Simulation of Battery Systems: Fundamentals and Applications 0128162120, 9780128162125

Simulation of Battery Systems: Fundamentals and Applications covers both the fundamental and technical aspects of batter

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Table of contents :
Contents
Preface
Acknowledgments
1 Battery technologies
1.1 Applications
1.1.1 Miniature batteries
1.1.2 Batteries for portable equipment
1.1.3 SLI batteries (starting lighting & ignition)
1.1.4 Vehicle traction batteries
1.1.5 Stationary batteries
1.1.6 Military & aerospace wide range
1.2 Terms and definitions
1.2.1 Capacity
1.2.2 Energy and power
1.2.3 Specific values versus density
1.2.4 State of charge and depth of discharge
1.2.5 Charge efficiency
1.2.6 Side reactions
1.3 Battery main challenges
1.3.1 Energy content
1.3.2 Power
1.3.3 Self-discharge
1.3.4 Shelf life
1.3.5 Cyclic life
1.3.6 Memory effect
1.3.7 Maximum charging/discharging current
1.3.8 Charging time
1.3.9 Performance
1.3.10 Safety and reliability
1.3.11 Cost
1.3.12 Recycling
1.4 Conventional battery technologies
1.4.1 Lead-acid (LA)
1.4.2 Nickel-cadmium (NiCad)
1.4.3 Nickel-metal hydrate (NiMH)
1.4.4 Lithium-ion (Li-ion)
1.4.5 Metal-air
1.5 Future technologies
1.5.1 Advanced Edison Ni-Iron battery
1.5.2 Advanced lead-acid battery
1.5.3 Solid-state lithium-ion
1.5.4 Gold nanowire batteries
1.5.5 Grabat graphene batteries
1.5.6 Sodium-ion batteries
1.6 Modeling and simulation
1.7 Life cycle assessment
1.7.1 History of LCA
1.7.2 LCA methodologies
Goals and scopes
Inventory analysis
Impact assessment
Interpretation
1.7.3 LCA benefits
1.7.4 LCA limitations
1.7.5 Future of LCA
1.8 Environmental impact assessment of battery technology
1.8.1 Lithium-ion batteries
1.8.2 Lead-acid batteries
1.8.3 Nickel-metal hydride
1.8.4 Conclusion
1.9 Summary
1.10 Problems
2 Fundamentals of batteries
2.1 Introduction
2.1.1 Primary and secondary batteries
2.1.2 Half-cell reactions and symbol
2.1.3 Cathode/anode versus positive/negative electrodes
2.1.4 Reference electrode
2.1.5 Battery capacity
Peukert law for lead-acid batteries
2.1.6 Series and parallel connection
2.2 Basics of electrochemistry
2.3 Faraday's law of electrolysis
2.4 Butler-Volmer law
2.5 Summary
2.6 Problems
3 Fundamental governing equations
3.1 Porous electrode theory and volume-averaging technique
3.1.1 Modeling domain
3.2 Governing equations
3.2.1 Conservation of mass and momentum
3.2.2 Conservation of species
3.2.3 Conservation of electrical charge
3.2.4 Energy balance
3.3 Volume-averaged governing equations
3.3.1 Conservation of mass
3.3.2 Conservation of momentum
3.3.3 Conservation of species
3.3.4 Conservation of electrical charge
3.3.5 Energy balance
3.4 Microscopic modeling
3.4.1 Specific active area
3.4.2 Species diffusion length
3.4.3 Microscopic Ohmic resistance
3.5 Summary
3.6 Problems
4 Heat sources
4.1 Fundamental laws
4.2 Heat of reactions
4.2.1 Reversible or entropic heat of reactions
Enthalpy of reaction
Entropy of reaction
Entropy of electron
4.2.2 Joule heating
4.3 General Joule heating concept
4.4 Heat dissipation
4.4.1 Convection
4.4.2 Conduction
4.4.3 Radiation
4.4.4 Exhausted enthalpy
4.4.5 Equivalent circuit model
4.5 Summary
4.6 Problems
5 Simulation of batteries
5.1 The spatial dimension
5.2 Simulation domain
5.3 Governing equations
5.4 Initial values and guesses
5.5 Boundary conditions
5.6 Mesh generation
5.7 Ill-posedness and compatibility equation
5.8 Famous numerical methods
5.9 Summary
5.10 Problems
6 Lead-acid batteries
6.1 Lead-acid battery components
6.1.1 Plates
6.1.2 Separators
6.1.3 Electrolyte
6.2 Lead-acid battery types
6.3 Electrochemistry of lead-acid batteries
6.4 Lead-acid battery applications
6.5 Governing equations
6.5.1 Conservation of mass and momentum
6.5.2 Conservation of energy
6.5.3 Conservation of charge
6.5.4 Conservation of species
6.5.5 Conservation of mass
6.6 Thermal runaway problem
6.7 Heat sources and sinks
6.7.1 Heat of reactions
6.7.2 Joule heating
6.7.3 Heat dissipation
6.8 One-dimensional model
6.8.1 Governing equations for one-dimensional model
6.8.2 Boundary conditions
Potential in solid and electrolyte
Chemical species
Cell temperature
6.9 Physico-chemical properties
6.9.1 Electrode electrical conductivity σ
6.9.2 Electrolyte ionic conductivity k
6.9.3 Diffusion coefficients
6.9.4 Open-circuit voltage U
6.9.5 Partial molar volumes of sulfuric acid and water
6.9.6 Thermodynamic properties of different species
6.9.7 Calculation of properties in porous medium
6.9.8 Temperature dependency of parameters
6.10 Numerical simulation of lead-acid batteries
6.10.1 One-dimensional simulation without side reactions
6.10.2 One-dimensional simulation including side reactions
6.10.3 Numerical simulation of electrolyte stratification using two-dimensional modeling
6.10.4 Simulation of thermal behavior of lead-acid batteries
6.11 Summary
6.12 Problems
7 Zinc-silver oxide batteries
7.1 Zinc-silver oxide battery components
7.1.1 Zinc electrode
7.1.2 Silver oxide electrode
7.1.3 Separator
7.1.4 Electrolyte
7.2 Types and applications
7.2.1 Button cells
7.2.2 High-power primary prismatic cells
7.2.3 High-power secondary prismatic cells
7.3 Electrochemical reactions
7.4 Governing equations of zinc-silver oxide batteries
7.5 Mathematical model
7.5.1 Conservation of electrical charge in solid and electrolyte phases
7.5.2 Conservation of species
7.5.3 Balance of energy
7.5.4 Conservation of mass
7.5.5 Electroactive area
7.5.6 Reaction rate
7.6 Heat sources and sinks
7.6.1 Heat of electrochemical reactions
7.6.2 Joule heating
7.6.3 Heat dissipation or sinks
7.7 Physico-chemical characteristics
7.8 Numerical simulation of zinc-silveroxide batteries
7.8.1 One-dimensional simulation
7.8.2 Two-dimensional isothermal model
7.8.3 Simulation of thermal behavior of zinc-silver oxide batteries
7.8.4 Two-dimensional simulation with water cycle
7.9 Summary
7.10 Problems
8 Lithium-based batteries
8.1 Lithium-based battery components
8.1.1 Positive electrode
8.1.2 Negative electrode
8.1.3 Separator
8.1.4 Electrolyte
8.2 Types and applications
8.2.1 Lithium-cobalt oxide, LCO (LiCoO2)
8.2.2 Lithium-manganese oxide, LMO (LiMn2O4)
8.2.3 Lithium-nickel-manganese-cobalt oxide, NMC (LiNiMnCoO2)
8.2.4 Lithium-iron phosphate, LFP (LiFePO4)
8.2.5 Lithium-nickel-cobalt-aluminum oxide, NCA (LiNiCoAlO2)
8.2.6 Lithium-titanate, LTO (Li4Ti5O12)
8.3 Electrochemical reactions
8.4 Governing equations of lithium-based batteries
8.4.1 Conservation of mass and momentum
8.4.2 Conservation of electrical charge
8.4.3 Conservation of chemical species
Conservation of lithium ions in electrolyte
Conservation of lithium ions in solid phase
8.4.4 State of charge
8.4.5 Conservation of energy
8.5 Boundary and initial conditions
8.6 Physical characteristics
8.6.1 Conductivity of electrolyte
8.6.2 Solid conductivity
8.6.3 Open-circuit voltage
8.7 One-dimensional model of lithium-based batteries
8.7.1 Conservation of electrical charge in solid phase
8.7.2 Conservation of electrical charge in electrolyte phase
8.7.3 Conservation of chemical species
8.7.4 Kinetic rate or Butler-Volmer equation
8.8 Simulation of lithium-based batteries
8.8.1 Cell voltage
8.8.2 Concentration of lithium ions in electrolyte
8.8.3 Concentration of lithium ions in solid particles and state of charge
8.8.4 Simulation result
8.9 Summary
8.10 Problems
9 Techno-economic assessment of battery systems
9.1 Introduction
9.1.1 Battery technology
9.2 Environmental effects of different types of batteries
9.2.1 Environmental impacts of Photovoltaic battery systems
9.2.2 Environmental effects of electric vehicle battery systems
9.3 Summary
9.4 Problems
A Experimental tests for lead-acid batteries
A.1 Cell-I
A.2 Cell-II
A.3 Cell-III
A.4 Cell-IV
A.5 Cell-V
B Experimental test for zinc-silveroxide batteries
C Experimental tests for lithium batteries
D Finite volume method
D.1 FVM in one dimension
D.2 FVM in two dimensions
Summary
E SIMPLE algorithm
Summary
F Keller-Box method
Summary
References
Index
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SIMULATION OF BATTERY SYSTEMS

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2020 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-816212-5 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Joe Hayton Acquisition Editor: Lisa Reading Editorial Project Manager: Andrae Akeh Production Project Manager: R. Vijay Bharath Designer: Mark Rogers Typeset by VTeX

Authors would like to dedicate this book to a cleaner future

Contents Preface Acknowledgments

1. Battery technologies 1.1. Applications 1.1.1. Miniature batteries 1.1.2. Batteries for portable equipment 1.1.3. SLI batteries (starting lighting & ignition) 1.1.4. Vehicle traction batteries 1.1.5. Stationary batteries 1.1.6. Military & aerospace wide range 1.2. Terms and definitions 1.2.1. Capacity 1.2.2. Energy and power 1.2.3. Specific values versus density 1.2.4. State of charge and depth of discharge 1.2.5. Charge efficiency 1.2.6. Side reactions 1.3. Battery main challenges 1.3.1. Energy content 1.3.2. Power 1.3.3. Self-discharge 1.3.4. Shelf life 1.3.5. Cyclic life 1.3.6. Memory effect 1.3.7. Maximum charging/discharging current 1.3.8. Charging time 1.3.9. Performance 1.3.10. Safety and reliability 1.3.11. Cost 1.3.12. Recycling 1.4. Conventional battery technologies 1.4.1. Lead–acid (LA) 1.4.2. Nickel–cadmium (NiCad) 1.4.3. Nickel–metal hydrate (NiMH) 1.4.4. Lithium-ion (Li-ion) 1.4.5. Metal–air 1.5. Future technologies 1.5.1. Advanced Edison Ni–Iron battery

xv xix

1 6 6 8 11 11 12 13 14 14 16 16 18 19 20 21 21 22 22 25 25 26 26 27 27 28 29 29 30 30 31 31 32 33 34 34 vii

viii

Contents

1.6. 1.7.

1.8.

1.9. 1.10.

1.5.2. Advanced lead–acid battery 1.5.3. Solid-state lithium-ion 1.5.4. Gold nanowire batteries 1.5.5. Grabat graphene batteries 1.5.6. Sodium-ion batteries Modeling and simulation Life cycle assessment 1.7.1. History of LCA 1.7.2. LCA methodologies 1.7.3. LCA benefits 1.7.4. LCA limitations 1.7.5. Future of LCA Environmental impact assessment of battery technology 1.8.1. Lithium-ion batteries 1.8.2. Lead–acid batteries 1.8.3. Nickel–metal hydride 1.8.4. Conclusion Summary Problems

2. Fundamentals of batteries 2.1. Introduction 2.1.1. Primary and secondary batteries 2.1.2. Half-cell reactions and symbol 2.1.3. Cathode/anode versus positive/negative electrodes 2.1.4. Reference electrode 2.1.5. Battery capacity 2.1.6. Series and parallel connection 2.2. Basics of electrochemistry 2.3. Faraday’s law of electrolysis 2.4. Butler–Volmer law 2.5. Summary 2.6. Problems

3. Fundamental governing equations 3.1. Porous electrode theory and volume-averaging technique 3.1.1. Modeling domain 3.2. Governing equations 3.2.1. Conservation of mass and momentum 3.2.2. Conservation of species 3.2.3. Conservation of electrical charge 3.2.4. Energy balance 3.3. Volume-averaged governing equations

35 35 35 36 36 36 40 41 41 44 45 46 46 47 49 51 52 53 54

55 55 56 57 59 60 61 65 67 72 72 80 80

83 83 84 87 88 89 90 91 93

Contents

3.3.1. Conservation of mass 3.3.2. Conservation of momentum 3.3.3. Conservation of species 3.3.4. Conservation of electrical charge 3.3.5. Energy balance 3.4. Microscopic modeling 3.4.1. Specific active area 3.4.2. Species diffusion length 3.4.3. Microscopic Ohmic resistance 3.5. Summary 3.6. Problems

4. Heat sources 4.1. Fundamental laws 4.2. Heat of reactions 4.2.1. Reversible or entropic heat of reactions 4.2.2. Joule heating 4.3. General Joule heating concept 4.4. Heat dissipation 4.4.1. Convection 4.4.2. Conduction 4.4.3. Radiation 4.4.4. Exhausted enthalpy 4.4.5. Equivalent circuit model 4.5. Summary 4.6. Problems

5. Simulation of batteries 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8. 5.9. 5.10.

The spatial dimension Simulation domain Governing equations Initial values and guesses Boundary conditions Mesh generation Ill-posedness and compatibility equation Famous numerical methods Summary Problems

6. Lead–acid batteries 6.1. Lead–acid battery components 6.1.1. Plates 6.1.2. Separators

ix

93 95 96 101 103 107 107 108 109 110 111

113 114 118 119 124 125 125 125 126 126 126 127 127 128

129 130 134 136 140 141 143 144 146 147 148

149 151 151 152

x

Contents

6.2. 6.3. 6.4. 6.5.

6.6. 6.7.

6.8.

6.9.

6.10.

6.11. 6.12.

6.1.3. Electrolyte Lead–acid battery types Electrochemistry of lead–acid batteries Lead–acid battery applications Governing equations 6.5.1. Conservation of mass and momentum 6.5.2. Conservation of energy 6.5.3. Conservation of charge 6.5.4. Conservation of species 6.5.5. Conservation of mass Thermal runaway problem Heat sources and sinks 6.7.1. Heat of reactions 6.7.2. Joule heating 6.7.3. Heat dissipation One-dimensional model 6.8.1. Governing equations for one-dimensional model 6.8.2. Boundary conditions Physico-chemical properties 6.9.1. Electrode electrical conductivity σ 6.9.2. Electrolyte ionic conductivity k 6.9.3. Diffusion coefficients 6.9.4. Open-circuit voltage U 6.9.5. Partial molar volumes of sulfuric acid and water 6.9.6. Thermodynamic properties of different species 6.9.7. Calculation of properties in porous medium 6.9.8. Temperature dependency of parameters Numerical simulation of lead–acid batteries 6.10.1. One-dimensional simulation without side reactions 6.10.2. One-dimensional simulation including side reactions 6.10.3. Numerical simulation of electrolyte stratification using two-dimensional modeling 6.10.4. Simulation of thermal behavior of lead–acid batteries Summary Problems

7. Zinc–silver oxide batteries 7.1. Zinc–silver oxide battery components 7.1.1. Zinc electrode 7.1.2. Silver oxide electrode 7.1.3. Separator 7.1.4. Electrolyte

154 154 156 158 159 163 165 165 166 167 168 171 171 173 173 175 175 176 179 179 180 180 181 181 183 183 184 184 185 189 193 201 213 214

217 219 219 220 220 220

Contents

7.2. Types and applications 7.2.1. Button cells 7.2.2. High-power primary prismatic cells 7.2.3. High-power secondary prismatic cells 7.3. Electrochemical reactions 7.4. Governing equations of zinc–silver oxide batteries 7.5. Mathematical model 7.5.1. Conservation of electrical charge in solid and electrolyte phases 7.5.2. Conservation of species 7.5.3. Balance of energy 7.5.4. Conservation of mass 7.5.5. Electroactive area 7.5.6. Reaction rate 7.6. Heat sources and sinks 7.6.1. Heat of electrochemical reactions 7.6.2. Joule heating 7.6.3. Heat dissipation or sinks 7.7. Physico-chemical characteristics 7.8. Numerical simulation of zinc–silveroxide batteries 7.8.1. One-dimensional simulation 7.8.2. Two-dimensional isothermal model 7.8.3. Simulation of thermal behavior of zinc–silver oxide batteries 7.8.4. Two-dimensional simulation with water cycle 7.9. Summary 7.10. Problems

8. Lithium-based batteries 8.1. Lithium-based battery components 8.1.1. Positive electrode 8.1.2. Negative electrode 8.1.3. Separator 8.1.4. Electrolyte 8.2. Types and applications 8.2.1. Lithium–cobalt oxide, LCO (LiCoO2 ) 8.2.2. Lithium–manganese oxide, LMO (LiMn2 O4 ) 8.2.3. Lithium–nickel–manganese–cobalt oxide, NMC (LiNiMnCoO2 ) 8.2.4. Lithium–iron phosphate, LFP (LiFePO4 ) 8.2.5. Lithium–nickel–cobalt–aluminum oxide, NCA (LiNiCoAlO2 ) 8.2.6. Lithium–titanate, LTO (Li4 Ti5 O12 ) 8.3. Electrochemical reactions 8.4. Governing equations of lithium-based batteries 8.4.1. Conservation of mass and momentum 8.4.2. Conservation of electrical charge

xi

220 221 221 221 222 223 225 227 228 229 231 231 232 233 234 235 236 236 237 237 241 246 254 260 261

263 264 265 265 265 266 266 266 267 267 268 269 269 271 272 275 275

xii

Contents

8.5. 8.6.

8.7.

8.8.

8.9. 8.10.

8.4.3. Conservation of chemical species 8.4.4. State of charge 8.4.5. Conservation of energy Boundary and initial conditions Physical characteristics 8.6.1. Conductivity of electrolyte 8.6.2. Solid conductivity 8.6.3. Open-circuit voltage One-dimensional model of lithium-based batteries 8.7.1. Conservation of electrical charge in solid phase 8.7.2. Conservation of electrical charge in electrolyte phase 8.7.3. Conservation of chemical species 8.7.4. Kinetic rate or Butler–Volmer equation Simulation of lithium-based batteries 8.8.1. Cell voltage 8.8.2. Concentration of lithium ions in electrolyte 8.8.3. Concentration of lithium ions in solid particles and state of charge 8.8.4. Simulation result Summary Problems

9. Techno-economic assessment of battery systems 9.1. Introduction 9.1.1. Battery technology 9.2. Environmental effects of different types of batteries 9.2.1. Environmental impacts of Photovoltaic battery systems 9.2.2. Environmental effects of electric vehicle battery systems 9.3. Summary 9.4. Problems

A. Experimental tests for lead–acid batteries A.1. A.2. A.3. A.4. A.5.

Cell-I Cell-II Cell-III Cell-IV Cell-V

278 281 281 282 283 283 284 284 287 288 288 289 289 290 290 294 305 305 307 308

311 312 314 314 315 327 350 351

353 353 353 355 356 356

B. Experimental test for zinc–silveroxide batteries

359

C. Experimental tests for lithium batteries

363

D. Finite volume method

365

D.1. FVM in one dimension

365

Contents

D.2. FVM in two dimensions Summary

E. SIMPLE algorithm Summary

F. Keller–Box method Summary References Index

xiii

369 370

373 384

385 392 393 399

Preface We are in an era that fossil fuel depletion and global warming are two major concerns. Renewable energies have proven themselves as one of the crucial solutions to mitigate global warming. Although the deployment of renewable energies has increased significantly during the last decades, they still face some problems such as their high cost and intermittency. Energy storage options can assist us in tackling renewable energies intermittency problems. Modern technologies hugely depend on advanced storage systems such as hydro-power, compressed air, flywheels, and of course batteries. When it comes to portable devices, electrochemical batteries become of great interest. Since the invention of the first rechargeable or secondary batteries by Gaston Planté, portable applications became more and more of interest, and many different batteries have been made, tested, and manufactured. Although there are lots of different battery technologies in the market, we are still far from an ideal case, meaning that the manufactured batteries cannot deliver their theoretical energy content. In fact, the theoretical energy content of commercially available batteries such as lead-acid, lithium-based, nickel-based, and other types is much higher. Consequently, these batteries can be improved further to reach their potentially available energy content. In some cases the theoretical values are about five times the actually available values. To achieve this theoretical potential, many efforts have been made in various branches of science. Some works have been conducted on improving materials including nanomaterials, composites, additives, and so on. Some works have been carried out on the configuration of the cells and working on different aspects of their manufacturing process. Some works are on battery simulation to have better understanding of physical phenomena. In all cases the main goals are: • Improving energy and power content of the cells, • Improving safety issues, especially the thermal-runaway, • Reducing the cost of the cells, • Enhancing the performance of the cells, which is important in working in cold and hot climate, • Improving cyclic life and also shelf life of the cells, • Minimizing self-discharge and memory effect, and • Reducing the cost of recycling. xv

xvi

Preface

Any improvement in the mentioned characteristic parameters leads to good achievements in battery industry. All manufacturers try to improve their cells according to these characteristics. As mentioned before, one of the scientific research branches in improving the battery cells is modeling and simulation. Through modeling and simulation, we can gain very deep understanding of the internal physical phenomena, which in turn is important in analyzing experimental results. It is known that theoretical and fundamental modeling alone does not result in production of a new battery; however, the results are a powerful tool for designing new experiments and analyzing the test results. In other words, such simulation and modeling are a fundamental tool for helping experiments. We can say that modeling and simulation of battery systems are a vital tool to predict and interpret the behavior of new battery technology. In some real cases, this tool is very important, as some physical parameters either cannot be measured experimentally or require very expensive instruments or state-of-the-art methods. Measuring such parameters, reducing the cost of investigations, and gaining deep understanding of the physical meaning of the involved phenomena are outcomes of theoretical modeling and simulation. This book is a research-oriented textbook and provides comprehensive coverage of fundamentals and main concepts, and can be used for modeling and simulation of battery energy systems. The book includes practical features in a usable format often not included in other solely academic textbooks. The book can be used by senior undergraduate and graduate students in mainstream engineering fields (such as mechanical, chemical and electrical engineering) and as well as specialized engineering programs on energy systems. This book consists of nine chapters. Chapter 1 talks about various types of batteries and their working principles. Major battery types are introduced and some advances in battery types for future usage are presented. Finally the life cycle assessment and environmental impacts are discussed. Chapter 2 deals with basic electrochemical fundamentals of batteries. The governing equations of battery dynamics are fully covered in Chapter 3. Since heat generation in batteries is very important, Chapter 4 is dedicated to describe heat sources and sink in batteries. Chapter 5 provides a complete discussion about the battery simulation. It should be noted that the objective of the book is dealing with solution of governing equations by means of solving partial differential equations. Hence the simulation domain and other concerning issues are fully discussed in this chapter. Chapter 6 focuses on lead-acid batteries. In this chapter, lead-acid batteries are first introduced.

Preface

xvii

Then the governing equations are obtained from the general governing equations of batteries discussed in Chapter 3. Finally, the equations are solved numerically with different algorithms and the results are discussed. Chapter 7 is dedicated to zinc-silver oxide batteries. Again their governing equations are obtained from the general formulation; then the equations are solved at different situations and the results are discussed. In Chapter 8, the lithium-based batteries are introduced, discussed and simulated. Just note that in Chapter 8, the governing equations are solved analytically instead of numerically. Since materials for battery production, their delivery, and their assembly requires high amount of energy, both economic and environmental issues should be considered. This is why Chapter 9, explores the techno-economic assessment of battery systems in more detail Incorporated throughout are many illustrative examples and case studies, which provide the reader with a substantial learning experience, especially in areas of practical application. Complete references are included to point the curious reader in the right direction. Information on topics not covered fully in the text can, therefore, be easily found. We hope this book brings a new dimension for the simulation of battery systems and helps the community implement better solutions for a better future. Farschad Torabi and Pouria Ahmadi September 2019

Acknowledgments

Completion of this book on Simulation of Battery Systems was challenging and could not have been completed without the support of others. In particular, some graduate students and colleagues have specific contributions, and their helps and supports are gratefully acknowledged. Among the students, we would like to thank Amin Aliakbar, Niloofar Kamyab, Farnaz Tajdari, Zahra Porhemmat, Syawash Azizi, Alireza Khoshnevisan, Mohammadhosein Kazemi, Iman Fakhari, and Ali Mosahebi. We would also like to express our thanks to Dr. Nader Javani for his scientific feedbacks on some chapters. Farschad Torabi warmly thanks his wife Sheeva, his son Arya, and his parents Fazlollah (RIP) and Zahra for their inspiration, love, and support. Finally, Pouria Ahmadi would like to thank his parents Morad and Batoul for their understanding and encouragement throughout his education.

xix

CHAPTER 1

Battery technologies Contents 1.1. Applications 1.1.1 Miniature batteries 1.1.2 Batteries for portable equipment 1.1.3 SLI batteries (starting lighting & ignition) 1.1.4 Vehicle traction batteries 1.1.5 Stationary batteries 1.1.6 Military & aerospace wide range 1.2. Terms and definitions 1.2.1 Capacity 1.2.2 Energy and power 1.2.3 Specific values versus density 1.2.4 State of charge and depth of discharge 1.2.5 Charge efficiency 1.2.6 Side reactions 1.3. Battery main challenges 1.3.1 Energy content 1.3.2 Power 1.3.3 Self-discharge 1.3.4 Shelf life 1.3.5 Cyclic life 1.3.6 Memory effect 1.3.7 Maximum charging/discharging current 1.3.8 Charging time 1.3.9 Performance 1.3.10 Safety and reliability 1.3.11 Cost 1.3.12 Recycling 1.4. Conventional battery technologies 1.4.1 Lead–acid (LA) 1.4.2 Nickel–cadmium (NiCad) 1.4.3 Nickel–metal hydrate (NiMH) 1.4.4 Lithium-ion (Li-ion) 1.4.5 Metal–air 1.5. Future technologies 1.5.1 Advanced Edison Ni–Iron battery 1.5.2 Advanced lead–acid battery 1.5.3 Solid-state lithium-ion 1.5.4 Gold nanowire batteries Simulation of Battery Systems https://doi.org/10.1016/B978-0-12-816212-5.00005-2

6 6 8 11 11 12 13 14 14 16 16 18 19 20 21 21 22 22 25 25 26 26 27 27 28 29 29 30 30 31 31 32 33 34 34 35 35 35 Copyright © 2020 Elsevier Inc. All rights reserved.

1

2

Simulation of Battery Systems

1.6. 1.7.

1.8.

1.9. 1.10.

1.5.5 Grabat graphene batteries 1.5.6 Sodium-ion batteries Modeling and simulation Life cycle assessment 1.7.1 History of LCA 1.7.2 LCA methodologies Goals and scopes Inventory analysis Impact assessment Interpretation 1.7.3 LCA benefits 1.7.4 LCA limitations 1.7.5 Future of LCA Environmental impact assessment of battery technology 1.8.1 Lithium-ion batteries 1.8.2 Lead–acid batteries 1.8.3 Nickel–metal hydride 1.8.4 Conclusion Summary Problems

36 36 36 40 41 41 42 43 43 44 44 45 46 46 47 49 51 52 53 54

Literarily, a battery means a system that contains a specific amount of energy. By this definition, a compressed spring, a rock on a mountain, compressed air, elevated water, or any other energy storage system can be considered as a battery. However, in modern technologies, battery is a device that converts chemical energy into electricity using electrochemical reactions. By means of electrochemical reactions the chemical energy stored in chemical bonds is converted to electrical energy. The electrochemical reactions occur in two different places that are known as electrodes. At one electrode, the active material is oxidized, and a specific amount of electrons is produced. At the other electrode, the produced electrons are consumed, and the active material is reduced. Hence an electrochemical battery consists of a pair of oxidation/reduction reactions. Since the overall reaction of a battery is through the electrochemical reactions that are normally taking place in low temperatures, the efficiency of these devices is higher than a direct combination of the active materials. Consequently, batteries are categorized as medium-quality devices or converters. The present state of technology strongly depends on the improvement of electrochemical batteries. In many cases, battery technology is the Achilles’ heel; for instance, when we talk about the advances in electric car industries, we notice that the main challenge is the energy and power

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Figure 1.1 Persian batteries (224–650 AD).

density of the batteries. Therefore many megafactories have made a lot of investments in improving different battery types and technologies. Technically, batteries are categorized as energy storage devices that are made in different shapes and types. They can be found from very tiny coin cells to very large and industrial shapes. Coin cells are used in low energy consuming or portable applications, including watches, toys, laser points, calculators, and so on. On the other hand, large industrial batteries can be found in megafactories for energy regulation and optimization. A large number of cells can be connected in series or parallel to fulfill the desired energy and power. For example, in power stations, arrays of lead–acid batteries (conventionally tubular types) are installed for storing Mega-Watthour of electrical energy. It is interesting to note that the first usage of batteries in human history backs to about 2200 years ago. Wilhelm König in 1938 found a small pot about 14 cm in size near the city Baghdad. The place was close to the metropolis of Ctesiphon, the capital of the Parthian (150 BC–223 AD) and Sasanian (224–650 AD), the third and fourth Persian empires. The explored pot was equipped with a copper cylinder and an iron rod, as shown in Fig. 1.1. The cylinder and rod were isolated at the top by bitumen or asphalt to avoid their contact. König transferred the pot to Germany for further investigation. Analysis of the pot revealed that the pot was filled with an organic acid such as vinegar or grape juice, and the effect of electrochemical reactions was observed. The evidence proved that the excavated pot was a battery. The explored cell was able to produce about 0.8 V and 250 mA. Since the pot historically belonged to the Persian empires, the battery was called Persian battery or Baghdad battery after the name of the city where it was found. The exploration of Persian battery arose a lot of

4

Simulation of Battery Systems

questions, some to approve and some to deny the device as a battery; unfortunately, the beginning of World War II masked the issue. The main challenge of exploration was the role of the device. Did Persians really use them as a battery? If yes, what for? And if human being invented a battery for such a long period, why the science was dimmed in history? The questions became more interesting when some golden coated metallic pieces were found near the same place and further in remains of Parthians and Sasanians. Did really ancient Persians use the device for electroplating? The investigations showed that the voltage of the cells was not high enough for this purpose; but did ancient Persians know how to connect the cells in series and parallel? As another hypothesis, some believed that electrical energy was known at that era and was used for physiotherapy. The idea came from historical sources, which reported that ancient nations knew how to cure the pain by eels surely producing electricity. The case is still under discussion, and there is no any idea about the origin or application of Persian batteries. However, after the fall of Sasanians, it took about two thousand years for the human being to reinvent the electrochemical cells. Luigi Galvani started the very first ideas and experiments that finally led to the invention of electrical cells. Galvani and his wife in a series of fundamental experiments found that the muscles of animals are moved by electrical signals. The idea was quite controversial at the time and was opposed to the accepted concept of air causing motions. Their experiments led to the invention of the scientific term of animal electricity, the effect also known as Galvani effect. Today the bioelectricity or Galvani effect is called electrophysiology. In his experiments, Galvani used to deal with static electricity for muscle contraction of some animals such as frog. In a surprising experiment, he noticed that instead of static electricity, if one touches the muscles of a dead frog by two dissimilar metallic rods, then the same effect is applied on frog’s muscles; in other words, the muscle contraction is observed, and even electrical sparks were evident. The works of the great Italian physicist Alessandro Volta fine-tuned the works of Galvani and finally invented an electrochemical cell or battery. Volta invented the first battery in 1800 and called it the voltaic pile. The pile consisted of some zinc and copper disks piled on top of each other and separated by cloth. The assembly was filled with brine as the electrolyte. Although it was Volta who invented the first pile, the electrochemical laws and relations were studied and formulated by the famous British physicist

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Michael Faraday. The works of Faraday invented a bridge between chemical energy and electricity. After the invention of the first pile, many scientists started working on different battery types. One of the biggest challenges at the time was to prevent hydrogen evolution. For instance, the British scientist John Frederic Daniell invented a solution for capturing the evolved hydrogen in another electrolyte reservoir. As another attempt for preventing hydrogen bubbles, the French electrical engineer Georges Leclanché invented a cell consisting of zinc metal as the negative electrode and a mixture of manganese dioxide and carbon pasted on a graphite plate as the positive electrode. The invention took place in 1866 and was a great success in telegraphy, signaling, and electric bell work. It was also the first step toward the dry cell invention. Without a doubt, the invention of the first rechargeable battery by the French scientist, Gaston Planté in 1859 was a great breakthrough in battery technology and history. Planté made the cell by lead and lead dioxide as the negative and positive electrodes filled by sulfuric acid. Both electrodes were rolled in a spiral shape and were put in a cylindrical container. The most important characteristic of the cell was that after usage, it could be recharged by passing the current in reverse direction. Planté’s lead–acid cells were used for powering up the lights in train carriages while stopped. The invented lead–acid batteries were able to provide 2.7 V and easy to be produced in mass production. The internal resistance of the cell was very low, meaning that the cell could provide very large electrical surges. To increase the mechanical characteristics of the plates, Planté used to make thick plates resulting in heavy batteries. Consequently, the lead–acid cells were useful in places where the weight was not a matter. Many other scientists in different countries invented different battery types. For example, Waldmer Jungner in Sweden and Thomas Edison in the US invented alkaline batteries such as nickel–cadmium and nickel–iron between 1895 to 1905. These batteries were rechargeable like lead–acid batteries but were the first alkaline cells. Using an alkaline electrolyte instead of commonly used acid electrolytes opened a new window for improving and inventing many different batteries because many materials are unstable in acid but stable in alkaline. One of the advantages of nickel– iron battery, also known as Edison’s cell, was that it was very tolerant in bad charging and environmental conditions. Moreover, its cyclic life is very high, meaning that a cell can be charged and discharged many times. The invention of different types of batteries improved the performance of the cells. More specifically, the more the energy and power content and

6

Simulation of Battery Systems

cyclic life, the less the self-discharge, less gas production, low memory effect, and other characteristics that identify the behavior of cells. Lighter batteries are vital for portable devices. The invention of nickel–metal hydride (NiMH) in 1990 and lithium-ion (Li-ion) batteries in 1991 were a big breakthrough in battery technology. They were very high in specific energy and also could provide high power. At the same time, their cyclic life is very high. The electrochemistry of lithium-ion batteries differs from other conventional batteries due to the fact that in Li-ion cells the charge and discharge processes are taken place through intercalation or insertion.

1.1 Applications Batteries are commonly manufactured and produced in many shapes and sizes for many purposes in different applications. For these reasons, there are many battery types with chemistry available and also many other types are under investigation. The diverse applications of batteries makes it difficult to give a complete classification. Here some important applications are categorized.

1.1.1 Miniature batteries Many portable devices and applications require very low current; hence, low energy and power cells are suitable for them. The required amount of energy is usually in the range of 100 mWh to 2 Wh, which can be provided by coin or button cells. Electric watches, calculators, implanted medical devices, toys, laser points, and so on are some examples of devices that require miniature batteries. Miniature batteries are usually primary cells, but rechargeable versions are also available. In most technologies, zinc acts as the negative electrode, and the positive counter electrode may be silver, nickel hydroxide, mercury, or air. Other than zinc-based cells, the lithium-ion batteries are also available in a coin shape. The cells can be found from 5 to 25 mm in diameter and 1 to 6 mm in height. International standard IEC 60086–3 defines an alphanumeric coding system for “Watch batteries”. The first letter indicates the chemistry of the cell, according to Table 1.1. The second letter is usually R, indicating that the cell has a round shape, that is, the cell is cylindrical. The package size of button cells is indicated by a two-digit code representing a standard case size, or a three- or four-digit code representing the cell diameter and height. The first one or two digits are the nominal diameter of the cell in

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Table 1.1 Common specification of coin cells [1]. Letter code

Common name

Positive electrode

Electrolyte

Negative electrode

Nominal voltage (V)

End-point voltage (V)

L S P C B G Z

Alkaline Silver Zinc–air Lithium

Manganese dioxide Silver oxide Oxygen Manganese dioxide Carbon monofluoride Copper oxide Manganese dioxide nickel oxyhydroxide Mercuric oxide

Alkali Alkali Alkali Organic Organic Organic Alkali

Zinc Zinc Zinc Lithium Lithium Lithium Zinc

1.5 1.55 1.4 3 3 1.5 1.5

1.0 1.2 1.2 2.0 2.0 1.2 ?

Alkali

Zinc

1.35/1.40

1.1

M, N

Nickel oxyhydroxide Mercury

Table 1.2 Diameter code for the first one or two digits for coin cells [1]. Number code

Nominal diameter (mm)

Tolerance (mm)

Number code

Nominal diameter (mm)

Tolerance (mm)

4 5 6 7 9 10 11

4.8 5.8 6.8 7.9 9.5 10.0 11.6

±0.15

12 16 20 23 24 44

12.5 16.0 20.0 23.0 24.5 5.4

±0.25

±0.15 ±0.10 ±0.15 ±0.15 ±0.20

±0.25 ±0.25 ±0.50 ±0.50 ±0.20

±0.20

millimeters rounded down; exact diameters are specified by the standard, as tabulated in Table 1.2. The last two digits are the overall height in tenths of a millimeter. Example 1.1. Define the characteristics of with code numbers CR2032, CR2025, and SR516. Answer. According to IEC 60086–3 standard, these cells have the following characterizations: CR2032 The first letter C indicates that it is a lithium cell with manganese dioxide as its positive electrode and lithium as its negative electrode. It uses an organic electrolyte with a nominal voltage of 3 V. It is considered to be finished when the voltage reaches 2 V. The cell is round or cylindrical (R as its second letter) and has diameter 20 mm (referring to Table 1.2) and height 3.2 mm by its last two digits. CR2025 This cell has the same characteristics as CR2032, but its height is 2.5 mm.

8

Simulation of Battery Systems

SR516 The first letter S indicates that the cell is a zinc–silver oxide battery with silver oxide as the positive electrode and zinc as its negative electrode. The cell uses an alkali electrolyte, usually potassium or sodium hydroxide. The nominal voltage of the cell is 1.55 V, and it is considered to be finished when the voltage reaches 1.2 V. The second letter R indicates that the cell is cylindrical. The first digit 5 means that its outer diameter is 5.8 mm, and the last two digits indicate that its height is 1.6 mm. Example 1.2. An application requires zinc–air button cells with diameter 9.5 mm and height 2.0 mm. What is its standard code in the market? Answer. Since we need a zinc–air cell, the first letter should be P. The required diameter and height suggest the code PR920.

1.1.2 Batteries for portable equipment When coming to portable equipment, we require higher energy in the range of 2 to 100 Wh. These devices, including flashlights, toys, power tools, cell phones, laptop computers, cordless devices, wireless peripherals, emergency beacons and so on, usually use alkaline cells such as NiCd and NiMH or may use Li-ion batteries. The high energy consumption requires that secondary or rechargeable cells are to be used in these devices, but primary cells are also used in huge amount. Since the portable devices use lots of energy, the coin or button cell types are not proper. The cells used in these applications are usually in A, AA, AAA sizes, whereas B, C, and D sizes are also available in the market. These cells have cylindrical shapes and are standardized by IEC. The specifications of some of the most popular types are tabulated in Table 1.3. Note that there are many other types available, namely F, N, and so on, but they are less common in the market for household uses. Other than cylindrical shapes, the prismatic and pouch cells are also widely used in portable devices such as cell phones. Since the diameters of tubular cells are too large for incorporating in a cell phone, pouch types are preferred in such devices. The main benefit of pouch cells is that they have no hard case, so they become very light and suitable for portable devices. These cells have no any standard for their dimension, so they can be found in any shape or size. Since the pouch cells are usually Li-polymer, and the case is not very hard, a safe space should be designed inside the device in which the cell is to be used.

