharge to ensure cooling throughout the discharge. The drop at

around 360 s in the liquid fraction during discharge mirrors the drop in the heat generation inside

the battery, as shown earlier in Fig. 7.5b.

0 100 200 300 400 500 6000.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time / s

Liq

uid

frac

tion

Figure 7.7: Liquid fraction of PCM vs time during discharge at 5 C-rate.

With this model, one can, e.g., optimize the thickness of the PCM whilst accounting for the

operating conditions of the battery.

Although the PCM minimizes the thermal gradient inside the battery pack, this concept has

some disadvantages. The volume and weight of the battery pack increases due to the PCM. Com-

plete melting of PCM during repeated charge/discharge cycles increases the thermal resistance due

to its low thermal conductivity. [15]

7.6. Conclusions 135

7.6 Conclusions

A two-dimensional coupled thermal-electrochemical model for a commercially-available spiral-wound

Li-ion battery has been presented and analyzed for discharge in terms of geometry and edge e¤ects

as well as in terms of passive thermal management with a PCM. The reduction in dimensionality

was justi�ed through scaling arguments and negligible heat �ux in the axial direction of the battery.

In summary, the active material is depleted to a larger extent at the outer end of the spiral for the

positive electrode and at the inner end of the spiral for the negative electrode than at the remaining

parts, where it is utilized uniformly. Further, reversible heat generation is the highest contributor

at lower discharge rates around 1 C-rate, whereas ohmic heating is the highest contributor among

the heat generation sources for higher discharge rates. The �ndings suggest that the battery design

can be optimized in order to, for example, reduce the ohmic heating by improving the electrolyte

conductivity and the design-adjustable like the thickness and porosity of the electrodes.

A passive thermal management system has been evaluated by wrapping a PCM around the

battery, which reduces the average temperature of the battery for higher discharge rates: here,

around 5 C-rate.

The model can be extended to include the e¤ect of current collecting tabs, other active materials,

and the axial dimension if the heat �ux is not negligible by extruding the geometry. Furthermore,

thermal runaway can be simulated by including the additional reaction heats and temperature

dependence of the reaction rates. Finally, the numerical procedure to implement the geometry and

mathematical formulation could be automated [136] to allow for wide-ranging parameter studies as

well as multi-objective optimization of a Li-ion battery cell with passive thermal management.


Nomenclature

ai constants in electrolyte conductivity expression

A speci�c interfacial area per unit volume, m�1

Ci electrode capacity, Ah m�2

Cth theoretical capacity of electrode material, mAh g�1


c0l initial electrolyte concentration, mol m�3

Cp e¤ective speci�c heat capacity, J (kg K)�1

c0s initial concentration of lithium in the active material, mol m�3

cs concentration of lithium in the active material, mol m�3

cavgs average concentration of lithium in the active material, mol m�3

cmaxs maximum concentration of lithium in the active material, mol m�3

csurfs surface concentration of lithium in the active material in the electrodes, mol m�3


Ds di¤usion coe¢ cient of lithium in the active material, m2 s�1

Ea;� activation energy for a variable �, kJ mol�1



h heat transfer coe¢ cient, Wm�2K�1








k e¤ective thermal conductivity, W (mK)�1

k0 reaction rate constant

L latent heat of PCM, J kg�1

L length of the spiral, m




n normal vector

ni; pi constants in the entropic heat term for negative and positive electrodes

Ni;Pi constants in the open circuit potential for negative and positive electrodes



R gas constant, J (mol K)�1

Rb radius of battery (type 18650), m

R radius of active material in the electrodes, m

t time, s


T temperature, K

Ts start temperature of phase change, K


Tl end temperature of phase change, K

Uref; i open circuit potential of the electrode i, V

x number of moles of Li in C6

y number of moles of Li in Mn2O4

Greek



"l volume fraction of electrolyte

"f volume fraction of conductive �ller additive

"p volume fraction of polymer phase

� overpotential

�ne state of charge of negative electrode

�pe state of charge of positive electrode


� emissivity of the outer can material

� e¤ective density, kg m�3

� Stefan-Boltzmann constant, W m�2 K�4


�s electronic conductivity of solid matrix, S.m�1



�0l initial liquid phase potential, V


�0s initial solid phase potential, V

Bruggemann constant (= 1:5)

�(T ) placeholder for a temperature dependent property

Chapter 8

Conclusions and Outlook

8.1 Summary and Conclusions

This thesis dealt with mathematical modeling of transport phenomena in electrochemical energy

systems, especially Li-ion batteries and electrochemical capacitors. The similarities in the physical

phenomena taking place in batteries and electrochemical capacitors are highlighted and utilized

in deriving the mathematical models. The mathematical model comprised the governing transient

conservation of charge and species in the solid and the electrolyte phase. For Li-ion batteries,

conservation of energy is also included in the mathematical model. Models were analyzed based

on scaling arguments justi�ed by numerical results and model reductions were achieved that led

to reduction in computational cost without compromising the accuracy of the predicted results.

The process of generating the geometry from single cell to stacks/packs and implementing the

mathematical model is automated to achieve a signi�cant reduction in setting up the model for

numerical simulations as well as eliminate the human errors during set up. Lastly, the practical use

of the model is demonstrated by designing a passive thermal management system for a spiral-wound

cylindrical Li-ion battery to keep the battery within the operating temperature ranges under various

operating conditions. Overall, four computational studies and investigations have been carried out

in this thesis.

In the �rst study described in chapter 4 of the thesis, a transient, isothermal model comprising

conservation of charges and species at the macro- and micro-scale that also accounts for double-

layer charging and a Faradaic reaction for electrodes made of RuO2 that exhibit pseudocapacitance

140

8.1. Summary and Conclusions 141

is studied. The model is calibrated and validated with experimental data. Then, scaling arguments

were employed to derive reduced models. The �delity of the reduced models were established

by comparison with the full model: overall, good agreement was obtained. Besides providing

rigorous arguments for model reductions, the secured scales captured the key characteristics and

the nondimensional numbers identi�ed the conditions which have to be satis�ed for reductions in

dimensionality at the micro- as well as macro-scopic scale. The reduction in the computational

cost was highlighted through the reduction in the DoF, solution time and the memory requirement

for three cases: a single cell, a 10-cell and a 100-cell stack. The reduced models can be employed

as a basic building block for a stack model through an automated procedure that was described in

chapter 6.

A Li-ion battery model was also analyzed on the same lines in the second study as that carried

out for the model of an electrochemical capacitor in the �rst study. This model also included the

conservation of energy. In this work, the model for a spiral wound cylindrical 18650 Li-ion battery

was presented and analyzed with scaling arguments. The nondimensional numbers provided criteria

for reduction in dimensionality and the scales provided the estimates of the various variables in the

model. The estimated scales were veri�ed by comparison with the numerical results: overall, good

agreement was obtained. The thermal analysis provided an estimate of the various sources of heat

generation in the battery under various discharge rates. Further, it established the dependence of

the heat generation on the various design parameters and physical properties of the materials. The

e¤ects of the various design adjustable parameters, materials and operating conditions could be

estimated using the secured scales with reasonable accuracy compared to the detailed numerical

simulations.

The �rst and second studies dealt with single cells but the third study was carried out for

a bipolar battery stack. In order to reduce the complexity and computational cost of battery

systems comprising more than one cell, mathematical model reformulation was carried out at the

microscale to reduce the number of dependent variables. The reformulated model was veri�ed

through comparison of its predicted results with that of the full model. The advantage of the

reformulation was highlighted through the reduction of the computational cost in terms of DoF,

computational time and memory requirement for simulating a battery module; e.g., solving a 50-

cell battery module took around 1:5 � 103 s for the reformulated model as compared to 2:6 � 103

8.1. Summary and Conclusions 142

s for the original model �a reduction in time near to half. As the number of cells in the module

increase to hundreds or even thousands, even larger gains were expected. Much more bene�ts were

achieved by automating model: a reduction in computational time of several orders of magnitude

and complete removal of human errors after veri�cation, as the entire pre-processing � drawing

of geometry, meshing, and introduction of governing equations, boundary and initial conditions

as well as constitutive relations at multiple length scales �for the numerical implementation was

automated; and an easy-to-use automated numerical framework, which allows for cost-e¤ective

multi-objective optimization of key features, such as the battery design, operating conditions, and

management strategies.

The �nal study illustrated the application of the mathematical model derived and analyzed in

the previous chapters. A two-dimensional coupled thermal-electrochemical model for a commercially-

available spiral-wound Li-ion battery was presented and analyzed for discharge in terms of geometry

and edge e¤ects as well as in terms of passive thermal management with a PCM. The reduction

in dimensionality was justi�ed through scaling arguments and negligible heat �ux in the axial di-

rection of the battery. The investigation showed that the active material was depleted to a larger

extent at the outer end of the spiral for the positive electrode and at the inner end of the spiral for

the negative electrode than at the remaining parts, where it was utilized uniformly. The various

heating sources in the battery were investigated and it was found that the reversible heating was

the source of heat generation at lower discharge rates (until 1 C-rate). A passive thermal manage-

ment system was evaluated by wrapping a PCM around the battery, which reduced the average

temperature of the battery for higher discharge rates: here, around 5 C-rate.

As mentioned in the introduction and the other Chapters, the model reduction obtained through

scaling analysis and model reformulation was the fundamental contribution from this thesis. In

addition, the development of automated model generation tool formed a part of this thesis that will

help in the design, analysis and optimization of the battery packs and electrochemical capacitor

stacks.

8.2. Recommendations for future work 143

8.2 Recommendations for future work

This thesis mainly dealt with the analysis of the mathematical models with a view to gain insights

about the physical processes, and assess the importance of the physical properties and operating

conditions on the performance of the electrochemical energy storage systems. The capacity fade of

Li-ion batteries at elevated temperatures is one of the important problems to be addressed before

it �nds extensive use in fully electric vehicles. The capacity fade occurs because of side reactions

that degrade the active material. The thermal e¤ects can be added in the cycle life modeling of

Li-ion batteries along with these side reactions to predict the cycle life of the battery with improved

accuracy. The reduced models could be employed for this purpose.

Newer materials can be investigated theoretically with these models to achieve improved en-

ergy and power densities for batteries and electrochemical capacitors. Practically the fabricated

electrodes will not have a uniform particle size and distribution. The assumption of uniform active

material particle size in the electrodes for deriving the model can be relaxed and the e¤ect of the

various particle size distributions can be studied and analyzed for optimizing the cell design. The

volume changes in the electrode can also be accounted for in the model and its e¤ects on the per-

formance of the cell can be analyzed. The model for an electrochemical capacitor can be extended

to account for the equation of change of energy and heat generation. The heat generation due to

the double layer formation can be investigated and added to the other heat generation sources like

ohmic heating and heating from faradaic reaction. Scaling arguments can be employed to identify

key scales and nondimensional numbers for the thermal aspects of an electrochemical capacitor.