Table 1.3 Common specification of cylindrical cells for portable devices [2]. Letter IEC code Typical capacity Nominal Size code mAh voltage Dia. × H. (V) (mm) AAA R03 (carbon–zinc) 1200 (alkaline) 1.5 10.5 × 44.5

AA

FR03 (Li–FeS2) HR03 (NiMH) KR03 (NiCd) ZR03 (NiOOH) R6 (carbon–zinc) FR6 (Li–FeS2) HR6 (NiMH) KR6 (NiCd) ZR6 (NiOOH)

A

R23 (carbon–zinc) LR23 (alkaline)

B

R12 (carbon–zinc) LR12 (alkaline)

540 (carbon–zinc) 800–1000 (NiMH) 500 (NiZn) 2700 (alkaline) 1100 (carbon–zinc) 3000 (Li–FeS2) 1700–2700 (NiMH) 600–1000 (NiCd) 1500 (NiZn)

8350 (alkaline)

1.5

14.5 × 50.5

1.5

17 × 50

1.5

21.5 × 60

Comments

Introduced 1911, but added to ANSI standard in 1959 Used in many household electronic devices

Introduced 1907, but added to ANSI standard sizes in 1947. Note: 14,500 Lithium Batteries are not AA as they are 3.7 V; though 1.5 V AA compatibles (achieved with an internal voltage regulator [specifically a buck converter]) have been available since 2014. Used in many household electronic devices More common as a NiCd or NiMH cell size than a primary size, popular in older laptop batteries and hobby battery packs Most commonly found within a European 4.5 volt lantern battery. Not to be confused with the vacuum tube B battery continued on next page

Table 1.3 (continued) Letter IEC code code

C

Sub-C

D

Typical capacity mAh

Nominal voltage (V)

LR14 (alkaline) R14 (carbon–zinc) HR14 (NiMH) KR14 (NiCd) ZR14 (NiOOH) KR22C429 (NiCd) HR22C429 (NiMH)

8000 (alkaline) 3800 (carbon–zinc) 4500–6000 (NiMH)

1.5

Size Dia. × H. (mm) 26.2 × 50

1200–2400 (NiCd) 1800–5000 (NiMH)

1.2

22.2 × 42.9

A common size for cordless tool battery packs. This size is also used in radio-controlled scale vehicle battery packs

LR20 (alkaline) R20 (carbon–zinc) HR20 (NiMH) KR20 (Ni-Cd) ZR20 (NiOOH)

12,000 (alkaline) 8000 (carbon–zinc) 2200–11,000 (NiMH) 2000–5500 (NiCd)

1.5

34.2 × 61.5

Introduced 1898 as the first flashlight battery

Comments

Can be replaced with AA cell using a plastic sabot (size adaptor), with proportional loss of capacity

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1.1.3 SLI batteries (starting lighting & ignition) Starting Lighting & Ignition (SLI) batteries are very common in automobile industries. All the available fossil fueled vehicles, including bikes, cars, trucks, and so on, have at least one SLI battery for starting. The nominal voltage for bikes and cars is 6 or 12 V, and in heavy industrial vehicles, it is up to 24 V. For SLI applications, the best technology choice is lead–acid batteries. The main advantages of lead–acid batteries in such applications are their low cost, ability to provide high power, and their potential for recycling. The huge number of produced cars need a technology that is environment friendly and has a good recycling capacity. Lead–acid batteries in this view are the best candidate and are superior over their competitors. Lead–acid batteries made for SLI purposes can be found from 100 to 1200 Wh in the market, though higher capacity can be easily provided by lead–acid cells. The older technology was flooded lead–acid batteries, but in recent decades, gelled and sealed types are more common. SLI batteries are used in cars, trucks, buses, lawn mowers, wheelchairs, and robots. It is interesting to note that even in modern electric cars, the need to SLI batteries is not eliminated. Since all the electrical devices and equipment of cars are fabricated according to 12-V power source, but the electric car voltage is about 230 V. Therefore, for electrical devices such as lights, radio, indicators, and other electronic parts, we need either a voltage regulator, which is expensive, or a 12-V SLI battery, which is the selected choice.

1.1.4 Vehicle traction batteries For a hybrid or fully electric car, batteries are extensively used to store and provide the required energy. Even in fuel cell cars, in which the main source of energy is provided by a fuel cell, batteries are needed to provide the required power for accelerating or hill climbing. Since the power demand is very high, usually NiMH or lithium batteries are the best candidates. Nowadays, even NiMH cells are not very common, and most modern cars are using lithium-based batteries. The traction battery cells are packed together to provide the necessary power and energy demand. The energy of a power pack may be in the range of 20 to 630 kWh, although any other requirement can be fulfilled by using more cells in series or parallel. The packs can provide electric energy for electric vehicles, hybrid electric vehicles, plug-in hybrid electric vehicles, fork lift trucks, mild floats, locomotive, and so on.

12

Simulation of Battery Systems

For small urban cars, traction lead–acid batteries are also available since they are less costly comparing to lithium cells. Moreover, the used cells can be changed by new ones with less cost because the lead–acid manufacturers almost totally recycle the batteries (over 90%). In this case, lead–acid traction batteries are economical and are used in small cars, where the engine does not require lots of energy. The only drawback of lead–acid batteries is that they are heavy, and if the engine power is very high, the weight of the required battery becomes a problem.

1.1.5 Stationary batteries Stationary batteries are usually used for voltage regulation or storage of a large amount of energy. Hence the best choice is lead–acid batteries due to their low cost, but recently lithium batteries are also becoming popular. Among different types of lead–acid batteries, tubular shapes are the best candidate for this purpose because the tubular lead–acid batteries have a very long life cycle and can be under service for more than 12 years. Large arrays of battery cells can be put together to store and deliver huge energy of 5 MWh or even more. Applications of stationary batteries are not limited to emergency power, local energy storage, remote relay stations, communication base stations, and uninterruptible power supplies (UPS). In critical stations, such as hospitals, data centers, monitoring and communication centers, weather stations, and so on, stationary batteries are crucial for power backup. In stationary applications, the weight is not a problem, but the main issue is the thermal management and charging control. In a megawatt-scale storage system in a long period of service life, not all the cells have the same age or even the same capacity. The array may contain different batteries from different manufacturers, different capacities, different ages, and even unequally utilized cells, all of which require special attention, and a robust controller should manage all the cells to ensure that each cell is under good charging and discharging conditions. Moreover, active monitoring system assists the technical team to have a proper control over the whole system. Nowadays that lithium batteries are becoming less and less expensive, and thus they become economical to use as stationary batteries. If the battery energy goes below 150 $/kWh, lithium batteries become practically viable in stationary purposes. Whether lithium batteries are economic or not, their charging management is much more important than that of lead–acid batteries because lithium batteries are more sensitive to charging algorithm and may explode if are not well treated.

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For stationary purposes in large scales, Edison (nickel–iron) battery is also a good candidate. The main advantage of Edison batteries is their very long cyclic life and also their tolerance in bad environmental conditions and even improper charging. Besides their good characteristics, they are more costly and less efficient than lead–acid batteries. In this case, economical issues become important to make a decision whether to choose lead–acid or nickel–iron technology. A great candidate for energy storage in large scales is vanadium batteries. Vanadium battery is a flow battery invented by Professor Maria SkyllasKazacos at the University of New South Wales in the 1980s. It has many advantages: • Flow batteries, including vanadium battery, can offer unlimited energy capacity because the energy content of the battery is independent of its power. A large tank of vanadium salt connected to the cell can be operational until the slat is available. The larger the thank, the more the energy content. • Vanadium salt can be stored for a long period without any considerable degradation even if it is completely discharged. • The electrolyte of the cell is aqueous; hence it is safe in comparison to lithium batteries. • Vanadium is not toxic. • Vanadium redox batteries can be discharged up to 90%. • The cyclic life of vanadium batteries are about 15,000 to 20,000, which is much more than that of conventional solid-state batteries (about 2000 cycles). The main disadvantage of vanadium redox batteries is that they are very heavy and have poor energy-to-volume ratio. In other words, the specific energy and energy density of the cell are low; hence, to store a large amount of energy, the device would be large and heavy. But its long cyclic life makes it very attractive for large-energy storage systems.

1.1.6 Military & aerospace wide range Some special batteries are available for applications where the cost is not an issue. For example, for space applications such as satellites and space probes, cyclic life is much more important than the cost of the battery. In this case, Nickel–Hydrogen (NiH2 ) is a good candidate. The cyclic life of NiH2 batteries exceeds 20,000 cycles, which have made them very attractive. In addition to their cyclic life, their specific power is about 220 W kg−1 , which

14

Simulation of Battery Systems

is larger than that of lead–acid types (typically 180 W kg−1 ) but less than of lithium batteries (between 250 to 340 W kg−1 ). For military purposes such as missiles or torpedoes, the cost is not important, but rather the energy and power density, as well as specific energy and power, are more critical. In these applications, zinc–silver oxide batteries are very common. Other types such as fuel cells, water activated batteries, and other alternatives are used.

1.2 Terms and definitions Some basic terms and definitions are important in studying the batteries. Here we briefly discuss these terms.

1.2.1 Capacity The capacity of a battery is the amount of electrical charge that can be stored and released by the cell. The unit of capacity is Ampere-hour or Ah. Determining the capacity of a cell is not easy because it depends on many parameters. The details of its measurement are postponed to Chapter 2. The capacity and energy of the cell have a close relationship but are not the same. The energy content of a battery is the product of its capacity and its voltage: E = C × V,

(1.1)

where E is the energy content in Watt-hours, C is the capacity of the battery in Ampere-hours, and V is the voltage of the battery in Volts. The conventional unit for measuring energy for batteries is Watt-hour, which is equal to 3600 Joules. The capacity of a pack of battery, which consists of many identical cells in series, is the same as the capacity of a single cell because the same amount of current passes through all the cells. However, the capacity of the pack increases if the cells were connected in parallel because in parallel connection the current passing through the pack is divided between the cells; hence a portion of the total current actually passes through each cell. Consequently, for increasing the capacity of a battery pack, the cells should be connected in parallel. In both cases the energy content of the pack increases because its energy content is the sum of the energy contents of the individual cells, regardless of their connection type. Example 1.3. Calculate the capacity and energy content of an alkaline battery cell with 1.2 V and the capacity of 2100 mAh.

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Answer. The capacity of the cell is 2100 mAh as expected. The energy content is E = 2.1 × 1.2 = 2.52 Wh. Example 1.4. If in a battery pack, ten pieces of the same alkaline cell of the previous example are connected in series, calculate its capacity and energy content. Answer. The capacity of the cell is again 2100 mAh because the cells are connected in series. However, the energy content is equal to the product of the capacity and the voltage of the pack. Since the cells are connected in series, the pack voltage is equal to Vpack = 10 × Vcell = 10 × 1.2 = 12 V. Then the energy content of the pack is Epack = 2.1 × Vpack = 2.1 × 12 = 25.2 Wh. It is clear that the energy content is the sum of the energy contents of all cells. Example 1.5. Repeat the previous example, but this time the cells are connected in parallel. Answer. Since the cells are connected in parallel, the capacity of the cell is the sum of the capacities of all cells, or Cpack = 10 × Ccell = 10 × 2.1 = 21 Ah. In a parallel connection the voltage of the pack is equal to the voltage of a single cell. Then the energy content is Epack = Cpack × Vpack = 21 × 1.2 = 25.2 Wh. We can see that the energy content of the pack is the same as in the previous example. In other words, the connection of the cells does not change the energy content.

16

Simulation of Battery Systems

1.2.2 Energy and power The energy and power are two completely different characteristics of any energy storage device. Sometimes these terms are mistakenly used interchangeably although the two terms are not related to each other. More specifically, the energy content of an energy storage device is a measure of its ability to store a specific amount of energy, whereas the power of the device is a measure of its internal resistance. A system with small internal resistance can provide lots of energy in a very short period. Technically, the unit of energy is Joule, and the unit of power is Joule per second or Watt. The definitions show that power is the amount of energy that a device can deliver in a unit of time. If the internal resistance of the system is very high, then it cannot provide a lot of energy in a unit of time; but the low internal resistance makes it possible for the device to deliver all its energy content in a short time. It should be kept in mind that if a system stores a lot of energy, then it is not necessarily a powerful storage system. The system may store a huge amount of energy but in a long period. Also, it may release the energy by a low rate in a long time. This system is considered a low power but with high energy content. The same system may be able to provide a very high power if its internal resistance is very low. On the other hand, a system with low energy content may be able to release all its energy content in a very short time. As a famous example, we can refer to electrical capacitors that store a small amount of energy, but they can release all of it with a spark in milliseconds. There are also low-energy storage devices that have low power. Summing up, the energy content and power of the system are two completely distinct quantities with no relation between them. There are many batteries that can store lots of energy but can provide low power. For an example, zinc–carbon AAA batteries have large internal resistance, and if we connect the positive and negative electrodes by wire, then its voltage drops, and the wire does not show any spark. However, even for the smallest lithium batteries, if you connect the poles by wire, then it sparks, and the wire melts. Now by connecting some AAA cells in series, the energy of the pack becomes larger and larger, but the internal resistance still prevents from delivering high current.

1.2.3 Specific values versus density Although the energy and power of a storage device are important parameters, they are almost meaningless. In fact, what is important is their

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associated specific value and density. The specific value and density of any system is defined as follows. Specific value of any quantity of a system is equal to that quantity divided by the mass of the system. Density of any quantity of a system is equal to that quantity divided by the volume of the system. Again, these parameters are quite independent of each other, and both are important by themselves. In practice, a battery having a specific amount of energy may be very heavy, but its volume may be very small, or the volume is big, but it may be very light. The best case is that the battery is both light and small, which is one of the main challenges of battery manufacturers. Example 1.6. For a 18,650 lithium cell with m = 50 g weight and C = 3250 mAh capacity, calculate the specific energy and energy density. Answer. Let us first calculate the energy content of the cell. From Eq. (1.1) we have E = C × V = 3.25 × 3.6 = 11.7 Wh. In this example, we assume that the open-circuit voltage of the lithium cell is V = 3.6 V. The specific energy of the cell is em =

E 11.7 = = 234 Wh kg−1 . m 0.05

For the calculation of energy density, first, we have to calculate the volume of the cell. As is defined by IEC standard, the 18,650 cells are cylindrical cells with diameter 18 mm and height 65 mm. Hence V – cell = π

1.82 × 6.5 = 16.54 cm3 . 4

Then the energy density of the cell is ed =

E 11.7 = = 0.7073 Wh cm−3 = 707.3 Wh L−1 . V – 16.54

Conventionally, the density of energy or power is expressed as the quantity per liter.

18

Simulation of Battery Systems

1.2.4 State of charge and depth of discharge Each battery cell has a specific capacity. This means that the cell cannot store more energy than its capacity. When the cell is discharged, the capacity of the cell decreases. The ratio of the remained capacity to the capacity of the cell at its maximum charged state is called the state of charge and denoted by SoC. Mathematically, SoC is defined as SoC =

Ca , Cn

(1.2)

where Ca is the available energy, and Cn is the nominal cell capacity. The nominal capacity is the value declared by the manufacturer and slightly differs from the actual capacity of the cell. By this definition, when the cell is fully charged, SoC = 1, and when the cell is completely discharged, SoC = 0. The reason for selecting the nominal cell capacity instead of the actual capacity is that the actual capacity of individual cells is not equal to each other even if they are manufactured in the same production line, because the manufacturing tolerances make differences in cell capacities. In most cases the nominal capacity is less than the actual capacity since the manufacturers need to guarantee their products. Therefore in some reports and researches, we see that SoC exceeds unity. Analogously to SoC, the depth of discharge or DoD is defined as the amount of used energy of a cell. In other words, when the cell is fully charged, DoD = 0, and when the cell is fully discharged, we have DoD = 1. By this definition we have DoD = 1 − SoC .

(1.3)

DoD is mostly used when the deliverable energy is to be calculated. For instance, in the calculation of the available energy of traction cells, DoD is very important. The deliverable energy of a cell completely depends on its technology and chemistry. For example, SLI batteries should not deliver DoD < 0.8, but sealed lead–acid batteries can reach DoD = 0.5. Lithium-based batteries can provide a large depth of discharge about DoD = 0.85. This means that lithium-based batteries can deliver 85%, but SLI lead–acid batteries can only provide 20 percent of their stored energy. In most battery technologies, the cell is considered to be discharged if DoD = 0.5. This means that we have to charge the battery after consuming 50% of its energy. If we do not pay attention and discharge the cell over that limit, the cyclic life of the cell dramatically decreases.

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Figure 1.2 Charging curve of Example 1.7. (A) Charge. (B) Discharge.

1.2.5 Charge efficiency Batteries are energy storage devices. By this definition we have to store a specific amount of energy in a battery and use that energy whenever it is required. However, the charging and discharging processes are not reversible. In other words, we lose some energy in charging and some other in discharging. From the balance of energy point of view, a battery does not deliver the same amount of energy that was used for its charging. The difference determines the charging efficiency, which mathematically can be written as Ecell ηch = , (1.4) Ein where Ecell is the delivering energy of the cell, and Ein is the total energy used for its charging. Example 1.7. A DC charger made of a USB port applies 0.87 A to a 18,650 lithium cell battery. The charging profile of the cell is shown in Fig. 1.2A. The cell is then discharged according to the curve shown in Fig. 1.2B. Calculate the charging efficiency of the cell. Answer. As stated, the charging efficiency is the ratio of the delivered energy to the energy used for charging. Since the charger is a USB port, its terminal voltage is VUSB = 5.0 V. The charging profile shown in Fig. 1.2A indicates that the cell took 3 hours for charging, knowing that the charger provides constant current equal to 0.87 A. Therefore the energy provided by the charger is EUSB = VUSB IUSB × t.

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Simulation of Battery Systems

Table 1.4 Approximate values for charging efficiency of some conventional batteries. Cell chemistry Charge efficiency

Lead–acid Nickel–iron Nickel–metal hydride Lithium cobalt oxide Lithium–titanate Lithium iron phosphate Lithium manganese oxide Lithium nickel cobalt aluminum oxide Lithium nickel manganese cobalt oxide

50–92 65–80 66 90 85-90 90 90 90 90

Substituting the data yields EUSB = 5 × 0.87 × 3 = 13.05 Wh. The delivered energy is the integral of the power of the cell. Since the current is constant, according to Fig. 1.2B, we can integrate the voltage curve and then multiply it by the current Icell = 2.5 A. For integration, we can use the famous trapezoidal rule since we have discrete data: 

Ecell = Icell 0

1.5



t=1.2



 1.5 − 0 Vcell dt = 2.5 V (0) + 2 V (t) + V (1.5) . 2×5 t=0.3

Substituting the data, we have Ecell = 2.5 × 0.15 (3.6 + 2(3.5 + 3.4 + 3.2 + 2.9) + 2.4) = 12 Wh. Thus the cell requires 13.05 Wh energy for charging and delivers 12 Wh energy. Consequently, the charging efficiency of the cell is ηch =

12 = 0.91954023, 13.05

which is approximately equal to 91.95%. Table 1.4 lists some typical values for conventional batteries.

1.2.6 Side reactions Each battery technology has a pair of electrochemical reactions at its both electrode upon which the battery is made. These are called the main or

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primary reactions. For many purposes, manufacturers mix many different materials as additives to main active materials. Therefore, since in the cell a lot of different chemical species exist, they may react with each other and produce unavoidable results. For example, in aqueous electrolytes, the electrolyte may be dissociated into hydrogen and oxygen. These types of reactions are called side reactions. Side reactions usually are not favorable and make problems. Manufacturers usually try to find solutions for avoiding side reactions since they not only consume some energy resulting into a less efficient cell but also they may be considered as some failure modes in some cases. Especially in rechargeable batteries, side reactions may cause electrode passivation or internal short circuits by making dendrites.

1.3 Battery main challenges Regarding batteries, there are some major challenges that the manufacturers are trying to solve. The challenges strongly depend on the application and differ from application to application. For example, the weight is a major problem for portable devices, but it is not a big deal concerning stationary purposes. In this section, we discuss some main challenges in more detail.

1.3.1 Energy content As a storage device, the main role of a battery is storing a specific amount of energy. It is quite obvious that the more energy a call can store, the better the technology is considered. The stored energy on both specific or density scales is important in all applications, including portable or stationary ones. Regardless the application, we prefer to have a cell that has a larger capacity. The energy of a battery pack can be achieved by any amount simply by connecting many cells in series or parallel. Therefore we can have a battery pack capable of storing as much energy as we need. Hence the amount of energy is not important itself. The matter is the specific energy and energy density. As an example, if we need a longer range of electric car travels, then we can install more batteries. However, the more batteries are used, the more space we need, and the heavier the pack becomes. Therefore we have to make a decision on battery weight and volume and the required energy. In addition to these problems, the cost of the battery pack is significant. Developing batteries with higher energy storage capacity, on both mass and volume scales, is one of the main challenges of battery manufacturers. The higher energy content means less weight, less volume, and maybe less

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Simulation of Battery Systems

cost. More specifically, this parameter is vital when coming to portable devices and electric vehicles. The energy contents of some conventional storage systems are tabulated in Table 1.5 together with some chemical fuels for better comparison. The data in the table clearly show why producing electrical cars is still far from being accepted by the car manufacturers. The energy density and specific energy of fossil fuels such as diesel and methane are much greater than any conventional battery. However, some batteries such as metal–air batteries and, more specifically, lithium–air batteries theoretically have the potential for storing the same energy content as diesel. Manufacturers are trying to develop more energy-dense cells to achieve the goal.

1.3.2 Power Power density and specific power are also important parameters for storage devices such as batteries. It was stated before that a cell with higher energy content not necessarily can provide high power. Thus making batteries with higher power is considered as another challenge for battery manufacturers. In many applications the power is very low, and the energy content is of more interest. As such applications, we can name watches, remote controls, clocks, and so on. But some applications require high power. In some cases, the energy is not a choice, but the power is the main issue. For example, safety batteries in Formula 1 cars for ejecting the door in case of an accident require very high power, and their energy should be discharged in some seconds; therefore such batteries do not need very high energy. Power is measured in units of Watts. When we need to express a specific power, we use the unit of Watts per kilogram or W kg−1 , and when we are dealing with power density, the unit is Watts per volume or W cm−3 .

1.3.3 Self-discharge One of the characteristics of any storage system is its rate of self-discharge, that is, how fast the device is discharged without being actually consumed. All the energy storage mechanisms lose their stored energy if they are left unused. For example, the water stored in a hydraulic dam evaporates or penetrates in the ground. Hence its stored water decreases even if the dam is out of operation. As another example, the energy stored in a compressed spring is lost even if the spring is not used. This is because the elasticity of the material used for making the spring changes when it is compressed. Other examples can also be presented for all the storage devices. In the case

Table 1.5 Approximate values for energy density of some conventional batteries [3]. Storage material Energy type Specific energy Energy density (kJ/kg) (kJ/L)

Hydrogen (liquid)

Chemical

142,000

10,000

Hydrogen (at 700 bar) Methane

Chemical Chemical

142,000 55,500

9170 22,200

Diesel Coal Methanol Li battery (Li–Po, Li–Hv)

Chemical Chemical Chemical Electrochemical

48,000 30,000 19,700 1800

35,800 3800 15,600 43,200

Li–ion battery

Electrochemical

360–875

900–2630

Alkaline battery NiMH battery Lead–acid battery Supercapacitor Electrolytic capacitor

Electrochemical Electrochemical Electrochemical Electrical Electrical

500 288 170 10–36 0.01–0.2

1300 504–1080 560 50–60 0.01–1

Uses

Rocket engines, Fuel Cells, H2 Storage/Transport Fuel Cells, Natural Gas Heating Supplement Cooking, home heating, electric power plants Automotive engines, electric power plants Electric power plants, home heating Fuel engines Portable electronic devices, flashlights, RC vehicles Automotive motors, portable electronic devices, flashlights Portable electronic devices, flashlights Portable electronic devices, flashlights Automotive engine ignition Electronic circuits Electronic circuits

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Simulation of Battery Systems

Table 1.6 Self–discharge of some conventional batteries [4]. Battery chemistry Rechargeable Typical self-discharge or shelf life

Lithium metal Alkaline Zinc–carbon Lithium–ion Low self-discharge NiMH Lead–acid Nickel–cadmium Nickel–metal hydride (NiMH)

No No No Yes Yes Yes Yes Yes

10 years shelf life 5 years shelf life 2–3 years shelf life 2–3% per month As low as 0.25% per month 4–6% per month 15–20% per month 30% per month

of electrical batteries, self-discharge is also very important. We all noticed that if a car is left for a long time (about six months or more), then it will have a problem with starting because its SLI battery has lost its energy or the battery is discharged. Self-discharge in electrochemical batteries is due to chemical and electrochemical reactions that happen inside the cell. An electrochemical cell is in fact a reactor in which lots of chemical compounds are interacting with each other. The result of the interaction is the conversion of active material to passive materials, which cannot contribute to useful electrochemical reactions that provide electricity. There are some parameters that affect the self-discharge process, including battery type, charging current, temperature, state of charge, the existence of different materials and chemical agents, and some other factors. In primary batteries or cells, self-discharge is usually called shelf life. They should have very low self-discharge because they may be put on shelf before being used. Typical values of self-discharge is tabulated in Table 1.6 for some conventional cells. The unit of self-discharge is percent per month. Self-discharge accelerates with temperature because the temperature has a direct effect on enhancing the rate of internal reactions. Therefore the storage of batteries in a cold environment enhances their shelf life. It is accepted that the state of charge is also important in losing capacity during storage life. For example, in researches, it was observed that lithium cells keep their energy content during a long period of shelf storage if they are charged at SoC = 0.58. Other batteries have a similar situation.

Battery technologies

Table 1.7 Shelf life of some conventional batteries. Battery type Self-discharge

Nickel Metal Hydride Nickel–Zinc Nickel Cadmium Rechargeable Alkaline Alkaline Lithium Carbon Zinc/Zinc Chloride

Slow-Medium Fast Fast Very Slow Very slow Very slow Fast

25

Temperature ◦ F (◦ C)

Shelf life in years

-4–122 (-20–50) -4–140 (-20–60) 22–140 (-30–60) -4–140(-20–60) 0–131 (-18–55) -40–140 (-40–60) 0–130 (-18–55)

5 1 5 4–7 10 7–15 2–3

1.3.4 Shelf life Even if a battery is not used, it ages. Therefore, if an expired cell is bought, then it may not have enough capacity as expected. The shelf life depends on battery chemistry, temperature, its state of charge during storage, and, of course, time. The storage temperature is very important and depends on the chemistry. Each cell should be stored in a temperature range; otherwise, its capacity decreases dramatically. A cold temperature may also harm the cell chemistry. Hence we have to pay attention to the storage temperature. Table 1.7 tabulates the temperature ranges suitable for storing some typical battery technologies. The data in the table are for comparison only, and the exact range depends on the brand since each manufacturer adds some additive to its products, which can considerably affect the storage life.

1.3.5 Cyclic life In addition to shelf life, the cyclic life is also very important in rechargeable batteries. Secondary batteries are produced for repeated usage: each battery technology can be charged and discharged within a limited cyclic number, which is known as the cyclic life. It is clear that a battery loses some of its active material in each cycle; hence, after some cyclic usage, the cell cannot accept the same energy as it did when it was fresh. The cyclic life depends on some factors such as battery technology, depth of discharge, cyclic condition, and temperature. Some batteries can provide very low cyclic life about 100 cycles, whereas some others can provide more cyclic numbers as much as 2000. It is quite clear that the higher the cyclic life, the better the cell. Hence the manufacturers are trying to increase the cyclic life as much as possible.

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Simulation of Battery Systems

It is quite accepted that cyclic life strongly depends on how deep the discharge takes place. The deeper the discharge, the lower the cyclic life. For this reason, the applications should be designed such that they try to recharge the batteries in lower DoDs.

1.3.6 Memory effect When a secondary battery is repeatedly partially discharged and charged, some of its active material becomes passive, and the cell loses its capacity. This effect is known as the memory effect, battery effect, or battery memory . The effect was first observed in Ni–Cd batteries used in aerospace applications but it is not limited to Ni–Cd cells. Other battery types suffer from memory effect, but their rates differ from technology to technology. Memory effect can be prevented by occasionally fully discharging the cell. In other words, let all active materials contribute to charging and discharging processes. Also, repeatedly programmed charge and discharge increase the memory effect. Hence applying a random program will reduce the capacity lost. In any case, it should be noted that the cells that have lost their capacity due to the memory effect can be recovered to their full capacity if they are discharged to very low voltages and then fully recharged. Finally, it should be noted that the memory effect should not be misunderstood with cyclic life. A battery that was charged and discharged for a long time becomes old and cannot be recovered. The cyclic life completely differs from the memory effect. Many battery types do not suffer from the memory effect. For instance, lead–acid batteries have no memory effect and should not occasionally deeply discharged. In contrast, if a lead–acid cell is deeply discharged, it may get damaged and cannot be charged again. Lithium-based batteries also have no memory effect. This means that we can safely partially charge and discharge them without worrying about losing their actual capacity.

1.3.7 Maximum charging/discharging current Any battery technology has the ability to be continuously discharged by a specific amount of electrical current. If the cell is discharged by a higher current, then it may become unstable and explode. The same argument is true for charging current. If the cell is charged with a high current, then it becomes thermally unstable and in turn may explode. In addition, the high temperature may lead to capacity loss or aging of the cell. The maximum delivering current of a cell is measured by its C-rating current. By definition, the rating current nC, also known as nI, is the cur-

Battery technologies

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rent by which the cell is fully discharged in about 1/n hours. For example, when a cell is discharged by 2C current or 2I, it takes approximately half an hour to become fully discharged. Example 1.8. For an SLI lead–acid battery with 70 Ah capacity, determine 1C, 0.5C, and 2C rating currents. Answer. Since the battery capacity is 70 Ah, discharging the battery with I = 70 A will take about 1 hour for becoming fully discharged. Hence the 1C-rating current is I = 70 A. By analogy the 2C-rating current is I = 140 A, and the 0.5C-rating current is I = 35 A. If n is less than 1, it may also be written in front of the letter C. In other words, it is denoted by Cn, which means the current by which the cell is fully discharged in n hours. By this definition the 0.5C-rating is the same as the C2-rating.

1.3.8 Charging time Charging time is one of the main challenges of secondary batteries. This parameter indicates how long it takes for a fully discharged cell to be fully charged. In portable applications, such as cell phones, laptops, and similar cases, the charging time is important. In some applications such as electric vehicles, charging time is one of the main challenges. In conventional fossil fuel vehicles, the charging time (filling an empty tank) is of the order of 5 minutes. However, the charging time of the currently available batteries is of the order of 3 hours. Battery manufacturers are trying hard to reduce the charging time and make it possible to charge a battery pack as fast as possible. The Achilles’ hill in fast charging is that a high input current causes thermal instability, which may lead to a battery explosion, and also if a battery is charged faster, then its cyclic life dramatically reduces. Hence fast charging reduces the cyclic life and safety of the cell. Huge research is performing to find suitable chemistry or configuration that is safe when it is under fast charging and does not affect the cyclic life.

1.3.9 Performance The performance of a battery is attributed to its ability to deliver its power and energy in different temperature ranges. The performance of any battery cell decreases in cold conditions. This means that any electrochemical battery delivers less power in a cold climate than at standard temperature. This

28

Simulation of Battery Systems

fact is very important for applications that are installed in cold climates. For example, the battery used in electric vehicles should deliver enough power when the ambient temperature is below zero Celsius. In fact, the performance of the conventional lithium batteries that are used in electric cars dramatically decreases, and the car cannot start in cold weathers. At the present state, car manufacturers design necessary equipment to keep the battery warm in a reasonable range. The ability of the cell in working at elevated temperature is also important. In many cases the batteries are used in places where the temperature is very high. Then the cell should perform adequately without being destroyed. For example, the batteries of electric cars that are working in deserts may experience temperature ranges over 50◦ C. If we add its temperature rise due to its electrochemical reactions, then the cell temperature may exceed 60◦ C. At this temperature the battery should have proper performance. Some batteries cannot withstand such a high temperature and fail. Therefore, for these applications, special attention should be paid so that the batteries be kept in good condition or selecting proper batteries that can withstand the high temperature.

1.3.10 Safety and reliability Since batteries are electrochemical cells, they contain chemical compounds, most of which are made from acid or base. Hence if they explode, then their materials cause damage to its surroundings, some of which are also toxic and may harm human life. Consequently, the safety of batteries is very important and should be confirmed by manufacturers. One of the most known failure modes of batteries is thermal runaway. When a battery is discharged or charged, heat is generated inside the battery due to (a) electrochemical reaction at electrode/electrolyte interface and (b) Joule heating. If the cell is properly designed, then the generated heat is dissipated into ambient, and the cell temperature reaches a moderate value. But if the generated heat cannot be dissipated, the cell temperature rises. Then the elevated temperature causes more electrochemical reactions, which in turn produce more heat. The interaction between the electrochemical reactions and temperature rise in contact with Joule heating will generate thermal runaway. In aqueous batteries such as lead–acid cells, thermal runaway will result in melting of the case which in turn causes sulfuric acid splashing out. This phenomenon is even more dangerous in lithium batteries since lithium

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metal is very explosive, and the explosion is powerful. This safety issue is why special regulations are passed not to allow lithium cells to be shipped by airplanes. It should be noted that increasing safety is usually in contrast to attaining more energy content. For example, the invention of lithium-ion and lithium polymer batteries is mainly due to safety reasons, whereas lithiumion batteries are safer than lithium batteries, but they have a less specific energy. Many pieces of research are conducting to develop safer batteries. These attempts are targeting the electrode chemistry, additive, membranes, and cell configurations.

1.3.11 Cost All the mentioned challenges have some solution, but what has made them not working is cost and economic issues. Achieving all the mentioned challenges has meaning only if the solution is economical. Otherwise, the market cannot accept the solution. There may be many theoretical solutions confirmed in labs, but they are not practically applicable since the cost of the cell becomes very high. For example, titanium oxides are used in some lithium batteries to increase the cyclic life, but the cost of the cell has become very high. The resulting cells have great performance but are expensive and are used in special applications. As another example, for better performance, some papers suggest that platinum is used in the lead–acid batteries. Although the tests show really better performance, but the cost of the used platinum is double the cost of the whole battery itself. Therefore the solution will not be applicable for lead–acid batteries. The customers prefer to buy a much cheaper battery with less performance.

1.3.12 Recycling The last challenge discussed in this chapter is recycling potential. Since the batteries are used in a huge number, recycling is quite important. One of the benefits of lead–acid batteries (despite its low energy content) is its recycling capability. About 90% of a lead–acid battery can be recycled and reused for making new batteries. In contrast, Ni–Cd and lithium-based batteries are not very efficient in recycling. Just to imagine how important battery recycling is, we can refer to statistical data in countries. For instance, 680 million batteries were used in the UK in 2001 [5], and most of these (89%) were general purpose batteries. Currently, only a very small percentage of consumer disposable

30

Simulation of Battery Systems

batteries are recycled (less than 2%) and most waste batteries are disposed of in landfill sites. The rate for recycling of consumer rechargeable batteries is estimated to be 5%. In the UK the average household uses 21 batteries a year, and the UK generates 20,000–30,000 tonnes of waste general purpose batteries every year, but less than 1000 tonnes are recycled. The same statistics can be obtained for other countries. The data show that recycling of batteries is very important, and if they are not recycled, then landfill sites become full of batteries in the near future. For instance, by electric cars becoming more and more practically available in the market, the need for battery recycling is felt more and more.

1.4 Conventional battery technologies Different battery technologies have been manufactured for different purposes. All the batteries can be categorized as primary (nonrechargeable) and secondary (rechargeable) types. According to the many challenges discussed before, some batteries have successfully entered the market, but many other batteries are still under investigation and will emerge in near or far future. In this section, we briefly introduce some important and famous already commercialized battery technologies.

1.4.1 Lead–acid (LA) Lead–acid battery (LAB) can be considered as one of the oldest successful inventions in battery history. The cell was invented by Gaston Planté in 1859 with a lead–dioxide as the positive and pure lead as the negative electrode and sulfuric acid as the electrolyte. The invention opened a new window to the world of rechargeable batteries. Since the invention of lead–acid batteries, the chemistry remained the same, and just some modifications have been made to increase the efficiency and performance of the cell. Nowadays, LABs are still in the first position in the number of production. Note that every produced vehicle needs a lead–acid battery and needs a new one every three years. Besides the automotive industry, stationary applications, large-scale energy storage, storage of electricity produced by renewable energy sources, and other applications require LABs. There are different types of lead–acid batteries for different purposes: SLI LAB Starting–Lighting–Ignition (SLI) batteries are used for the automotive industry. Each cell consists of several thin positive and negative electrodes with separators in between. This configuration enables the cell to produce a high current, which is suitable for cranking.

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31

Sealed LAB Sealed lead–acid batteries are used for deep discharge, which is applicable in uninterrupted power supplies, solar systems, and so on. The electrolyte of these batteries is either confined in absorbed glass mats (AGM) or is gelled by adding silica particles to the electrolyte. Tubular LAB Tubular LABs are extensively used in large energy storage systems. In tubular LABs the positive electrode is made of tubes in which positive material is filled. The tubular plates are positioned in front of flat negative plates. This configuration prevents the shedding of the positive electrode and enhances its service life.

1.4.2 Nickel–cadmium (NiCad) Nickel–cadmium or NiCad batteries are very famous due to their low cost and durable characteristics. They can be seen in many high-power applications, including portable vacuum cleaner, shavers, cameras, music players, wireless telephones, toys, and other similar devices. One of the main advantages of NiCad batteries is that their charging time is very low. Another advantage is that they can operate quite well in low temperatures. The nominal voltage of a NiCad cell is about 1.2 V. The positive electrode of NiCad batteries is made of Ni(OH)2 /NiOOH, the negative electrode of Cd(OH)2 , and the electrolyte is made of KOH. In some batteries, a porous nickel electrode is used to enhance the energy density of the cell. NiCad batteries have some disadvantages including: • Energy density and specific energy of NiCad batteries are lower than other candidates such as NiMH and Li-ion cells. • The memory effect of NiCad batteries is very high. Hence the applications in which NiCad batteries are used should occasionally be fully discharged. • Cadmium is a toxic metal, and hence producing NiCad batteries in huge amount has environmental effects.

1.4.3 Nickel–metal hydrate (NiMH) The Japanese company Sanyo, introduced NiMH cells in 1990, which became a competitor for NiCad batteries. The chemistry of NiMH batteries is very close to NiCad, and the operational voltage is in the same range at about 1.2 V. Since the energy content of NiMH batteries is higher than that of NiCad cells, NiMH cells are used in cell phones, notebooks, shavers, and other portable applications where the energy content is very important. NiMH cells were also used in the driving first generation of electric

32

Simulation of Battery Systems

vehicles, but nowadays, lithium batteries are the choice of interest in this field. The positive electrode of NiMH batteries is the same as in NiCad ones, but the negative electrode is made of a metal hydride instead of cadmium. The electrolyte is also the same as used in NiCad cells. In general, two groups of metals are used in making the negative electrode, namely AB2 such as titanium and zirconium and AB5 group that mainly consists of earth metals such as nickel and lanthanum. Nowadays, all the NiMH cells are made of AB5 metals due to their better performance. There are some differences between NiMH and NiCad batteries, including: • The energy content of NiMH is greater than that of NiCad cells. • The environmental effect of NiMH cells is lower than that of NiCad batteries. • NiCad batteries have less self-discharge rate than NiMH ones. In NiMH cells the produced hydrogen releases, and the more hydrogen is released, the greater rate of self-discharge. • The NiMH cells almost do not suffer from memory effect, whereas the NiCad batteries are prone to memory effect. • NiCad batteries are more tolerant to overcharging. If a NiMH cell is overcharged, then its active material becomes destroyed, but NiCad cells do not show such a behavior. • The charging process in NiMH batteries is exothermic, and the cell temperature rises during charge, but in NiCad batteries the charging process is not exothermic, and the cell temperature remains almost constant.