To exploit the advantages of batteries that have higher energy density and capacitors that

have higher power density, hybrid systems are considered as e¤ective power sources. These type

of hybrid systems can be analyzed by combining the models for an electrochemical capacitor and

Li-ion battery. The reduced models developed will be quite useful for evaluating the performance

of these systems and the automation procedure can be extended to implement various design of

such systems.

Also, tools were developed that can be employed for multi-objective optimization, statistical

modeling, perturbation studies, thermal runaway studies of an entire battery module and wide-

ranging parameter studies for various applications, such as battery and thermal management, de-

8.2. Recommendations for future work 144

sign, and overall optimization. As an example for statistical modeling, the e¤ect of the distribution

of the various sizes of the active material in the electrodes can be analyzed using the automated

procedure by assuming a particle size distribution pro�le. Various driving cycles for an electric

vehicle or a hybrid electric vehicle can be studied using the automated model for analyzing the

thermal characteristics of battery packs under various operating conditions. The automation pro-

cedure can be extended to include other types of cell designs, such as spiral-wound batteries, as

well as additional physical phenomena, such as �uid dynamics for cooling purposes.

References

[1] Doyle, M. and Newman, J., �Comparison of Modeling Predictions with Experimental Data

from Plastic Lithium Ion Cells,�J. Electrochem. Soc., Vol. 143, No. 6, 1996, pp. 1890�1903.

[2] Srinivasan, V. and Wang, C. Y., �Analysis of Electrochemical and Thermal Behavior of Li-ion

cells,�J. Electrochem. Soc., Vol. 150, No. 1, 2003, pp. A98�A106.

[3] Botte, G. G., Johnson, B. A., and White, R. E., �In�uence of Some Design Variables on

the Thermal Behavior of a Lithium Ion Cell,� J. Electrochem. Soc., Vol. 146, No. 3, 1999,

pp. 914�923.

[4] Subramanian, V. R., Boovaragavan, V., Ramadesigan, V., and Arabandi, M., �Mathematical

Model Reformulation for Lithium-Ion Battery Simulations: Galvanostatic Boundary Condi-

tions,�J. Electrochem. Soc., Vol. 156, No. 4, 2009, pp. A260�A271.

[5] Reynier, Y., Graetz, J., Swan-Wood, T., Rez, P., Yazami, R., and Fultz, B., �Entropy of Li

Intercalation in LixCoO2,�Phys. Rev. B , Vol. 70, No. 174304, 2004.

[6] Winter, M. and Brodd, R. J., �What Are Batteries, Fuel Cells, and Supercapacitors?�Chem.

Rev., Vol. 104, 2004, pp. 4245�4269.

[7] Zheng, J. P., Cygan, P. J., and Jow, T. R., �Hydrous Ruthenium Oxide as an Electrode Ma-

terial for Electrochemical Capacitors,�J. Electrochem. Soc., Vol. 142, No. 8, 1995, pp. 2699�

2703.

[8] David Linden, T. B. R., editor, Handbook of Batteries, McGraw-Hill, 3rd ed., 2002.

[9] http://dellbatteryprogram.com/ .

145

References 146

[10] http://www.cpsc.gov/cpscpub/prerel/prhtml06/06245.html .

[11] Kotz, R. and Carlen, M., �Principles and applications of electrochemical capacitors,�Elec-

trochim. Acta, Vol. 45, 2000, pp. 2483�2498.

[12] Bard, A. J. and Faulkner, L. L., editors, Electrochemical Methods Fundamentals and Appli-

cations, Wiley-Interscience, 2nd ed., 2001.

[13] Ultracapacitor, The Honda FCX, http://world.honda.com/FuelCell/FCX/ultracapacitor/ .

[14] van Schalkwijk, W. A. and Scrosati, B., editors, Advances in Lithium-Ion Batteries, Ch.12,

Mathematical Modeling of Lithium Batteries by K.E. Thomas, J. Newman, and R.M. Darling ,

Kluwer Academic Publishers, 2002.

[15] Bandhauer, T. M., Garimella, S., and Fuller, T. F., �A Critical Review of Thermal Issues in

Lithium-Ion Batteries,�J. Electrochem. Soc., Vol. 158, No. 3, 2011, pp. R1�R25.

[16] Newman, J., Thomas, K. E., Hafezi, H., and Wheeler, D. R., �Modeling of lithium-ion

batteries,�J. Power Sources, Vol. 119-121, 2003, pp. 838�843.

[17] Newman, J. and Tiedemann, W., �Porous-Electrode Theory with Battery Applications,�

AIChE J., Vol. 21, No. 1, 1975, pp. 25�41.

[18] Newman, J. and Alyea, K. E. T., editors, Electrochemical Systems, John Wiley Sons, Inc.,

3rd ed., 2004.

[19] Doyle, M., Fuller, T. F., and Newman, J., �Modeling of Galvanostatic Charge and Discharge

of the Lithium/Polymer/Insertion Cell,�J. Electrochem. Soc., Vol. 140, No. 6, 1993, pp. 1526�

1533.

[20] Thomas F. Fuller, Marc Doyle, J. N., �Simulation and Optimization of the Dual Lithium Ion

Insertion Cell,�J. Electrochem. Soc., Vol. 141, No. 1, 1994, pp. 1�10.

[21] Bernardi, D., Pawlikowski, E., and Newman, J., �A General Energy Balance for Battery

Systems,�J. Electrochem. Soc., Vol. 132, No. 1, 1985, pp. 5�12.

[22] Pals, C. R. and Newman, J., �Thermal Modeling of the Lithium/Polymer Battery I. Discharge

Behavior of a Single Cell,�J. Electrochem. Soc., Vol. 142, No. 10, 1995, pp. 3274�3281.

References 147

[23] Song, L. and Evans, J. W., �Electrochemical-Thermal Model of Lithium Polymer Batteries,�

J. Electrochem. Soc., Vol. 147, No. 6, 2000, pp. 2086�2095.

[24] Gu, W. B. and Wang, C. Y., �Thermal-Electrochemical Modeling of Battery Systems,� J.

Electrochem. Soc., Vol. 147, No. 8, 2000, pp. 2910�2922.

[25] Arora, P., Doyle, M., and White, R. E., �Mathematical Modeling of the Lithium Deposition

Overcharge Reaction in Lithium-Ion Batteries Using Carbon-Based Negative Electrodes,�J.

Electrochem. Soc., Vol. 146, No. 10, 1999, pp. 3543�3553.

[26] Gang Ning, Bala Haran, B. N. P., �Capacity fade study of lithium-ion batteries cycled at

high discharge rates,�J. Power Sources, Vol. 117, 2003, pp. 160�169.

[27] Ramadass, P., Haran, B., Gomadam, P. M., White, R., and Popov, B. N., �Development of

First Principles Capacity Fade Model for Li-Ion Cells,�J. Electrochem. Soc., Vol. 151, No. 2,

2004, pp. A196�A203.

[28] Ning, G., White, R. E., and Popov, B. N., �A generalized cycle life model of rechargeable

Li-ion batteries,�Electrochim. Acta, Vol. 51, 2006, pp. 2012�2022.

[29] Santhanagopalan, S., Guo, Q., Ramadass, P., and White, R. E., �Review of models for

predicting the cycling performance of lithium ion batteries,� J. Power Sources, Vol. 156,

2006, pp. 620�628.

[30] Mellgren, N., Brown, S., Vynnycky, M., , and Lindbergh, G., �Impedance as a Tool for

Investigating Aging in Lithium-Ion Porous Electrodes I. Physically Based Electrochemical

Model,�J. Electrochem. Soc., Vol. 155, No. 4, 2008, pp. A304�A319.

[31] Brown, S., Mellgren, N., Vynnycky, M., and Lindbergh, G., �Impedance as a Tool for In-

vestigating Aging in Lithium-Ion Porous Electrodes II. Positive Electrode Examination,�J.

Electrochem. Soc., Vol. 155, No. 4, 2008, pp. A320�A338.

[32] Safari, M., Morcrette, M., Teyssot, A., and Delacourt, C., �Multimodal Physics-Based Aging

Model for Life Prediction of Li-Ion Batteries,� J. Electrochem. Soc., Vol. 156, No. 3, 2009,

pp. A145�A153.

References 148

[33] Hallaj, S. A., Maleki, H., Hong, J., and Selman, J., �Thermal modeling and design consider-

ations of lithium-ion batteries,�J. Power Sources, Vol. 83, 1999, pp. 1�8.

[34] Baker, D. R. and Verbrugge, M. W., �Temperature and Current Distribution in Thin-Film

Batteries,�J. Electrochem. Soc., Vol. 146, No. 7, 1999, pp. 2413�2424.

[35] Arora, P., Doyle, M., Gozdz, A. S., White, R. E., and Newman, J., �Comparison between

computer simulations and experimental data for high-rate discharges of plastic lithium-ion

batteries,�J. Power Sources, Vol. 88, 2000, pp. 219�231.

[36] Wu, M.-S., Liu, K., Wang, Y.-Y., and Wan, C.-C., �Heat dissipation design for lithium-ion


[37] Gomadam, P. M., White, R. E., and Weidner, J. W., �Modeling Heat Conduction in Spiral

Geometries,�J. Electrochem. Soc., Vol. 150, No. 10, 2003, pp. A1339�A1345.

[38] Shin-Chih Chen, Chi-Chao Wan, Y.-Y. W., �Thermal analysis of lithium-ion batteries,� J.

Power Sources, Vol. 140, 2005, pp. 111�124.

[39] Smith, K. and Wang, C. Y., �Power and thermal characterization of a lithium-ion battery

pack for hybrid-electric vehicles,�J. Power Sources, Vol. 160, 2006, pp. 662�673.

[40] Onda, K., Ohshima, T., Nakayama, M., Fukuda, K., and Araki, T., �Thermal behavior

of small lithium-ion battery during rapid charge and discharge cycles,� J. Power Sources,

Vol. 158, 2006, pp. 535�542.

[41] Chen, S.-C., Wang, Y.-Y., and Wan, C.-C., �Thermal Analysis of Spirally Wound Lithium

Batteries,�J. Electrochem. Soc., Vol. 153, No. 4, 2006, pp. A637�A648.