1.4.4 Lithium-ion (Li-ion) The first lithium-ion battery was introduced by the Japanese company Sony in 1991. The chemistry of this type of battery completely differs from other types, since in lithium cells the electrochemical reactions take place in an intercalation or insertion process. The cell voltage of Li-ion batteries is about 3.6 V, but it strongly depends on the chemistry of the cathode and anode materials. The high voltage of the cell limits the electrolyte to nonaqueous ones since the aqueous electrolytes dissociate in such a high voltage. Li-ion cells have high specific energy and power since lithium is a light metal. Therefore the cells are used in electric vehicles, notebooks, cell phones, and any other portable devices. In many cases, NiMH batteries

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33

are replaced by Li-ion ones because they provide higher energy density and specific energy. The negative electrodes of Li-ion batteries are usually made of graphite, but many different materials are also under investigation. The positive electrode is made of some metal oxides that can provide enough space for the intercalation process. These metal oxides include LiCoO2 , LiMn2 O4 , and LiNiO2 , but many other materials are experienced each day to find a better chemical compound. The main advantages of Li-ion batteries are: 1. The energy density and specific energy of Li-ion cells are higher than of any conventional batteries. 2. They do not suffer from the memory effect. 3. Their cyclic life is very high. The main disadvantages of Li-ion batteries are: 1. They are sensitive in charging algorithm, and if overcharged, they tend to explode. 2. Due to their tendency to explosion, a sophisticated controller should be assembled for their charging. 3. The temperature of the cells should be monitored for safety reasons. 4. They are more costly than NiMH cells.

1.4.5 Metal–air Metal–air batteries are a mature family of primary and secondary cells. In metal–air batteries the positive electrode is carbon–based covering with some precious metals for reacting with oxygen. The other electrode is made of a metal such as zinc, aluminum, magnesium, and lithium. Since in these batteries, the air is flowing through the cell, they are sometimes categorized as fuel cells. Among metal–air batteries, lithium–air has unique characteristics in concerning the fact that its energy content is almost equal to diesel fossil fuel (see Table 1.5), in other words, about 100 times the present conventional lithium–ion batteries. This fact makes lithium–air attractive since by having a 50 kg battery pack, we can drive about 500 km with only one charge! That sounds very interesting. At the present stage, lithium–air batteries with 6000 kJ kg−1 have been demonstrated that its energy contents are five times the current conventional lithium–polymer versions. However, their cyclic lives are low and cannot be incorporated into electric cars.

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Simulation of Battery Systems

1.5 Future technologies In addition to commercialized batteries, different research groups are working on different aspects of batteries to overcome the main challenges discussed before and deliver a battery with better performance. Since there are very different parameters affecting battery performance, and some of which are in contrast to each other, the research is very vast. Different materials are being tested, different configuration, additives, different technology, and many different aspects are being tested each day. The final goal of these efforts is to enhance all the characteristics discussed. Here we introduce some advanced cells that are tested and verified in labs. These batteries are not still available in the market since they are either expensive or have some problem that makes them unattractive for customers. Scientists are trying hard to make them economically affordable and enter them into the market. It is obvious that the technologies that are listed here are just some examples of huge researches that are continuously taking place at different labs.

1.5.1 Advanced Edison Ni–Iron battery Edison battery or Ni–Iron (NiFe) was one of the first battery cells that were used in electric vehicles before the industrialization of internal combustion engines. Edison believed that NiFe is the best battery because it is very tolerant to charging and discharging cycles, and its cyclic life is unlimited. In fact, NiFe batteries made by Edison never died, and he was not able to obtain their cyclic life. However, the batteries were heavy and were not able to deliver high power. Consequently, to provide the necessary power, many cells were used to be connected in parallel. This arrangement made the overall battery pack even much heavier. In recent years, Stanford university used nanowires plated by Ni and Fe as the electrodes. The result was a NiFe battery, which was able to be charged in only 2 minutes! In addition, it can provide a very high power such that the cell can be fully discharged in only 30 seconds. The scientists believe that the created cell is also cheaper than the current advanced batteries, but the only disadvantage is that it decays like Li batteries! The characteristics of this battery make it absolutely a good candidate for electric vehicles. From all the challenges, this battery targets the charging time, cost, and specific power. No other information still exists about other specifications.

Battery technologies

35

1.5.2 Advanced lead–acid battery To increase the specific energy of lead–acid batteries, scientists at Australia’s Commonwealth Scientific and Industrial Research Organisation have combined LAB with a graphite plate that acts as a capacitor. The result is a low-cost, fast charging LAB with a higher cyclic life. The battery is called the ultra battery and is entering the market. One drawback of ultra battery is its rapid voltage drop, which is attributed to the capacitor effect of the negative electrode.

1.5.3 Solid-state lithium-ion Scientists at Toyota published a paper in which a solid-state lithium battery is introduced. In this battery a superionic conductor is used as the conventional separator that makes the battery operate at super capacitor levels. A complete charge or discharge of the battery can be done in just seven minutes. This means that the cell has a very high power density and can be used in electric cars. If this happens, it takes just 7 minutes for an empty car to become fully charged just like currently available fossil fueled cars. In addition, scientists claim that the invented cell is far more stable and safer than current lithium–based batteries, and also it can operate at −30 up to 100 degrees Celsius. The invention shows that scientists are targeting various aspects of battery challenges namely (a) charging time, (b) cyclic life, (c) performance, and (d) safety. However, the invention is still suffering stability, and the electrolyte materials still pose challenges. Therefore they are not expected to be seen in the near future.

1.5.4 Gold nanowire batteries To increase the cyclic life of batteries, scientists at The University of California Irvine have used golden nanowires that are a thousand times thinner than a human hair. Conventionally, nanowires tend to break down when recharging, but the use of golden nanowires showed that they could withstand plenty of recharging. This invention leads to batteries that do not die! This discovery uses gold nanowires in a gel electrolyte to avoid break down. These batteries were tested recharging over 200,000 times in three months and showed no degradation at all. The invention has excellent potential for a replacement for current batteries. However, it is not used in a real battery.

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Simulation of Battery Systems

1.5.5 Grabat graphene batteries Graphene batteries have the potential to be one of the most superior available. Grabat company has developed graphene batteries delivering up to 500 miles on a charge. Moreover, the batteries can be charged in just a few minutes and have the potential to be discharged 33 times faster than lithium-ion cells.

1.5.6 Sodium-ion batteries Scientists in Japan have used sodium instead of lithium and made sodiumion battery to reduce the cost. Another advantage of sodium ion battery over lithium-based ones is that sodium is one of the most common materials on the planet rather than rare lithium. By using salt the sixth most common element on the planet, batteries can be made much cheaper. Commercializing of the batteries is expected to begin for smartphones, cars, and more in the next five to 10 years.

1.6 Modeling and simulation Developing new ideas in battery field requires scientific research that is carried on in two different ways, namely (a) experiments and tests and (b) modeling and simulation. Currently, the experiment is the most common way of studying and developing batteries. Each day, many batteries are tested in different labs all over the world, including cyclic tests, endurance tests, capacity tests, temperature tests, and so on. Although the main advances are the results of experimental tests, they are very costly and in some cases are not repeatable. Some tests such as cyclic life test are very time-consuming and require several months for conducting a single test. Moreover, some physical values either cannot be measured or are very hard and expensive. As another tool that is a complement for experimental devices is the theoretical investigation of batteries or, in other words, modeling and simulation. Based on fundamental laws of physics, modeling and simulation can give powerful tools for battery developers and manufacturers. Some important and critical parameters that are hard to experience can be obtained through simulation. There are two common misconceptions about modeling and simulation:

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1. Experimentalists think that simulation is a competitor and substitution for experimental tests. Hence they do not accept the results of numerical simulations and do not give value to simulations. 2. Some people who conduct simulations overbelieve their results without conducting experiments. In fact, both ideas are incorrect. The fact is that modeling and simulation should be considered as another lab tool or device. For conducting an experiment, there are many high-tech instruments available, each of which is calibrated for a specific purpose. Modeling and simulation should be considered as another high-tech device or instrument for battery testing and obtaining valuable data that are sometimes very hard to obtain. Up to now, many different modelings have been proposed and developed, each having its own pros and cons and can fulfill only some aspects of battery behavior. In general, the modeling targets can be categorized as follows. Design These models are important for battery developers and manufacturers. The main goal of models that are in this category is to increase battery performance in different aspects such as reducing weight, increasing energy and power, reducing cost, and studying some physical characteristics such as thermal runaway and electrolyte stratification. There are many parameters that can be tuned via such modelings including: 1. Material and types of electrodes. 2. The thickness of electrodes. 3. Size and geometry of the cell. 4. Electrolyte composition. 5. Case and cap of the batteries. 6. Material and specification of separators. 7. Grid size, shape, and material. 8. Battery setup. The tuning parameters dictate that the only modelings appropriate for this purpose are those that are able to consider the above-mentioned parameters. For example, models based on governing equations of battery dynamics are quite applicable. These models are usually simulated using numerical methods available in computational fluid dynamics (CFD) sources. Control In some applications, the model is used for control purposes. For example, models that are used in electric vehicles, load balancing, and so on do not worry about the type of the cell. But they deal with

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the behavior of the cell and the role of the cell as a component in a very large assembly. These models should be simple and very fast since real-time modeling is always a case. There are many different types of dynamic models that are suitable for these purposes, such as black boxes or neural network models. For this purpose, CFD-based models are not appropriate because they are very time-consuming. Monitoring Many battery models are used for monitoring purposes. These models must be very fast since monitoring is always a realtime problem. The models based on black–box or electrochemical model are quite appropriate for this purpose. There are many mathematical models available in open literature, each suitable for some purposes. Each method has its own pros and cons and should be used with care. Here we discuss some common models in more detail. Equivalent circuit modeling (ECM) Any energy system can be modeled with an equivalent circuit model or ECM. Batteries as energy systems can also be modeled using ECM. In this modeling, each physical phenomenon such as electrical resistance, thermal resistance, capacity, and electrical double-layer is modeled using an equivalent electrical element such as a resistor, a capacitor, an inductor, or a combination of them. The whole system then resembles the battery behavior if the value of each element is properly obtained. To obtain the parameters of electrical elements, we need some experimental tests, including current interrupt or electrochemical impedance spectroscopy (EIS). Therefore lots of tests are required to obtain reliable values for electrical elements. ECM is a very simple model, and lots of circuits can fit a battery; hence different values are obtained for each element in different models. One of the advantages of the model is that once the parameters of the model are obtained, the model can predict the cell in a very short time. Hence ECM is adequate for control systems and monitoring. On the other hand, since the model depends on the experimental tests and the value of elements is obtained using currently available cells, the model is not usually used for design purposes and cannot be used for developing new batteries. Black-box modeling In black-box modeling, the battery is viewed as a black box as is clear from its name. The black-box has an equivalent transform function with several inputs and several outputs. The

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function should be trained by an experimental test to obtain a proper transfer function. Usually, intelligent high-level methods such as neural networks are used in black-box models. Black-box models are extensively used in control systems and applied in different battery technologies such as lead–acid, lithium-based, and NiMH ones. Like ECM, this method also depends on the experimental test on an available battery and has a very fast response time. Hence black-box methods are also useful for monitoring and control purposes. Another advantage of the model is that by using sophisticated models the effect of many complex input parameters such as operational conditions, cell temperature, and state of charge of the cells can be trained. The disadvantage of the models is that the blackbox models give accurate results if the inputs of the cell are within the trained range and cannot guarantee to give the correct answer if the inputs of the system are out of the range. Fundamental governing equations Modeling on the basis of fundamental governing equations is another approach. In this model the governing equations of battery are derived based on basic physical phenomena and the fundamental laws. Governing equations can describe real physical phenomena by means of transport equations in the form of partial differential equations (PDEs). For studying each parameter, such as electrical potential or electrolyte concentration, one transport equation should be added to the model. The result is a system of PDEs whose solution gives the dynamic behavior of the battery as well as many distributed parameters, including porosity change along the electrodes, temperature gradient, electrolyte concentration gradient, and so on. The fundamental governing equations include: 1. Conservation of mass. 2. Conservation of momentum for simulation of electrolyte movement. 3. Balance of energy to obtain temperature gradient. 4. Conservation of chemical species for obtaining chemical species’ concentration gradient. 5. Conservation of electrical charge to obtain electrical potential inside solid and electrolyte phases. Since the governing equations are PDEs, numerical methods should be used for simulation. Fortunately, different numerical methods and schemes have been developed and documented in CFD-related books

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and references. The same methods have been used for simulation of governing equations of battery dynamics. Typically, numerical methods are very time-consuming and require a lot of computational resources. Hence the methods are not appropriate for online applications such as control and monitoring. However, the results of this type of simulation give very accurate information about internal physical phenomena and open new windows in the understanding of what is happening inside the cells. Thus the numerical simulation of governing equations can be viewed as a virtual lab beside the actually available labs.

1.7 Life cycle assessment With the increasing awareness of human societies about the environment, efforts have increased to control social activities that have serious impacts on the environment. The life cycle assessment (LCA), also known as life-cycle analysis or cradle-to-grave analysis, is a method to assess environmental impacts associated with all the stages of a product life cycle. This includes extraction of raw materials, transportation, processing, manufacturing, distribution, use of the product, repairing and maintenance, and recycling. Life cycle assessment is a successful and growing tool that, despite the absence of a single method and the need to consider assumptions and parameters for doing it, provides reliable and effective environmental results [6]. According to the international standard organization (ISO), LCA is defined as studies of the environmental aspects and potential impacts throughout a product’s life from raw material acquisition through production, use, and disposal [7] The general categories of the environmental effects needing consideration include resource use, human health, and ecological consequences [8]. In addition, the main objective of LCA is to compare the full range of environmental effects assigned to products and services by quantifying all inputs and outputs of material flows and assessing how these material flows affect the environment [8]. This information is used to improve processes, support policy, and provide a sound basis for informed decisions. The main boundaries for LCA can be classified as: • Cradle-to-grave: From raw material extraction through product use and disposal [9]. • Cradle-to-cradle: specific kind of cradle-to-grave from raw material extraction through product use and disposal and recycling, where recycled materials are a new product.

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• Cradle-to-gate: From raw material extraction to the factory gate. • Gate-to-gate: From a specific point of the life cycle to another particular

point in the life cycle. LCA has been a potential tool to assess the environmental impacts for various industries ranging from fossil fuel power generation systems, renewable energy systems to alternative fuels for transportation. The main intention of using LCA in several applications is to have better environmental insights when a real comparison is required. Compression of emission from electric vehicles (EVs) with gasoline-powered vehicles is a good example where LCA can significantly assist. In several studies, EVs are considered as zero emission vehicles in comparison with conventional gasoline-powered vehicles. Zero emission for EVs is just in the operation phase, not fuel production, fuel dispense, and material production. In this case, LCA can provide better results for EVs environmental impact assessment.

1.7.1 History of LCA The first studies on the life cycle assessment began in the late 1960s, focusing on issues such as energy efficiency and the residual of materials. Primary LCA focus was on energy resource rather than on pollution. The society of environmental toxicology and chemistry (SETAC) had a significant role in the development of the LCA method. Although there is so much debate about the life cycle assessment methodology so far, its overall structure has been steady since 1997 with the appearance of the ISO 14040 [10]. The first LCAs consisted of life cycle inventory and usually a primary impact assessment. SETAC developed the first structure, SETAC triangle, in 1990 [11]. Two years later, they reviewed the LCA structure, and the “goal and cope” step was added.

1.7.2 LCA methodologies In order to use LCA, there are four major steps that should be fully considered. They are as follows: 1. Goal definition and scoping. 2. Life-cycle inventory. 3. Environmental Impact analysis. 4. Interpretation of results. The LCA framework including the four major steps is shown in Fig. 1.3 and Fig. 1.4.

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Figure 1.3 LCA evolution.

Goals and scopes The initial step of an LCA study consists of defining the goal and scope of the study under investigation. This step identifies the LCA purpose, the products of the study, and determination of the boundaries. The goal definition is a declaration made by the organization. The depth and accuracy of the study have to be considered during the goal definition. According to the ISO 14044, the goal and scope of an LCA will be clearly defined and consistent with the intended application. Due to the iterative nature of LCA, the scope may have to be refined during the study [8]. The scope definition determines which product is evaluated and how it is evaluated. In addition, the scope should determine the unit of analysis, system boundaries, and types of impact to consider any level of essential details. The goal and scope can be reconsidered for various reasons, such as the availability of information, the inability of some data to affect the outcome, and some unforeseen problems.

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Figure 1.4 Framework of LCA.

Inventory analysis Life cycle inventory analysis (LCI analysis) involves identifying and quantifying all resources used to produce the product, such as energy, water, raw materials and processed materials, and all substances released into the environment, such as the emission of pollutants into the air, soil and water, and losses resulting from the production and consumption of the product. The source and reference data collection has a vital role in the reliability of data and their completeness. Managing all data with the highest possible quality for all processes is a tough and costly task. So, concerning the most important goal we are pursuing, we should devote most of our time to the sectors that have the most significant impact on our goals.

Impact assessment Life cycle impact assessment (LCIA) is the step in which the practitioner uses impact categories, category indicators, characterization models, equivalency factors, and weighing values to realize the potential impact of data that are obtained in LCI analysis as shown in Fig. 1.5. In practice, this step is usually carried out with LCA software, and the practitioner only chooses the method and some other details. The impact assessment may include a duplicate review of the purpose and scope of the study of life cycle assessment to determine whether the objectives of the study have been met, or if not, whether the purpose and scope of the application should be corrected.

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Figure 1.5 LCIA framework.

Interpretation The fourth step of the life cycle assessment is interpretation. In this step the results of other steps are considered together and analyzed in the light of the uncertainties of the applied data and the assumptions that have been made and documented throughout the study [8] The outcome of the interpretation should be conclusions or recommendations that (1) respect the intentions of the goal definition and the restrictions imposed on the study through the scope definition and (2) take into account the appropriateness of the functional unit and system boundaries. Interpretation can reflect the fact that the LCIA results are based on a relative approach to determine the potential environmental impacts and not to predict actual outcomes at the end point of the category. It is worth noting that the practitioner should do three main activities in this step: • Finding environmental issues. • Evaluation. • Conclusion and recommendations.

1.7.3 LCA benefits Life cycle assessment has many advantages over other environmental assessment tools. Some of the most important benefits: • This method can provide a systematic assessment of the environmental consequences of the product. • The methodology of life cycle assessment, based on new information and scientific advancements, is open to today.

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• Data shows how environmental impacts are transmitted from one cycle • • • • •

• •

to another. Quantify the effects of emissions into the air, water, and land at each stage of the life cycle [12]. Identification of opportunities to improve the environmental performance of products at different points in their life cycle [13]. Informing decision makers in industries (i.e., government agencies or nongovernmental organizations). LCA can provide footprinting data. The life cycle assessment results can be used to assess the ecological and human health effects of consumables on the environment on local, regional, and global scales. With LCA, a practitioner can notice that if investment in one part of a cycle can make enough improvement. By this method it is possible to compare the health and ecological effects of two products or similar processes to choose the optimal option.

1.7.4 LCA limitations Apart from all the benefits, life cycle assessment is not a complete tool without any fault, and some of its disadvantages and limitations are: • Performing this process may not lead to a product or process selection that is cheaper and more efficient. • Since weighing in this method involves the collection of values, there is no scientific basis for reducing the life cycle assessment to a single score or number. • In this analysis, we do not have information for workers direct impact; for example, their travel for work is not included in the study. • The life cycle assessment is costly and time-consuming. Depending on how life cycle assessment is performed, data collection and access to the required information are the most difficult stages of the assessment. Therefore, the availability of required information and time and financial resources are among the essential factors in how to do an LCA. • There is no single method for the life cycle assessment; each organization chooses to carry out this assessment by its standards and objectives. • The life cycle assessment may not be the most appropriate and best way to use for all situations and projects because LCA focuses on the environmental aspects of products and says nothing about their economic and social aspects [13].

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1.7.5 Future of LCA LCA is widely used in many organization as a tool for environmental analysis, but nowadays LCA has also broadened to include life cycle costing (LCC) and social LCA (SLCA) covering all three dimensions of sustainability (i.e., people, planet, and prosperity). With these developments, LCA has broadened from merely environmental assessment to a more comprehensive life cycle sustainability assessment (LCSA). LCSA works with many models and guides, selecting the proper ones, given a specific sustainability question. In this analysis the main challenge is structuring, selecting, and making a practically available model about different types of LCSA. Although it is mentioned in the definition of LCA in ISO that there is not only one specific method, this is a tangible difference compared to the LCA. The broaden to economic and social issues in this analysis is another disagreement with LCA, which is limited to the environmental issue only [12]. So far, LCSA has had very little use. SLCA has been neglected in the past but is now beginning to be developed. One of the challenges is how to relate the social indicators (i.e., social impact assessment) to the functional unit of the product system [10]. The LCSA (assessment) approach is the most developed but has been criticized for a mechanistic perspective that prevents full understanding of the mutual interdependencies of the three pillars of sustainability [8].

1.8 Environmental impact assessment of battery technology In today’s life, most of the human activities depend on the use of batteries due to the growth of the use of portable tools that consume electrical energy. This increase in the applications of batteries in our life has been so far due to the use of electronic devices, such as mobile phones, laptops, toys, remote control devices, and recently hybrid, plug in, and full battery electric vehicles have been added to this list. The total power capacity of batteries was 800 MW in 2014, which has increased to more than double (i.e., 1720 MW) within two years (2016), and it is likely to go up to 4000 MW by 2022 [14]. Every year billions of batteries are disposed of, all containing toxic or corrosive materials. One of the ways that has always been proposed to reduce environmental pollution is the electric transport of vehicles, but often it is not taken into account that the pollution comes from the production and use of batteries in these vehicles. The rechargeable batteries that were initially used were nickel–cadmium, which was replaced by nickel–cadmium hybrid systems, and these batteries are being replaced

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by lithium-ion batteries. In recent years, with the use of more chemical batteries, safety issues have become more critical. In particular, the presence of corrosive electrolyte and highly flammable materials or explosives in cells under certain conditions has become a crucial issue in the battery industry. Metals in cells, such as lead, cadmium, mercury, nickel, cobalt, chromium, vanadium, lithium, manganese, and zinc, and acid or alkaline electrolytes can have adverse effects on human health and the environment. The specific compounds of this material and their relative proportions create the dangers of batteries made from such materials. Manufacturers are continually struggling to replace batteries with current batteries that have a more recyclable structure with less toxic substances used to make them. Despite many advances that have been made to recycle rechargeable batteries, there is still a long way to putting into practice reducing environmental pollution. In this section, we will discuss the environmental effects of battery life and investigate the methods for improvement.

1.8.1 Lithium-ion batteries In portable electronic devices the high energy density is critical. Since 1991, when Sony Inc. commercially released the first lithium-ion battery, the use of these batteries was further expanded [15]. Electric vehicles (EVs) have always been mentioned as one of the ways to reduce fuel consumption and reduce environmental pollution. With industrial progress, many electric vehicles (EVs) have been built, all of which are battery-dependent. Therefore the proper understanding of the environmental effects of electric vehicle batteries can help us to make decisions about selecting a vehicle from internal combustion vehicles or electric vehicles. One of the major concern we will be facing is the depletion of lithium sources if penetration of EVs in transportation sector increases drastically. This high demand for lithium has led to a sharp rise in the price of lithium, more than doubling between 2016 and 2018, and this rise in prices for countries like Chili and Bolivia, which have the world’s largest sources of lithium, are persuading to overexploit this source of land. In addition to emptying the Earth resources, this excessive extraction has other harmful effects. In dry weather, such as the weather of areas with significant lithium resources, to produce a ton of lithium, 500,000 gallons of water is needed [16]. This can also affect the water resources in the regions where lithium exists. Lithium-ion batteries have fewer greenhouse gases than NiMH cells, and less metal and raw materials are needed to make it, and fewer fossil

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Table 1.8 Lithium-ion battery ingredients [19]. Component Materials Percentage (%)

Cathodes

Anodes Electrolyte

Separator Case

Li2 Co3 LiCoO2 LiMn2 O4 LiNiO2 LiFePO4 LiNi1/3 Mn1/3 Co1/3 O2 LiNi0.8 Co0.15 Al0.05 O2 Graphite(LiC6 ) Li4 Ti5 O12 Ethylene carbonate Diethyl carbonate LiPF6 LiBF4 LiClO4 Polypropylene Steel

15–27

10–18 10-016

3–5 40

fuels are used [14]. Considering the fact that NiMH batteries are one of the best cells for the environment, we can say that the use of lithium-ion batteries has the least destructive effects. The lithium-ion battery ingredients are different depending on the application they are supposed to use. In high-power cases, such as battery electric vehicles, which have no internal combustion engine, battery specific energy should be maximized, but in a plug-in hybrid vehicle (PHEV) the battery specific power is more important [17]. Understanding the combinational form of lithium-ion batteries can help us to investigate them better. Table 1.8 shows the percentage of materials for lithium-ion batteries. In this table, it should be noted that these percentages change with the weight of the electrode, and as the battery capacity increases, the cathode and anode also increase because more reaction is required [18]. The importance of recognizing the generation of lithium-ion batteries is one of the most devastating effects on the environment. The most significant results of this section are ionization radiation and water eutrophication. The battery operation stage has massive impact on ionization radiation, which affects human health. The most destructive effect is the recycling of these batteries, which poisons water resources. Before discussing the recycling of lithium-ion batteries, we need to point out that many studies have shown that retired electric vehicles with

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lithium-ion batteries still have 80% of their initial capacity. Thus, both economically and environmentally, it is good to use these batteries in another area where energy storage is needed [20]. Before recycling, we need to consider all aspects because recycling is not always the best choice for the environment. Even though recycling of lithium-ion batteries has many benefits to the environment, these benefits highly depend on the ingredients of this battery and the way it is intended for recycling. As discussed in the previous section, the growth of lithium-ion batteries could damage lithium minerals and other minerals. By recycling these batteries the use of lithium mines reaches a balanced state, and for production, recycled materials can be replaced by part of the raw material we extract from mines. Usually, recycled materials are lower than the original raw material, so using recycled materials to produce lithium-ion batteries can reduce the cost of providing these batteries. With the benefits of recycling lithium-ion batteries, some factors should be taken into consideration when recycling these batteries. One of these factors to be considered is the possibility of collecting a large number of cells for recycling at a reasonable cost. Another problem with the recycling of lithium-ion batteries is the storage of this high amount of materials, which are harmful to human health and may also have dangers such as fire. The safekeeping of these materials is one of the factors that cause increased recycling costs.

1.8.2 Lead–acid batteries Electrochemical energy storage has always helped humans to grow the industry because it is compact and portable, and in many industries, it can save energy. One of the electrochemical batteries used today is lead–acid batteries. For more than 130 years, lead–acid batteries have been used in various activities. One of the primary sources of energy in the industry today is lead–acid batteries, and it is expected that with the rapid growth of the economy, the use of these batteries will also be expanded. Lead–acid batteries compared to other cells are the cheapest ones. For vehicle use, this battery is the predominant battery used and is also the most widely used battery for industrial electric [21]. The main advantage of the lead–acid battery is its low cost compared to other battery types. In the same way, as in the previous step, we first studied the materials of the lithium-ion batteries, where we are trying to see the materials used in each part of the lead–acid batteries as shown in Table 1.9.

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Table 1.9 Lead-acid batteries ingredients. Component Materials

Inorganic lead compound

Electrolyte Case material

Other

Lead Antimony Arsenic Calcium Tin Sulfuric acid (H2 SO4 /H2 O) Polypropylene Polystyrene Styrene Acrylonitrile Acrylonitrile Butadiene Styrene Styrene Butadiene Polyvinylchloride Polycarbonate, hard rubber, Polyethylene Silicon Dioxide (Gel batteries only) Sheet Molding Compound

Percentage (%)

60–70 2 0.2 0.04 0.2 10–30 5–10

1–5

Lead and antimony are both toxic and harmful to humans and the environment, but both play a key role in batteries and are often solid in the cell. Thus the possibility of leakage and spreading during the transfer and maintenance phase is very low. However, the electrolyte is liquid in the battery, so it can leak because it is corrosive, and it is possible to leak electrolyte by corrosion of the cell. Therefore the greatest danger to the environment of lead acid in the transfer, production, and processing phase is due to the presence of this electrolyte and the possibility of leakage. Sulfuric acid is the main ingredient in the electrolyte, which, depending on the concentration, can have a great deal of corrosion on the metals. Also, the contact of this acid with humans can also cause irreparable damage [22]. Lead is the most efficiently recycled commodity metal, and in the EU and the USA, more than 99% of lead-based cells are collected and recycled in a closed-loop system [23]. One of the ways people and the environment are exposed to lead is recycling of lead–acid batteries. In almost all methods of recycling of lead–acid batteries, the following stages occur: • Collect and transfer consumed batteries to a place to be recycled. • Separate different parts of the cell from each other. • Melt the lead components.

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Table 1.10 Nickel–metal hydride batteries ingredients. Component Materials Percentage (%)

Hydrogen Absorbing Alloy Nickel–Cobalt–Zinc oxide Nickel Iron Carbon Black Potassium Hydroxide Sodium Hydroxide Lithium Hydroxide

20-40 15-25 5-15 20-40 0-1 0-15 0-15 0-15

• Clean and melt plastic parts. • Perform repurification of electrolyte and sulfuric acid. • Dispose the waste.

One of the main ways humans and the environment are exposed to lead is recovering the lead–acid battery. This happens at all recycling stages. Smoke containing lead enters the air, and, in addition to breathing damage to human health, this lead can enter the soil and water. Also, disposal of lead–acid waste recycling must be done with great care, so that this happens appropriately because the waste contains plenty of lead that can pollute the environment and cause soil and water poisoning.

1.8.3 Nickel–metal hydride The Nickel–Metal Hydride battery represents an evolution from the Nickel–Hydrogen battery. NIH2 has a high specific energy and a decent lifetime. The main problem of NiH2 was the high volume required for hydrogen gas. NiMH batteries resolved this problem. NiMH cells are widely used in the world today, from small appliances to hybrid vehicles. Since these batteries have an energy density of almost double magnitudes of nickel–cadmium batteries [24], they quickly replaced nickel-cadmium batteries. Due to its high energy density and long life, the NiMH battery is superior to most other secondary batteries. The main parts of NiMH cells are anode, cathode, electrolyte, separator, and the steel case. In the same way as before, Table 1.10 shows the materials percentage in the Nickel–metal hydride batteries. The percentage of nickel purity in the ore is usually low, thus making it is highly energy intensive to extract and refine the metal. This leads to high emissions of greenhouse gases into the atmosphere, and fossil fuels supply the energy, so many fossil fuels are also consumed [25]. Some believe that

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the discharge of nickel Earth resources is the hidden cost that the land pays for the construction of electric machines. Now, the expansion of the use of rechargeable electronic devices with a steep slope is increasing. Also NiMH batteries and various applications they can have are batteries of many hybrid vehicles, and with the growth of demand for hybrid vehicles, there is a need to provide as much material as possible by recycling batteries. Considering the increasing global need for nickel, one of the main ways to reduce land mines depletion is recycling of NiMH batteries.

1.8.4 Conclusion This part examines the environmental impacts of the three most used batteries in the industry. It has been observed that each of these batteries affects the environment at each stage of their life. One of the primary uses of these batteries is in the automotive industry for electric and hybrid vehicles. Given the information here about the amount of pollution and the damaging effects that these batteries can have on the environment when comparing the environmental impacts of internal combustion engines with electric vehicles, the differences in their battery life should be taken into account. As explained before, considering the importance of the recycling, these batteries at the end of their lifespan, to preserve land mines, more extensive steps should be taken to collect used batteries. In addition, due to the an increase in the use of cells and the impact on land resources, it might be better to restrict the use of land resources in the future. Due to the toxicity of a large part of the batteries, there should be an excellent place to dispose their waste in areas that cannot be recycled. Lithium-ion and NiMH batteries have less harmful environmental effects. Therefore, as much as possible, these two batteries should be replaced by other cells. Materials for battery production, battery service life, collection efficiency, and the material recovery rate are some of the factors that will affect the amount of mentioned environmental impacts. To minimize the environmental impacts of battery technology assessment, using recycled materials and batteries, increasing battery life, and increasing energy density should be a priority for manufacturers. Moreover, regulations should be implemented to reduce the need for virgin alloys. A higher efficiency of batteries in charging and discharging can also result in a more efficient battery and will be useful in reducing environmental impacts. In summary, Table 1.11 shows the materials used in various batteries with their danger for the environment.

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Table 1.11 Various metrails percentage used in batteries with their dangers [26]. Component Lead–acid NiMH Li-ion Classification (Wt%) (Wt%) (Wt%)

Nickel La, Ce, Nd, Pr Cobalt Iron Electrolyte (KOH) Aluminum Copper LiPF6 Solvent (organic) Carbon Lead Electrolyte (H2 SO4 ) Plastics, etc.

– – – – – – – – – – 65–70 22–30 8

30–42 8–10 3–4.5 22–25 5–10 0.5–2 – – – – – – 4

– – 16–18 – – 14–16 6–7 2–3 16–18 18–20 – – 2–4

Harmful – Harmful – Corrosive, Irritant – – Toxic Flammable – Toxic Corrosive, Irritant –

1.9 Summary Battery industry is dealing with many different cell types, and many other materials are tested each day by different scientists. All the efforts are performed to improve the performance of the cell, but the problem is not very easy. Because there are many different challenges available, many of them conflict to each other. For example, improving cyclic life may increase the cost of the cell. In this chapter, the present state of the battery technology is first investigated. Then the main challenges are discussed in more detail. Finally, some promising technologies are introduced and briefly explained. Life cycle assessment (LCA) as an environmental management tool, especially in recent decades, has received a special status for helping decisionmakers. LCA is the most critical methodological platform for understanding the environmental impact of different production systems. Despite the advantages and uses that LCA have in today’s societies, it has a great potential for progress. These improvements must be both scientifically accurate and practicable. The ability to quantify in LCA is one of the most critical features of this analysis, and it is worth noting that if extended to the LCSA, this feature is maintained. Although this tool is not able to identify methods, processes, and products with more economical costs, it can detect the option with the lowest load and environmental impact and change the final result. This advantage and increasing the importance of the environment in the choice of the alternative have expanded its use.

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1.10 Problems 1. Discuss the difference between capacity and energy content. 2. A battery type has large amount of energy, whereas another cell has lower energy. Which of them can deliver higher power? 3. By having low power density cells, how can we obtain a powerful battery? 4. A lead–acid battery has six cells in series. Each cell has 2.5 Ah capacity. Calculate the energy content of the battery. 5. If the cells of the previous battery were connected in parallel, recalculate the energy content. 6. For 18,650 Li-ion LTO cell with m = 45 g and C = 2800 mAh, calculate the specific energy and energy density. Note that the typical voltage for LTO cells is about 2.4 V.

CHAPTER 2

Fundamentals of batteries Contents 2.1. Introduction 2.1.1 Primary and secondary batteries 2.1.2 Half-cell reactions and symbol 2.1.3 Cathode/anode versus positive/negative electrodes 2.1.4 Reference electrode 2.1.5 Battery capacity Peukert law for lead–acid batteries 2.1.6 Series and parallel connection 2.2. Basics of electrochemistry 2.3. Faraday’s law of electrolysis 2.4. Butler–Volmer law 2.5. Summary 2.6. Problems

55 56 57 59 60 61 63 65 67 72 72 80 80

2.1 Introduction Batteries and fuel cells are categorized as electrochemical systems. The main characteristic of an electrochemical system is the transfer of electron. Although in a chemical system, electron transfer occurs, there are some differences between electrochemical and chemical systems including: • While chemical reactions can take place in bulk, electrochemical reactions require a surface to happen and cannot be done in bulk. • The electron transfer mechanism differs in electrochemical and chemical reactions. In an electrochemical reaction, an electron flows through an external circuit, which is not the case for chemical reactions. • Electrochemical reactions happen in different places, whereas chemical reactions should be carried on in one place. For example, in an electrochemical reaction the whole reaction can be done in two different places by two half-cell reactions. The concept of an electrochemical cell comes from the fact that different materials have different potentials to release or accept electrons. When two different materials are placed in a container, filled with electrolyte, because of their different potential in electron releasing, a potential difference takes place between them. This phenomenon makes the concept of batteries. Simulation of Battery Systems https://doi.org/10.1016/B978-0-12-816212-5.00006-4

Copyright © 2020 Elsevier Inc. All rights reserved.

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Simulation of Battery Systems

To compare electron producing ability of different materials, they are all compared with a reference material by their electrical potential value. There are many different reference materials, also known as the reference electrodes, for different purposes. One of the most famous ones is hydrogen whose voltage is regarded as the set point of voltage calculation. If hydrogen flows over a platinum plate at standard conditions (at 1 atm pressure and 25◦ C temperature) the electrode is called Standard Hydrogen Electrode or SHE. Since a potential field has no reference point, the potential of SHE is normally used as the reference point, and the potential of all the other materials are compared regarding SHE. It is customary to tabulate materials according to their potential versus SHE in a table called Electrochemical Table. Such a table is shown in Table 2.1, where some materials have a positive potential, and some have a negative potential. The sign of the potential has no physical meaning because the reference point of the table is chosen arbitrarily. From the electrochemical table we conclude that the more we go to the top of the table, the more tendency have the materials to release their electrons and become ions. The materials at the bottom of the table, in contrast, tend to accept electrons and become reduced. Consequently, the top materials can make good anodes, whereas the materials at the bottom are better cathodes.

2.1.1 Primary and secondary batteries Batteries are made in two ways: they can be recharged after use, or they can be recycled. The rechargeable batteries are called secondary batteries, whereas nonrechargeable ones are called primary batteries. Primary batteries are widely used in watches, remote controls, toys, and many other applications, whereas secondary batteries are used in cell phones, notebooks, shavers, and so on. Many battery technologies have both versions, but some others are made either as primary or secondary ones. The main reason for making primary batteries is that they are cheaper and usually have more energy density than their secondary versions. The reason for more energy content is that for converting a primary battery to secondary version, some facilities should be added. For example, they should be equipped with more sophisticated separators, sealing, and so on.

Fundamentals of batteries

57

Table 2.1 Potential of some selected material. Oxidizer Reducer Potential electron accepter electron donater [V] Li + Li(s) −3.01 Rb(s) −2.98 Rb + + Cs(s) −2.92 Cs K+ K(s) −2.92 Ca(s) −2.84 Ca 2+

Na + Mg + Zn 2+ Fe 2+

Na(s) Mg(s) Zn(s) Fe(s)

−2.71 −2.38 −0.76 −0.40

In 3+ Cd + Pb + 2H+ AgCl +

In(s) Cd(s) Pb(s) H2 (g) Ag(s) + Cl − (aq)

−0.34 −0.40 −0.13 0.00 +0.22

Hg2 Cl2 (s) Cu 2+ Hg 2+ I2 (s) Fe 3+

2 Cl − (aq) + Hg(l) Cu(s) 2 Hg(s) 2I− Fe 2+

+0.27 +0.34 +0.80 +0.54 +0.77

Ag + Br2 (g) O2 (g) Cl2 (g) F2 (g)

Ag(s) 2 Br(s) 2 H2 O(l) 2 Cl − 2F−

+0.80 +1.07 +1.23 +1.36 +2.87

2.1.2 Half-cell reactions and symbol One of the main characteristics of an electrochemical reaction that distinguishes it from a chemical reaction is that an electrochemical reaction takes place at two different places. When a battery is discharged, one electrode is oxidized and converts to ions according to the following reaction: M1 (s) −→ M1n+ (aq) + ne− ,

(2.1)

where M1 (s) stands for reduced material, and M1n+ (aq) represents the oxidized ion. At the same time a reduction reaction occurs at the counter

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Simulation of Battery Systems

electrode according to M2n+ (aq) + ne− −−→ M2 (s).