[42] Kim, G.-H., Pesaran, A., and Spotnitz, R., �A three-dimensional thermal abuse model for

lithium-ion cells,�J. Power Sources, Vol. 170, 2007, pp. 476�489.

[43] Kumaresan, K., Sikha, G., and White, R. E., �Thermal Model for a Li-Ion Cell,� J. Elec-

trochem. Soc., Vol. 155, No. 2, 2008, pp. A164�A171.

References 149

[44] Williford, R. E., Viswanathan, V. V., and Zhang, J.-G., �E¤ects of entropy changes in anodes

and cathodes on the thermal behavior of lithium ion batteries,�J. Power Sources, Vol. 189,

2009, pp. 101�107.

[45] Fang, W., Kwon, O. J., and Wang, C.-Y., �Electrochemical½Uthermal modeling of automotive

Li-ion batteries and experimental validation using a three-electrode cell,�Int. J. Energy Res.,

Vol. 34, 2010, pp. 107�115.

[46] Gerver, R. E. and Meyers, J. P., �Three-Dimensional Modeling of Electrochemical Perfor-

mance and Heat Generation of Lithium-Ion Batteries in Tabbed Planar Con�gurations,�J.

Electrochem. Soc., Vol. 158, No. 7, 2011, pp. A835�A843.

[47] Zhang, X., �Thermal analysis of a cylindrical lithium-ion battery,�Electrochim. Acta, Vol. 56,

2011, pp. 1246�1255.

[48] Jeon, D. H. and Baek, S. M., �Thermal modeling of cylindrical lithium ion battery during

discharge cycle,�Energy Convers. Manage., Vol. 52, 2011, pp. 2973�2981.

[49] Hallaj, S. A. and Selman, J., �A Novel Thermal Management System for Electric Vehi-

cle Batteries Using Phase-Change Material,� J. Electrochem. Soc., Vol. 147, No. 9, 2000,

pp. 3231�3236.

[50] Khateeb, S. A., Farid, M. M., Selman, J. R., and Al-Hallaj, S., �Design and simulation of a

lithium-ion battery with a phase change material thermal management system for an electric

scooter,�J. Power Sources, Vol. 128, 2004, pp. 292�307.

[51] Sabbah, R., Kizilel, R., Selman, J., and Al-Hallaj, S., �Active (air-cooled) vs. passive (phase

change material) thermal management of high power lithium-ion packs: Limitation of tem-

perature rise and uniformity of temperature distribution,�J. Power Sources, Vol. 182, 2008,

pp. 630�638.

[52] Kizilel, R., Sabbah, R., Selman, J. R., and Al-Hallaj, S., �An alternative cooling system to

enhance the safety of Li-ion battery packs,�J. Power Sources, Vol. 194, 2009, pp. 1105�1112.

[53] Hatchard, T. D., MacNeil, D. D., Basu, A., and Dahn, J. R., �Thermal Model of Cylindrical

and Prismatic Lithium-Ion Cells,�J. Electrochem. Soc., Vol. 148, No. 7, 2001, pp. A755�A761.

References 150

[54] Wang, C. Y., Gu, W. B., and Liaw, B. Y., �Micro-Macroscopic Coupled Modeling of Batteries

and Fuel Cells,�J. Electrochem. Soc., Vol. 145, No. 10, 1998, pp. 3407�3417.

[55] Subramanian, V. R., Ritter, J. A., and White, R. E., �Approximate Solutions for Galvanos-

tatic Discharge of Spherical Particles I. Constant Di¤usion Coe¢ cient,�J. Electrochem. Soc.,

Vol. 148, No. 11, 2001, pp. A444�A449.

[56] Subramanian, V. R., Diwakar, V. D., and Tapriyal, D., �E¢ cient Macro-Micro Scale Coupled

Modeling of Batteries,�J. Electrochem. Soc., Vol. 152, No. 10, 2005, pp. A2002�A2008.

[57] Qi Zhang, R. E. W., �Comparison of approximate solution methods for the solid phase

di¤usion equation in a porous electrode model,�J. Power Sources, Vol. 165, 2007, pp. 880�

886.

[58] Boovaragavan, V., Harinipriya, S., and Subramanian, V. R., �Towards real-time (millisec-

onds) parameter estimation of lithium-ion batteries using reformulated physics-based mod-

els,�J. Power Sources, Vol. 183, 2008, pp. 361�365.

[59] Cai, L. and White, R. E., �E¢ cient Reformulation of Solid-Phase Di¤usion in Physics-Based

Lithium-Ion Battery Models,�J. Electrochem. Soc., Vol. 156, No. 3, 2009, pp. A154�A161.

[60] Ramadesigan, V., Boovaragavan, V., J. Carl Pirkle, J., and Subramanian, V. R., �E¢ cient

Reformulation of Solid-Phase Di¤usion in Physics-Based Lithium-Ion Battery Models,� J.

Electrochem. Soc., Vol. 157, No. 7, 2010, pp. B854�B860.

[61] Renganathan, S. and RalphE.White, �Semianalytical method of solution for solid phase dif-

fusion in lithium-ion battery electrodes: Variable di¤usion coe¢ cient,� J. Power Sources,

Vol. 1, 2011, pp. 442�448.

[62] Ramadesigan, V., Chen, K., Burns, N. A., Boovaragavan, V., Braatz, R. D., and Subra-

manian, V. R., �Parameter Estimation and Capacity Fade Analysis of Lithium-Ion Batteries

Using Reformulated Models,�J. Electrochem. Soc., Vol. 158, No. 9, 2011, pp. A1048�A1054.

[63] Northrop, P. W. C., Ramadesigan, V., De, S., and Subramanian, V. R., �Coordinate Trans-

formation, Orthogonal Collocation, Model Reformulation and Simulation of Electrochemical-

References 151

Thermal Behavior of Lithium-Ion Battery Stacks,� J. Electrochem. Soc., Vol. 158, No. 12,

2011, pp. A1461�A1477.

[64] Liaw, B. Y., Nagasubramanian, G., Jungst, R. G., and Doughty, D. H., �Modeling of lithium

ion cells°UA simple equivalent-circuit model approach,� J. Power Sources, Vol. 175, 2004,

pp. 835�839.

[65] Dubarry, M. and Liaw, B. Y., �Development of a universal modeling tool for rechargeable

lithium batteries,�J. Power Sources, Vol. 174, 2007, pp. 856�860.

[66] Moss, P. L., Au, G., Plichta, E. J., and Zheng, J. P., �An Electrical Circuit for Modeling

the Dynamic Response of Li-Ion Polymer Batteries,�J. Electrochem. Soc., Vol. 155, No. 12,

2008, pp. A986�A994.

[67] Dubarry, M., Vuillaume, N., and Liaw, B. Y., �From single cell model to battery pack simu-

lation for Li-ion batteries,�J. Power Sources, Vol. 186, 2009, pp. 500�507.

[68] Conway, B., �Transition from "Supercapacitor" to "Battery" Behavior in Electrochemical

Energy Storage,�J. Electrochem. Soc., Vol. 138, No. 6, 1991, pp. 1539�1548.

[69] Conway, B., Birss, V., and Wojtowicz, J., �The role and utilization of pseudocapacitance for

energy storage by supercapacitors,�J. Power Sources, Vol. 66, 1997, pp. 1�14.

[70] Conway, B. E., editor, Electrochemical Capacitors: Scienti�c Fundamentals and Technological

Applications, Kluwer Academic/Plenum, 1999.

[71] Burke, A., �Ultracapacitors: why, how, and where is the technology,� J. Power Sources,

Vol. 91, 2000, pp. 37�50.

[72] Simon, P. and Gogotsi, Y., �Materials for electrochemical capacitors,� Nature Materials,

Vol. 7, 2008, pp. 845�854.

[73] Garche, J., editor, Encyclopedia of Electrochemical Power Sources, Elsevier, 2009.

[74] Posey, F. and Morozumi, T., �Theory of Potentiostatic and Galvanostatic Charging of the

Double Layer in Porous Electrodes,�J. Electrochem. Soc., 1966, pp. 176�184.

References 152

[75] Johnson, A. and Newman, J., �Desalting by Means of Porous Carbon Electrodes,�J. Elec-

trochem. Soc., Vol. 118, No. 3, 1971, pp. 510�517.

[76] Pillay, B. and Newman, J., �The In�uence of Side Reactions on the Performance of Electro-

chemical Double-Layer Capacitors,� J. Electrochem. Soc., Vol. 143, No. 6, 1996, pp. 1806�

1814.

[77] Srinivasan, V. and Weidner, J. W., �Mathematical Modeling of Electrochemical Capacitors,�


[78] Lin, C., Ritter, J. A., Popov, B. N., and White, R. E., �A Mathematical Model of an

Electrochemical Capacitor with Double-Layer and Faradaic Processes,�J. Electrochem. Soc.,

Vol. 146, No. 9, 1999, pp. 3168�3175.

[79] Lin, C., Popov, B. N., and Ploehn, H. J., �Modeling the E¤ects of Electrode Composition

and Pore Structure on the Performance of Electrochemical Capacitors,�J. Electrochem. Soc.,

Vol. 149, No. 2, 2002, pp. A167�A175.

[80] Kim, H. and N.Popov, B., �A Mathematical Model of Oxide/Carbon Composite Electrode

for Supercapacitors,�J. Electrochem. Soc., Vol. 150, No. 9, 2003, pp. A1153�A1160.

[81] Venkat R. Subramanian, Sheba Devan, R. E. W., �An approximate solution for a pseudoca-

pacitor,�J. Power Sources, Vol. 135, 2004, pp. 361�367.

[82] Verbrugge, M. W. and Liu, P., �Microstructural Analysis and Mathematical Modeling of

Electric Double-Layer Supercapacitors,�J. Electrochem. Soc., Vol. 152, No. 5, 2005, pp. D79�

D87.

[83] Schi¤er, J., Linzen, D., and Sauer, D. U., �Heat generation in double layer capacitors,� J.


[84] Ph. Guillemet, Y. Scudeller, T. B., �Multi-level reduced-order thermal modeling of electro-

chemical capacitors,�J. Power Sources, Vol. 157, 2006, pp. 630�640.

[85] Farahmandi, C., Proceedings of the Symposium on Electrochemical Capacitors II, F. M. Del-

nick, D. Ingersoll, X. Andrieu, and K. Naoi, Editors, The Electrochemical Society Proceed-

ings Series, Pennington, NJ, 1997.

References 153

[86] Fritts, D. H., �An Analysis of Electrochemical Capacitors,� J. Electrochem. Soc., Vol. 144,

No. 6, 1997, pp. 2233�2341.