(2.2)

Note that in a cell the number of transferred electron is the same for the two reactions. Each of the previous reactions is called a Half-cell reaction. In any electrochemical reaction, it is convenient to show reaction (2.1) by the symbol M1 (s)|M1n+ (aq)

(2.3)

where the vertical line is the symbol of half-cell reaction and indicates that M1 (s) in solid form is converted to M1n+ (aq) ions and n electrons are produced. Similarly, the half-cell reaction (2.2) is M2n+ (aq)|M2 (s).

(2.4)

A battery cell made of two half-cell reactions is written as M1 (s)|M1n+ (aq)||M2n+ (aq)|M2 (s).

(2.5)

Double vertical lines indicate the cell definition. Example 2.1. If a battery is made of zinc and copper, write the half–cell reactions. Answer. According to Table 2.1, zinc is above copper, and hence if a battery is made of zinc and copper electrodes, then Zn is most favorable to release electrons according to the following reaction: Zn −−→ Zn++ + 2 e− . Thus the half-cell reaction of zinc electrode is Zn|Zn++ . Meanwhile, at copper electrode the following reaction occurs: Cu++ + 2 e− −−→ Cu, whose half-cell reaction symbol is Cu++ |Cu.

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59

The whole cell then can be shown as Zn|Zn++ ||Cu++ |Cu.

2.1.3 Cathode/anode versus positive/negative electrodes According to the definition, cathode is the electrode at which reduction occurs or electrons are consumed, and anode is the one at which oxidation occurs or electrons are produced. By this definition, when the cell is charged, reactions (2.1) and (2.2) reverse and take place from right to left. At this state, reaction (2.1) consumes electrons, and the electrode is called cathode, whereas reaction (2.2) produces electrons, and the electrode is called anode. Consequently, during charge or discharge, cathode and anode change, and we cannot label an electrode as cathode or anode. By contrast the positive electrode is the electrode at which electron is consumed during discharge, and the counterelectrode is the anode. By this definition the positive and negative electrodes do not change during charge or discharge. This is the main reason why the battery manufacturers mark positive or negative poles on their productions but not cathode and anode. Example 2.2. If a battery is made of zinc and copper, determine cathode, anode, positive and negative electrodes. Answer. According to Example 2.1, the cell reaction during discharge is Zn|Zn++ ||Cu++ |Cu. The cell reaction shows that during discharge, Zn electrode produces electron and hence is oxidized, and Cu consumes electrons and is reduced. Consequently, during discharge, Zn is the anode, and Cu is the cathode. But when the cell is charged, the above reactions reverse, and the cell should be written as Cu|Cu++ ||Zn++ |Zn. This cell reaction clearly indicates that during the charging process, the cathode is the zinc electrode, and the copper electrode is the anode. However, in both charge and discharge, Cu is the positive electrode, and Zn is the negative electrode. Reemphasizing that the positive and negative electrodes do not change by charging or discharging process.

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Simulation of Battery Systems

Figure 2.1 Schematic of SHE.

2.1.4 Reference electrode Electrical potential, like any other potential field, has no standard reference. Therefore, for each application, a reference state should be defined. For each electrochemical system, a proper material is used whose electrical potential is defined as the reference state. The electrodes made of these materials are called Reference Electrodes. There are many reference electrodes for different applications and environments. The main factors required for an electrode to become a good reference electrode are: 1. The electrode should be stable in the operating environment. 2. The voltage of the material should be stable in a good level. 3. The electrode should be made in a practical shape. Among many different reference electrodes, Standard Hydrogen Electrode, also known as SHE, is one of the most famous ones. This electrode shown in Fig. 2.1 is made of a platinum plate over which hydrogen gas flows. The hydrogen is in standard conditions (at 1 atm and 25◦ C) with the following half-cell reaction: Pt| 12 H2 (g)|H+ (aq)|| · · · .

(2.6)

Pt is just a catalyst and does not contribute to electrochemical reactions; hence the generated potential is attributed to hydrogen with the following reaction: 1 −− + 2 H2 (g) −− H

+ e− .

(2.7)

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61

At standard conditions with pH = 0, the potential of SHE is arbitrarily chosen as zero. Most electrochemical tables are based on SHE in which the potential of hydrogen is zero. For example, Table 2.1 gives the potential of materials versus SHE. Other than SHE, Ag|AgCl is another famous reference electrode. The half-cell reaction of this electrode is Ag|AgCl(s)|Cl− (aq)|| · · · ,

(2.8)

where Cl − is a binary salt of Cl such as NaCl or KCl. As another practical reference electrode, we can name Calomel. The electrode is made of Hg and Hg2 Cl2 by the following half-cell reaction: Hg|Hg2 Cl2 (s)|KCl · · ·.

(2.9)

In any case, it should be noted that each of the above-mentioned reference electrodes are useful in some applications and may be not practical in others. Finally, there are many other reference electrodes available for practical purposes.

2.1.5 Battery capacity Battery capacity is a tricky term and is a matter of debate. From a fundamental point of view, the capacity is simply the total amount of electrical charge stored in a battery and can be obtained using the relation 

Q=

Idt.

(2.10)

The battery capacity (with the unit of Coulomb) is a measure of its active material. At first glance, Eq. (2.10) looks very simple, and for measuring the capacity, all you need is to discharge a battery and record its current versus time. Integrating the resulting data will give the battery capacity. For instance, if the discharging process is constant current, then the capacity is Q = I × t.

(2.11)

However, in practice, some points need to be kept in mind. In each electrochemical battery system, some side reactions are accompanying the main reactions. Other than the side reactions, there are some other undesirable reactions such as crystallization, oxidation, and so on. The higher the current density is, the more energy serves to drive undesirable processes, and a

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Simulation of Battery Systems

part of input energy converts to the irreversibility of the system. Therefore, if one discharges a battery with a very high current density, then a part of the internal capacity of the cell is converted to heat, side reactions, crystallization, and so on. Consequently, what we measure as the output of the cell is not exactly its overall capacity. In addition, the temperature also plays an important role in determining the capacity of the cell. When the cell is cooled down, its internal resistance increases, leading to greater irreversibility of the cell. The higher the temperature, the more measurable the capacity. Thus, in measuring the capacity of the cell, we should take care of the temperature and temperature variation during the discharging process. Theoretically, the capacity is an indicator of the moles of active material; hence it has a physical interpretation because it is quite clear that the moles of active material inside the batter are fixed. The mentioned problem is the measurement of the active materials as discussed before. For example, if a particular cell is discharged with I1 and the measured time is t1 and if we repeat the test with I2 = 0.5I1 with the measured time t2 , then usually the following relation exists between the results: Q1 = I1 × t1 < Q2 = I2 × t2 .

(2.12)

In practical tests, it is observed that the lower the current density, the better the obtained measurement. This means that if we choose a very small value for I1 and I2 , then Eq. (2.12) converts to Q1 = I1 × t1 ∼ Q2 = I2 × t2 .

(2.13)

For conventional batteries, if t1 > 10 hr, then Eq. (2.13) becomes valid. The current by which the cell is fully discharged around n hours is denoted In . For example, for current I10 , the cell is discharged around 10 hours, which in practice is a good indicator of cell capacity. Correspondingly, the capacity measured by In is denoted Cn . If the current is too high so that the discharge time is less than one hour, then the current is defined as nI, which means that by this current the battery is discharged around one nth of an hour. Similarly, the corresponding measured capacity is nC. Since the capacity of a battery does not have a unique value, the manufacturers write an approximate value on their products. The approximate value is called Nominal Capacity and does not mean that it is the exact capacity of the cell. Fig. 2.2 shows a typical lithium battery used for cell phones. As it is indicated on the cover of the cell, it has Qn = 3500 mAh capacity.

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63

Figure 2.2 A typical lithium-ion battery.

This means that if we measure its capacity, then the value is around that value. In addition, in a production line, it is not practical to make identical cells with infinite accuracy. Therefore the printed value on the cells is just an indicator of its capacity and is not an exact value. In scientific reports, however, the experimental condition at which the value is obtained should also be emphasized. The test temperature, discharge current, and the accuracy of the devices are all affecting the result. Example 2.3. For a conventional lithium battery shown in Fig. 2.2, determine I1 , I2 , I10 , 2I, and 10I. Which one is the best choice to find the capacity of the battery? Answer. The nominal capacity of the cell is Qn = 3500 mAh, meaning that if it is discharged by 3500 mA, then it takes about one hour for full discharge. Thus I1 = 3500 mA, I2 = 1750 mA, I10 = 350 mA, 2I = 7000 mA, and 10I = 35000 mA. According to the previous discussion, the current by which it is most probable to obtain Q = 3500 mAh is I10 .

Peukert law for lead–acid batteries Peukert’s law was proposed by the German scientist Wilhelm Peukert in 1897 to address the capacity relation to current density in the lead–acid batteries. As discussed before, the higher the discharge current, the lower the obtained capacity. In practice, I10 is an almost good choice for obtaining the correct capacity. However, I10 means that the test should take about 10 hours, which is too much for practical purposes. To overcome this challenge, Peukert suggested that the capacity of lead– acid batteries can be related to discharge current by the relation I nt = C.

(2.14)

By this equation we can carry on two successive tests by I 1 and I 2 obtaining the corresponding times t1 and t2 . (Note that the numbers are not pre-

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Simulation of Battery Systems

sented in subscript so that the symbols do not conflict with the definition of In .) According to Peukert’s law, we can write n

I 1 t1 = C , n I 2 t2 = C .

(2.15) (2.16)

Taking the logarithm of these above equations results in n log I 1 = log C − log t1 , n log I 2 = log C − log t2 .

(2.17) (2.18)

From this equation system we can obtain n: n=

log t2 − log t1 . log I 1 − log I 2

(2.19)

Finally, we can obtain the true capacity from Eq. (2.15) or (2.16). Example 2.4. A lead–acid battery is fully discharged by I 1 = 10 A within one hour. In another test, the same battery was discharged by I 2 = 5 A and it took 2.25 hours for complete discharge. What is the true capacity of the battery? Answer. From the experiments we have: I 1 = 10, t1 = 1, I 2 = 5, t2 = 2.25. Hence we have n=

log t2 − log t1 log 2.25 − log 1 = ∼ 1.17. log I 1 − log I 2 log 10 − log 5

Consequently, the true capacity of the battery is C = I n t = 101.17 × 1 = 51.17 × 2.25 = 14.788 Ah. Example 2.5. In the previous example, at which current density do you expect to obtain the correct value for capacity by an experiment? Answer. The results show that the battery capacity is about 15 Ah. This means that I10 = 1.5 A. Therefore, by a 10-hour experiment with I = 1.5 A we can obtain the true capacity via an experiment. It is worth noting that

Fundamentals of batteries

65

Figure 2.3 Illustration of cells in series and parallel combination. (A) Serial configuration. (B) Parallel configuration.

the advantage of Peukert’s law is that we can obtain the true value in about 3 hours of discharge. If we add an hour for recharging in between, then the whole test will take only 4 hours. This is very important in industrial tests.

2.1.6 Series and parallel connection In many portable applications, a unit cell is enough for providing the required power. For example, in watches, cell phones, and so on, we need just one cell. However, in high-power applications, such as electric cars, UPS, and so on, a single cell is not enough to provide all the necessary power and energy. In these cases, we need to use many cells in series and in parallel as is shown in Fig. 2.3. A combination of batteries is called a battery pack. A battery pack may contain many cells in series, or in parallel, or in a combination of both. Fig. 2.3A shows the cells in series in which the positive pole of one cell is connected to the negative pole of the next cell, and so on. In this case, the voltage of the whole pack is the sum of voltages of the individual cells, but all the electrical current passes through all the cells. Fig. 2.3B shows the cells in parallel where the positive poles are connected, and so are the negative poles. In this case the voltage of the pack is the same as that of a single cell, but the current passing the pack is divided between the cells, and only a portion of current passes through each cell. Consequently, the battery pack has the ability to deliver longer power than a single cell.

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Simulation of Battery Systems

Technically, this means that the capacity of the pack is increased by parallel cells. We should kept in mind that in both cases the energy content will be the same and is equal to the sum of the energy content of the whole cells. It is because the power output of the pack is the product of its voltage and current. It is clear that in both series or parallel configuration the product remains constant. Example 2.6. Suppose that three cells shown in Fig. 2.2 are available. In case I, we connect the cells in series, and in case II, we connect them in parallel. If a current of I = 0.35 A is passing through the packs, determine: 1. What is the voltage of the packs? 2. How long does it take to fully discharge the packs? 3. What is the capacity of each pack? 4. What is the energy content of each pack? Answer. The cell shown in Fig. 2.2 is a 3500 mAh cell with I10 = 0.35 A. Therefore we can assume that by this current all the available energies of the cells are delivered and available. By this assumption we can answer the questions. 1. When the cells are connected in series, the voltage Vpack,s of the pack is the sum of all voltages Vcell of the cells. In this case the pack voltage is Vpack,s =

3 

Vcell = 3 × 3.7 = 11.1 V.

1

The voltage of the parallel pack Vpack,p is equal to the voltage of a single cell, that is, Vpack,p = Vcell = 3.7 V. 2. When the cells are in series, the applied current passes through all the cells at the same time. Hence, the capacity of the pack is equal to the capacity of a single cell, which means that the pack is drained out at the same time a single cell is fully discharged. Since the applied current is I10 , it will take about 10 hours to fully discharge the pack. In the parallel connection the applied current is divided between the cells. Therefore, one-third of the applied current is passing through each cell. This means that the applied current for every single cell is I30 , and it makes 30 hours for each individual cell to be fully discharged.

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67

Figure 2.4 A Zn–Cu cell.

3. From the results of the previous item the capacity of serial pack Cs is Cs = I × t = 0.35 × 10 = 3.5 Ah, which obviously is the same as the capacity of a single cell. In parallel connection the capacity Cp is Cp = I × t = 0.35 × 30 = 10.5 Ah, which is triple the capacity of a single cell. 4. In a serial pack the energy content Ws is Ws = Cs × Vpack,s = 3.5 × 11.1 = 38.85 Wh, and in a parallel pack the energy content Wp is Wp = Cp × Vpack,p = 10.5 × 3.7 = 38.85 Wh. It is clear from this example that the energy content does not vary in different configurations.

2.2 Basics of electrochemistry Consider an electrochemical cell shown in Fig. 2.4 including zinc electrode Zn as negative and copper electrode Cu as its positive electrode, which are placed in a binary electrolyte. During discharge, the following half-cell reaction occurs at the negative electrode: Zn −−→ Zn++ + 2 e− ,

(2.20)

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where the reduced Zn material is oxidized into Zn++ ions and releases two electrons per mole. At the same time, the following reaction occurs at the positive electrode: Cu++ + 2 e− −−→ Cu,

(2.21)

in which copper ions that are in the electrolyte accept electrons and are reduced to metallic copper. Therefore the overall reaction is Zn + Cu++ −−→ Zn++ + Cu.

(2.22)

According to the basics of thermodynamics, each reaction generates a specific amount of heat known as enthalpy of reaction or H. By considering the enthalpy of reaction the last equation can be rewritten as Zn + Cu++ −−→ Zn++ + Cu + H

(2.23)

Enthalpy of reaction is the whole available energy in any reaction. However, this amount of energy cannot be completely used as useful energy. This is because a specific amount of energy is lost due to the irreversibility of the system. The net useful energy is called Gibbs free energy indicated by G: G = H − T S,

(2.24)

where T is the reaction temperature, and S is the entropy difference of the reaction. It is clear that G is the useful energy that can be converted to electricity in electrochemical systems such as batteries and fuel cells. On the other hand, the electrical work Wel is equal to the product of the electrical field E and electrical charge qel , that is, Wel = Eqel .

(2.25)

For a general electrochemical reaction in the form of M + N n+ −−→ M n+ + N

(2.26)

for one mole of reactants and n moles of electrons is released; According to Faraday’s law of electrolysis (see Section 2.3), the amount of electrical charge can be calculated as qel = nF .

(2.27)

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69

Using Faraday’s law, Eq. (2.25) can be rewritten as Wel = nEF .

(2.28)

Equating the electrical work (Eq. (2.28)) with G of reaction, Wel = −G,

(2.29)

we can obtain the necessary formulation for calculation of open circuit voltage: Vocv =

−G

. (2.30) nF This equation is frequently used for calculation of any battery open-circuit voltage. It is quite clear that in the case of an operational battery that is charged or discharged, the voltage of battery changes. Eq. (2.30) is very helpful for calculation of battery open-circuit voltage at different pressures or temperatures, because of the dependence of Gibbs free energy on temperature and pressure; the only thing to do is to calculate G in any state. From the fundamentals of thermodynamics it is clear that G decreases by increasing the temperature. Consequently, the open-circuit voltage of any battery decreases as the temperature increases.

Example 2.7. For lead–acid batteries, calculate the enthalpy of reaction, Gibbs free energy, and open-circuit voltage under standard conditions. Answer. The primary reaction of a lead–acid battery is as follows: Pb + PbO2 + 2 H2 SO4 −−→ 2 PbSO4 + 2 H2 O + H . The enthalpy of reaction H for a reaction is defined as H = 2(hf )H2 O + 2(hf )PbSO4 − 2(hf )H2 SO4 − (hf )PbO2 − (hf )Pb .

At standard conditions, that is, one molar concentration, 1 atm pressure, and 25◦ C, H is calculated from the enthalpy of formation of materials, which can be found in thermodynamic handbooks. Using standard values for materials contributing in the lead–acid battery reaction, H can be calculated: H = 2(−285.830) + 2(−919.936) − 2(−813.989) − (−277.4) − 0 = −506.154 kJ mol−1 .

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The negative sign indicates that the reaction is exothermic. Similarly, G = 2(gf )H2 O + 2(gf )PbSO4 − 2(gf )H2 SO4 − (gf )PbO2 − (gf )Pb = 2(−813.202) + 2(−237.178) − 0 − (−217.36) − 2(−690.100) = −503.2 kJ mol−1 .

Note that in lead–acid reactions, for each mole of reaction, two moles of electrons are released; hence n = 2. Using Eq. (2.30), the open-circuit voltage of lead–acid batteries under standard conditions is Vocv =

−G

nF

=

503200 J mol−1 = 2.607 V. 2 × 96485 As mol−1

In practical lead–acid batteries the electrolyte concentration is about 5 M, and hence the open circuit voltage of an industrial lead–acid battery differs from the obtained value. Eq. (2.30) gives a fundamental basis for the calculation of open-circuit voltage at any state. For instance, to calculate the open-circuit voltage of a battery in any temperature or pressure, we just have to calculate G at that state. The dependency of H and S on the temperature is defined by the following thermodynamical relations: 

h T = hf + sT = sf +

T

cp dT

(2.31)

298.15  T

1 cp dT , T 298.15

(2.32)

where hf and sf are enthalpy and entropy of formation at reference temperature (i.e., T = 298.15 K), respectively, and hT and sT are their corresponding values and any temperature T. When the temperature difference is not too large, we can assume that cp is almost constant. Consequently, the above integrals can be solved, and the equations reduce to hT = hf + cp (T − 298.15), T . sT = sf + cp ln 298.15

(2.33) (2.34)

For an electrochemical device, the efficiency is defined as the ratio of G to H. This is because by definition G is the useful work, whereas

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71

H is the total amount of available energy. Mathematically, the efficiency

is written ηeff =

G . H

(2.35)

Dividing the numerator and denominator of the equation by −nF, we get G −G − nF Vocv ηeff = = = .  H −H Vth −

(2.36)

nF

The numerator indicates the open-circuit voltage of the battery, and the denominator is a fictitious voltage called the thermoneutral potential. The thermoneutral potential is an indicator of the theoretical maximum potential. The importance of Eq. (2.36) is that it can be used in actual cases where the battery is under discharge because under discharge conditions, the voltage drop (or polarization) shows itself in decreasing the Gibbs free energy. Therefore the efficiency of a battery can be calculated by dividing its voltage, whether the open circuit or under discharge, by its thermoneutral potential. Example 2.8. For lead–acid batteries, calculate its thermoneutral potential and find the efficiency at open-circuit voltage. Answer. Thermoneutral potential by definition is Vth =

−H

nF

=

506154 J mol−1 = 2.623 V 2 × 96485 As mol−1

Using this value, the efficiency is η=

Vocv 2.607 = = 0.994 Vth 2.623

or 99.4%, which indicates that lead–acid batteries have very good theoretical efficiency. However, in practical cases, the acid concentration is about 5 M instead of 1 M, which results in less open circuit about 2.2. In this case the efficiency is Vocv 2.2 ηeff = = = 0.838, Vth 2.623 which is less than its theoretical value.

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Simulation of Battery Systems

2.3 Faraday’s law of electrolysis Faraday’s law of electrolysis focuses on the relationship between mass and amount of electrical charge. Consider an arbitrary half-cell reaction like Eq. (2.37): M −−→ M++ + ne− .

(2.37)

In this reaction, for each mole of material M, exactly n moles of electrons are released. Consequently, the amount of electrical charge is equal to the amount of electrical charge of n moles of electrons. Mathematically, the electrical charge of one mole of electrons then can be obtained: F = NA × qe ,

(2.38)

where NA = 6.022140857 × 1023 is the Avogadro’s number, and qe = 1.60217662 × 10−19 Coulombs is the charge of a single electron. The charge of one mole of electrons is called Faraday’s Constant and is denoted by F; Faraday’s constant is equal to F = NA qe = 6.022140857 × 1023 × 1.60217662 × 10−19 = 96485. (2.39) The unit of F is Coulombs per mole, C mole−1 . The main advantage of Faraday’s law is that it connects the produced mass with the generated electrical charge.

2.4 Butler–Volmer law When an electrode is at its equilibrium state, its potential reaches a maximum value known as the reversible potential. However, when current passes through the electrode, its potential drops due to its electrochemical ability. Some materials have faster kinetics in releasing and absorbing electrons, and some have slower. The faster the electrode, the less its potential drop. As an example, lithium has faster kinetics than iron in releasing and absorbing electron in discharge and charge, respectively. Consequently, when a battery has lithium electrodes, it experiences less potential drop under a certain amount of electrical current than a battery with iron electrodes. To determine the ability of an electrode in releasing and absorbing electrons, its kinetic rate should be determined. Note that the kinetic rate is individually related to each electrode; hence the following discussion is regarding a single electrode.

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Figure 2.5 A single electrode.

Before further discussing, it is worth mentioning that in kinetic analysis of electrodes, it is customary to carry on all the relations based on the current density instead of the current itself. It is because it is evident that a larger plate can produce more current. This means that we cannot compare a very small electrode with a large one from the kinetic point of view. To have a consistent comparison and analysis, all the relations are performed based on the current density defined as i=

I , A

(2.40)

where I is the total current, A is the electrode surface area, and i is the current density. Assume that a single metallic electrode is dipped into a vane of electrolyte as shown in Fig. 2.5. At the very beginning of insertion, the metal starts dissolving into the electrolyte via the reaction Red −−→ Ox + ne − .

(2.41)

In this reaction, Red means the reduced metal, and Ox indicates the oxidized metal or ion. As an example, assume that the metallic electrode is made of zinc resulting in the reaction Zn −−→ Zn ++ + ne − .

(2.42)

The process stops when there are enough electrons deposited on the metallic rod. The number of deposited electrons depends on the material of the electrode. The more active the material is, the more electrons will be deposited. However, it should be noted that, at the equilibrium process, Eq. (2.42) does not really stop, but enters a reversible active condition. In

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Simulation of Battery Systems

Figure 2.6 A single electrode. (A) Electron extraction. (B) Electron injection.

other words, at equilibrium state, some metal ions tend to be reduced by the reaction Ox + ne − −−→ Red.

(2.43)

At that time, for each reaction equation (2.41), there exactly exists one reaction equation (2.43). Therefore, at each time, one mole of metal is dissolved, and at the same time, one mole of ions is reduced. The final process is an equilibrium state defined by the reaction kf

−−  Ox + ne −  − − Red.

(2.44)

kb

For an electrode in equilibrium, the rates of generation and consumption of electrons are equal resulting in zero net currents. However, if we can draw the electrons out of the electrode as shown in Fig. 2.6A, then reaction (2.44) starts moving from right to left, and if the electrons are injected to the electrode as illustrated in Fig. 2.6B, then reaction (2.44) tends to go from left to right. From the basics of chemistry it is known that the rate of consumption of reactants j is proportional to the concentration of active materials at the reaction site. Therefore, for forward jf and backward jb reactions in Eq. (2.44), the reaction rates are jf = kf COx jb = kb CRed ,

(2.45) (2.46)

where kf and kb are the forward and backward reaction rates, respectively, and COx and CRed are their corresponding concentrations of active materials. In an equilibrium state, where the net rate of current is zero, jf = jb . But

Fundamentals of batteries

75

when a net current is passing through the cell, at cathode the forward rate dominates, and at anode the backward reaction is superior to the forward reaction. From Faraday’s law the current density is proportional to the mass production rate. Therefore the net current density produced by a single electrode is defined by i = nF (jf − jb ) = nF (kf COx − kb CRed ).

(2.47)

When the net current is zero, the forward and backward reactions are equal; note that it does not mean that forward and backward reaction rates are zero. The equal forward and backward current density at equilibrium state is called the exchange current density and denoted by i◦ . The exchange current density is an important physical property of each material. Any material that has a larger exchange current density is more active than the material with lower current density. Physically, the larger i◦ means a faster electrode, which can produce more electrons in a short time. The rate constants strongly depend on the temperature and G of reaction. The dependency is expressed by 

k=



G kB T , exp − h RT

(2.48)

where kB = 1.38 × 10−23 J K−1 and h = 6.626068 × 10−34 J s are Boltzmann and Planck constants, respectively. Clearly, R and T indicate the universal gas constant and absolute temperature, respectively. In chemical reactions the Gibbs free energy is calculated by the difference between the chemical Gibbs energies of reactant species. However, in an electrochemical reaction, the Gibbs energy contains both chemical and electrochemical energies. Therefore, in electrochemical reactions, G is defined as G = Gch + α nFV ,

(2.49)

where Gch accounts for the energy of chemical components, and α nFV is the electrical energy of reaction and electrons. In this equation, α is called the charge transfer coefficient and plays an important role in electrochemical systems. Comparing heat transfer, the heat transfer coefficient is an indicator of the ability of the medium in transferring heat, and the larger the heat transfer coefficient, the better the system handles the heat. Charge

76

Simulation of Battery Systems

transfer coefficient is an indicator of current transferring ability and the larger the coefficient is, the higher current can be handled by the medium. Using Eq. (2.49), the Gibbs free energy of the oxidation and reduction reactions are as follows: G = Gch − αRed nFV ,

(2.50)

G = Gch + αOx nFV .

(2.51)

Substituting these equations into Eq. (2.48), the forward and backward reaction rate constants can be obtained: 



kB T Gch + αRed FV exp − kf = , h RT   kB T Gch − αOx FV exp − kb = . h RT

(2.52) (2.53)

The above equations can be further simplified to 

kf = k◦,f exp −

αRed nFV



RT   αOx nFV kb = k◦,b exp + , RT

(2.54)

,

(2.55)

where all the constants and chemical parts of the Gibbs free energy are factored in k◦ coefficients. Applying Eqs. (2.54) and (2.55) to Eq. (2.47), we can obtain the total current density: 



i = nF k◦,f COx exp −

αRed nFV

RT



  αOx nFV − k◦,b CRed exp + . (2.56)

RT

Eq. (2.56) gives a proper relation between the current density and potential of a single electrode. In other words, if we want to calculate the potential of an electrode subjected to an amount of current, then Eq. (2.56) is an appropriate tool. We just have to pay attention that for calculation of Eq. (2.56), all the parameters should be measured at the reaction site. This is not a simple task, especially in experimental tests. It can be shown that using equilibrium state, this equation can be further simplified to obtain a better and more practical version. At equilibrium state, the potential of the electrode reaches its maximum (the value that is obtained from electrochemical tables). Here, for clarity,

Fundamentals of batteries

77

we denote the equilibrium or reversible potential by Vr . In this state, as mentioned before, the rates of forward and backward reactions are equal; we call it the exchange current density and show it with the symbol i◦ . From Eq. (2.56) we can calculate i◦ : 

i◦ = nFk◦,f COx exp −

αRed nFVr



RT

  αOx nFVr (2.57) = nFk◦,b CRed exp +

RT

Combining Eqs. (2.56) and (2.57), we have: 



i = i◦ exp −

αRed nF (V − Vr )

RT



 − exp +

αOx nF (V − Vr )

RT

 .

(2.58)

This equation is called the Butler–Volmer equation and gives the kinetic rate of each electrode. From the Butler–Volmer equation we can conclude that at equilibrium state V = Vr , the rates of production and consumption of electron at a single electrode are equal, and we can judge that no current is produced or consumed by the electrode. However, when the electrode is charged or discharged, either forward or backward reaction dominates, and V differs from its reversible value. The difference of the actual potential and its reversible value (η = V − Vr ) is called the overpotential and is in fact the driving force that causes the electrons to enter or exit from the electrode. Note that the Butler–Volmer equation is separately written for each electrode. Since a battery cell consists of positive and negative electrodes, we should write two distinct Butler–Volmer equations for each electrode. For example, for anode, we write: 



ia = i◦,a exp −

αRed,a nF (Va − Vr ,a )



RT

  αOx,a nF (Va − Vr ,a ) − exp + ,

RT

(2.59) and for cathode, we have: 



ic = i◦,c exp −

αRed,c nF (Vc − Vr ,c )

RT



  αOx,c nF (Vc − Vr ,c ) − exp + .

RT

(2.60) In these equations, subscripts a and c respectively indicate anode and cathode. It is quite evident that the only parameter that is equal in the previous two equations is the amount of net current with a negative value. In other words, ia = −ic .

(2.61)

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Simulation of Battery Systems

Table 2.2 Electrochemical properties of Example 2.9. αc i◦ A cm−2 Electrode αa Zn 0.5 0.5 0.01 Cu 0.5 0.5 0.0001

The negative sign means that the net current leaves one electrode but enters the counter one. Example 2.9. An electrochemical cell is made of zinc and copper electrodes whose electrochemical properties are tabulated in Table 2.2. If the cell is operating at T = 300 K, draw the characteristics curve up to i = 0.5 A cm−2 . Answer. In the current cell, the half-cell reactions are Zn −−→ Zn ++ + 2 e− at anode and Cu ++ + 2 e− −−→ Cu. These half-cell reactions show that the number of transferred electrons for this cell is n = 2. For and electrochemical cell, the Butler–Volmer equation is used to determine the electrode voltage versus current density. It should be emphasized that the Butler–Volmer equation is individually written for each electrode. Also note that αa = αc for both electrodes. A closer look at Eqs. (2.59) and (2.60) with αa = αc suggests that these equations can be simplified to 



0.5nF (Va − Vr ,a ) ia = 2i◦,a sinh , RT   0.5nF (Vc − Vr ,c ) ic = 2i◦,c sinh . RT Substituting the input parameters in these equations, we have: 



−ia RT sinh−1 , F 2i◦,a   −ic RT −1 sinh . V c = Vr , c − F 2i◦,c

V a = Vr , a −

(2.62) (2.63)

Fundamentals of batteries

79

Figure 2.7 Characteristic curve of Example 2.9.

The reversible or equilibrium potential of zinc and copper can be obtained using electrochemical tables. These values versus SHE are: Vr ,a = −0.76,

Vr ,c = +0.34.

Hence at equilibrium state, when no electrical charge is passing the external circuit, the cell voltage is Vcell = Vr ,c − Vr ,a = +0.34 − (−0.76) = +1.10 V. Now suppose that this cell is under discharge with i = 0.05 A cm−2 . In this case, we have to set ic = −ia = −0.05 and separately calculate the voltage of each electrode:  −0.05 Va = −0.76 − 0.02585065 sinh ≈ −0.717 V, 0.02   0.05 Vc = +0.34 − 0.02585065 sinh−1 ≈ +0.179 V, 0.0002 −1



(2.64) (2.65)

resulting in cell voltage of Vcell = Vr ,c − Vr ,a = +0.179 − (−0.717) ≈ +0.897 V. Similarly, the same procedure can be repeated for each current density to obtain the cell potential. The result is shown in Fig. 2.7. From the figure it is clear that both electrodes experience some voltage drop, but the amount of voltage drop for Cu electrode is larger than for the zinc electrode. This is because the zinc electrode is more active than its counter electrode by having higher i◦ . Example 2.10. In Example 2.9, if we want a battery to produce a current of 5 A with a voltage of 0.85 V, determine the size of electrodes.

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Simulation of Battery Systems

Answer. From Fig. 2.7 we deduce that at voltage V = 0.85 V, the current density is about i = 0.125 A cm−2 . Hence from Eq. (2.40) we have: i=

I I 5 =⇒ A = = = 40 cm2 . A i 0.125

Therefore the area of plates should be 40 cm2 ; for example, a plate with 5 × 8 cm is adequate.

2.5 Summary In this chapter, we presented the basic fundamental laws of electrochemistry. The fundamental laws will help to develop the governing equations of battery dynamics, which will be used in the simulation. Understanding the basics of electrochemistry is crucial both in simulation and analyzing the simulation results. The fundamental laws discussed in this chapter can be applied on any battery technology; however, each individual battery has a different chemistry and requires special attention when that technology is examined. In the following chapters the governing equations are applied in some traditional and industrial battery types, and the differences are discussed in more detail.

2.6 Problems 1. A battery cell is made of Mg and Fe. (a) Write the half-cell reactions and battery symbols. (b) Determine cathode and anode, positive and negative electrodes. (c) What is the open circuit voltage? 2. A lithium battery is discharged with I10 = 2.1 A. (a) How long does it take for being fully discharged? (b) What is its SoC and DoD after one, three, and five hours? (c) What is the estimated capacity of the cell? (d) How large is the energy content of the cell? 3. For a lead–acid battery, the following results are available: No.

Current (A)

Duration (hr)

1. 2.

22 35

0.75 0.4

Fundamentals of batteries

81

According to Peukert’s law, determine the true capacity of the battery. For the battery of 12 V sealed type, calculate its energy content. 4. For the lead–acid battery of problem 3, determine its currents 2I, I10 , and I20 .

CHAPTER 3

Fundamental governing equations Contents 3.1. Porous electrode theory and volume-averaging technique 3.1.1 Modeling domain 3.2. Governing equations 3.2.1 Conservation of mass and momentum 3.2.2 Conservation of species 3.2.3 Conservation of electrical charge 3.2.4 Energy balance 3.3. Volume-averaged governing equations 3.3.1 Conservation of mass 3.3.2 Conservation of momentum 3.3.3 Conservation of species 3.3.4 Conservation of electrical charge 3.3.5 Energy balance 3.4. Microscopic modeling 3.4.1 Specific active area 3.4.2 Species diffusion length 3.4.3 Microscopic Ohmic resistance 3.5. Summary 3.6. Problems

83 84 87 88 89 90 91 93 93 95 96 101 103 107 107 108 109 110 111

3.1 Porous electrode theory and volume-averaging technique Modern high-energy batteries are made of porous electrodes. Technically, porous electrodes contain more energy in comparison with solid flat ones since they have a more specific active area. Since electrochemical reactions take place at solid surfaces, increasing specific area (available surface area per volume) means a more intensive reaction in the same physical volume. From the simulation point of view, the theory of porous media should be applied for modeling and further simulating the electrodes. According to porous media modeling concepts, all the governing parameters should be averaged with volume as the main weight function. This means that the governing equation of battery dynamics should be volume-averaged. Simulation of Battery Systems https://doi.org/10.1016/B978-0-12-816212-5.00007-6

Copyright © 2020 Elsevier Inc. All rights reserved.

83

84

Simulation of Battery Systems

Figure 3.1 A typical lead–acid cell [modified from Ref. [27]]. (A) Battery section view. (B) Cell top view.

3.1.1 Modeling domain In Fig. 3.1, a typical lead–acid battery is shown. A cross section of a single cell can be seen in Fig. 3.1A. As can be seen, the cell consists of some positive and negative electrodes with proper separators in between. The top view of the cell shown in Fig. 3.1B clearly indicates that each positive electrode is surrounded by two negative electrodes. Consequently, each positive electrode makes two electrochemical cells by its two neighbors. The same argument is true for negative electrodes. In this point of view, we can imagine that each electrode can be divided into two half electrodes, each of which works against its own counter half-electrode. By this definition, for each cell set, there are many electrochemical cells that start from the center of the positive electrode to the center of the negative electrode. The above illustration implies that for each electrode, except for two boundary electrodes, a symmetric condition is valid. This assumption makes a negligible error in computation; as it is clear from the figures, there are normally many electrochemical cells inside each single cell set. The only exceptions are the two boundary electrodes for which the symmetric condition is not valid. Although this assumption introduces some errors in modeling, the amount of the error is negligible. As mentioned before, the symmetric condition suggests that the proper domain for modeling and simulation is the area starting from the center

Fundamental governing equations

85

Figure 3.2 A typical electrochemical cell model.

of the positive electrode and ends at the center of the negative electrode. Fig. 3.2 shows the domain of a typical cell model. The domain of interest, as discussed before, begins at the center of the positive electrode and ends at the center of the negative electrode. A separator is sandwiched between the electrodes. The separator may contain some ribs (as is typical in the lead–acid batteries shown in Fig. 3.1B) leaving a bulk volume of electrolyte next to one or both electrodes. In Fig. 3.2 the bulk electrolyte exists only near the positive electrode, but in some cases, the free electrolyte may also exist at the negative side. In the porous electrode model, each part of the cell is called a region. For example, the positive electrode, separator, and negative electrode in Fig. 3.2 are the regions of the cell. The regions are made of porous electrodes and separator that are filled with a specific electrolyte. In this figure, just a cross section of the cell is shown. Although it seems that the solid parts are not connected, they are actually connected in other sections. The current collector is located at the center of the positive electrode, and the other current collector is located at the center of the negative electrode. The positive electrode is made of positive active materials (PAM), and the negative electrode is made of negative active material (NAM). Concerning any porous media, the governing equations should be obtained according to the laws of porous media. In porous media theory, all governing equations should be volume-averaged inside the porous medium. A typical porous medium is shown in Fig. 3.3, which contains solid, gas, and electrolyte phases. This volume could be a representative electrode volume (REV) used for volume-averaging purposes. It is clear that each property (i.e., such as potential, concentration, etc.) is not uniform across

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Simulation of Battery Systems

Figure 3.3 Illustration of a sample volume containing solid, gas, and electrolyte phase.

the volume. For example, the potential on solid phase differs from each solid particle in the sample REV. If the REV is considered to be small enough, then we can assign a unique solid-phase potential to the whole REV. This unique value is obtained from averaging the solid-phase potential all over the REV. However, since the averaging depends on the volume of the solid phase with respect to the volume of the REV, the averaging should be a volume-averaging. This fact is true for any characteristic parameter, which we denote by  . Assume that V◦ is the volume of the REV in Fig. 3.3 with solid phase (k = s), electrolyte phase (k = e), and even gas phase (k = g). General volume averaging theory for any property k in phase k is defined by the following equations: 

  ∂k ∂ k  1 − k w  k · nk dA = ∂t ∂t V◦ Ak

(3.1)

for temporal averaging and ∇k  = ∇ k  +

1 V◦

 k n k dA

(3.2)

Ak

for spatial averaging. In these equations, w k is the velocity of the boundary of the phase k, and nk is the normal unit vector pointing outward the phase boundaries. The integral in these equations is taken over all the phase surface areas or Ak that is in contact with other phases such as m; hence, the area is defined

Fundamental governing equations

as Ak =



Akm ,

m = k.

87

(3.3)

m

Finally, we define two intrinsic averaging operators, volume averaging and phase volume averaging, as 

1 Xk k d∀, V◦ V◦  1 k k = Xk k d∀. Vk V◦ k  =

(3.4) (3.5)

In these operators, Xk is a phase operator equal to unity inside phase k and zero otherwise. Moreover, Vk is defined as the volume of phase k inside the volume of REV or V◦ . By this definition the volume averaging and phase volume averaging operators are related to each other by the relation k  = εk k k ,

(3.6)

where εk is the volume fraction of phase k in V◦ . In other words, εk is the fraction of volume occupied by phase k and is called the phase porosity: εk =

Vk . V◦

(3.7)

Eqs. (3.1) and (3.2) provide necessary tools for obtaining volumeaveraged governing equations out of normal equations. In this chapter, we first introduce the governing equations of battery dynamics. Then using the mentioned volume-averaging technique, all the governing equations are averaged, which gives us the proper form of the equations, which is useful for numerical simulation.

3.2 Governing equations Electrochemical reactions occur at the electrode/electrolyte interface. For each electrode, there can be more than one electrochemical reaction, either primary or side reactions. In general, we can describe each reaction by the equation 

species p=1

z

sp Mp p = nj e− ,

(3.8)

88

Simulation of Battery Systems

where the summation takes place over active species in reaction j for which sp is the stoichiometric coefficient of the active species Mp , and nj is the charge number or number of electrons that participates in the reaction. According to the kinetic rate, an electrical charge is produced or consumed by electrochemical reactions with rate rse , which can be obtained from the equation rse = −

  sj j

nj F



(3.9)

ij ,

where sj is the stoichiometric coefficient for the reaction, F is Faraday’s constant, and ij is the reaction rate described by the Butler–Volmer equation      αaj F αcj F ij = i◦j exp ηj − exp − ηj .

RT

RT

(3.10)

The subscript se in Eq. (3.9) indicates that the source term should be considered at electrode/electrolyte interface. It should be noted that bulk values differ from their corresponding values at the reaction site, due to the available sources or sinks. In the Butler–Volmer equation (3.10), anodic and cathodic charge transfer coefficients are αa and αc , respectively, which are correlated to each other by the relation αaj + αcj = n.

(3.11)

In Eq. (3.10) the overpotential ηj of reaction j is defined as ηj = φs − φe − Uj ,

(3.12)

which indicates the driven force by which the electrochemical reactions take place.

3.2.1 Conservation of mass and momentum The Electrolyte and active materials of many battery technologies are liquid or gas, whose movement can be described by fluid mechanics laws. These laws are the conservation of mass or continuity equation for each phase k, which is shown in the equation ∂ρk + ∇ · (ρk vk ) = 0, ∂t

(3.13)

Fundamental governing equations

89

and the equation of conservation of linear momentum, ∂ (ρk vk ) + ∇ · (ρk vk vk ) = −∇ pk + ∇ · τk + Bk . ∂t

(3.14)

In these equations, the subscript k refers to phase k, and τ and B are the stress tensor and body force, respectively. Full discussions about fluid motion can be obtained from various classical fluid mechanics textbooks.

3.2.2 Conservation of species In an electrochemical cell, various species exist, some of which are neutral, and some others are reactive. The concentration of these species is crucial for battery simulation. According to Fick’s law, the following equation is valid for each species in phase k: ∂ ck  k, = −∇ · N ∂t

(3.15)

where c is the molar concentration of that species, and N describes the species flux vector. It is worth mentioning that in an interface the species may jump from one level to another level. In an interface like k − m, we have the following relation between the concentration of each species:  k − ck w  m − cm w (N  k ) · n k + (N  m ) · nm = −rkm .

(3.16)

This equation clearly indicates that across an interface, species may jump  and/or interface movement w.  To use due to generation rkm , species flux N,  and interthe equation, we need to pay attention to the velocity vector w, face normal vector n must be considered outward from the phase boundary. Finally, we should emphasize that the generation rate rkm not only depends on electrochemical reactions at k − m interface, but also on physical phenomena such as evaporation, solidification, and so on contribute to its value.  k . In an electroAn important part of Eq. (3.15) is flux of species or N chemical cell, in general, species transport depends on diffusion, migration, and convection and can be written as  k = − D k ∇ ck + N

tk  ik + ck vk , zF

(3.17)

where Dk and tk are the diffusion coefficient and transference number of that species in phase k, respectively. It is clear that the transference number

90

Simulation of Battery Systems

is calculated with respect to a specific velocity. Although different references can be defined for the velocity, it is better to take the averaged mass velocity into account. By this manner the velocity field obtained from Eq. (3.14) is used in all the equations and vastly simplifies numerical calculations. In solid electrolytes the velocity field is zero, and mass transfer occurs only via diffusion and migration. Finally, it is worth mentioning that mass transfer by migration is almost zero for solid species. In other words, ts ∼ 0.

(3.18)

3.2.3 Conservation of electrical charge Electrochemical reactions occur at active material and electrolyte interface; consequently, no electrical charge is produced or consumed inside any phase k. This indicates that the net electrical charge production inside a single phase is zero and can be mathematically expressed as ∇ · ik = 0.

(3.19)

This equation is called electroneutrality, which is valid inside any phase. However, electrochemical reactions contribute to the production or consumption of electrons governed by the equation ik · nk = −



inj .

(3.20)

j

In these equations, ik is the current density in phase k. For the solid phase, the electrical charge is transferred by electrons whose transport is governed by Ohm’s law: is = −σ ∇φs ,

(3.21)

where is and φs are the current density and potential in the solid phase, respectively, and σ is the electrical conductivity of the phase. On the other hand, in the electrolyte phase, no matter if it is solid or liquid, electrical charge is transferred by ions. Ion transfer takes place by two different mechanisms: 1. Diffusion of ions from more or less concentrated locations governed by Fick’s law. 2. Migration of ions due to the electrical field. Consequently, Ohm’s law in the electrolyte phase should be modified to govern both effects. This form is sometimes is called the modified Ohm’s

Fundamental governing equations

91

law and is frequently used for simulation of current density for electrolyte: ie = −k∇φe − kD ∇(ln cei ),

i = + or − .

(3.22)

In this equation, k and kD are the effective electrolyte conductivity and diffusion conductivity, respectively. The diffusion conductivity accounts for the share of current density due to the electrolyte concentration gradient.

3.2.4 Energy balance Energy is described by the general equation  ρk cpk

  ∂ Tk

k ∇ · Jk , + vk ∇ Tk = −∇ · qk + H ∂t species

(3.23)

where ρ and cp are the density and specific heat at constant pressure respectively, T is the temperature, v is the velocity vector, q is the heat flux

vector, J is the molar flux of species due to diffusion and migration, and H is the partial molar enthalpy of species. In all these parameters, the subscript k refers to phase k. By these definitions the second term at the right-hand side of the equation determines the heat transfer due to the diffusion and migration of species. This is why the summation takes place over all species. In general, the heat flux vector contains the effect of Fourier conductive heat transfer, heat flux due to species diffusion, and Dufour heat flux, which is the energy flux due to a mass concentration gradient occurring as a coupled effect of irreversible processes. Usually, the Dufour heat flux can be neglected in battery systems; hence we can describe the heat flux vector q as qk = −λk ∇ Tk +



k Jk , H

(3.24)

species

where λk is the heat transfer coefficient in phase k. Combining Eq. (3.24) and continuity in phase k, that is, ∇ · vk = 0,

(3.25)

  ∂ Tk

k . + ∇ · (vk Tk ) = ∇ · (λk ∇ Tk ) − Jk · ∇ H ∂t species

(3.26)

we obtain:  ρk cpk

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Simulation of Battery Systems

From the thermodynamic relation

k = μk − T H

∂μk ∂T

(3.27)

and the definition of chemical potential, μk = μ0k + RT ln

ak

ak,ref

+ zF φk ,

(3.28)

where μ0k and ak respectively are the standard chemical potential and species activity in phase k, and ak,ref is species activity at standard conditions, we obtain ⎡



ak



⎢ ∂ ln ak,ref ⎢

k = −R ∇ ⎢ T 2 ∇H ⎢ ∂T ⎣

⎥ ⎥ ⎥ + zF ∇(φk − T ∂φk ). ⎥ ∂T ⎦

(3.29)

Eq. (3.29) is valid without any assumption, but in practical cases, some appropriate assumptions simplify the equation. For instance, the first term at the right-hand side is mixing enthalpy and is neglected in practical applications. Furthermore, we can ignore the dependency of phase potential on temperature. Consequently, Eq. (3.29) reduces to

k = zF ∇φk . ∇H

(3.30)

Substituting Eq. (3.30) into Eq. (3.26) yields  ρk cpk

  ∂ Tk + ∇ · (vk Tk ) = ∇ · (λk ∇ Tk ) − zF Jk · ∇φk . ∂t species

(3.31)

By the electro-neutrality assumption the current density in phase k takes place by diffusion and migration. In other words, ik =



zF Jk .

(3.32)

species

Therefore Eq. (3.31) can be rewritten as  ρk cpk

 ∂ Tk + ∇ · (vk Tk ) = ∇ · (λk ∇ Tk ) − ik · ∇φk . ∂t

(3.33)

Fundamental governing equations

93

The second term on the right-hand side is called the Joule heating and acts as a heating source term in electrical systems such as batteries. At any electrode/electrolyte interface the balance of energy becomes λe ∇ Te · ne + λs ∇ Ts · n s = ¯in + ¯in η.

(3.34)

This is because electrochemical reactions occur at the interface. In this equation, n is the phase normal vector pointing outward the phase, and subscripts s and e indicate the solid and electrolyte phase. Moreover, in is the current density due to electrode reactions. The right-hand side of the equation accounts for the generated heat due to electrode reactions. The generated heat is divided into the reversible and irreversible parts. The first term on the right-hand side indicates the reversible part and changes sign when the battery is charged or discharged, whereas the second term accounts for the irreversible part and is always positive. Summing up, Eq. (3.31) with interface balance shown by Eq. (3.34) constitutes a system of equations for the balance of energy. However, this form of energy balance is not suitable for mathematical simulation and like other governing equations, should be volume-averaged.

3.3 Volume-averaged governing equations As discussed before, the governing equations obtained do not apply to porous media. Since almost all electrochemical batteries are made of porous electrodes and separators, we have to apply volume-averaging technique to these governing equations to obtain their volume-averaged forms. The volume-averaged form applies to porous electrodes and will be used in other chapters to simulate different batteries. In the following sections of the chapter, for generality, we discuss all governing equations. For each specific battery technology, these equations should be simplified according to its own specific physical phenomena. For example, for solid electrolyte batteries (such as lithium-ion batteries), the conservation of momentum is not used; for low-current batteries such as zinc–silveroxide button batteries, we may drop energy equation since the battery does not get warm during operation.

3.3.1 Conservation of mass Applying Eqs. (3.1) and (3.2) to Eq. (3.13) results in

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Simulation of Battery Systems

 ∂(εk ρk ) + ∇ · (εk ρk vk k ) =

km , ∂t m

where 1

km = V◦

m = k,

(3.35)

 ρk (w  k − vk ) · n k dA.

(3.36)

Akm

In these equations, km is the rate of phase change from phase m to k at electrode/electrolyte interface. Using the mean theorem of integration, we can replace the integral of Eq. (3.36) with a production of a mean flux and an average surface area. Therefore Eq. (3.36) becomes

km = akm ρk w¯ nkm ,

(3.37)

where akm is called the specific interface area at the interface between phase m-k and is obtained by the equation akm = Akm /V◦ .

(3.38)

Also, V◦ is the volume of REV with unit of cm2 cm−3 , and w¯ nkm is the normal velocity of the interface of phases m − k with respect to phase k pointing outward phase k. Concerning porous electrodes, w¯ nkx or interface velocity is the consequence of phase or volume change due to all electrochemical reactions. Just for an example, we can refer to lead–acid batteries in which porosity of the electrode changes during discharge because porous electrodes (Pb in negative and PbO2 in positive electrodes) convert to lead sulfate, which occupies more space. Similarly in Ni−Cd batteries, cadmium converts to Cd(OH)2 , causing volume change resulting in surface velocity. Another example is hydration and dehydration of metals in Ni−Cd batteries. At the electrode/electrolyte interface, the interface velocity is mostly due to the structure change of electrodes, which is a direct result of species reactions. The reactions cause the species conversion with different molecular weight and density. Therefore, a different amount of space is occupied during charge and discharge resulting in interface movement, and hence w¯ nse =

   sj j

species

nj F

 ¯inj V¯ s .

(3.39)

Fundamental governing equations

95

In Eq. (3.39), ¯inj is the exchange current density of reaction j obtained from the Butler–Volmer equation (3.10) with overpotential defined by η¯ j = φ¯ se − φ¯ es − Uj .

(3.40)

Also, the partial molar volume of a species in the solid phase is denoted by V¯ s . Combining Eq. (3.39) with Eq. (3.37) results in

se = ase ρs

   sj j

species

nj F

 inj V¯ s .

(3.41)

Applying Eq. (3.35) in solid phase (k = s) and knowing that the solid surface velocity is equal to zero result in     sj ∂(εs ρs ) inj V¯ s = aes ρs ∂t nj F j species

(3.42)

with the mean partial molar volume V¯ s = MW /ρ,

(3.43)

where MW and ρ are the molecular weight and density of species, respectively. Since the porosity of an electrode ε is related to the solid phase porosity εs by the relation ε = (1 − εs ),

(3.44)

we can use Eq. (3.42) for calculation of electrode porosity change.

3.3.2 Conservation of momentum Volume-averaging of momentum equation results in     k    ∂  εk ρk vk k + ∇ · εk ρk vk k vk k = −εk ∇ pk + ∇ · τk  + τkt ∂t   d

Mkm + Mkm (3.45) , + εk Bk k + m

where   t  τk = (vk − vk k )(vk − vk k ) ,  1 d = τk · n k dA, Mkm

V◦

Akm

(3.46) (3.47)

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Simulation of Battery Systems

1 Mkm = V◦

 ρk vk (w  k − vk ) · n k dA.

(3.48)

Akm

 

d

and Mkm In these equations, τkt is the dispersion shear stress tensor, Mkm respectively are the momentum transfer rates due to viscous drag and interface movement. The macroscopic shear and dispersion stress in Eq. (3.45) can be simplified by using an effective mean viscosity:

  τk  + τkt = μ∗k εk ∇ vk k ,

(3.49)

where μ∗ is the total transport property containing both effects of microscopic property and the effect of microstructures in the porous medium. Since the velocity inside the porous electrodes is very small, for the first estimation, we may choose μ∗ equal to its microscopic value, neglecting the effect of velocity dispersion due to the microstructures of porous media. In porous media problems, it is customary to use the modified Darcy law for accounting the effect of interface drag, which includes both effects d

of Mkm and Mkm . The modified Darcy law in this case is 



d

Mkm + Mkm = −εk2

m

μk

Kkrk

· vk k .

(3.50)

In this equation, K is the absolute permeability, and krk is the relative permeability of phase k, including the effect of decreasing of the area due to fluid vortexes that occupy the pores. The relative permeability is a function of the volumetric phase ratio and is obtained from experimental data [28]. Substituting Eqs. (3.49) and (3.50) into Eq. (3.45), the macroscopic equation of momentum can be obtained as       k ∂  εk ρk vk k + ∇ · εk ρk vk k vk k = −εk ∇ pk + ∇ · μ∗k εk ∇ vk k ∂t μk · vk k . (3.51) + εk Bk k − εk2

Kkrk

3.3.3 Conservation of species Applying volume-averaging technique to Eq. (3.15) results in   ∂ ck  1 k − = −∇ N ∂t V◦

 Ak

 k − ck w (N  k ) · nk dA.

(3.52)

Fundamental governing equations

97

In comparison   with Eq. (3.17), the volume-averaging of species flux in  k can be determined by the equation phase k or N 



 k = − Dk ∇ ck  + N

tk   ik + ck vk  , zF

(3.53)

where the coefficient tk /zF is the result of volume averaging. This value is almost constant in volume V◦ . From the physical point of view, the first term in the right-hand side of Eq. (3.53) determines the macroscopic diffusion of species, which normally is obtained using the effective mean diffusion coefficient: Dk ∇ ck  = Dkeff ∇ ck k ,

(3.54)

where Dkeff is the effective diffusion coefficient. Since the electrolyte is confined in porous electrodes and separator, this coefficient is modified so that the effect of porosity and tortuosity of the porous media is considered. The convective term in Eq. (3.53) (the third term in the right-hand side) normally is replaced by the product of the mean concentration and mean velocity plus an additional term, which accounts the effect of hydrodynamic dispersion causing from electrolyte movement inside the microscopic pores. In mathematical form the convective term becomes ck vk  = εk ck k vk k − Da ∇ ck k ,

(3.55)

where Da is called the dispersion coefficient, which includes the axial dispersion caused by electrolyte movement near the porous electrode walls. The dispersion coefficient is not a fundamental property of fluid and depends on fluid movement. When the electrolyte is stagnant (e.g., in solid electrolytes), this term is zero. Substituting Eqs. (3.54) and (3.55) into Eq. (3.53), we obtain 







 k = − Dkeff + Da ∇ ck k + N

tk   ik + ε ck k vk k . zF

(3.56)

Hence Eq. (3.52) can be rewritten as     ∂(εk ck k ) + ∇ · εk ck k vk k = ∇ · (Dkeff + Da )∇ ck k ∂ t      tk   1 1 tk  −∇ · ik + Dk ∇ ck · nk dA − ik · nk dA +

1 V◦



zF

V◦

Ak

ck (w k − vk ) · nk dA. Ak

V◦

Ak

zF

(3.57)

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Simulation of Battery Systems

By applying the averaging technique to Eq. (3.19), Eq. (3.57) can be further simplified. The result is    1 ik · nk dA = 0.  ∇ · ik +

V◦

(3.58)

Ak

Examining this equation reveals that a part of right-hand side of Eq. (3.57) is canceled out by the fourth term. Hence this equation reduces to ∂(εk ck k ) + ∇ · (εk ck k vk k ) = ∇ · [(Dkeff + Da )∇ ck k ] ∂t      tk d

+ (Jkm + Jkm ) − ik · ∇ ,

zF

m

(3.59)

where 

1 = Dk ∇ ck · nk dA, V◦ Akm  1

= ck (w k − vk ) · nk dA. Jkm V◦ Akm

d Jkm

(3.60) (3.61)

d

and Jkm represent the surface transport of a species in The terms Jkm phase k due to microscopic diffusion and surface movement, respectively. Eq. (3.59) determines the conservation of species in solid and liquid electrolytes. If transport equation of surface species in Eqs. (3.60) and (3.61) are accurately modeled, then we can use this equation to determine the macroscopic equations from microscopic ones. For this purpose, by considering the mean value theorem in integration the transport of species term due to the surface movement can be replaced by the product of the mean surface concentration and mass flux at the interface:

= c¯km km . Jkm

(3.62)

Similarly, the integral of Eq. (3.60) can be replaced by the product of an interfacial specific area and a mean diffusion flux at the interface. From physical point of view, the mass transfer term at interface represents the diffusion phenomena by which concentration gradient occurs. The diffusion flux is proportional to its own driving force, which is the concentration

Fundamental governing equations

99

Figure 3.4 Illustration of species diffusion length and Ohmic drop at an interface [29].

gradient. On the other hand, the flux is inversely proportional to the diffusion length l. The diffusion length determines the resistance to diffusion; hence we have d Jkm

 ∂ ck  = akm Dk  ∂ n k 

= akm Dk km

¯ckm − ck k

lkm

.

(3.63)

Hence mathematically, the diffusion length of species in phase k can be written as lkm =

¯ckm − ck k  . ∂ ck  −  ∂ n k 

(3.64)

km

In these equations, Dk is the diffusion coefficient of species in phase k, and ¯ckm is the mean concentration of species at m − k interface. Fig. 3.4 schematically illustrates the microscopic species distribution at solid and electrolyte interface. The interface shown in this figure is just a tiny portion of what is shown in Fig. 3.3. In this figure the definition of diffusion length is illustrated. The diffusion length defined in Eq. (3.64) is a very complex function of microscopic phenomena, where its determination requires specific atten-

100

Simulation of Battery Systems

tion. In the following sections, we discuss this characteristic length in more detail. The species transport equations are correlated to each other by means of surfaced balancing. To carry out the balance, Eq. (3.16) is integrated at the interface of phases k and m inside volume V◦ , which results in 

1 tk  (−Dk ∇ ck + ik + ck vk − ck w k ) · nk dA+ V◦ Akm zF  1 tm  (−Dm ∇ cm + im + cm vm − cm w m ) · nm dA = −akm r¯km , V◦ Akm zF

(3.65)

where rrm is the mean reaction rate at Akm interface. By Eqs. (3.60) and (3.61) and the relations given by Eq. (3.21), Eq. (3.65) can be rewritten as 





d

d

Jkm + Jkm + Jmk + Jmk







tk + tm  ¯ ⎠ inj . = akm ⎝r¯km − zF j

(3.66)

In this equation, two important case are to be considered: 1. When the m − k interface is electrochemically active at which ions exist only at one side (e.g., in phase k). Therefore Eq. (3.66) can be simplified as 

d

Jkm + Jkm



⎡ ⎤    tk s j ¯inj ⎦ . + = −akm ⎣ j

zF

nj F

(3.67)

To obtain this equation, Eq. (3.9) is averaged over the surface area. This equation not only can be used directly, but also can be used to obtain surface species concentration ¯ckm . To do this, we should use the constitute equations defined by Eqs. (3.62) and (3.63) in Eq. (3.67), which results in ⎡ ⎤      tk s akm Dk akm Dk j ¯inj ⎦ , (3.68) ck k − akm ⎣ ¯ckm km + + =

lkm

lkm

j

zF

nj F

where the phase change rate km is obtained from Eq. (3.41). Note that Eq. (3.68) is coupled with the Butler–Volmer equation so that ¯inj can be obtained. This is because the exchange current density and reversible potential are strong functions of species surface concentration c¯km . 2. The second case is where the m − k interface is electrochemically inactive. For an example, we can refer to soluble hydrogen or oxygen at

Fundamental governing equations

101

liquid or gas phases. In this situation the balance of species is reduced to 







d

d

Jkm + Jkm + Jmk + Jmk = 0.

(3.69)

Substituting Eqs. (3.62) and (3.63) into this equation, we have akm Dk amk Dm (¯ckm − ck k ) + (¯cmk − cm m ) lkm lmk

km = , ¯cmk − c¯km

(3.70)

where the mass balance mk = − km at interface is used. From Eq. (3.70) the phase change rates ¯ckm , c¯mk at the interface can be determined.

3.3.4 Conservation of electrical charge Macroscopic conservation of electrical charge is obtained by applying volume-averaging to Eq. (3.19):    ∇ · ik − Ikm = 0,

m = k.

(3.71)

m

In this equation the interfacial current with the unit of A cm−3 is calculated by the relation  1 ik · n k dA. (3.72) Ikm = − V◦ Akm Ohm’s law can be used to relate the electrical potential and current density in the solid phase. This law in proper vector form is ∇ · (σ eff ∇ φe e ) +



Ism = 0,

m = s.

(3.73)

m

Similarly, for the electrolyte phase, Ohm’s law indicates:   e e ∇ · (keff ∇ φe e ) + ∇ · (keff Iem = 0, D ∇ ln ci ) +

m = e.

(3.74)

m

We can use the Taylor expansion rule to obtain 

e  e ln cei = ln cei ,

(3.75)

and substituting into Eq. (3.74) yields   i e ∇ · (keff ∇ φe e ) + ∇ · (keff Iem = 0, D ∇ ln ce ) + m

m = e.

(3.76)

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Simulation of Battery Systems

The current density Ikm at electrode/electrolyte interface is Ise = −Ies = ase



¯inj .

(3.77)

j

Similarly, for any interface between phases m and k, the interface current density can be modeled as Ikm = akm

φ¯ km − φk k

Rkm

,

(3.78)

where Rkm with units of  cm−2 is defined as Rkm =

φ¯ km − φk k  . ∂φk  −σ  ∂ n k 

(3.79)

km

This is the Ohmic resistance between phases k and interface m − k. In Eq. (3.78), it is assumed that phase k is a solid phase in which Ohm’s law is valid. For liquid phases at the interface, the effect of migration should also be considered. By this assumption Ohm’s law becomes Ikm = akm

φ¯ km − φk k

Rkm

+ akm

 kD  c¯km − ck k , ¯ckm lkm

(3.80)

where phase k is liquid. By combining Eq. (3.78) or Eq. (3.80) with Eq. (3.77) we can obtain the surface potential φ¯ se or φ¯ es . These potentials are used to calculate the surface overpotential, which is necessary for calculation of the overpotential used in the Butler–Volmer equations. In practical applications the electrode conductivity of electrodes σ is usually very high. This implies that φ¯ se must be equal to its averaged value φs s . However, this argument is not true for liquid electrolytes. In other words, the conductivity of electrolytes is relatively low; hence φ¯ se and its phase mean value φe e are not equal especially in high-current applications. This means that we cannot use φe e instead of φ¯ se to calculate the overpotential used in the Butler–Volmer equation. The relation between surface potential and volume-averaged potential is illustrated in Fig. 3.4.

Fundamental governing equations

103

3.3.5 Energy balance Volume averaging from energy equation (3.23) results in # ρk cpk

$   ∂εk Tk k k k k   + ∇ · (εk Tk  vk  ) = ∇ · (λeff + λ )∇ T , a ,k k k ∂T   Joule d

Qkm + Qkm (3.81) − ik  · ∇ φk k + Qk , m

where 

1 λk ∇ Tk · nk dA, V◦ Akm  1

= ρk cpk Tk (w  k − vk ) · nk dA, Qkm V◦ Akm

d Qkm =

(3.82) (3.83)

and QkJoule

 k 1  = − ik ·

1 − V◦



V◦

(φk − φk k )nk dA

Akm

  ik − ik k · ∇(φk − φt k )d∀.

(3.84)

Vk

In these equations, λeff k is the effective thermal conductive coefficient in phase k, and λa,k is called the dispersion coefficient in phase k. Like other properties, λeff k should be modified so that the effect of porosity and its tortuosity is included. Normally, these effects are included using the Bruggman relation [30] 1.5 λeff k = λk ε k .

(3.85)

In contrast to λk , which is an intrinsic property of any material, λa,k accounts for hydrodynamic dispersion effects, which obviously depend on the velocity and temperature of the fluid. In other words, λa,k is a flow property and is zero when the electrolyte is stagnant. The second term in the right-hand side of Eq. (3.81) is the total sum

of heat transfer at the interface; Qkm indicates conductive heat transfer,

whereas Qkm is the heat transfer due to interface movement. Using the

mean value theorem in integrals, we can replace Qkm by the product of the mean velocity value and the mean temperature of the interface:

Qkm = cpk T¯ km km ,

(3.86)

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Simulation of Battery Systems

where T¯ km is the mean surface temperature at m − k interface. The last term in the right-hand side of Eq. (3.81), QkJoule , comes from volume averaging from Eq. (3.33). This term vanishes when phase k is electrically at equilibrium or φk = φk k . This means that the electrical potential is uniform all over phase k. When the electrical conductivity is not very high or the current density is too high, the equilibrium condition is not valid, and QkJoule should be considered. Such a case happens when semiconductor electrodes are in use like NiOOH electrode with electrical conductivity about 10−5 S/cm. Summation of Eq. (3.81) over all contributing phases (i.e., solid, liquid, and gas phases) yields %

#

$& ∂εk Tk k k k ρk cpk + ∇ · (εk Tk  vk  ) ∂t k ( '  k = ∇ · (λeff k + λa,k )∇ Tk 



k

+

 1  k=m m

V◦

     ¯ ik · ∇ φk k . λk ∇ Tk · nk dA + cpk Tkm km −

Akm

k

(3.87) The second sum in the right-hand side of Eq. (3.87) comes from the electrochemical reactions at solid/electrolyte interface. From the energy point of view, to obtain the sum, we need to write heat balance at the interface, that is, λe ∇ Te · ne + λs ∇ Ts · ns = ¯in η + ¯in ,

(3.88)

where n is the normal surface unit vector pointing outward from the phase, and the subscripts e and s as usual indicate the electrolyte and solid phases, respectively. Moreover, in is the local transfer current density, which is the result of electrochemical reactions. The right-hand side of Eq. (3.88) is the overall generated heat due to electrochemical reactions that can be divided into irreversible (the first term) and reversible (the second term) parts. Torabi and Esfahanian [31,32] presented a full detailed discussion about reversible and irreversible parts of the generated heat and introduced the concept of general Joule heating because the irreversible part of the generated heat is of Joule heating type. While the irreversible part is always positive, the reversible part changes the sign in charge and discharge conditions, the fact that is very obvious

Fundamental governing equations

105

from their definition. In other words, the irreversible part is the product of the local current density in and the local overpotential η, whose values are always positive since both parameters simultaneously change the sign in charge and discharge. However, the reversible part is the product of the local current density and local Peltier coefficient j . The Peltier coefficient was first introduced by Newmann [33], where by neglecting Duffour energy flux it is related to the entropy of reaction by the relation j =

T Sj . nj F

(3.89)

Since the entropy Sj for each reaction is a known value, during charge and discharge, it is fixed, and its value does not change. Therefore, the product of the Peltier coefficient and local current density changes sign in charge and discharge resembles a reversible process. It should be strongly pointed out that the represented heat balance is carried out only for heat of electrochemical reactions and does not include the effect of any physical phenomena such as evaporation, condensation, crystallization, and so on. If we need to include these effects, an additional source therm should be added to the right-hand side of Eq. (3.88). Hence for a general formulation, we can express Eq. (3.88) as  λk ∇ Tk · nk + λm ∇ Tm · n m =

asj¯inj (ηj + j ) + (hk − hm ) km

j

akm

,

(3.90)

where km is the phase change rate from phase m to phase k, asj is the specific active surface for electrochemical reactions, and akm is the specific surface area at interface between phases k and m. Finally, h indicates the enthalpy with respect to phases k or m according to its subscript. Applying Eq. (3.90) to Eq. (3.87) results in %

#

j

  − ( ik · ∇ φk k ).

$&   ∂εk Tk k k k k ρk cpk + ∇ · (εk Tk  vk  ) = ∇ · (λeff k ∇ Tk  ) ∂t k k  ) * ¯ (hk − hm ) km + (cpk − cpm )T¯ km km + asj inj (η¯ j + j ) +



k=m m

k

(3.91)

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Simulation of Battery Systems

When local thermal equilibrium exists, that is, Tk k = Tm m = T¯ km = T¯ mk = T ,

(3.92)

we can drop averaging symbols; hence Eq. (3.91) is simplified to ∂(ρ cp T ) + ∇ · (vT ) = ∇ · λ∇ T + q, ∂t

(3.93)

where ρ cp =



εk ρk cpk ,

(3.94)

εk ρk cpk vk k ,

(3.95)

k

v =

 k

 λ= (λeff k + λa,k ),

(3.96)

k

and the heat source is defined as q=



asj¯inj (η¯ j + j ) +



     ik · ∇ φk k h∗ km −

k=m m

j

(3.97)

k=e,s

with h∗ = (hk − hm ) + (cpk − cpm )T ,

(3.98)

the mean volumetric current density at electrolyte phase    i ik = −keff ∇ φe e − keff D ∇ ln ce i,

and at solid phase

  is = −σ eff φs s .

i = + or −,

(3.99)

(3.100)

In these equations, kD is ionic conductivity at phase k, and σ is the solid conductivity. The superscript eff like before indicates that these values are corrected to account the effect of the porous media. Moreover, note that no current flows through the gas phase (k = g), meaning that the gas acts as an insulator. Eq. (3.97) shows that heat sources due to Joule heating, electrochemical reactions, and phase change exist both in solid and electrolyte phases. Since this equation is described in vector form, we can make use of local current density to obtain the temperature distribution for systems that are not

Fundamental governing equations

107

in thermal equilibrium. Finally, Eq. (3.97) is a temporal equation, which means that by its solution the temperature variation concerning time can be obtained. The heat sources and sinks in any battery system play a vital role in determining its thermal behavior. According to the importance of source term, that is, q in Eq. (3.97), an entire chapter is dedicated for a full discussion about the issue. Hence a detailed information is available in Chapter 4.

3.4 Microscopic modeling The battery model discussed here is a macroscopic model. This means that all fundamental parameters are averaged; hence the microscopic distribution of these parameters are rendered. However, three important microscopic parameters are crucial for the evaluation of accurate surface properties in the macroscopic model. In other words, the effect of microscopic surface phenomena that plays an important role in determining battery behavior is to be included in the model. These effects can be added to the model by accurate modeling of the following three parameters: 1. Specific active area akm . 2. Diffusion length lkm . 3. Ohmic resistance Rkm . Accurate evaluation and implementation of these parameters extremely affect the accuracy of the model. These parameters are usually very hard to obtain, and their modeling requires complex analysis of microscopic tortuosity of the porous media and analysis of the diffusion of species and charge transfer at the solid interface. Here an acceptable model for considering the effect of these parameters is presented. It will be shown that by the present modeling the effect of microscopic phenomena such as solid diffusion and ohmic resistance in semiconductors can be modeled and added to the overall macroscopic model.

3.4.1 Specific active area As it was previously shown, the specific active area or electrode/electrolyte interface area is defined as ase = Ase /V◦

(3.101)

and has a crucial role in modeling interfacial properties. In practice the interface area contains much important information about the electrochem-

108

Simulation of Battery Systems

ical reactions but unfortunately are canceled in volume-averaging process. The canceled information is really important; thus their effect should be added to the model by constitute equations. The specific active area is normally obtained by adsorbing methods (like BET test) or by double-layer charging method. However, here an estimation method for calculation of specific active area is presented. The specific active area for electrodes that are made of spherical particles (like porous electrodes made of powders) having radius rs can be calculated by a◦se =

3εs 3(1 − ε◦ ) = , rs rs

(3.102)

where ε◦ is the porosity of the electrodes. Note that the solid phase porosity and electrode porosity are related such that εs = 1 − ε◦ .

3.4.2 Species diffusion length In practical cases, battery active materials are made of planar, cylindrical, or spherical particles. Each geometry has its own relation for calculation of species diffusion length. In cylindrical geometries the active materials are supposed to be uniformly pasted on a substrate. Since in making the electrodes a very thin layer of active material is used, the effect of electrode curvature is negligible, and a parabolic equation is enough for modeling of species diffusion length: cs = a◦ + a1 r + a2 r 2 ,

(3.103)

where r is the radial distance. Boundary conditions depict ∂ cs =0 ∂r cs = c¯se

at

r = r◦ ,

(3.104)

at

r = rs ,

(3.105)

where r◦ is the radius of the core, and rs is the radius of electrode/electrolyte interface. The local mean species concentration cs s is defined as cs s =

1 Vs



cs rdr = Vs



2 rs2

− r2 ◦

rs

cs rdr . r◦

(3.106)

Fundamental governing equations

109

Using these three equations, we can obtain the coefficients a◦ , a1 , and a2 in Eq. (3.103). Substituting the coefficients into Eq. (3.103) results in (rs − r )(rs − 2r◦ + r ) ¯cse − cs = . c¯se − cs s rs2 − r◦2 2 4r◦3 − r s r◦ + 2 3 3(rs + r◦ )

(3.107)

Consequently, using Eq. (3.64), we easily calculate the diffusion length: lse =

r s + r◦ rs r◦ 3r◦3 − + . 4 3(rs − r◦ ) 3(rs2 − r◦2 )

(3.108)

Using this equation when r◦ = 0, we simplify Eq. (3.108) rs lse = . 4

(3.109)

This equation shows that the species diffusion length is proportional to the species particle size and has almost the same order of magnitude. Similarly, we can calculate the species diffusion length for Cartesian and spherical cases. For planar electrodes with a layer of active materials with a half thickness of rs , rs (3.110) lse = , 3 and for spherical particles with radius of rs , rs lse = . 5

(3.111)

3.4.3 Microscopic Ohmic resistance By the definition given in Eq. (3.79), the Ohmic resistance from electrode/electrolyte and electrode/substrate interfaces to the bulk of active materials are respectively defined as Rse =

φ¯ se − φs s  , ∂φs  −σs  ∂r 

(3.112)

r =rs

φ¯ b − φs s  , Rsb = ∂φs  −σ◦  ∂r  r =r◦

(3.113)

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Simulation of Battery Systems

where the subscript b refers to the substrate, and φs is the solid phase potential in active materials. Note that the electrical conductivity σ is a function of SoC; consequently, it is variable across the electrode surface. The symbols σ◦ and σs show the electrical conductivity of electrode/substrate and electrode/electrolyte interfaces, respectively. It is assumed in this equation that the electrical current flows radially and the assumption of electroneutrality is valid. Integration of the equation over the active material layer results in     ∂φs ∂φs = σ ·r . σ ·r ∂ r r =r◦ ∂ r r =rs

(3.114)

Two other boundary conditions are: % φs =

at at

φb φ¯ se

r =r◦ , r =rs .

(3.115)

Assuming a quadratic profile for potential distribution in the solid phase of the form φs = a◦ + a1 (r − r◦ ) + a2 (r − r◦ )2 ,

(3.116)

the coefficients a◦ , a1 , and a2 are to be obtained from Eqs. (3.114) and (3.115). After obtaining the coefficients and substituting into Eqs. (3.112) and (3.113), we calculate the Ohmic resistances: 





r s r s − r◦ rs + 3r◦ 3rs + 5r◦ + , 12 rs + r◦ ) σ◦ r◦ σs rs    r ◦ r s − r◦ 5rs + 3r◦ 3rs + r◦ + Rsb = . 12 rs + r◦ σ◦ r◦ σs rs

Rse =

(3.117) (3.118)

It is obvious that these resistances are proportional to species diffusion length and are inversely proportional to electrical conductivity, σ .

3.5 Summary In summary, Eqs. (3.42), (3.45), (3.59), (3.73) or (3.74) and (3.93) constitute a complete set of governing equations for obtaining macroscopic behavior of batteries. The solution of the system of equations gives a variation of the following unknowns: 1. Electrode porosity εk . 2. Velocity vector of electrolyte vk k .

Fundamental governing equations

Table 3.1 Governing equations at interfaces. Transport terms at interface balance at interface

km = akm ρk w¯ nkm

km + mk =0 d = a Dk (¯c − c k ) Jkm km k lkm km

= c J Jkm km km

Ism = asm Iem = aem

zF

φ¯ sm − φs s

Rsm

φ¯ em − φe

Rem

d + J ) + (J d + J ) (Jkm mk km mk ⎛ ⎞  t + t m k ¯inj ⎠ = akm ⎝r¯km −

e

111

related equation

mass species

j

Ism + Ims = 0

electrode charge

Iem + Ime = 0

electrolyte charge

3. Concentration of chemical species ck k . 4. Potential of solid and electrolyte phases φk k . 5. Temperature Tk k . Note that Eq. (3.59) should be repeated for each active species; hence the number of concentration equations and related unknowns are as many as the number of species under consideration. Besides the governing equations, boundary conditions at any interface are necessary. These conditions are tabulated in Table 3.1, as well as the balance of parameters at the same interface. Using these equations, we can obtain the necessary information used for obtaining surface properties and, consequently, the rate of reaction at the interfaces.

3.6 Problems 1. Using the balance of mass, obtain a proper relation for porosity change during charge and discharge for NiCd batteries. 2. Using the balance of mass, obtain a proper relation for porosity change during charge and discharge for Li-ion batteries. 3. In a lead–acid factory, the size of lead particles used for making the electrodes is rs ≈ 2.5 × 10−4 cm. Calculate the approximate active area versus porosity and plot the result. 4. Discuss on mass transport mechanisms in a Li-ion battery. Which mechanisms are dominant and which we can ignore. 5. In a gelled lead–acid battery the rate of heat generation is assumed to be dependent only on Joule heating. Using Eq. (3.93), give a proper relation for energy equation.