[87] Dunn, D. and Newman, J., �Predictions of Speci�c Energies and Speci�c Powers of Double-

Layer Capacitors Using a Simpli�ed Model,� J. Electrochem. Soc., Vol. 147, No. 3, 2000,

pp. 820�830.

[88] R.A. Dougal, L. Gao, S. L., �Ultracapacitor model with automatic order selection and capac-

ity scaling for dynamic system simulation,�J. Power Sources, Vol. 126, 2004, pp. 250�257.

[89] Moss, P., Zheng, J., Au, G., Cygan, P., and Plichta, E., �Transmission Line Model for

Describing Power Performance of Electrochemical Capacitors,�J. Electrochem. Soc., Vol. 154,

No. 11, 2007, pp. A1020�A1025.

[90] Sakka, M. A., Gualous, H., Mierlo, J. V., and Culcu, H., �Thermal modeling and heat

management of supercapacitor modules for vehicle applications,�J. Power Sources, Vol. 194,

2009, pp. 581�587.

[91] Gu, W. B. and Wang, C. Y., Lithium Batteries edited by S. Surampudi, R. A. Marsh, Z.

Ogumi, and J. Prakash, The Electrochemical Society Proceedings Series, Pennington, NJ,

2000.

[92] Pillay, B., Design of Electrochemical Capacitors for Energy Storage, PhD Dissertation, Uni-

versity of California, Berkeley, 1996.

[93] Sikha, G., White, R. E., and Popov, B. N., �A Mathematical Model for a Lithium-Ion

Battery/Electrochemical Capacitor Hybrid System,� J. Electrochem. Soc., Vol. 152, No. 8,

2005, pp. A1682�A1693.

[94] Zuleta, M., Electrochemical and Ion Transport Characterisation of a Nanoporous Carbon

Derived from SiC , Doctoral Thesis, KTH-Royal Institute of Technology, Sweden, 2005.

[95] Malmberg, H., Nanoscienti�c investigations of electrode materials for supercapacitors, Doc-

toral Thesis, KTH-Royal Institute of Technology, Sweden, 2007.

References 154

[96] Toupin, M., Brousse, T., and Belanger, D., �Charge Storage Mechanism of MnO2 Electrode

Used in Aqueous Electrochemical Capacitor,�Chemistry of Materials, Vol. 16, No. 16, 2004,

pp. 3184�3190.

[97] Zhiru Ma, Jim P. Zheng, R. F., �Solid state NMR investigation of hydrous ruthenium oxide,�

Chemical Physics Letters, Vol. 331, 2000, pp. 64�70.

[98] Hu, C.-C., Chen, W.-C., and Chang, K.-H., �How to Achieve Maximum Utilization of Hy-

drous Ruthenium Oxide for Supercapacitors,� J. Electrochem. Soc., Vol. 151, No. 2, 2004,

pp. A281�A290.

[99] Brumbach, M. T., Alam, T. M., Kotula, P. G., McKenzie, B. B., and Bunker, B. C., �Nanos-

tructured Ruthenium Oxide Electrodes via High-Temperature Molecular Templating for Use

in Electrochemical Capacitors,�ACS Applied Materials and Interfaces, Vol. 2, No. 3, 2010,

pp. 778�787.

[100] Zheng, J., Huang, J., and Jow, T., �The Limitations of Energy Density for Electrochemical

Capacitors,�J. Electrochem. Soc., Vol. 144, No. 6, 1997, pp. 2026�2031.

[101] Lobo, V. M. M., editor, Electrolyte Solutions: Literature Data on Thermodynamic and Trans-

port Properties, Coimbra, Portugal, 1984.

[102] Kim, H., Development of Novel Nanostructured Electrodes for Supercapacitors and PEM Fuel

Cells, PhD Dissertation, University of South Carolina, 2003.

[103] Comsol Multiphysics 3.5a, http://www.comsol.com.

[104] Ly, H., Birgersson, E., and Vynnycky, M., �Asymptotically Reduced Model for a Proton

Exchange Membrane Fuel Cell Stack: Automated Model Generation and Veri�cation,� J.


[105] McKeown, D. A., Hagans, P. L., Carette, L. P. L., Russell, A. E., Swider, K. E., and Roli-

son, D. R., �Structure of Hydrous Ruthenium Oxides: Implications for Charge Storage,�J.

Phy.Chem. B , Vol. 103, 1999, pp. 4825�4832.

[106] Cho, Y. I., Frank, H., and Halpert, G., �Computer Simulation of Thermal Modeling of

Primary Lithium Cells,�J. Power Sources, Vol. 21, 1987, pp. 183�194.

References 155

[107] Evans, T. and White, R., �A Thermal Analysis of a Spirally Wound Battery Using a Simple

Mathematical Model,�J. Electrochem. Soc., Vol. 136, No. 8, 1989, pp. 2145�2152.

[108] Harb, J. N. and LaFollette, R. M., �Mathematical Model of the Discharge Behavior of a

Spirally Wound Lead-Acid Cell,�J. Electrochem. Soc., Vol. 146, No. 3, 1999, pp. 809�818.

[109] Verbrugge, M. W. and Koch, B. J., �Electrochemistry of Intercalation Materials Charge-

Transfer Reaction and Intercalate Di¤usion in Porous Electrodes,� J. Electrochem. Soc.,

Vol. 146, No. 3, 1999, pp. 833�839.

[110] Johnson, B. A. and White, R. E., �Characterization of commercially available lithium-ion


[111] Doyle, M., Meyers, J. P., and Newman, J., �Computer Simulations of the Impedance Response

of Lithium Rechargeable Batteries,�J. Electrochem. Soc., Vol. 147, No. 1, 2000, pp. 99�110.

[112] Somasundaram, K., Birgersson, E., and Mujumdar, A. S., �Thermal-electrochemical model

for passive thermal management of a spiral-wound lithium-ion battery,� J. Power Sources,

Vol. 203, 2012, pp. 84�96.

[113] Bird, R. B. and Warrren E. Stewart, E. N. L., editors, Transport Phenomena Second Edition,

John Wiley Sons, Inc., 2002.

[114] Gene Berdichevsky, Kurt Kelty, J. S. and Toomre, E., The Tesla Roadster Battery System,

2006.

[115] Botte, G. G., Subramanian, V. R., and White, R. E., �Mathematical modeling of secondary

lithium batteries,�Electrochim. Acta, Vol. 45, 2000, pp. 2595�2609.

[116] Gomadam, P. M., Weidner, J. W., Dougal, R. A., and White, R. E., �Mathematical modeling

of lithium-ion and nickel battery systems,�J. Power Sources, Vol. 110, 2002, pp. 267�284.

[117] Smith, K. A., Rahn, C. D., and Wang, C.-Y., �Control oriented 1D electrochemical model of

lithium ion battery,�Energy Convers. Manage., Vol. 48, 2007, pp. 2565�2578.

[118] Wang, C.-W. and Sastry, A. M., �Mesoscale Modeling of a Li-Ion Polymer Cell,� J. Elec-

trochem. Soc., Vol. 154, No. 11, 2007, pp. A1035�A1047.

References 156

[119] Lee, D. H., Kim, U. S., Shin, C. B., Lee, B. H., Kim, B. W., and Kim, Y.-H., �Modelling

of the thermal behaviour of an ultracapacitor for a 42-V automotive electrical system,� J.


[120] Ramandi, M., Dincer, I., and Naterer, G., �Heat transfer and thermal management of elec-

tric vehicle batteries with phase change materials,�Heat and Mass Transfer , Vol. 47, 2011,

pp. 777�788.

[121] Marsh, R. A., Russell, P. G., and Reddy, T. B., �Bipolar lithium-ion battery development,�

J. Power Sources, Vol. 65, 1997, pp. 133�141.

[122] Berger, T., Dreher, J., Krausa, M., and Tubke, J., �Lithium accumulator for high-power

applications,�J. Power Sources, Vol. 136, 2004, pp. 383�385.

[123] Prentice, G., editor, Electrochemical Engineering Principles, Prentice Hall, 1991.

[124] Matlab R2008a, www.mathworks.com..

[125] Mills, A. and Al-Hallaj, S., �Simulation of passive thermal management system for lithium-ion

battery packs,�J. Power Sources, Vol. 141, 2005, pp. 307�315.

[126] Mills, A., Farid, M., Selman, J., and Al-Hallaj, S., �Thermal conductivity enhancement

of phase change materials using a graphite matrix,� Appl. Thermal Eng., Vol. 26, 2006,

pp. 1652�1661.

[127] Zalba, B., Marin, J. M., Cabeza, L. F., and Mehling, H., �Review on thermal energy storage

with phase change: materials, heat transfer analysis and applications,�Appl. Thermal Eng.,

Vol. 23, 2003, pp. 251�283.

[128] Kalu, E. and White, R., �Thermal Analysis of Spirally Wound Li/BCX and Li/SOCl2 Cells,�


[129] Py, X., Olives, R., and Mauran, S., �Para¢ n/porous-graphite-matrix composite as a high

and constant power thermal storage material,� Int. J. Heat and Mass Transfer , Vol. 44,

2001, pp. 2727�2737.

References 157

[130] Pollard, R. and Newman, J., �Mathematical Modeling of the Lithium-Aluminum, Iron Sul�de

Battery I. Galvanostatic Discharge Behavior,� J. Electrochem. Soc., Vol. 128, No. 3, 1979,

pp. 491�502.

[131] Bark, F. H. and Alavyoon, F., �Convection in electrochemical systems,� Appl. Sci. Res.,

Vol. 53, No. 1-2, 1994, pp. 11�34.

[132] Bark, F. H. and Alavyoon, F., �Free convection in an electrochemical system with nonlinear

reaction kinetics,�J.Fluid Mech., Vol. 290, 1995, pp. 1�28.

[133] Borg, K., Birgersson, K., and Bark, F., �E¤ects of non-linear kinetics on free convection in an

electrochemical cell with a porous separator,� J. Appl. Electrochem., Vol. 37, No. 11, 2007,

pp. 1287�1302.

[134] Ozisik, M. N., editor, Heat Conduction, Wiley-Interscience, 2nd ed., 1993.

[135] AutoCAD 2011 - STUDENT VERSION, http://students.autodesk.com/ .

[136] Ly, H., Birgersson, E., Vynnycky, M., and Sasmito, A., �Validated Reduction and Acceler-

ated Numerical Computation of a Model for the Proton Exchange Membrane Fuel Cell,�J.


Documents

Mathematical Modeling of Transport Phenomena in Karthik ... · Karthik Somasundaram, Erik Birgersson, Kenneth Teo Hua Yeong, and Arun Sadashiv Mu-jumdar, Development of a Mathematical