CHAPTER 4

Heat sources Contents 4.1. Fundamental laws 4.2. Heat of reactions 4.2.1 Reversible or entropic heat of reactions Enthalpy of reaction Entropy of reaction Entropy of electron 4.2.2 Joule heating 4.3. General Joule heating concept 4.4. Heat dissipation 4.4.1 Convection 4.4.2 Conduction 4.4.3 Radiation 4.4.4 Exhausted enthalpy 4.4.5 Equivalent circuit model 4.5. Summary 4.6. Problems

114 118 119 119 120 123 124 125 125 125 126 126 126 127 127 128

Usually a battery, just like any other systems from the thermodynamical point of view, is a system with fixed mass. However, in some cases the battery is considered as a control volume because we have mass transport to or from the battery. Metal–air batteries are of control volume type, whereas lead–acid, lithium-ion, nickel-based batteries, and many other technologies are of system type. Whether they are considered as a system or control volume, they exchange heat with ambient and also act as a heat generator or in some rare cases as sinks. To obtain the battery temperature, we have to deal with thermodynamical laws and relations. Any thermodynamical system is subjected to generation, dissipation, and storage of energy. To predict its temperature distribution, the first law of thermodynamics should be applied on it. In a multiphase multicomponent medium like batteries, special care should be paid to obtain accurate results. In such a system, heat generation and dissipations are due to: 1. Heat generation due to electrochemical reactions. 2. Material phase change. 3. Mixing. 4. Change in heat capacity of the system due to the material change. Simulation of Battery Systems https://doi.org/10.1016/B978-0-12-816212-5.00008-8

Copyright © 2020 Elsevier Inc. All rights reserved.

113

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Simulation of Battery Systems

5. Electrical work. 6. Heat exchange with surroundings. The traditional methods for prediction of thermal behavior of batteries yield to the estimation of a uniform temperature for the whole cell. This assumption is made because in traditional methods a lumped model is used, in which the battery cell is considered as a uniform system with high thermal conductivity resulting in a uniform temperature. If we like to obtain a temperature gradient inside the battery, we need to introduce a model that estimates both generation and dissipation inside the cell. Such models are based on fundamental governing equations in vector form. The fundamental laws are in the form of convective–diffusive transport equations that include the effect of transient parts, heat convection, heat conduction, and heat sources and sinks. A proper system of governing equations with proper boundary conditions gives an appropriate tool for prediction of temperature profile inside any battery cell. In this chapter, we fully discuss the fundamental thermodynamical equations that govern the energy balance of a battery system. We discuss in detail heat sources and sinks that are the main issue in temperature profile.

4.1 Fundamental laws For a multicomponent multiphase medium in which the energy content is changing and different phases are in contact with each other, the first law of thermodynamics can be written as the equation dHtot = q − IV , dt

(4.1)

in which Htot is the total enthalpy of all the phases: Htot =

 j

Vj







ci,j H¯ i,j d∀,

(4.2)

i

where i and j describe the ith species and the jth reaction, respectively. In Eq. (4.1) the rate of heat generation and dissipation is shown by q, and IV represents the electrical work. For simplicity, we can assume a mean composition for each phase; hence we have 

dHtot d   = ni,j H¯ iavg ,j + dt dt j i





Vj





ci,j H¯ i,j − H¯ iavg d∀ . ,j

(4.3)

Heat sources

115

The first term on the right-hand side of Eq. (4.3) shows the rate of enthalpy change of the system when all the species are in an equilibrium state and the second term accounts for their deviation from equilibrium. The first term can be divided into three parts:

avg

¯ i ,j ∂H

¯ . =C pi,j avg

∂T

(4.4)

p

The first part in Eq. (4.3) can be calculated as  d j

i

dt

¯ i ,j ) = (ni,j H avg

 j

ni,j C¯ piavg,j

i



dni,j dT ¯ iavg +H . ,j dt dt

(4.5)

As explained earlier, at each electrode a set of electrochemical reactions occurs, all of which can be defined by the general formula 

νi,l Mizi = nl e− .

(4.6)

i

This equation is written so that each species i is in a single phase. Therefore diffusion phenomena from one phase to another phase should be separately considered. Using the balance of chemical species, we can calculate the mole of each species by the equation dni,m  νi,l¯il  dni,j = − . dt nl F j, j=m dt l

(4.7)

The first term on the right-hand side of the equation represents the rate of generation of consumption of species i due to electrochemical reactions. Moreover, the partial current ¯il is the necessary current that each reaction requires. This current is positive for cathodic reactions and negative for anodic reactions. The second term of Eq. (4.7) indicates the change of mole of species i due to phase change. Integration of Eq. (4.7) results in ni,m = n◦i,m −



(ni,j − n◦i,j ) +

j, j=m

 νi ,l  l

nl F



t

¯il dt.

(4.8)

The partial molar enthalpy is defined by ◦ 2 H¯ iavg ,m = Hi,m − RT

d avg ln(ai,m ). dT

(4.9)

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Simulation of Battery Systems

On the other hand, the theoretical open-circuit voltage for reaction l in an averaged composition with respect to a reference electrode is ◦ + Ui,avg = Ul◦ − URE

RT  RT  avg νi,RE ln(aRE νi,l ln(ai,m ). (4.10) i )− nRE F i nl F i

Using the Gibbs–Helmholtz relation, we can calculate the standard enthalpy of any reaction as a function of cell voltage:  νi ,l

nl F

i

Hi◦,m

d =T dT



2

Ul◦ . T

(4.11)

According to Eqs. (4.5) and (4.7), we conclude that the share of electrochemical reactions in enthalpy generation can be calculated by the equation  m

H¯ iavg ,m

 νi,l ¯il

i

=

nl F

l

  ¯il  m

l

nl F

H¯ iavg ,m νi ,l .

(4.12)

i

This value can be expressed as a function of potential of electrode reactions using Eqs. (4.9) and (4.11). The result is   ¯il  m

l

nl F

H¯ iavg ,m νi ,l

=

i





It

l

d T dT 2





Ul◦ RT 2  d avg avg ln(ai,m )νi,j . − T nl F i dT (4.13)

Since we need to express the potentials with respect to a reference electrode, we rewrite Eq. (4.13) using Eq. (4.10):

¯ iavg   H  Ul,avg ,m νi ,l 2 d ¯il ¯ = il T . l

m

i

nl F

l

dT

T

(4.14)

The quantity that is multiplied by ¯il in both sides of Eq. (4.14) is called the enthalpy voltage of reaction l. Applying Eqs. (4.7), (4.8), and (4.14) in Eq. (4.5) and using the results in Eq. (4.3) give the general form of enthalpy change of the system as a

Heat sources

117

function of time, dHtot /dt; using this value, Eq. (4.1) yields: q − IV =

 d Ul,avg ¯il T 2 l

Enthalpy of reaction

dT T



 d









γ i ,j ∂ ln avg d∀ ∂ T γ i ,j Vj j  i 

avg  γ dn d i , j i , m Hij◦→m − RT 2 − ln avg dT dt γ i ,j j,j=m i  t ⎛ ¯il dt   dT ⎜ ◦ ¯ avg ◦ ⎜ + ni,j Cpi,j + Cpl dt ⎝ nl F −

dt

j

+

 j,j=m

i

ci,j RT 2

l

Enthalpy of mixing Phase change



¯ avg − C ¯ avg )(ni,j − n◦ )⎠ , (C pi,j pi,m i ,j

Heat capacity

i

(4.15) where Cpl =



¯ avg νi ,l C pi,m

(4.16)

i

and Hij◦→m = Hi◦,m − Hi◦,j .

(4.17)

It should be noted that all the composition properties in Eq. (4.15) are defined with respect to the activity coefficient ai,j = xi,j γi,j . This definition emphasizes the fact that if the activity coefficient of composition is known, then the thermodynamical of that composition is also known. Moreover, in Eq. (4.15) the heat capacity of the battery should be accurately calculated. Since a battery is a multicomponent system, we need to consider the heat capacity of all its elements. Since the heat capacity of the battery changes due to material change, in many practical cases, we can assume a constant mean value for the heat capacity of the battery in all cases. Finally, in Eq. (4.15), we need to consider the effect of heat dissipation to ambient. The majority of heat dissipation is due to convection of heat to ambient, which can be calculated by the equation q = −hA(T − TA ).

(4.18)

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Simulation of Battery Systems

4.2 Heat of reactions Neglecting enthalpy of mixing, phase change, and change of heat capacity, we reduce Eq. (4.15) to

  Ul,avg 2 d ¯ il T q = IV + .

dT

l

T

(4.19)

In discharge the chemical energy is directly converted to the electrical work. The maximum electrical work is obtained if the reaction is reversible. In that case the reversible electrical work is IVrev =

   ¯il Ul,avg .

(4.20)

l

The difference between the real and reversible voltages is called the overpotential: η = V − Vrev .

(4.21)

The overpotential is an indicator of the irreversibility of the system, such as Ohmic drop, charge transfer overpotential, and mass transfer limitations. The product of overpotential and current is called the heat of polarization and consists of Joule heating and energy loss inside electrodes. Beside polarization heat, the enthalpy of reaction includes entropic heat defined as   dUl,avg ¯il T (4.22) . qrev = − dT l Eqs. (4.20) and (4.22) are generated by power and heat in a reversible reaction, respectively. The reversible work can be obtained from changes in the Gibbs free energy. Inserting Eqs. (4.20) and (4.22) into Eq. (4.19) results in 

q = IV −

 l

 ¯il Ul,avg +

  dUl,avg ¯il T . l

dT

(4.23)

This is another expression for Eq. (3.34) and shows that the generated heat of any reaction can be divided into two parts: 1. The generated heat due to chemical bonds of species that are involved in electrochemical reactions and are of a reversible type. The heat changes sign in charge and discharge, meaning that if it is exothermic in charge, then it becomes endothermic in discharge and vice versa.

119

Heat sources

2. The irreversible part of heat generation is always positive (both in charge and discharge) and hence always contributes in temperature build–up. The reversible or entropic heat of reactions requires further inspection, whereas in electrochemical reactions, we are dealing with ions and electrons as well as with chemical compounds. The following sections cover this issue.

4.2.1 Reversible or entropic heat of reactions Entropy change in any half-cell reaction gives good information about the distribution of heat generation and consumption inside a battery. The entropy of different chemical compounds is tabulated in thermodynamic tables, which can be used for estimation of reversible heat of the cell for complete reactions. However, concerning the half-reactions, we are dealing with ions other than chemical compounds since any half-cell reaction contains some chemical compound and ions. Therefore we need a way to calculate the entropy of ions. Lampinen and Fomino [34] introduced a proper way as will be discussed here.

Enthalpy of reaction The standard molar enthalpy of a compound material (Hfi◦ ) is equal to enthalpy change of its elements for a unit mass of compound i in standard conditions. Its elements should also be maintained in standard conditions (i.e., standard temperature and pressure). In this case and under standard conditions, the standard enthalpy of formation h◦i is defined as h◦i = Hfi◦ .

(4.24)

Under nonstandard conditions (i.e., in any temperature T and pressure p), the enthalpy of the unit mass of compound i is calculated from the equation hi (T , p, x◦ ) = Hfi◦ (T◦ , p◦ , x◦ ) +



T

Cpi dT +

T◦

 p p◦

vi − T

∂ vi dp. (4.25) ∂T

To obtain Eq. (4.25), we used the following thermodynamic relations: ∂ hi = Cpi , ∂T ∂ hi ∂ vi = vi − T , ∂p ∂T

(4.26) (4.27)

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Simulation of Battery Systems

vi =

∂V . ∂ ni

(4.28)

Moreover, the partial mole of species is defined as x = (x1 , . . . , xm ),

(4.29)

ni xi =  . nk

(4.30)

where

Since the net electric charge of an electrochemical reaction is constant and does not change during the reaction, we cannot experimentally measure the thermodynamic properties of ions, but we can measure the whole thermodynamic property of a reaction such as 1 Na(s) + H+ (aq) → Na+ (aq) + H2 (g). 2

(4.31)

In this reaction, sodium ions are produced from the main element, and hydrogen gas is released from the solution under standard conditions. In Eq. (4.31) the net electric charge remains unchanged. On the other hand, the standard scale defined for standard enthalpy of formation of ion in water indicates that when the net electric charge is constant in standard conditions, that is, T = 298.15 K and p = 1 bar for ions with activity coefficient of unity that is an indicator of unit molality, we have Hf◦ [Na+ (aq)] = H ◦

[Eq. (4.31)].

(4.32)

According to Eq. (4.32), for hydrogen ion, we obtain Hf◦ [H+ (aq)] = 0.

(4.33)

Consequently, using Eq. (4.25), we can write h◦ [H+ (aq)] = Hf◦ [H+ (aq)] = 0,

h◦ [H2 (g)] = Hf◦ [H+ (aq)] = 0. (4.34)

Entropy of reaction The third law of thermodynamics indicates that the entropy has an absolute value. This scale is called the absolute entropy. By that statement any pure

121

Heat sources

element has the absolutely zero entropy at absolute zero. For example, [Na(s), T = 0 K] = 0.

(4.35)

The absolute entropy of elements at higher temperatures above absolute zero is obtained from the integral of heat; as an example, 

298.15

sa [Na(s), T = 298.15 K] = 0

dQ = 51.17 J/mol K. T

(4.36)

However, the main problem is finding the absolute entropy of charged particles in solutions, which is not easy to obtain. For example, to calculate sa [Na+ (aq)], we need to use the entropy of reaction. Just for an example, for Eq. (4.31), we can write 1 2

Sf◦ [Na+ (aq)] = s◦a [Na+ (aq)] + s◦a [H2 (g)] − s◦a [Na(s)] − s◦a [H+ (aq)].

(4.37) The absolute entropies s◦a [H2 (g)] and s◦a [Na(s)] are well known, but the value of s◦a [H+ (aq)] should be calculated. Note that s◦a [H+ (aq)] = 0 but Sf◦ [Na(aq)] = 0. Hence the absolute entropy of sodium ions cannot be calculated using Eq. (4.37) unless s◦a [H+ (aq)] is determined in some way. The values that are tabulated as absolute entropies in thermodynamic tables are defined as follows: 1 s◦ [Na+ (aq)] = Sf◦ [Na+ (aq)] − s◦a [H2 (g)] + s◦a [Na(s)]. 2

(4.38)

Here another zero scale is defined for entropy such that s◦ [H+ (aq)] = 0,

(4.39)

where the subscript that indicates the absolute entropy is dropped out, so that it can be distinguished from absolute entropy. In other words, s◦ [Na+ (aq)] = s◦a [Na+ (aq)].

(4.40)

Since s◦a [H+ (aq)] = 0, the new value for entropy can be called the semiabsolute entropy because although it is absolute in some point of view, its zero value differs from the absolute zero.

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Simulation of Battery Systems

Not we have the absolute and semiabsolute entropy values for each element at temperature T and pressure p, which can be obtained from the following relations: ◦





sai (T , p, x◦ ) = sai + s◦i (T , p, x◦ ) = s◦i +

T

T◦  T T◦

 



p Cpi ∂ vi dT + dp, − T ∂ T p◦  p Cpi ∂ vi dp. − dT + T ∂T p◦

(4.41) (4.42)

In these equations, we used the following thermodynamic relations: cpi ∂ sai ∂ si = = , ∂T ∂T T ∂ sai ∂ si ∂ vi = =− . ∂p ∂p ∂T

(4.43) (4.44)

Now the main problem is that how we can use different entropy scales to calculate the absolute entropy? To answer the question, consider a system with differently charged particles (such as ions and electrons) whose entropy is defined by semiabsolute scale, i = 1, . . . k, and some noncharged particles whose entropy is defined by absolute entropy, i = k + 1, . . . , m. The net entropy of the system then can be calculated as S(T , p, n1 , . . . , nm ) =

k 

nisi +

i=1

m 

nisai .

(4.45)

i=k+1

However, from the absolute entropy point of view, the absolute entropy of the system is S (T , p, n1 , . . . , nm ) =

m 

ni sai .

(4.46)

i=1

Using Eqs. (4.38) and (4.39), we see that for charged particles (zi = 0, i = 1, . . . , k), sai = si + zi s◦ai [H+ (aq)],

(4.47)

and according to Eqs. (4.45) to (4.47), we obtain ◦

+

S − S = sai [H (aq)]

k  i=1

ni zi = 0.

(4.48)

Heat sources

123

This is because according to the balance of electric charge, the net electric charge is zero, that is, q=F

k 

ni zi = 0.

(4.49)

i=1

Therefore Eqs. (4.48) and (4.49) show that the simultaneous usage of absolute and semiabsolute entropies for calculation of the absolute entropy will give the same results if 1. all the semiabsolute scales are defined according to a single zero, 2. the entropy of electron is used in calculations. As a result, for calculation of the entropy of electrochemical and chemical reactions, we need to have a value for the entropy of electron. In the succeeding subsection, we discuss this issue.

Entropy of electron According to thermodynamic relations, the entropy of a system can be calculated using the Gibbs free energy that contains electrostatic energy as follows:

∂G . (4.50) S=− ∂ T p,ni ,...,nm According to this equation and Eq. (4.45), the partial entropy of noncharged particles (zi = 0, μ˜ i = μi , where chemical and electrochemical potentials are equal) is sai =

∂S ∂ ∂G ∂ ∂G ∂ μ˜ ∂μ =− =− =− =− , ∂ ni ∂ ni ∂ T ∂ T ∂ ni ∂T ∂T

where

μ˜ =

∂G ∂ ni

(4.51)

.

(4.52)

T ,p,n1 ,...,ni −1,ni +1,...,nm

In the same manner, for charged particles (zi = 0), si =

∂S ∂ μ˜ i =− . ∂ ni ∂T

(4.53)

In an equilibrium state for standard hydrogen, we have μ◦ [H2 (g)] = 2μ˜ ◦ [H+ (aq)] + 2μ˜ ◦ [e− (Pt)].

(4.54)

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Simulation of Battery Systems

Figure 4.1 A resistance model for Joule heating.

Differentiating these equations and using Eqs. (4.51) and (4.53), we obtain s◦a [H2 (g)] = 2s◦ [H+ (aq)] + 2s◦ [e− (Pt)].

(4.55)

Substituting Eq. (4.39) into Eq. (4.55) yields 1 s◦ [e− (Pt)] = s◦a [H2 (g)] = 65.29 J/mol K. 2

(4.56)

Eq. (4.56) gives a tool for calculation of entropy of electron with the same zero scale.

4.2.2 Joule heating In general, the applied current should be conserved throughout a cell. This current enters the cell from an external circuit and enters the solid phase. As can be seen schematically in Fig. 4.1, the current enters the electrolyte through surface reactions with rates determined by the kinetics of the reactions. At the end of the solid phase, all the current enters the electrolyte phase and is carried by ions to the other electrode. On the other electrode surface, this current enters the solid phase again in a reverse manner. This description can be understood better by considering the resistance model, as shown in Fig. 4.1. The electrical resistance of the cell consists of the resistance of the positive and negative electrodes, electrolyte, and separator. In each region the current passes through different phases. Hence Joule heating should be considered in all the phases. Therefore Joule heating can be written as qJoule =



|φk · ik |.

(4.57)

k

In this equation, the absolute value signs should be present since Joule heating is always positive and contributes to a rise in temperature.

Heat sources

125

The last point about the Joule heating is that although the summation should be carried on all phases, we can neglect the current through the gas phase since it is negligible compared to the current through the solid and liquid phases.

4.3 General Joule heating concept Splitting the heat of reactions into reversible and irreversible parts show that the irreversible part is very similar to Joule heating (i.e., Eq. (4.57)), where the quantity is a product of current and a voltage difference. Consequently, the irreversible part of heat generation has a Joule heating nature and is positive both in charge and discharge in both electrodes. This fact leads to a new concept known as the generalized Joule heating, which is the sum of the classical Joule heating and irreversible part of heat generation due to electrochemical reactions, that is, qGJH = qJoule + qirrev =



|φk · ik | + qirrev .

(4.58)

k

This concept was first introduced by Torabi and Esfahanian [32]. They noted that the generalized Joule heating plays an important role in thermal behavior of a battery, especially in thermal runaway.

4.4 Heat dissipation In addition to heat sources in a battery, there are a variety of procedures ending up by releasing an extra produced heat into the ambient. These mechanisms can dissipate the excess heat in the form of different heat fluxes.

4.4.1 Convection One of the most common transferred heat to the ambient is done by convection mechanism, the equivalent resistance of which is demonstrated in the following equation, where the term h∞ indicates heat transfer coefficient: Rconv =

1 . h∞

The role of convection resistance is illustrated in Fig. 4.2.

(4.59)

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Simulation of Battery Systems

Figure 4.2 Heat sinks.

4.4.2 Conduction Conduction heat transfer occurs in solid parts of battery case leading heat from inside to the outer surface of the battery. Therefore, for calculating the heat dissipation by convection mechanism, heat conduction from cell to cell and to cell case (as shown in Fig. 4.2) should be calculated. The conductive resistance is calculated according to the following equation, where Lcase is the thickness of the case wall, and λcase is its heat conductivity: Rcond =

Lcase λcase

.

(4.60)

4.4.3 Radiation Another mechanism responsible for heat dissipation is radiation, and the amount of transferred heat can be calculated by applying the following equation, where σs is the Stefan–Boltzmann constant: 4 4 − Tamb ). qrad = σs A(Tcase

(4.61)

Since the operational temperature of a ZSOB does not increase too much, the radiation part is negligible.

4.4.4 Exhausted enthalpy In systems with external flows, another heat sink is available and required to be taken into account. This heat can be defined as the exhausted heat

Heat sources

127

through exhausting gas flow, and its amount can be obtained by using the following equation, where m˙ shows the mass flux of the exhausting gas: qexh = mh ˙

(4.62)

4.4.5 Equivalent circuit model In general, dissipation mechanisms are capable of being simulated based on their equivalent electrical resistance. Most conventional batteries are operating at moderate temperatures (usually less than 100◦ C and are sealed so that no or negligible mass transfer occurs). Consequently, heat dissipation due to radiation and mass transfer can be neglected. In other words, conduction and convection are the only existent heat dissipation mechanisms. In Fig. 4.2, heat sinks are represented for a conventional battery in more detail. This battery consists of three cells in series. There are three important points about this picture [31]. First, there is a big difference between the amount of heat that an internal cell is capable of dissipating to the ambient in comparison with the cells that are located at the sides of the battery. Indeed, a side-cell has more surface area for dissipating heat to the ambient in comparison with an internal cell. Thus the critical cell that has higher temperature can be considered to be an internal cell. Second, the resistance in the upper parts of an internal cell itself is greater than that in the lower parts. As a result, the top areas in an internal cell have the potential of becoming warmer. Finally, there is a symmetric condition in the amount of heat transferred from an internal cell to its neighbors in the form of conduction. According to this symmetry, we can assume that the heat fluxes are required to be zero in the borders in contact with neighbors. As a result, an internal cell can be considered to be isolated from each side, and the only available heat fluxes are those from the upper and lower parts of the cell.

4.5 Summary Heat generation inside a battery is very important because battery performance strongly depends on temperature. Therefore, for a good simulation, the heat sources and sinks should be properly modeled. The general formulation that was given in the present chapter enables us to obtain proper formulation for any simulation. It should be noted that heat sinks are also important in analyzing the thermal behavior of the batteries. The thermal management of batteries

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Simulation of Battery Systems

depends mainly on designing efficient heat sinks so that the battery pack temperature remains within a specific limit.

4.6 Problems 1. Explain the difference between reversible and irreversible heat sources. 2. Is Joule heating reversible? 3. Try to figure out how to model heat sinks in a one-dimensional model.

CHAPTER 5

Simulation of batteries Contents 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8. 5.9. 5.10.

The spatial dimension Simulation domain Governing equations Initial values and guesses Boundary conditions Mesh generation Ill-posedness and compatibility equation Famous numerical methods Summary Problems

130 134 136 140 141 143 144 146 147 148

Simulation of a battery or cell aims to obtain its characteristics curve or in other words, to obtain its temporal variation of voltage under a specified current or vice versa. The cell voltage means the difference between the solid potential of the positive and negative current collectors, which depends on electrolyte potential and active material concentration. Since the electrolyte potential and material concentration are themselves functions of the solid potential, we come to realize that governing parameters of a battery are highly coupled to each other, and consequently, the process of solving pertinent equations to get the distribution of these key parameters must occur simultaneously. Furthermore, the temperature as an influential factor impacts reaction rates, active material concentration, cell potential, and other operational characteristics of a cell. This little introduction is sufficient to realize that battery modeling deals with solving nonlinear coupled partial differential equations whose solutions are not easy to obtain. One interesting issue about simulation is that obtaining the characteristic curve gives lots of information about the internal processes of the simulated battery. In many cases, the obtained information is much more informative than the characteristic curve itself. Moreover, the obtained information is very important because many of them cannot be obtained experimentally, and some of them are very hard and expensive to obtain. The obtained information gives us much physical sense, which is very important in analyzing and improving battery characteristics. Simulation of Battery Systems https://doi.org/10.1016/B978-0-12-816212-5.00009-X

Copyright © 2020 Elsevier Inc. All rights reserved.

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One of the drawbacks of numerical simulation of battery governing equations is that we are dealing with a system of nonlinear differential equations, which in many cases are ill-posed and stiff. Therefore specific numerical methods should be used to ensure acceptable results. For instance, for simulation of a constant current test, we are dealing with an ill-posed system of partial differential equations (PDEs), which may have no answer or may have infinitely many answers. Hence we need to check the existence and uniqueness of the results. Since we are dealing with a physical phenomenon, we are sure that the governing equation has an answer or that the solution exists. However, we are encountering many different answers, or in other words, the governing system of equations has many answers. Thus to ensure the uniqueness of the answers, we need to apply some physical condition, which is mathematically called the compatibility equation. The compatibility equation is the mathematical formulation of conservation of electrical charge or electro-neutrality. This physical law ensures the uniqueness of the solution and should be kept in mind during the simulation. When dealing with numerical simulation, besides the existence and uniqueness, we have to be sure that the numerical simulation is valid and consistent with real values. The spatial dimension of the governing equation, the type of numerical method, the generated mesh, the time-step used in the simulation, and other parameters all affect a proper simulation. In this chapter, we consider the main concerns of a proper numerical simulation.

5.1 The spatial dimension As we discussed in the previous chapters, since the governing equations are given in vector form, they can be simulated in one, two, or three dimensions. Fig. 5.1A shows a real three-dimensional model of a single cell. The cell consists of a separator that is sandwiched between the positive and negative electrodes. In this configuration the applied current Iapp enters an electrode from one lug and leaves the cell from the lug of the counter electrode. The direction of the current changes during charge and discharge. In practice the thickness of the negative electrode, the separator, and the positive electrode that are respectively shown by dn , ds , and dp are very small in comparison with the cell width w and height h. In most modelings the voltage drop due to the lugs is neglected, and hence the lugs are not considered in simulation. Instead of lugs, as is illus-

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Figure 5.1 Illustration of a three-dimensional cell. (A) Real model with lugs. (B) Threedimensional model.

trated in Fig. 5.1B, boundary conditions are applied. Therefore the domain of simulation consists of three cubic blocks connected to each other. It should also be noted that in real cases, as stated before, the height h and width w of the cells are much larger than their thickness. Such a condition suggests that one- and two-dimensional models may have accurate results. Therefore, one- and two-dimensional models are more practical and available in applied researches and papers. Fig. 5.2 shows the situation where a two-dimensional model is constructed upon. In a two-dimensional model, it is assumed that all the properties and parameters have negligible variation in the width direction. Therefore, as shown in Fig. 5.2A, a unit length portion of the cell is enough for simulation as the results can be extended to the whole width of the cell. The concept of a two-dimensional model is shown in Fig. 5.2B, where the cell has exactly the same height as the three-dimensional model but has a unit length in width direction. In the two-dimensional model, we have to take care about the amount of the current used in modeling. Since we are dealing only with a unit length, we have to modify the actually applied current. According to Fig. 5.2, we have the following relation between the three- and two-dimensional models: Iw =

Iapp , w

(5.1)

where Iw is the current that passes through the unit length of the cell. It is worth mentioning that the conventional length dimension in battery

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Figure 5.2 Illustration of two dimensional modeling. (A) Three dimensional model. (B) Two dimensional conceptual model. (C) Two dimensional model.

modeling is centimeter. In other words, in almost all simulations, the unit length is 1 cm. However, we should really pay attention to reading papers since other units such as inch or meter are sometimes used. The last thing about two-dimensional modeling is that in this type of modeling, we have virtually accepted that lugs are located all over the top face of the electrodes. This is because the model is quite symmetric in width direction. It should be noted that this assumption poses very little error in the final results. Practically, a two-dimensional model is drawn as is illustrated in Fig. 5.2C. The dimensions shown on the figure make it clear to understand the figure. Just note that the lugs are not modeled and are just shown for clarity of the figure. The same argument is true for the one-dimensional model shown if Fig. 5.3. In the one-dimensional model, we assume that the current density is equal all over the electrode surface. Hence we can simulate only a unit square of the cell as shown in Fig. 5.3A. The conceptual model is shown in Fig. 5.3B, and it should be noted that in this model the outer surfaces act as current collectors. The unit square (conventionally 1 × 1 cm square) in its practical format is shown in Fig. 5.3C. As is illustrated, the one-dimensional model is shown according to its axis only since it is the only important axis. Again in the one-dimensional format, it is understood that the whole outer surfaces of the electrodes are assumed to be the collectors or the

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Figure 5.3 Illustration of one-dimensional modeling. (A) Three-dimensional model. (B) One-dimensional conceptual model. (C) One-dimensional model.

Figure 5.4 Different Model Types. (A) One-Dimensional Model. (B) Two-Dimensional Model.

electrode center lines. The numerical results show that this assumption gives very accurate results, especially in low current density applications. To provide a better understanding of the structural characteristics, oneand two-dimensional battery models are compared in Fig. 5.4, and the differences are discussed accordingly. By using a one-dimensional model, as shown in Fig. 5.4A, we can assume that the electrical current uniformly enters one electrode, passes through the separator, and exits from the other electrode. Since the current pattern is assumed to be in the x-direction, all quantities will be uniform in the y- and z-directions. Therefore the cell lugs should be placed on the current collectors as shown by solid black lines in the figure.

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By considering the inaccuracies imposing on the simulation as a result of this single dimension assumption, it seems that in the two-dimensional model (shown in Fig. 5.4B) the assuming sketch far more resembles the real battery condition. We can see that the positive and negative lugs are located at the top of the cell and the current entering an electrode is distributed along the cell height. After resulting in a nonuniform current distribution inside the separator, the whole current enters the opposite electrode and leaves it from the other lug. As can be understood from this comparison, in low current density applications, since different current components (i.e., i1 , i2 , . . . , in shown in Fig. 5.4B) have small values, which are very close to each other, a onedimensional simulation suffices. However, by increasing I these current components (i.e., i1 , i2 , . . . , in ) differ in amount, and therefore the assumption of the uniform current distribution cannot be valid anymore. Hence, in this case, a two-dimensional modeling is inevitably necessary for simulation. Moreover, regardless of the current rate, if operational conditions are not uniform, then a one-dimensional model will fail to provide accurate results. For example, if the thermal conditions differ at the top and bottom of the cell, then a temperature gradient occurring in the vertical direction significantly affects the electrochemical reactions. This discussion shows that selection of one- or two-dimensional simulation is important depending on the operational condition of the cell. However, as a rule of thumb, we have to recognize that in any case the numerical simulation of a battery by solving the governing equations is very accurate even if we use a one-dimensional model. Just note that in many applied practices the battery is modeled as a single component with bulk values, which is very less accurate comparing even with a one-dimensional model.

5.2 Simulation domain After analyzing the simulation dimension, we need to determine the simulation domain. In practice, a battery cell is made of either a single electrode as shown in Fig. 5.1A or a set of successive negative and positive electrodes with separators in between. The latter one is a cell set and is schematically illustrated in Fig. 5.5A. In a cell set, depending on the design of the cell, all the negative cells are interconnected, and this is also true for the positive electrodes. For a better understanding, a projected view of the cell

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Figure 5.5 Illustration of simulation domain. (A) A battery set. (B) Projected view of the set. (C) Simulation domain.

is shown in Fig. 5.5B. The negative, separator, and positive electrodes respectively have thicknesses dn , ds , and dp . From the projected view we can see that each positive electrode reacts with two negative electrodes on both sides. The same argument is true for the negative electrodes. Thus a cell set consists of many single cells. This configuration shows symmetry in electrochemical reactions for all the plates except the two end electrodes. The two end plates react only with one electrode and hence do not have the same chemistry as the internal ones. However, if we assume that a symmetric electrochemical reaction takes place for all the plates, the error will be negligible since we have only two end plates and lots of internal ones. By this assumption we need to simulate a single cell as is shown by dashed lines in Fig. 5.5B. For clarity, the single cell is shown in Fig. 5.5C. From the figure it is clear that the single cell starts from the center of the negative electrode and ends at the center of the positive electrode. Therefore the simulation domain for a cell set is made of half positive electrode, half negative electrode, and a full separator in between. This is why the thickness of the electrodes is divided by two in Fig. 5.5C, but the thickness of the separator is not. The simulation domain shown in Fig. 5.5C is very common in the simulation of batteries such as lead–acid battery, in which cell sets are made of many positive and negative electrodes. From the mathematical point of view, the left and right boundary conditions are of symmetric type. In a symmetric boundary condition, all the fluxes are zero, or in other words, we have zero transport of any phenomena such as mass, electrical charge, heat, and so on.

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The situation described is valid for cell sets; however, if the cell is made of one pair of negative and positive electrodes, then we need to use fullwidth electrodes in our model. In that case the thickness of the negative and positive electrodes shown in Fig. 5.5C should respectively be dn and dp . Moreover, the boundary conditions at the left and right sides of the cell are changed to wall instead of symmetry.

5.3 Governing equations There are two different methods in expressing the governing equations of battery dynamics, namely a) Regional and b) Single–domain methods. Recall that a region is a generic name for each part of the cell. In other words, each of the positive electrode, the separator, and the negative electrode are called a region. By this definition regional methods are methods that deal with each region separately. In these methods, for each region including electrodes, separator and electrolyte reservoir, a separate system of equations is introduced. For example, the system of governing equations for the positive electrode differs from those of the separator. From the physical point of view, this method is quite compatible since the physical phenomena in the electrode and separator are completely different. In regional methods, we need not only to express a proper system of equations for each region but also to specify a specific system of equations for each boundary between the regions. This is because mathematically, each region is solved independently; hence, we need boundary conditions for each region. Consequently, in regional methods, we have to specify one system of governing equations for each region and one system of equations for their boundaries. Example 5.1. For Figs. 5.2C and 5.3C, determine how many systems of equations are required if we are dealing with a regional approach? Answer. As discussed, for each region, we need one system of equations. The number of systems does not depend on the order of spatial dimension. Therefore, for both cells, we need the same number of systems. Since the cells consist of three regions, namely the positive electrode, the separator, and the negative electrode, we need three systems of equations for the regions. On the other hand, we need one system of equations for each boundary. Clearly, there is one boundary between any two regions; hence we need two other systems of equations for the internal boundaries.

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However, for outer boundaries, we have to specify proper boundary conditions. For each transport equation, we have to exactly determine one boundary condition on each boundary. Therefore the number of boundary conditions differ in one- and two-dimensional models. In the onedimensional model (Fig. 5.3C), we have two external boundaries, namely two current collectors shown by dark circles. Therefore we need two systems of boundary conditions. However, for the two-dimensional model, we have eight outer boundaries; hence it requires eight systems of boundary conditions. The regional method treats each region as a single numerical field with its own boundaries. The connection between the regions is accomplished by the system of equations of internal boundaries. In contrast to the regional method, Wang et al. [29] and Gu et al. [35, 36] presented a general and complete mathematical model of batteries and fuel-cells based on their previous experiences (e.g., [37]). This model is transient, multidimensional, and fully coupled with the Navier–Stokes equations and also includes the effect of porosity and heat generation due to the electrochemical reactions. Later, Gu, and Wang [30] introduced the energy equation to their proposed model and obtained a thermal– electrochemical model of battery, which can be applied on any battery and fuel-cell system. The main advantage of this model is that in a single– domain formulation a unique set of governing equations is applied to the whole domain, and we need to apply boundary conditions only at the outer boundaries. This means that all the battery regions (electrodes, separator, etc.) are fully coupled with each other, and there is no need to define any additional interior regional boundary condition. The concept of single-domain modeling greatly simplifies the formulation, programming, and simulation. Torabi and Esfahanian [31,32] used the same model to develop a general formulation, which is required for simulation of thermal runaway in any kind of battery system. The singledomain formulation has been applied to many battery systems, including lead–acid, [37–39], lithium-based batteries, [36] nickel-based batteries, [35, 30] fuel cells, [40–42], and zinc silver–oxide batteries [43]. The singledomain method is not limited to these examples, and there are many other papers available in the open literature. Example 5.2. Considering Example 5.1, how many systems of equations we need in case of the single-domain approach?

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Answer. In a single-domain formulation, we need just one system of equations for the whole domain. The difference between the regions is automatically considered by different properties. For example, the electrical conductivity is high but different in the electrodes, and it is zero in the separator. Note that internal boundaries in this approach are meaningless since the boundaries are part of the solution domain and are solved automatically. The outer boundaries are treated as we did in the regional approach. Therefore, for each boundary, we need one system of boundary conditions exactly as discussed in Example 5.1. Reviewing the literature, the regional methods were commonly used long ago, and by emerging the single-domain approach they became less interesting because the single-domain methods have many advantages over the regional methods. For instance: • From numerical calculation point of view, simulation of a single set of governing equations on a single domain is much easier than programming a multiregional or multiblock code. • Treatment of boundaries is much simpler. We do not have interior boundaries. • The whole domain becomes fully coupled. • No special routines are required for numerical simulation as is the case in regional approaches. Regardless of the modeling approach, all the governing equations of battery dynamics can be cast into a general transport equation   ∂ tk   + ∇ · (F ) + ∇ · i = ∇ · (∇) + S .        ∂t zF     source term convection conduction

unsteady

(5.2)

migration

The general transport equation combines different modes by which the quantity of a variable  changes. Unsteady part The unsteady term gives the temporal variation of the variable. This term is omitted if the phenomenon under study has a steady-state behavior. For example, the governing equations of electrical charge in solid or electrolyte (Eqs. (3.73) and (3.76)) are steady state and hence do not include this term. However, other governing equations such as conservation of chemical species or Eq. (3.59) contain the unsteady term. Convection The convection accounts for the variation of  due to convection or fluid flow. The flux vector is defined as F = ρ V with ρ and

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V being the fluid density and velocity field, respectively. This term vanishes if the convection is not present. Migration A charged particle such as ions in the electrolyte phase moves when is placed in a potential gradient. The transport phenomenon is known as migration, which is important in battery modeling. The migration depends on transference number of the species tk and its charge number z, and also on the current vector, which is another interpretation of the potential field. Conduction The mass transport due to concentration gradient is called diffusion or conduction. Conduction mechanism influences the variation of  and the other discussed mechanisms. Diffusion depends on the concentration gradient of  and its diffusion coefficient  . Source term A source term directly affects the variation of . The source can be a producer or consumer (sink) of the variable and may be a function of variable  and other parameters. Example 5.3. Compare the equation of potential in the electrolyte with the general transport equation and discuss. Answer. The equation of potential in electrolyte phase is described by Eq. (3.74):

 e e ∇ · (keff ∇ φe e ) + ∇ · (keff Iem = 0, D ∇ ln ci ) +

m = e.

m

This equation is steady-state without any convective term. Moreover, migration does not appear in this equation. Therefore the equation contains only diffusion and source terms. This means that the distribution of the potential field in electrolyte depends on diffusion and source terms. Comparing with Eq. (5.2), the convective term is ∇ · (∇) = ∇ · (keff ∇ φe e )

with  = keff being the diffusion coefficient and  = φe e . The source term is defined as 

e

e Sφe e = ∇ · (keff D ∇ ln ci ) +



Iem ,

m = e.

m

Note that in this equation, migration also does not appear because migration is a mass transport mechanism.

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Since all the governing equations of battery dynamics can be cast into a single general transport equation, a general solver can be developed for simulation of different batteries. Moreover, commercial or open-source multipurpose programs can be used for simulation of such systems of partial differential equations.

5.4 Initial values and guesses Iterative methods are commonly used for simulation of governing equations because of the stiffness of the system and ease of programming. In iterative methods, we need to specify initial guesses for different parameters to start the iterative procedure. From a mathematical point of view, initial values are crucial for transient equations, but we do not need any initial value for steady-state equations. For steady-state equations such as equation for electrical potential distribution, we have to specify initial guesses instead of initial values. Since initial guesses and values contribute to the iterative procedure, an improper estimation will result in divergence of the solver and will not lead to a correct answer. Therefore, for both transient and nontransient parameters, we have to define proper and reasonable initial values and guesses. Let us divide the transport equations into transient and steady-state equations. The conservation of chemical species and heat transfer (energy equation) are transient, and the equations of potential distribution in solid and electrolyte phases are steady-state. For transient equations, a uniform distribution of parameters is a natural choice since conventionally the simulations are performed from a steady-state condition. At a steady-state condition the parameters become uniform all across the cell by diffusion mechanism. For instance, for any species concentration such as C, the proper initial condition is C = C◦ , where C◦ is a uniform value obtained from the physical observations. For energy equation, putting T = T◦ with T◦ being a uniform temperature is a natural selection of the initial value. Example 5.4. For the simulation of a conventional lead–acid battery, what would be the best choices for initial conditions? Answer. Initial conditions are defined as the stable condition of the cell before starting the test. Conventionally, lead–acid batteries for the automotive industry use 5 M sulfuric acid as their electrolyte. If a cell is put in room temperature, then the initial natural conditions are c = 5 M and T = 298.15 K.

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For steady-state equations, we do not need to specify initial conditions, but for iterative methods, we have to input some initial guesses. As stated before, an improper initial guess will cause instability in numerical simulation. Moreover, since we are dealing with exponential terms in source terms (like the Butler–Volmer equations), an improper initial guess would cause very large source terms, or even the value of the source term may exceed computer floating size; hence the numerical code produces error and stops. Consequently, imposing a proper initial guess is of great importance. There are two ways to find a proper guess for solid and electrolyte phase potential: 1. We can solve the steady-state equations with uniform values for transient terms such as the electrolyte concentration and temperature. In this case the transient PDEs are not solved numerically, and hence the solver is not transient, so that the stability is much better, and we will obtain a proper value for steady-state parameters. 2. The whole system of equations are solved including both transient and steady-state equations with a very small time step like t = 10−8 s. The small time step increases the stability of the code and makes it converge to proper values. In the latter case, in fact, we have marched t = 10−8 s, and the main solution starts from that level, which in practical cases can be treated as zero time without any important deviation from the real case.

5.5 Boundary conditions In addition to initial conditions, numerical simulation of the governing equations requires proper boundary conditions. As discussed about Fig. 5.5C, current collectors are either the centers of the electrodes or solid walls (such as lithium cells). In either case, zero mass flux is a proper choice for mass transport equations. Hence for all species with concentration C, in collectors we have ∂C = 0, (5.3) ∂n where n is the normal unit vector of the boundary pointing outward. In addition to current collectors, Eq. (5.3) can be applied on all the outer boundaries since we do not have any mass flux to or from the cell. In general, zero mass flux should be applied on all the external boundaries. For energy balance, however, we have to have a better look. If the collectors are of symmetric type, then the zero heat flux is a proper choice,

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but if the collectors are of wall type, then we have to specify a suitable equation for heat dissipation. Therefore, for symmetric types, we have ∂T = 0, ∂n

(5.4)

∂T = qdiss , ∂n

(5.5)

and for wall types, we have −λ

where qdiss is the dissipation rate form the current collectors and can be evaluated using the resistance method introduced in Chapter 4. For other boundaries other than the current collector, Eq. (5.5) should be applied on all the external boundaries with the resistance method of Chapter 4. The proper boundary condition for electrical potential in electrolyte phase is ∂φe =0 ∂n

(5.6)

on all external boundaries since no ionic transport exists either on the external walls or symmetric boundaries. For electrical potential in the solid phase, we have to specify a reference point since potential has no unique value and requires a reference point. Conventionally, the potential of the negative electrode lug is assumed to be zero or φs,n = 0. On the positive electrode lug, however, we may impose different boundary conditions regarding the operating conditions. If we are dealing with a constant voltage discharge or charge, then the proper boundary condition is φs = V◦ ,

(5.7)

where V◦ is the applied voltage. However, if we apply a constant current load on the cell, then we have −σ eff

∂φs = Iapp . ∂n

(5.8)

The left-hand side of Eq. (5.8) is the definition of electrical current, and the right-hand side is the practically applied current. From Eq. (5.8) we can find the proper boundary condition, which is of Newman type. The applied current Iapp is negative for discharge and positive for the charging conditions. Other than the lugs, we have the zero flux boundary condition

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on all the outer boundaries, that is, ∂φs = 0. ∂n

(5.9)

The boundary conditions described here can be applied on three- and one-dimensional models without any modifications. Hence for the onedimensional model shown in Fig. 5.3C, all the boundary conditions can be applied; just notice that the normal vector n should be replaced by x.

5.6 Mesh generation Mesh generation in numerical methods is very important and should pass some characteristics. First, the mesh size contributes to numerical error. The larger the size of the numerical cells, the greater the numerical error. Therefore the domain of interest should be divided into a very fine grid so that we can guarantee that the final result does not include the errors arising from the computational grid. It should also be noted that the finer the grid, the greater the computational cost; hence we have to choose a proper grid size. Second, the shape of the cells should be as harmonic as possible. In other words, if the mesh shapes are rectangular, then the more the grid is similar to a square, the better the results. Third, we have to have a proper grid distribution across the domain. In other words, if the successive cells suddenly change in size, then numerical errors occur. Putting all the mentioned criteria, the best choice for the numerical grid is a nonuniform structured grid as shown in Fig. 5.6. We can see that this type of grid generation fulfills all the mentioned characteristics. First of all, the size of the grid is controllable for different regions. Secondly, the structured grid is very harmonic, and by the clustering the grid at all the exterior and interior boundaries we can guarantee that we have enough mesh where the properties have a large gradient. Finally, this grid form has a very smooth transition from small cells to large cells in successive neighboring cells. The generated numerical grid is nonuniform to minimize the computational cost and increase the accuracy of the results where the gradients of parameters are very high. Usually, the parameters have a great gradient at boundaries and interfaces. Therefore the shown generated grid is very suitable for this purpose. Since the numerical process requires a lot of time, a grid study should be carried on to obtain the best configuration for computation. Stringent grid

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Figure 5.6 A proper numerical grid for two-dimensional simulation.

Figure 5.7 A proper numerical grid for one-dimensional simulation.

study must be performed to ensure that the computations are independent of the mesh size. Analogous to two-dimensional simulation, for one-dimensional cases, all the characteristics of a good grid should be fulfilled. A proper mesh for one-dimensional modeling is shown in Fig. 5.7. As it is quite clear, the mesh is a nonuniform structured grid with proper clustering at all the boundaries. In a good quality mesh the grid size should have almost the same size at both sides of an interior boundary. Again grid study should be performed to ensure that the results are grid independent.

5.7 Ill-posedness and compatibility equation From a mathematical point of view, an elliptic partial differential equation with Newman boundary condition on all its boundaries either has no solution or has infinitely many solutions. Such a problem, when discretized, results in an ill-posed system of equations, or in other words, the determinant of the resulting matrix becomes zero. The stationary equations of battery governing equations, that is, the equations of solid and electrolyte

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potential distribution are of elliptic type. Of these equations, the boundary conditions for potential distribution in electrolyte phase or Eq. (5.6) are of Newman type on all boundaries. When the operating condition is defined as a predefined current, the equation governing the potential distribution in the solid phase also becomes ill-posed since, on all the boundaries, the boundary condition becomes of Newman type according to Eq. (5.8). In such cases, we have infinitely many solutions, one of which is the correct answer. Thus we have to pick the correct answer out of infinitely many answers. The key to distinguish the correct answer is called the compatibility equation. The compatibility equation, being another translation of electroneutrality, says that the correct answer is that for which the current in the positive electrode is exactly equal to the current in the negative electrode. In the mathematical form, we can write

j(V )dV = −



pos.electrode

j(V )dV ,

(5.10)

neg.electrode

where j(V ) is the current density distribution inside the electrodes. The negative sign means that the current gets out of one electrode and enters the counter electrode. Using Eq. (5.10), the procedure for removing illness from Eqs. (6.49) and (6.50) is replacing one of the boundary conditions with a guessed Dirichlet one. By try-and-error and usage of Eq. (5.10) (i.e., compatibility equation) we will find a proper solution for which all the boundary conditions are fulfilled. For solid phase potential, it is clear that we can choose a reference state for the potential that is usually the current collector of the negative electrode. This greatly simplifies the try-and-error procedure because we have and exact value for one of the equations. Then the tryand-error procedure is conducted for potential in the electrolyte phase. To be more clear, let us examine a one-dimensional model. The compatibility equation in the one-dimensional space becomes



j(x)dx = − pos.electrode

j(x)dx,

(5.11)

neg.electrode

where j(x) is the current density distribution along the electrodes’ width. To remove the ill-posedness from solid phase potential distribution equation, we change the boundary conditions to ∓σ eff

∂φs = I (t), ∂x

x = 0,

(5.12)

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φs = 0,

x = l.

(5.13)

By this substitution the problem becomes well-posed because, in one of the boundaries, the Newman type is replaced by the Dirichlet type. For the electrolyte phase, however, we need to make try and errors. This means that we have to change Eq. (5.6) to φl = φguessed

(5.14)

in one boundary only. Then we solve the system of equations. If Eq. (5.11) is satisfied, then the guessed value is OK. Otherwise, we have to change the guessed value and repeat the procedure. This procedure shows that the solution of the system of equations is extremely time-consuming even for a one-dimensional model. Many efforts have been made to reduce the simulation time, one of which is the work of Esfahanian et al. [44], where a very novel method was used, in which Kirchoff ’s laws were also used to choose a proper estimation for φguessed . The method extremely accelerates the solution.

5.8 Famous numerical methods The solution of partial differential equations can be obtained using many famous numerical methods. Among the famous ones, we name the following methods: Finite Difference Method (FDM) FDM is one of the most commonly used methods for solving ordinary and partial differential equations. In this method the domain of interest is divided into small control points, and the derivatives are written as discrete quantities of dependent and independent variables. The result is a system of algebraic equations that can be solved using many different methods. Finite Volume Method (FVM) In the FVM the domain is divided into some control volumes. Then using the divergence theorem, the differential equations are converted to surface integrals, and the surface fluxes at each finite volume are evaluated. The method is very famous because it can be applied to unstructured meshes; hence many industrial applications can be simulated. Finite Element Method (FEM) Similar to FDM and FVM, in the FEM the domain of interest is divided into some elements. Then using variational methods from the calculus of variations, an approximation to the solution is obtained by minimizing a defined error

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function. Similar to FVM, one strength of the FEM is that it can be applied on unstructured grids resulting in many industrial programs. The mentioned methods are very common in the simulation of partial differential equations, including the governing equations of battery dynamics. Other than the mentioned methods, there are many numerical techniques such as the spectral method, meshless methods, and panel methods, which are commonly used in practice. Selection of the method is absolutely arbitrary, but it should be kept in mind that the results should be independent of the applied method. For example, simulation of a specific problem by FDM should come to the same result as FVM, FEM, or any other numerical scheme within a very small tolerance. The tolerance arises from the fact that different methods have different accuracy. In the present book, the finite volume method is the method of choice in solving the governing equations of battery dynamics. The details of FVM in one- and two-dimensional forms are fully discussed in Appendix D. When dealing with fluid flow motion, the standard FVM requires some modification to resolve the problems that arise from the pressure field. The resulting method is known as SIMPLE and first was introduced by Patankar [45]. Since for batteries with a liquid electrolyte such as lead–acid batteries, the movement of electrolyte is important, the SIMPLE algorithm is discussed in Appendix E. In addition to standard FVM, a method called the Keller–Box method, which is a very natural choice for battery simulation, is also discussed in Appendix F. Why Keller–Box is a natural choice for simulation is discussed in different chapters.

5.9 Summary Numerical simulation is very tricky and requires a lot of experiences. If some details are missed, then the result will not be reliable. Before starting a new simulation, we have to keep in mind many parameters. First, the problem should be physically analyzed. The domain of interest, initial values, boundary conditions, and modeling dimensions must be carefully identified. Second, a proper numerical method should be selected. It should be kept in mind that not all the numerical methods can be applied to simulation of all partial differential equations. Third, according to the selected numerical method, a proper grid should be generated. There are many parameters associated with grid generation that should be considered. The generated grid has a very close relation to the selected time step. Hence we

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have to choose a proper time step as well. Finally, the operational conditions should be applied properly, and the results should be analyzed with enough care. The importance of the mentioned parameters on simulation of battery systems was discussed in this chapter, and the main issues were studied by more detail. It is clear that for the simulation of different batteries, special treatment should be done. Since each battery has its own characteristics, the specific details are considered individually for each technology.

5.10 Problems 1. Compare the equation of potential in solid with the general transport equation and discuss. 2. Compare the equation of chemical species with the general transport equation and discuss. 3. Compare the equation of balance of energy with the general transport equation and discuss. 4. Inspecting numerical methods, there are lots of published papers on grid generation, one of which is called algebraic grid generation. Write a programming code for generating a proper mesh in one-dimensional case as is shown in Fig. 5.7. 5. Extend the one-dimensional code for mesh generating on a twodimensional case as shown in Fig. 5.6.

CHAPTER 6

Lead–acid batteries Contents 6.1. Lead–acid battery components 6.1.1 Plates 6.1.2 Separators 6.1.3 Electrolyte 6.2. Lead–acid battery types 6.3. Electrochemistry of lead–acid batteries 6.4. Lead–acid battery applications 6.5. Governing equations 6.5.1 Conservation of mass and momentum 6.5.2 Conservation of energy 6.5.3 Conservation of charge 6.5.4 Conservation of species 6.5.5 Conservation of mass 6.6. Thermal runaway problem 6.7. Heat sources and sinks 6.7.1 Heat of reactions 6.7.2 Joule heating 6.7.3 Heat dissipation 6.8. One-dimensional model 6.8.1 Governing equations for one-dimensional model 6.8.2 Boundary conditions Potential in solid and electrolyte Chemical species Cell temperature 6.9. Physico-chemical properties 6.9.1 Electrode electrical conductivity σ 6.9.2 Electrolyte ionic conductivity k 6.9.3 Diffusion coefficients 6.9.4 Open-circuit voltage U 6.9.5 Partial molar volumes of sulfuric acid and water 6.9.6 Thermodynamic properties of different species 6.9.7 Calculation of properties in porous medium 6.9.8 Temperature dependency of parameters 6.10. Numerical simulation of lead–acid batteries 6.10.1 One-dimensional simulation without side reactions 6.10.2 One-dimensional simulation including side reactions 6.10.3 Numerical simulation of electrolyte stratification using two-dimensional modeling Simulation of Battery Systems https://doi.org/10.1016/B978-0-12-816212-5.00010-6

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151 151 152 154 154 156 158 159 163 165 165 166 167 168 171 171 173 173 175 175 176 176 178 179 179 179 180 180 181 181 183 183 184 184 185 189 193 149

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6.10.4 Simulation of thermal behavior of lead–acid batteries 6.11. Summary 6.12. Problems

201 213 214

Lead–acid batteries were introduced by Gaston Planté in 1859. He found out that a pair of lead–lead dioxide electrodes in sulfuric acid can produce electricity. This invention made a significant breakthrough in the history of battery technology. The positive electrode of lead–acid batteries are made of PbO2 , whereas the negative counter electrode is made of pure lead or Pb. The electrolyte of this type batteries is made of sulfuric acid. Since the invention of lead–acid batteries, the primary chemistry has remained unchanged, and only some modifications have been made to improve its performance. Using paste instead of flat plates in the construction of electrodes was one of the significant improvements. Pasted electrodes are able to provide more energy than foils or solid ones. Another major modification of the original design was usage of absorbed glass mat (AGM) separators instead of conventional separators. This invention resulted in invention of sealed lead–acid batteries, in which oxygen recombination occurs, and electrolyte concentration remains constant during its operational time. Hence the battery requires less maintenance. The most advantages of lead–acid batteries are their ability to supply high surge current and their cost. These batteries can provide a lot of power in a short time, the property that made them attractive in car industries for starting a car. In this view, lead–acid batteries have a high power-to-weight ratio, or more scientifically, they have a high specific energy. In addition to a high specific energy, lead–acid batteries are very inexpensive in comparison with other industrial technologies. Their economical cost has made them very attractive in various industries. Although a high surge current is not a matter, lead–acid batteries are a good candidate for electric storage. For example, in UPS, industrial electric cars such as lift trucks, in small cheap electric cars, and any other industry where the cost is more important than the battery weight, lead–acid batteries are the primary candidate. Another advantage of lead–acid batteries is their ability for recycling. A lead–acid battery can be recycled above 95%, and the recycled materials can be used to make new batteries. The recycling plants are very mature, and many technologies have been developed to make them environment friendly and economic. This advantage makes lead–acid batteries even more attractive for industry.

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Figure 6.1 Conventional lead–acid batteries.

Although lead–acid batteries have many advantages, there are some drawbacks that make other technologies to be their potential competitors in the market. This type batteries have a very low energy density and specific energy. This means that they occupy a lot of space and are heavy. Hence in applications where volume and weight are essential, lead–acid batteries are not a choice of interest. For example, for car industries where the driving energy of the car is stored in batteries, lead–acid batteries are not good nominates.

6.1 Lead–acid battery components Like any other batteries, lead–acid batteries are composed of plates, separator, and electrolyte. Here we briefly discuss each component. Details of a conventional lead–acid battery is shown in Fig. 6.1.

6.1.1 Plates The plates of lead–acid batteries are usually made in three different shapes: 1. Flat plates are the most conventional type of lead–acid batteries, where the plates are pasted on a flat grid made of lead. The grid may contain different additives to improve its performance and enhance its operational life. 2. Tubular plates are another major battery type, in which the positive plates are put in some cylinders or tubes. The main advantage of this

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shape is that in the tubular lead–acid batteries the active surface of the positive electrode is larger than of flat plates because the electrolyte is in greater contact with active materials; hence the battery is suitable for high current applications. However, tubular batteries have less active material comparing to flat plate types, meaning that they store less energy than flat plate types. Moreover, they are more expensive in manufacturing than flat plate batteries. 3. In another technology the battery plates are rolled and made in a spiral shape. The positive and negative plates are sandwiched in AGM separators and rolled to shape the spiral. They showed better performance and stability, but they are more expensive than flat plates. In all the configurations the positive electrodes are made of PbO2 , and the negative electrode is made of Pb. Various additives are added to the positive and negative plates so that they show better performance in different situations, for example, to have a good performance in very cold or very hot climates or to increase battery cycling life. Some additives are also added to current collectors made of pure lead. The additives differ for the positive and negative electrodes. Bismuth, calcium, silver, and many other additives are tested and practically added to improve the performance and durability of the grids.

6.1.2 Separators Separators between the positive and negative plates prevent short-circuit through physical contact, mostly through dendrites (“treeing”), but also through the shedding of the active material. Separators allow the flow of ions between the plates of an Electro-chemical cell to form a closed circuit. Wood, rubber, glass fiber mat, cellulose, and PVC or polyethylene plastic have been used to make separators. Wood was the original choice but deteriorated in the acid electrolyte. Rubber separators are stable in battery acid and provide valuable electrochemical advantages that other materials do not. An effective separator must possess several mechanical properties such as permeability, porosity, pore size distribution, specific surface area, mechanical design and strength, electrical resistance, ionic conductivity, and chemical compatibility with the electrolyte. In service the separator must have excellent resistance to acid and oxidation. The area of the separator must be a little larger than the area of the plates to prevent material shorting between the plates. The separators must remain stable over the battery operating temperature range.

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In the absorbed glass mat (AGM) design the separators between the plates are replaced by a glass fiber mat soaked in electrolyte. There is enough electrolyte in the mat to keep it wet, and if the battery is punctured, then the electrolyte will not flow out of the mats. Principally, the purpose of replacing liquid electrolyte in a flooded battery with a semisaturated fiberglass mat is a substantial increase of the gas transport through the separator; Hydrogen or oxygen gas produced during overcharge or charge (if the charge current is excessive) is able to freely pass through the glass mat and reduce or oxidize the opposing plate, respectively. In a flooded cell the bubbles of gas float to the top of the battery and are lost to the atmosphere. In AGM configuration, however, the transport mechanism causes the produced gas to recombine and make water again. The additional benefit of the semisaturated cell is providing no substantial leakage of electrolyte upon physical puncture of the battery case that allows the battery to be completely sealed, which makes them useful in portable devices and similar roles. Additionally, the battery can be installed in any orientation, though if installed upside down, the acid may be blown out through the overpressure vent. Reducing the water loss rate, the plates are alloyed with calcium. However, gas build-up remains a problem when the battery is deeply or rapidly charged or discharged. To prevent overpressurization of the battery casing, AGM batteries include a one-way blow-off valve and are often known as “valve regulated lead–acid” (VRLA) designs. Another advantage of the AGM design is that the electrolyte becomes the mechanically strong separator material. This advantage allows the plate stack to be compressed together in the battery shell, slightly increasing the energy density compared to liquid or gel versions. AGM batteries often show a characteristic “bulging” in their shells when built in common rectangular shapes due to the expansion of the positive plates. The mat also prevents the vertical motion of the electrolyte within the battery. When a normal wet cell is stored in a discharged state, the heavier acid molecules tend to settle to the bottom of the battery, causing the electrolyte to stratify. When the battery is then used, the majority of the current flows only in this area, and the bottoms of the plates tend to wear out rapidly. This is one of the reasons a conventional car battery can be ruined by leaving it stored for a long period and then used and recharged. The mat significantly prevents this stratification, eliminating the need to periodically shake the batteries, boil them, or run an “equalization charge” through them to mix the electrolyte. Stratification also causes the upper layers of the battery to become almost completely water, which can freeze

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in cold weather; therefore AGMs are significantly less susceptible to damage due to low-temperature use. Although AGM cells do not permit watering (typically, it is impossible to add water without drilling a hole in the battery), their recombination process is fundamentally limited by the usual chemical processes. Hydrogen gas will even diffuse right through the plastic case itself. Some have found that it is profitable to add water to an AGM battery, but this must be done slowly to allow the water to mix via diffusion throughout the battery. When a lead-acid battery loses water, its acid concentration increases, increasing the corrosion rate of the plates significantly. AGM cells already have a high acid content in an attempt to lower the water loss rate and increase standby voltage, and this brings about shorter life compared to a lead-antimony flooded battery. If the open-circuit voltage of AGM cells is significantly higher than 2.093 volts, or 12.56 V for a 12 V battery, then it has a higher acid content than a flooded cell; although this is normal for an AGM battery, it is not desirable for long life. AGM cells intentionally or accidentally overcharged will show a higher open-circuit voltage according to the water lost (and acid concentration increased). One Ah of overcharge will liberate 0.335 grams of water; some, but not all, of this liberated hydrogen and oxygen will recombine.

6.1.3 Electrolyte The electrolyte of lead–acid batteries are made of sulfuric acid. For different purposes, many different additives are added to the electrolyte. In lead–acid batteries, sulfuric acid or the electrolyte is an active material. In other words, the electrolyte is consumed during discharge to produce lead–sulfate and produces water. During the charging process, the reverse reaction converts lead–sulfate on both electrodes into sulfuric acid. In gelled lead–acid batteries, sulfuric acid is mixed with silica gelling agent to make a gelled lead–acid battery. The gelled electrolyte requires less maintenance, but it reduces ion mobility, which in turn reduces battery power. Therefore gelled lead–acid batteries are frequently used in energy storage devices where surge current capability is not an issue.

6.2 Lead–acid battery types From application-based viewpoint, there are two basic types, namely starting and deep cycle lead–acid batteries.

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155

Starting batteries are those used in starting cars, motor cycles, boats, or any other vehicles. The main characteristic of such batteries is that in starting time, a high surge current is required. Starting batteries should not be discharged very deep, or in other words, they should work so that their state of charge is always greater than 80 percent, SoC > 80%. The reason is that their plates are thin, and if they become discharged in values higher than the limit, then lead sulfate covers all the plate area and decreases the plate conductivity. As a result, the battery cannot be charged again and looses its cyclic life. Deep cycle batteries are those used for power regulation, energy storage, and low-current applications. The plates are thicker than starting batteries; hence they can be discharged more than starting types. Consequently, they will provide more energy in a longer period. However, they are not able to provide high surge current and thus are not suitable for starting applications. From another point of view, lead–acid batteries can be divided into wet or flooded and sealed or valve-regulated types. Wet Cell (flooded) batteries are used in most automotive industries. In these batteries the electrolyte is a liquid solution of sulfuric acid where during charge and discharge is dissociated into hydrogen and oxygen. The produced gases are vented to prevent the explosion. Value Regulated (VRLA) batteries have the same configuration as flooded types, but they are sealed, so the produced gases are not able to escape from the battery. The evolved gases increase the internal pressure, which is dangerous if it exceeds some limits. To prevent the damage, some pressure valves are located at the top of each cell, and if the cell pressure reaches a designed value, the valves open and decrease the pressure. VRLA batteries require less maintenance and are commonly composed of two types: 1. AGM (Absorbed Glass Mat) In this type the electrolyte is confined in an AGM separator, so it becomes immobilized. The evolved oxygen moves from the positive electrode to the negative electrode and recombines with hydrogen ions. The oxygen cycle maintains the electrolyte concentration at its original level. The AGM cells show less internal resistance than the other types since acid migration is faster in AGM batteries. As a result, during discharge and charge, the battery can deliver higher current than any other sealed batteries.

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2. Gelled Cell In gelled cells the electrolyte is converted into a gel by addition of Silica gel and becomes a solid mass. Consequently, the electrolyte does not move, and this prevents its spillage. These batteries are charged with lower voltage to prevent evolving excess gas that eventually will damage the cell. Gel batteries are best used in very deep cycle application and may last a bit longer in hot weather.

6.3 Electrochemistry of lead–acid batteries In a lead acid batteries the following reaction takes place: discharge

−−  PbO2(s) + HSO−4 + 3 H+ + 2 e−  −− −− −− − − PbSO4(s) + 2 H2 O

(6.1)

charge

at the positive electrode and discharge

+ − −  Pb(s) + HSO−4 −  −− −− −− − − PbSO4(s) + H + 2 e

(6.2)

charge

at the negative electrode. Then the whole battery reaction is written as discharge

−−  Pb + PbO2 + 2 H2 SO4  −− −− −− − − 2 PbSO4 + 2 H2 O.

(6.3)

charge

It is remarkable that both electrodes convert to lead sulfate during discharge and recover again when they are charged. Also, in the lead–acid batteries, electrolyte acts as an active material. In many battery technologies, this is not the case. In other words, the electrolyte is just a medium for ionic transportation and does not contribute to electrochemical reactions. In the case of lead–acid batteries the energy content depends on the volume and concentration of the electrolyte and on the mass of positive and negative electrode active materials. When a lead–acid cell is discharged, as discussed before, both electrodes convert to lead sulfate and recover when it is charged. In an ideal case, when the battery is fully charged, all lead sulfates are converted to original materials. However, it is not a real case in practice. Actually, when a battery cell is charged, some lead sulfate remains intact and does not recover and forms a passive layer on the electrode surface. This fact repeats in succeeding cycles, and in each cycle the amount of passive lead sulfate increases. The passivation process continues until the cell becomes unusable.

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The passivation process takes place due to many reasons. It should be noted that there are two common lead sulfate crystals α and β . It is known that α crystals are needle-shaped and resistive; hence they do not contribute in the charge and discharge process. In contrast, β crystals have soft and round edges and contribute to charge and discharge. The more α crystals are generated, the more passivated the electrode becomes. Although the main reason for the generation of α crystals is not yet clearly known, it is observed in practice that by increasing the discharge current density the rate of its generation becomes faster. Also, deep discharge has a positive effect on α crystal generation. Besides the main reactions, some side reactions occur in the lead–acid batteries. For the positive electrode, the following side reactions accompany the main reaction: discharge

+ − −  2 H2 O −  −− −− −− − − O2 + 4 H + 4 e ,

(6.4)

charge

discharge

+ − −  H2 −  −− −− −− − − 2H + 2e ,

(6.5)

charge

and the following side reactions accompany the main reaction of the negative electrode: discharge

−−  O2 + 4 H+ + 4 e−  −− −− −− − − 2 H2 O,

(6.6)

charge

discharge

−  2 H+ + 2 e− −  −− −− −− − − H2 .

(6.7)

charge

Side reactions are important when SoC > 0.6 and can be ignored otherwise. Especially, they play an essential role in overcharge when SoC  1 and the cell is still being charged. The importance of side reactions in battery dynamics will be discussed in more detail in Section 6.10. As a result of side reactions, hydrogen and oxygen release. The mixture is an explosive material and should be vented from the battery. In the flooded lead–acid types, small flame arrester equipped orifices are located at the top of each cell to vent the mixture. Flame arresters are for further safety because the hydrogen/oxygen mixture is highly explosive. Fig. 6.2A shows the schematic gas evolution in a flooded lead–acid battery, where evolved oxygen bubbles are not able to move through the separator and reach the negative electrode. In contrast to flooded lead–acid batteries, in AGM or gelled lead–acid types the electrolyte does not let the evolved gases to rise to the headspace,

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Figure 6.2 Hydrogen and oxygen evolution in lead–acid batteries. (A) Flooded lead– acid batteries. (B) Sealed lead–acid batteries.

but the evolved oxygen moves toward the negative electrode, where recombination takes place to produce water according to Eq. (6.6). The oxygen recombination cycle compensates water dissociation, and the electrolyte composition remains intact. Fig. 6.2B shows the moving direction of evolved gases in a sealed lead–acid battery.

6.4 Lead–acid battery applications Without a doubt, the automobile industry is one of the largest markets of lead–acid batteries since each car has a lead–acid battery for its starting,

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lighting, and ignition; due to the applications, the battery is called SLI battery. SLI batteries are made of thin plates that can produce very high surge current about 500 A or more. Flooded, gelled, and spiral wound batteries are made as SLI batteries. Valve-regulated lead–acid (VRLA) and tubular batteries are commonly used to supply the necessary power for large factories, telephone and computer centers, and for off-grid energy storage. Especially with the growth of renewable energy sources such as solar cells or wind turbines, VRLA batteries are getting more and more attention. Traction batteries are used in industrial cars such as lift–trucks, submarines, emergency devices, and so on. Also, to power up small electric cars, traction lead–acid batteries are the choice because of their low cost and proper recycling. Many other applications use lead–acid batteries mostly because of their inexpensive cost, especially where the weight and volume is not a matter. For instance, in many playing devices such as small cars, toys, and so on, lead–acid batteries are good choices.

6.5 Governing equations Lead-acid batteries types and applications were studied in the previous chapter. Although the technology is relatively mature and the batteries are produced in huge numbers, they are still not satisfactorily optimized. Traditionally, the optimization process is based on experimental tests that are costly and time-consuming. To complement the conventional trial-anderror method of improving the performance and cycle life of the lead-acid battery, mathematical models have been developed to predict discharge and charge behavior and the effect of cycling. Many efforts have been made to improve mathematical methods so that they could be applied on lead–acid batteries with minimum error. The first comprehensive study of porous electrodes was developed by Newman and Tiedemann [46]. They applied the theory to simulate the discharge behavior of a lead–acid cell. To correct the uniform acid concentration assumption considered in that study, Sunu [47] extended the model to the case of nonuniform acid concentration in the electrolyte reservoir. The first model that was able to simulate both discharge and charge process of a lead–acid cell was introduced by Gu et al. [48]. The model was extended so that it was able to predict the cell behavior during the rest and charge cycles in addition to the discharge cycle. Moreover, the dynamic behavior

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of acid concentration, the porosity of the electrodes, the state of charge (SoC) of the cell, and the dependency of cell performance on electrode thicknesses and operating temperature were also investigated. They solved the obtained system of equations using a finite volume scheme. Later, Esfahanian and Torabi [49] applied the Keller–Box method to the coupled one-dimensional electrochemical transport equations and showed that the method is very efficient in solving such a system of equations. Although it was approved that electrolyte movement makes acid stratification in the lead–acid batteries, none of the previous modelings was able to simulate electrolyte movement. To overcome the weakness, for the first time, Alavyoon et al. [50] developed a two-dimensional mathematical model for mass transfer and electrolyte motion during charge process. Later, Gu et al. [37] introduced a model with an integrated formulation for battery dynamics. In this approach the whole battery was considered as a model volume, and the transport equations including electrochemical, mass, and fluid flow were derived for the whole cell volume. Also, the model was capable of predicting the transient behavior of the battery during discharge, rest, and charge. Esfahanian et al. [44] introduced an improved and efficient mathematical model for the simulation of flooded lead-acid batteries based on computational fluid dynamics and equivalent circuit model (ECM). This model inherited not only the accuracy of the CFD model, but also the physical understanding of ECM, which made it quite suitable for real-time simulations. Torabi and Esfahanian [31,32] studied thermal-runaway in batteries theoretically and, as a result, presented a general set of governing equations by which the thermal behavior of batteries could be obtained. Since the proposed system of equations was general, it can be used for any battery systems. This literature review shows in short the efforts that have been done on simulation of lead–acid batteries. In all the literature the system of governing equations was applied to a cell model, and the system was solved using different numerical schemes. It is necessary to note that the numerical scheme should be chosen such that the numerical stability is provided because the obtained system of equations is nonlinear with exponential source terms resulting in a stiff system of algebraic equations. Also, it should be noted that the selection of a numerical scheme should not affect the results, meaning that if we choose two different methods, then we have to obtain the same results.

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The governing equations of lead–acid batteries result in an ill-posed system of equation as discussed in Chapter 5. In addition to ill-posed behavior of the governing equations discussed before, simulation of the system of nonlinear governing equations requires some attention. The problem of illness of the systems is tackled by the compatibility equation as discussed before. In addition to the compatibility equation, it is remarkable that the potential field has no unique value and is a relative quantity. Hence a reference point should be selected as the reference state in all the simulations. In a battery, usually, the potential of the current collector or lug of the negative electrode is taken as the reference state. By this selection the potential of the positive electrode is the voltage of the cell. However, we can assume that the reference point of potential is the current collector of the positive electrode. This assumption leads us to negative values for the potential of the negative electrode. In any case the selection of reference point should not make any difference in results. In the related literature, both assumptions are used without any problem. Another difficulty arises from the fact that a battery cell is a multiregion domain, where special care should be taken to resolve proper numerical values at regions’ boundaries. Therefore, for an accurate numerical simulation with reasonable numerical cost, a nonuniform mesh should be generated with specific attention on the uniformity of mesh size at the boundaries. An inappropriate mesh results in inappropriate numerical values. The governing equation of lead–acid batteries can be obtained from the general governing equations of battery systems discussed in Chapter 3. A typical lead–acid cell is shown schematically in Fig. 6.3; it consists of the following regions: a lead–grid collector at x = 0, which is at the center of the positive electrode; a positive PbO2 electrode, an electrolyte reservoir for providing more electrolyte for the positive electrode; a porous separator; a negative Pb electrode; and finally a lead–grid collector at x = l, which is at the center of the negative electrode. The reservoir region in practice is made by constructing some ribs on one side of the separator. It is worth noting that in some designs the separator has ribs on both sides, meaning that the separator provides reservoir for both electrodes. Thus the model should contain electrolyte reservoir on both sides. The positive and negative electrodes consist of porous solid matrices whose pores are flooded by a binary sulfuric acid H2 SO4 and of a gaseous phase made up of O2 and H2 . During charge and discharge, the following electrochemical reactions occur:

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Figure 6.3 A Typical lead–acid cell model.

• The positive electrode (PbO2 /PbSO4 ): discharge

−  PbO2(s) + HSO−4 + 3 H+ + 2 e− −  −− −− −− − − PbSO4(s) + 2 H2 O,

(6.8)

charge

discharge

+ − −−  2 H2 O  −− −− −− − − O2 + 4 H + 4 e ,

(6.9)

charge

discharge

+ − −−  H2  −− −− −− − − 2H + 2e .

(6.10)

charge

• The negative electrode (Pb/PbSO4 ): discharge

+ − −−  Pb(s) + HSO−4  −− −− −− − − PbSO4(s) + H + 2 e ,

(6.11)

charge

discharge

−  O2 + 4 H+ + 4 e− −  −− −− −− − − 2 H2 O,

(6.12)

charge

discharge

−  2 H+ + 2 e− −  −− −− −− − − H2 .

(6.13)

charge

Eqs. (6.8) and (6.11) are the main reactions of a lead–acid battery, and the other equations are side reactions. Side reactions are not important in the flooded lead–acid batteries unless they are in overcharge mode. However, in valve–regulated, sealed, and gelled lead–acid batteries, side reactions play a significant role in oxygen recombination. Therefore in the flooded lead–acid batteries, only main reactions exist, and all the side reactions are neglected. In studying side reactions, it is customary to neglect hydrogen reactions (i.e., Eqs. (6.10) and (6.13)) because of their poor kinetics.

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According to the main electrochemical reactions of lead–acid batteries, sulfuric acid is produced and consumed in the positive and the negative electrodes during charge and discharge. However, the rate of production and consumption in a positive electrode is much faster than the negative electrode due to its higher stoichiometric coefficient. Because of this fact, it is customary to build some ribs on the separator to provide more electrolyte on the positive side. The rib space is called “Reservoir” as shown in Fig. 6.3. In some cases, ribs are designed in either side of the separator; hence in both sides, we have to consider one reservoir. In the presence of side reactions, a lead–acid cell is a three-phase system consisting of the solid matrix, the liquid electrolyte, and a gas phase. During charge and overcharge, oxygen is generated at the PbO2 /electrolyte interface and evolves into the gas phase after exceeding its solubility limit in the electrolyte. The generated oxygen moves to the negative electrode via the liquid and gas phases. At the negative electrode the oxygen gas reduces at the electrode interface. This process forms an internal oxygen cycle in VRLA cells. Governing equations of lead–acid batteries are deduced from general governing equations explained in Chapter 3. In this chapter, physical characteristics of lead–acid batteries are applied to the general governing equations, and a specific set of equations is obtained suitable only for lead–acid batteries. The obtained system of equations is defined in vector form that can be used in one-, two-, or three-dimensional modelings. In the next section, we derive a simplified system of equations for the one-dimensional case, since one-dimensional modeling is very common and is used in many cases.

6.5.1 Conservation of mass and momentum Electrolyte movement occurs in a lead–acid cell mostly due to the interaction of gravity and concentration gradient in a form of natural convection. Since electrolyte is confined in porous media, usually, the electrolyte movement can be neglected. However, in the rib space or reservoir region the electrolyte movement cannot be neglected; in fact, it has a very strong role in producing stratification. Electrolyte movement is modeled using the Navier–Stokes equations in the form of porous media as is shown in the following equation:   μ ∂ρv + v · ∇(ρv) = −∇ p + ∇ · (μ∇v) + g 1 + β(c H − c◦H ) + (εv). (6.14) ∂t K

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Eq. (6.14) is the famous Navier–Stokes equation with two additional terms on its right-hand side. These terms are added to the original equations to include the effect of porosity. The third term adds the effect of buoyancy force induced due to the concentration difference from a reference state c◦ . The term clearly shows that in places where electrolyte concentration is greater than c◦ , it moves toward gravity, and where it is less than c◦ , it moves upward. The last term in Eq. (6.14) indicates the exerted drag force due to the porous media on electrolyte. This term, also known as the Darcy term, is proportional to the velocity v and porosity ε. The parameter μ is the viscosity of the electrolyte, and K indicates the permeability coefficient and is calculated from the Kozeny–Carman relation K=

ε 3 d2 , 180(1 − ε)2

(6.15)

where d is the size of solid particles that are making the porous medium. Example 6.1. Calculate the permeability of a porous electrode with ε  0.5 and particle size d  0.1 µm. Answer. From the Kozney–Carman relation or Eq. (6.15) we have K=

0.53 × (10−7 )2 ε 3 d2 =  3 × 10−17 . 180(1 − ε)2 180(1 − 0.5)2

The value indicates that the porosity of a porous electrode is such that it exerts a lot of force on electrolyte, and we can assume that the electrolyte is nearly stagnant inside the electrodes. The example shows that stratification only happens in separator rib regions or inside the separators with larger porosity and particle size. In VRLA batteries the whole electrolyte is either confined in porous media (electrodes and separator) or immobilized using a gelled electrolyte. In either case, either the electrolyte is stagnant, or its velocity is very low. In this paper the movement of the electrolyte is neglected. Hence the momentum and continuity equations are not considered. This means that the convective mass transfer is neglected because of very small velocity field of the electrolyte.

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6.5.2 Conservation of energy Neglecting the electrolyte movement, the general heat balance over a representative element results in ρ Cp

∂T = ∇ · λ∇ T + q, ∂t

(6.16)

where ρ and Cp are the density and specific heat capacity, respectively, T is the temperature, t is time, λ is the thermal diffusion coefficient, and q represents the total heat source. Eq. (6.16) is defined in spatial form by which the temperature distribution inside a battery can be obtained. To do this, the heat source should also be expressed in distributed form. This term will be discussed in more detail in the following section.

6.5.3 Conservation of charge Conservation of charge in solid and liquid phases is represented according to the following relations: • Conservation of charge in solid: ∇ · (σ eff ∇φs ) = Amain Jmain + AO2 JO2 + AH2 JH2 .

(6.17)

• Conservation of charge in liquid: H ∇ · (keff ∇φe ) + ∇ · (keff D ∇(ln c )) = −(Amain Jmain + AO2 JO2 + AH2 JH2 ),

(6.18) where Jmain , JO2 , and JH2 are exchange current densities from the solid matrix toward electrolyte associated with main, oxygen, and hydrogen reactions, respectively. These exchange current densities can be obtained from the well-known Butler–Volmer equation and are represented for different reactions as follows [38]:  main Jmain = i◦, ref

 O2 JO2 = i◦, ref

cH H cref cH

H cref

γ main   main    αa F main αcmain F main η η exp − exp − ,

RT

γ O2   O2  αa F O2 η exp −

RT

RT



ceO2 2 ceO,ref

δO2

(6.19)

⎤   O 2 α F O2 ⎦ exp − c η ,

RT

(6.20)

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Simulation of Battery Systems

 H2 JH2 = −i◦, ref

cH H cref

γ H2

 exp −

αcH2 F

RT

 η

H2

.

(6.21)

In these equations, ηmain = φs − φe − U main , ηO2 = φs − φe − U O2 , and = φs − φe − U H2 represent the overpotentials that act as the driven force for driving each reaction. It should be noted that the coefficients and open-circuit potentials U are different at positive and negative electrodes. Eq. (6.21) is simplified regarding the fact that the anodic reaction related to recombination of hydrogen ions is extremely poor and with good accuracy can be neglected.

ηH2

6.5.4 Conservation of species Other conservation equations are obtained using the mass balance for hydrogen ions in the electrolyte phase, the dissolved oxygen gas in the electrolyte phase, and the oxygen gas in the gas phase. These equations are well known as the conservation of species equations: H H ∂(εe c H ) = ∇ · Deff ∇ c + a1 Amain Jmain + a2 (AO2 JO2 + AH2 JH2 ), ∂t   ∂(εe ceO2 ) O2 2 = ∇ · DeO,eff ∇ ceO2 + a3 Amain Jmain − Jeg , ∂t   ∂(εg cgO2 ) O2 2 = ∇ · DgO,eff ∇ cgO2 + Jeg . ∂t

(6.22) (6.23) (6.24)

In these equations, a1 , a2 , and a3 are the coefficients related to each species at each electrode, and JegO2 represents the evaporation rate of oxygen at the electrolyte/gas interface. It should be noted that in the separator region, Eqs. (6.22), (6.23), and (6.24) can be used knowing that Jmain , JO2 , and JH2 are zero. However, since JegO2 is not related to the electrochemical reactions and just represents the thermophysical equilibrium at the electrolyte/gas interface, it is not zero in the separator region and acts as a source term in Eqs. (6.23) and (6.24). A good correlation for modeling the interfacial evaporation rate of dissolved oxygen from liquid phase to the gas phase can be presented as JegO2 = k(ceO2 − H  cgO2 ).

(6.25)

In this equation, k is the interfacial mass transfer coefficient related to dissolved oxygen referred to electrolyte side, which can be obtained using the concept of diffusion length proposed by Wang et al. [29], and H  is

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Lead–acid batteries

the Henry constant [38]. In the equilibrium condition, H  cgO2 , which represents the interfacial concentration of dissolved oxygen, is equal to ceO2 , which is the average concentration of oxygen in the gas phase. Every reduction or enhancement of oxygen in either liquid or gas phase causes a net mass transfer at the interface.

6.5.5 Conservation of mass To account the porosity change of the electrodes, conservation of mass for solid phase can be used. This balance results in ∂εs = a4 Amain Jmain ∂t

(6.26)

∂εe = a5 Amain Jmain + a6 AO2 JO2 + a7 AH2 JH2 ∂t

(6.27)

for solid electrodes and

for electrolyte. Again in Eqs. (6.26) and (6.27) the coefficients a4 to a7 are constant values but different at different electrodes [38]. The porosity of gas the phase εg is a function of solid and liquid porosity satisfying the relation εe + εs + εg = 1

(6.28)

The effective or specific area of each reaction is approximated using simple correlations: • During discharge for main reaction in both electrodes, Amain = Amain,max × SoC ξ .

(6.29)

• During charge for main reactions in both electrodes,

Amain = Amain,max × (1 − SoC ξ ).

(6.30)

• For secondary reactions during both charge and discharge processes,

Aj = Aj,max × SoC ξ ,

(6.31)

where the index j refers to either O2 or H2 . In Eqs. (6.29) to (6.31), SoC is the local state of charge of a microscopic volume inside the electrodes, and its value is limited to one when the whole active material in that volume is

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Simulation of Battery Systems

Figure 6.4 Typical thermal behavior of a lead–acid battery [54]. (A) Thermal Rise. (B) Thermal Runaway.

fully charged. The rate of SoC at each point can be explained as follows: ∂ SoC Amain Jmain =± . ∂t Qmax

(6.32)

In this equation, Qmax is the maximum theoretical capacity of electrodes, and the positive and negative signs correspond to PbO2 and Pb electrodes, respectively.

6.6 Thermal runaway problem Thermal behavior of lead–acid batteries is of great interest. When a battery is charged (usually under a float charge at constant voltage), its temperature rises due to the internal chemical and electrochemical reactions and Joule heating. When the generated heat is balanced by heat dissipation to its surrounding, the temperature rise stops at a moderate temperature as shown in Fig. 6.4A. This is the normal behavior of a VRLA battery, and this fact is called temperature rise (TR). However, at some floating voltages the internal heat generation exceeds the heat dissipation, and as a consequence, the temperature of the battery increases dramatically and out of control. Consequently, the temperature may exceed 60◦ C as illustrated in Fig. 6.4B [51–53]. At this stage and under some critical conditions (e.g., electrolyte saturation) the battery could go into a unstable state, which triggers the spontaneous rise in cell temperature and current. As a result, the battery undergoes a nonstationary and self-accelerating state at which the temperature of the battery rises out of control. This phenomenon is known as thermal-runaway or TRA. TRA is usually considered to be the result of positive feedback of current (chemical and electrochemical reactions) and temperature when a cell

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169

Figure 6.5 Different mechanisms of oxygen reduction in negative electrode [54]. (A) Pure electrochemical reaction. (B) Electrochemical and chemical reactions.

is under float charge at a constant potential. The initial float current flowing through the cell causes the cell temperature to increase, which in turn causes an increase in current, which further increases the temperature until both current and temperature reach high values. This phenomenon continues until the self-accelerating conditions are interrupted. Then the cell temperature and current decrease, and the battery reaches a new stationary state. To understand the cause of TR or TRA in a valve-regulated lead–acid batteries, it would be essential to study the involved physical phenomena. During the charge process [54], at the states of charge (SoC) over 0.6, water dissociated at the positive electrode through a successive rather complex mechanism on which no common mechanism exists. However, we can consider the overall reaction with its initial and end products: H2 O −−→ 2 H+ + 2 e− + 12 O2 ,

H = +285.8 kJ mol−1 . (6.33)

The generated oxygen moves toward the negative electrode and reduces by a set of chemical and electrochemical reactions for which two different accepted mechanisms exist as illustrated in Fig. 6.5. The first mechanism shown in Fig. 6.5A states that the oxygen reduces through the following pure electrochemical reaction: 1 2 O2

+ 2 H+ + 2 e− −−→ H2 O.

(6.34)

The second accepted mechanism shown in Fig. 6.5B states that at the negative electrode, oxygen reduces by the following chemical and elec-

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Simulation of Battery Systems

Figure 6.6 Self accelerating heat generation mechanism [54].

trochemical mechanism: Pb + 12 O2 −−→ PbO, PbO + H2 SO4 −−→ PbSO4 + H2 O,

H = −219.08 kJ mol−1 , −1

H = −172.71 kJ mol .

(6.35) (6.36)

The negative sign in enthalpy change indicates that these are exothermic reactions and increase the battery temperature. It is worth noting that when the evolved oxygen flows from the positive to the negative electrode through the separator micropores, it pushes the electrolyte out causing the separator dry-out. During the TRA, the separator dry–out enhances the oxygen flow toward the negative electrode. This means that more reaction occurs at the negative electrode, and hence more heat is released. Moreover, separator dry-out means increasing the internal resistance, which in turn means increasing the Joule heat. The first step on studying TRA is determining the origin of heat sources in a battery cell. A review of the literature reveals that there is no unanimous idea about the heat generation inside the battery. Some [51–53, 55,54,56] believe that heat generation inside the battery is due to the oxygen reduction reactions at the negative electrode (according to reactions (6.34), (6.35), and (6.36)). Besides that, Joule heating increases the battery temperature. The increase in battery temperature accelerates the rate of electrode reactions, which in turn increases the battery temperature more and more. This phenomenon constitutes a self-accelerating mechanism seen in Fig. 6.6. This viewpoint is accepted by most researchers, and let us call it the first paradigm. Unlike the first paradigm, the second paradigm [57,58] states that the closed oxygen cycle cannot produce any heat. Since it is a closed system and according to Hess’ law, the net generated heat during a closed cycle is

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171

Figure 6.7 Born–Haber cycle illustrating the essential parts of the closed oxygen cycle [57].

zero regardless of the intermediate steps. For example, drawing the Born– Haber diagram for the second mechanism of oxygen reduction, as shown in Fig. 6.7, we can conclude that the net generated heat is zero. Accepting this statement, the only mechanism remained in a battery is the Joule heating. Therefore this paradigm states that the only mechanism that can contribute to the temperature rise is the Joule heating. To validate the second paradigm, Catherino [57] has made some tests on TRA and showed that in some cases the first paradigm cannot explain the cause of TRA, but the second paradigm is capable of explaining TRA in all the situations. This is the conflict point between the first and second paradigms. The second paradigm states that the enthalpy of the reaction cannot produce any heat, because there is a cycle in the battery and water is dissociated at the positive electrode and recombined at the negative electrode. Therefore, the only mechanism which is responsible for temperature rise is Joule heating. Torabi and Esfahanian [32] studied the mentioned conflict and showed that by dividing the generated heat into reversible and irreversible parts a new concept called the generalized Joule heating can be defined that is the sum of classical Joule heating and the irreversible part of electrochemical reaction. Then they showed that generalized Joule heating is the only mechanism responsible for thermal rise in lead–acid batteries. Details of the study are given in Section 6.10.

6.7 Heat sources and sinks The heat source q in Eq. (6.16) is the sum of heat generation and dissipation. The generation is due to the electrochemical reactions and Joule heating.

6.7.1 Heat of reactions The heat generation inside the battery is due to the chemical and electrochemical reactions. As it was discussed before, each reaction (Eqs. (6.8)

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Simulation of Battery Systems

to (6.13)) produces a specific amount of heat, which can be divided into reversible and irreversible parts: qrev = Jmain T

dU main dU O2 dU H2 + JO2 T + JH2 T dT dT dT

(6.37)

and qirrev = −Jmain U main − JO2 U O2 − JH2 U H2 ,

(6.38)

where J and U are the current density and local cell voltage, respectively. In calculating the irreversible heat, it should be noted that the following thermodynamic relationship between the open circuit voltage U and the entropy change s for each reaction can be used: s = nF

dU dT

(6.39)

On the other hand, Lampinen and Fomino [34] showed that to calculate the entropy change of a half-cell reaction, we can use an absolute scale for noncharged species together with a semiabsolute scale for charged species considering the entropy of an electron. They used this semiscale to calculate the entropy of an electron: 1 s[e− (Pt)] = sa [H2 (g)] = 65.29 J mol−1 K−1 , 2

(6.40)

where s and sa are semiabsolute and absolute entropies, respectively. For example, consider the half main reaction at the positive electrode (for a charge process): charge

PbSO4 (s) + 2 H2 O −−−→ PbO2 (s) + HSO4− + 3 H+ + 2 e− .

(6.41)

The entropy change is + − s =sa [PbO2(s) ] + s[HSO− 4 ] + 3s[H ] + 2s[e ]

− sa [PbSO4(s) ] − 2sa [H2 O].

(6.42)

In Table 6.2 the entropy of all the necessary charged and noncharged species are given.

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173

6.7.2 Joule heating Besides the heat generation due to the chemical and electrochemical reactions, Joule heating is also a source of heat generation in the lead–acid batteries. The resistance of the battery can be obtained in the same manner as discussed in Section 4.2.2. From the concept of Joule heating defined in that section we can calculate the Joule heating part of heat generation as qJoule = |φ · ik |.

(6.43)

The net heat generation is the sum of reversible, irreversible, and the Joule heats: qgen = qrev + qirrev + qJoule .

(6.44)

6.7.3 Heat dissipation In addition to heat generation in an electrochemical cell, heat dissipation plays a vital role in its thermal behavior. There are different heat dissipation mechanisms in a battery cell: 1. Heat dissipation to surrounding by convection. 2. Radiation to ambient. 3. Dissipated heat by exhausting gas. The convected heat to the surrounding from the battery outer walls is compensated by heat conduction through battery walls as well as electrical connections and supporting mechanical structure. Therefore the resistance model developed in Section 4.2.2 can be used to account for all these mechanisms. The total heat dissipation can be calculated as qdiss. = qconv. + qrad. + qexhaust .

(6.45)

In the case of lead–acid batteries the radiation can be neglected since the cell temperature is not very high. Moreover, it is assumed that the present battery model is sealed and there is no exhausting gas. Hence we also omit the dissipated heat by exhausting gas. Consequently, only the convective heat transfer to the ambient is simulated. When the battery case temperature exceeds the ambient temperature, the convection starts to dissipate the thermal energy. The amount of this dissipation can be obtained from the formula qconv. = hA(Tcase − Tambient ),

(6.46)

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Simulation of Battery Systems

Figure 6.8 Illustration of thermal resistances in the modeled battery.

where h is the convection coefficient, A is the area of the case, and Tcase and Tambient are the case and ambient temperatures, respectively. This equation shows that the convective heat dissipation is proportional to the temperature difference between the battery case and the ambient. The temperature of the case surface usually differs from the bulk temperature and can be found by solving the energy equation inside the battery case. Moreover, the Navier–Stokes equations should also be solved inside the headspace, since this space is filled with gaseous material. However, in most situations the captured air at the headspace is almost stagnant, and the thickness of the battery case is small. Therefore we can use the one-dimensional thermal convective and conductive resistance models illustrated by Fig. 4.2 for headspace and battery cases, respectively. The values of these resistances illustrated in Fig. 6.8 can be obtained from the equations Rconv. =

1 , h∞ d

Rcond. = , λ

(6.47) (6.48)

where h∞ is the convection heat coefficient of the gas, d is the thickness of the case, and λ is the conduction heat coefficient of the battery case. As can be seen from Fig. 6.8, the middle cells have less dissipation, which results in higher temperatures. Therefore the assumption of a bulk temperature for the whole battery is not valid in all cases. Moreover, it should be noted that heat can also be transferred from one cell to its neigh-

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Lead–acid batteries

bor cell. To calculate the amount of this heat transfer, we can use the same resistance methodology described before. The amount of thermal resistance can also be calculated from Eq. (6.48).

6.8 One-dimensional model The electrodes of a lead–acid battery usually have a very high aspect ration. This means that the width of the plates are of order of 1 mm, whereas their other dimensions are about 10 cm. In other words, the aspect ratios of the plates are of order of 100. As a consequence, the assumption of onedimensional model usually fits well, and we may apply a one-dimensional model for simulations. The literature survey shows that in most cases, this assumption has proven to be accurate enough for many purposes including optimization loops and design. The assumption of one-dimensional model greatly simplifies the governing equation. For instance, in a one-dimensional model, the fluid flow is dropped out of the equations. This fact means that the natural convection and stratification phenomena cannot be captured by a one-dimensional model.

6.8.1 Governing equations for one-dimensional model Applying the previous assumption results in the following system of governing equations: 1. Conservation of charge in solid electrodes:    ∂ ∂φs (Aj) = 0. σ eff − ∂x ∂x All

(6.49)

reactions

2. Conservation of charge in electrolyte:      ∂ ∂ eff ∂φe eff ∂ ln c k kD (Aj) = 0. + + ∂x ∂x ∂x ∂x All

(6.50)

reactions

3. Conservation of chemical species for electrolyte concentration H+ :  H ∂(εe c H ) AjH 1 − t+◦  ∂ H ∂c = Deff + (Aj). + a2 ∂t ∂x ∂x 2F F Side reactions

(6.51)

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Simulation of Battery Systems

4. Conservation of chemical species for dissolved oxygen in concentraO tion ce 2 :  O2  ∂ ∂(εe ceO2 ) 1 O2 ∂ ce O2 = De,eff (Aj)O2 − Jeg . + ∂t ∂x ∂x 4F

(6.52) O

5. Conservation of chemical species for dissolved oxygen in gas phase cg 2 : ∂(εg cgO2 ) ∂t

 O2 ∂ O2 ∂ cg O2 + Jeg = Dg,eff . ∂x ∂x

(6.53)

6. Energy balance: ∂ρ Cp T ∂φs ∂φe 1 = − is − ie + ∂t ∂x ∂x Vc

 Vc

 All



A η−T

∂U ∂T



reactions

+

hAc (T − T∞ ) . Vc

(6.54)

These equations are the main fundamental governing equations that depict a one-dimensional model of a battery cell. The solution of the system by numerical techniques will simulate the battery subjected to the fact that proper boundary conditions are provided.

6.8.2 Boundary conditions The solution of any system of differential equations strongly depends on initial and boundary conditions. Here we discuss the appropriate boundary conditions for a one–dimensional model. It should be noted that Eqs. (6.49) and (6.50) are steady state and require only boundary conditions; however, Eqs. (6.51) to (6.54) are transient, and initial conditions should be defined in addition to boundary conditions.

Potential in solid and electrolyte Although Eqs. (6.49) and (6.50) are steady state and do not require any initial conditions, a proper initial guess is strongly recommended for potential distribution both in solid and electrolyte phases. The reason is that with a badly defined initial guess, source terms in all the equations may become very large, which in turn results in unstable numerical calculations. Therefore we need to have good initial guesses for these values.

Lead–acid batteries

177

There are two main methods for defining proper initial guesses for solid and electrolyte potentials φs and φe : a) neglecting other equations and solving only Eqs. (6.49) and (6.50) with a specific constant value for each species or b) simulating the whole system of equations with uniform concentration for all species with a very small time step (e.g., 10−8 s). These methods will give a proper initial guess for potential fields that will be used for simulation. Note that these are referred to as initial guesses instead of initial values since a steady-state differential equation does not require initial values. For boundary conditions, we need to define proper values at the center of the positive electrode, x = 0, and at the center of the negative electrode, x = l. For φl , the proper values for both sides are ∂φl = 0. ∂x

(6.55)

This condition indicates that the current in the electrolyte phase is zero at both boundaries due to the symmetry of the domain. In other words, at the boundaries, the whole current is in the solid phase, and no current is in the electrolyte phase. The boundary conditions defined by Eq. (6.55) cause some illness in the system of governing equations. Basically, from the mathematical point of view, an elliptic differential (ordinary or partial) equation with Newman boundary condition on all its boundaries is an ill-posed equation. Eqs. (6.49) and (6.50) are mathematically elliptic and hence defining all the boundaries of Newman type (i.e., Eq. (6.55)) results in an ill-posed equation and will have infinitely many solutions, where only one of which is the correct answer. To find the correct answer, we need to impose another constraining equation together with the original equations. This constraining equation is called the compatibility equation and will be discussed in more detail in this section. Compatibility equation is a mathematical description of electroneutrality, which means that no charge is produced or consumed in electrochemical reactions. For solid potential, we can define different boundary values depending on operational conditions. For a voltage defined operation, we have: φs = V (t), φs = 0,

x = 0, x = l.

(6.56) (6.57)

Eq. (6.57) indicates that the zero level of potential is the potential of the current collector of the negative electrode. On the other hand, Eq. (6.56)

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Simulation of Battery Systems

shows that the potential of the current collector of the positive electrode can be changed with time. This boundary condition enables us to define any defined voltage discharging or charging operation. If the operation condition is defined with a defined current function, we have to impose the following boundary conditions: ∓σ eff

∂φs = I (t). ∂x

(6.58)

The left-hand side of Eq. (6.58) is the current in the solid phase that enters the current collector, and the right-hand side is the applied operation value. The positive and negative signs also indicate the current at the negative and positive current collectors, respectively. It is also clear that in predefined current density the equation of potential in solid phase again becomes ill-posed. As discussed before, we need to impose the compatibility equation to overcome the ill-posed condition.

Chemical species When the battery is on rest and the electrochemical cell is not charging or discharging, the concentration of any chemical species becomes uniform across the cell because of mass diffusion. Consequently, the proper initial condition for chemical species (i.e., Eqs. (6.51) to (6.54)) can be defined as follows: c H = c◦H , O2

c O2 = c◦ , H2

c H2 = c◦ ,

(6.59) (6.60) (6.61)

where the zero subscript indicates a uniform value. Symmetric boundary conditions properly define physical boundary conditions for these equations: ∂ cH = 0, ∂x ∂ c O2 = 0, ∂x ∂ c H2 = 0. ∂x

(6.62) (6.63) (6.64)

Lead–acid batteries

179

Cell temperature Since the energy equation is transient, we need an initial condition and proper boundary conditions. For a battery staying in a uniform temperature, the proper initial condition is T = T◦ .

(6.65)

Symmetry boundary conditions are proper physical choices because the domain of solution has symmetry on both sides. In other words, ∂T = 0. ∂x

(6.66)

Note that since transient equations Eqs. (6.51) to (6.54) are parabolic rather than elliptic, defining Newman boundary condition on all the boundaries will cause no illness. Therefore, Eqs. (6.62)–(6.64) and (6.66) do not impose illness to the system, and the system can be solved properly.

6.9 Physico-chemical properties Physico-chemical properties of lead–acid batteries are temperature dependent. As stated before, the dependency of the parameters are described by Arrhenius’ equations or Eq. (6.85). Besides temperature, some properties are functions of electrolyte concentration as well. In this section, properties of lead–acid batteries are summarized.

6.9.1 Electrode electrical conductivity σ The electrical conductivity of electrodes is defined in units of S cm−1 . The electrical conductivity of electrodes varies during charge and discharge due to the conversion of lead and lead-dioxide into lead-sulfate and also porosity change. Typical values for pure materials of the electrode are: σ = 500 σ = 4.8 × 10

4

for PbO2 ,

(6.67)

for Pb.

(6.68)

For porous electrodes, these value should be corrected according to porosity. The relation is given by Brugmann [29]: σ eff = σ ε 1.5 .

(6.69)

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Simulation of Battery Systems

The exponent 1.5 is normally used in literature, but in general, it is a case of study. In all the governing equations proposed in this chapter the effective conductivity is used instead of pure conductivity of materials.

6.9.2 Electrolyte ionic conductivity k The ionic conductivity of sulfuric acid is a function of concentration and temperature. Newman and Tiedeman [59] expressed the value of ionic conductivity as an experimental relation in units of cm2 s−1 : 



3916.95 7.2186 × 105 − k = c exp 1.1104 + T T2    4 9.9406 × 10 2 c − 16097.781 c . + 199.475 − T

(6.70)

6.9.3 Diffusion coefficients The diffusion coefficient of sulfuric acid is a function of acid concentration, where its value at T = 25◦ C is [60] DH = (1.75 + 260 c ) × 10−5 .

(6.71)

To include the effect of temperature on the diffusion coefficient, Newman and Tiedeman [59] gave the following relation: DH = (1.75 + 260 c ) × 10−5 exp





2174.0 2174.0 − . 298.15 T

(6.72)

It is quite evident that Eq. (6.72) gives a relation for sulfuric acid binary diffusion in a nonporous medium. In a porous medium the value should be corrected according to the Brugmann relation, that is, Eq. (6.84). The diffusion coefficient of dissolved oxygen in electrolyte is [61] O

De 2 = 0.8 × 10−5 .

(6.73)

The activation energy for this property is equal to EO2 = 14000 J mol−1 .

(6.74)

Therefore, the diffusion coefficient for dissolved oxygen is O

De 2 = 0.8 × 10−5 exp





14000 1 1 − R 298.15 T

 .

(6.75)

Lead–acid batteries

181

6.9.4 Open-circuit voltage U Open-circuit voltage of a lead–acid battery is not unique since in general potential does not have an absolute value. Therefore we have to define a reference state for potential measurement. Normally, the potential of the negative electrode is chosen as the reference state, and any other voltage is measured with this scale. For lead–acid batteries, Bode [62] gave the following relation for calculation of open-circuit voltage at T = 25◦ C: UPbO2 = 1.9228 + 0.147519 log m + 0.063552 log2 m + 0.073772 log3 m + 0.033612 log4 m,

(6.76)

where m is the molality of the electrolyte at T = 25◦ C. The molality and concentration of sulfuric acid is also given by Bode in the same reference: m = 1.00322 × 103 c + 3.55 × 104 c 2 + 2.17 × 106 c 3 + 2.06 × 108 c 4 . (6.77) By selecting the negative electrode as the reference point, the opencircuit potentials of side reactions are: UH2 = 0.356,

(6.78)

UO2 = 1.649.

(6.79)

These values can be modified by the Arrhenius relation (6.85) to include the effect of temperature. The activation energy for these values in kJ mol−1 are: EactH2 = 54,

(6.80)

EactO2 = 70.

(6.81)

6.9.5 Partial molar volumes of sulfuric acid and water Partial molar volumes Ve and V◦ of sulfuric acid and water, respectively, are given by Bode [62]. A part of these values that are functions of acid concentration are given in Table 6.1. To have a better understanding of these values, the data are plotted in Figs. 6.9 and 6.10. For numerical simulation, it is better to give a relation between partial molar volume values, instead of tabulated data. Using a curve fitting, we have

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Simulation of Battery Systems

Table 6.1 Partial molar volumes of sulfuric acid and water as functions of acid molality. Partial molars Density Molality Ve V◦ ρ m cm3 mol−1 cm3 mol−1 kg l−1 l3 kg−1 35.10 18.07 0.9970 − 39.44 18.03 1.0300 0.526 39.49 18.01 1.0640 1.133 17.94 1.1365 2.549 40.85 43.45 17.76 1.2150 4.370 17.57 1.2991 6.797 45.29 45.90 17.47 1.3911 10.196

Figure 6.9 Partial molar volume of sulfuric acid.

Figure 6.10 Partial molar volume of water.

⎧ 3 −1 ⎪ ⎨ (0.02538165 − 5.4367785e − 5 m ) Ve = 34.877248 + 2.962056 m ⎪ ⎩ −0.262232 m2 + 0.007625 m3

m 2.549.

Lead–acid batteries

Table 6.2 Thermodynamic properties of different species. species G◦ cP◦ S◦ − 1 − 1 − 1 − 1 (kJ mol K ) (kJ mol−1 ) (kJ mol K )

183

H ◦ (kJ mol−1 )

H2 H+ O2 H2 O(g) H2 O(l)

6.889 0 7.016 8.025 17.995

31.208 0 49.003 45.107 16.710

0 0 0 −54.634 −56.687

0 0 0 −57.796 −68.315

OH − SO41− HSO4− H2 SO4 Pb

−35.5 −70

−2.57 4.4 28.45 37.501 15.49

−37.594 −177.83 −180.55 −164.936

−54.970 −217.32 −212.08 −194.548

0

0

2.5 15.9 35.509 22.1 18.26

−5.83 −45.16 −194.36 −51.94 −52.34

0.4 −52.34 −219.82 −63.52 −66.12

Pb 2+ PbO PbSO4 α − PbO2 β − PbO2

33.20 6.32 10.95 24.667 −

15.44

⎧ 2 ⎪ ⎨ 18.042475 − 0.019430 m − 0.0081488 m V◦ = 18.210753 − 0.105352 m ⎪ ⎩ −0.0015226 m2 + 0.000464 m3

m 2.549. (6.83)

The tabulated data and the curve fittings are plotted in Figs. 6.9 and 6.10. The figures show that the curve fittings have good accuracy for calculations.

6.9.6 Thermodynamic properties of different species In many calculations, thermodynamic properties of different species are required. The necessary values for different species that are involved in lead–acid battery reactions can be found in different thermodynamics handbooks. Table 6.2 summarizes sum of the values.

6.9.7 Calculation of properties in porous medium In all the above equations, the physicochemical properties like the diffusion coefficient and ionic conductivity of electrolyte should be modified because of the porosity of the media.

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Simulation of Battery Systems

Figure 6.11 Geometry of a lead–acid battery.

In the case of porous media, all the properties are modified such that the effect of porosity comes into account. For any property such as  , the effective value and its pure value are related to each other according to the Brugmann equation  eff = εjξ .

(6.84)

In this relation, j indicates phase j, and ξ is an exponent that shows the tortuosity of the porous medium. In battery systems, ξ = 1.5 is usually chosen for all the simulations.

6.9.8 Temperature dependency of parameters The temperature dependency of physicochemical properties is obtained using the Arrhenius equation [30] 

Eact,φ  = ref exp R



1 1 − Tref T

 .

(6.85)

In this equation, ref is the physicochemical property at the reference temperature (including the effect of tortuosity), and Eact,φ is the activation energy of the evolution process of . The activation energy is a measure of the sensitivity of the parameter to temperature. The higher the value, the more the parameter is sensitive to temperature.

6.10 Numerical simulation of lead–acid batteries Just like other battery systems, a lead–acid cell is shown in Fig. 6.11, where the domain of solution consists of a current collector at the center x = 0

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of the positive electrode, which is a porous medium made of lead dioxide PbO2 , an electrolyte region near the positive electrode, a porous separator, the negative electrode that is made of pure, and finally the negative current collector at the center x = l of the negative electrode. The whole system is filled with sulfuric acid, which reacts with electrodes to produce lead sulfate. The electrolyte region attached to the positive electrode is an extra space to provide more electrolyte for the positive electrode since the stoichiometric of the reaction of the positive electrode requires three times more electrolyte than the negative electrode. This extra space is created by attaching some ribs on the separators. In some separators the rib is located at both sides of the separator, and hence the electrolyte space should be also considered at the negative electrode side. Although the governing equations of lead–acid batteries are expressed in vector form and are valid in three dimensions, one- and two-dimensional simulations are more common in practice. This is because one- and twodimensional assumptions are accurate enough for many purposes. In some cases, one-dimensional simulation is even more than enough because many distributed parameters are not required for those applications. Consequently, depending on the application, we can choose the desired dimension and start solving the equations with proper physical assumptions. For example, for simulation of acid stratification, a two-dimensional model is appropriate, whereas for monitoring purposes, in many applications or for optimization and design purposes, a one-dimensional model is sufficient. In this chapter the simulation of governing equations is discussed in one and two dimensions separately, and the main issues and assumptions are discussed in more detail. Since three-dimensional simulation is rarely required, it is not considered in this chapter, but the same procedure discussed here can be applied for three-dimensional simulations.

6.10.1 One-dimensional simulation without side reactions In some batteries, except in overcharge situations, side reactions are not involved in discharge or even in charge. Hence in these situations the battery can be simulated without side reactions, that is, by dropping Eqs. (6.52) and (6.53) from the system of equations. The resulting system of nonlinear equations can be solved using different numerical techniques including the finite volume method (FVM), the finite difference method (FDM), the finite element method (FEM), or any other known methods. Each method has advantages and drawbacks. Although FVM is a popular method amongst battery simulation teams, Esfahanian

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and Torabi [49] showed that the Keller–Box method is a good choice for these types of numerical computations. To show the ability of the method, here the system of equations is solved using the Keller–Box method. In the next section, the FVM method is used for simulation of battery behavior when side reactions are present. The main advantages of Keller–Box method can be summarized as follows: 1. In the Keller–Box method, both function and its derivative are presented in the final system of equations. Hence in battery systems, where both function values and their derivatives are present at all boundaries, the Keller–Box method is a good choice. 2. The Keller–Box method provides second-order accuracy both in time and space, whereas the normally used numerical methods provide the first-order accuracy in time. 3. The resulting coefficient matrix in the Keller–Box method is always tridiagonal, which means that the Thomas algorithm can be used for its solution. This advantage gives a fast algorithm for numerical integration. 4. Mixed boundary conditions such as the simultaneous implementation of potential and current at boundaries are possible. 5. Since in the Keller–Box method, each point depends on its previous numerical point, a nonuniform grid can be easily applied to the numerical domain without any difficulty. Other characteristics of the Keller–Box method can be found in CFD books such as [63,64] and is vastly used in boundary layer theories [65]. The method and its implementation to system of nonlinear partial differential equations (PDEs) are discussed in Appendix F. By neglecting side reactions in a constant temperature situation, the system of governing equations is limited to Eqs. (6.49)–(6.51). By solving the system we are looking for the values of the solid potential φs , the potential distribution in electrolyte phase φe , and the electrolyte concentration distribution c H . In Keller–Box method, we have to convert the second–order PDEs to a system of first-order PDEs. To do that, we introduce the following assumptions: ∂φs = φsx , ∂x ∂φe = φex , ∂x

(6.86) (6.87)

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∂ cH = cx . ∂x

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(6.88)

Substituting Eqs. (6.86) to (6.88) into Eqs. (6.49) to (6.51), we get:  ∂ eff σ φsx − (Aj) = 0, ∂x All

(6.89)

   ∂ eff ∂ ln c ∂ k φex + keff (Aj) = 0, + D ∂x ∂x ∂x All

(6.90)

∂ H ∂(εe c H ) AjH 1 − t+◦  = Deff cx + a2 + (Aj). ∂t ∂x 2F F Side

(6.91)

reactions

reactions

reactions

Eqs. (6.86)–(6.91) constitute a system of first–order nonlinear PDEs with six unknowns φs , φsx , φe , φex , c H , and cx . The nonlinear source terms in these equations should be linearized so that the resulting numerical matrix becomes stable. The details can be found in Appendix F. Cell-II from Appendix A is chosen for simulation. The parameters of the cell are tabulated in Table A.3 and Table A.4; the other required lead– acid characteristics are taken from Section 6.5. The results of the simulation are shown in Fig. 6.12. Fig. 6.12A shows the cell voltage versus time, where the simulation results are compared with the results of other researchers. It is clear from the figure that the present simulation has a good agreement with the results of others, and also it is obvious that the cell reaches the cut-off voltage at t = 105 s. This indicates that the chosen cell is a starter battery. The variation of electrolyte concentration in times 60 and 105 s is shown in Fig. 6.12B. As we can see, at t = 105 s, where the battery reaches the cutoff voltage, electrolyte concentration reaches zero at the positive electrode. Considering the amount of active materials at both electrodes as shown in Fig. 6.12C, it can be definitely concluded that electrolyte is the limiter of the cell voltage. Although we still have plenty amount of active material in both electrodes, the lack of acid concentration causes the cell voltage to drop. Also from Fig. 6.12B it is clear that the net electrolyte concentration in all the cell regions is not zero since there is still enough acid concentration at the negative electrode. We can conclude that if we put the cell in rest for a while, then the electrolyte becomes uniform again due to diffusion, and the cell can become discharged again. Fig. 6.12C also indicates that the active material in the positive electrode is consumed more than the negative counterpart. Also, it is evident that in

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Figure 6.12 Simulation results for Cell-II. (A) Cell voltage vs time. (B) Electrolyte concentration in different time levels. (C) Variation of active material in different time levels. (D) Electrolyte potential distribution.

both electrodes, about 80% of active materials are still available in both electrodes, indicating that the state of charge of the battery is SoC  0.8. Again it emphasizes that we are dealing with a starter battery in which active materials should not be consumed more than 20–30%. From Figs. 6.12B and 6.12C we can conclude that the design of the battery is not appropriate since while the positive electrode stops delivering power, the negative electrode is still alive and has potential to provide more energy. By a better design the thickness of the negative electrode can be minimized to reduce the battery weight resulting in less cost. The variation of electrolyte potential distribution is shown in Fig. 6.12D, in which it is clearly indicated that the electrolyte potential drops in all re-

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gions resulting in cell potential drop. It is also evident that the electrolyte potential is a little bit higher in the negative electrode in comparison with the positive counter electrode.

6.10.2 One-dimensional simulation including side reactions When side reactions are present, we need to simulate the whole system of Eqs. (6.49)–(6.54). The system can be solved using any numerical methods including the Keller–Box method, as explained in the previous section or any other methods. Choosing the numerical method is quite arbitrary; just we have to keep in mind that the numerical method should not alter the final results. In other words, regardless of the numerical method, the same results should be obtained. In this section, to show the capability of FVM, we used this method to solve the system of governing equations. The details of FVM in one dimension is explained in Appendix D; it is based on the method of Patankar [45]. To verify and validate the numerical scheme, Cell-V from Appendix A is chosen. The cell belongs to a sealed lead–acid battery, in which side reactions become important in the overcharge period. The cell is under constant current charge until side reactions become important, and after that, the charging process continues. The results of simulation are shown in Figs. 6.13 and 6.14. The simulated cell voltage during charge is compared and shown in Fig. 6.13A. It takes about 15 hours for the cell to become fully charged. The difference between the simulation results of different researchers is because they used different values for simulation due to the lack of information about the original test. The experiment carried out by Bernardi [67] shows that during the charging process, the cell voltage rises to a pick value at about t = 8 hr and falls to a specific value about 2.4 V. From then on, the cell voltage reaches a plateau, which is an indicator of a steady state. All the results of the other researchers including present simulation predict the same behavior, but the value differs a little bit from the experimental data. The difference can be translated to different simulation parameters used since the real values are not properly tabulated in the open literature. The cell voltage behavior in charge process is attributed to the side reactions. Without simulation of side reactions, the voltage decrease at t = 8 hr cannot be resolved. Fig. 6.13B illustrates the charging process with and without the simulation of side reactions. We can see that neglecting side reactions causes the cell voltage to rise exponentially when the active

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Figure 6.13 Simulation results for Cell-V. (A) Cell voltage during charge. (B) Cell voltage with and without side reactions. (C) Current density of reactions at t = 0 hr. (D) Current density of reactions at t = 8 hr. (E) Current density of reactions at t = 10 hr. (F) Current density of reactions at t = 13.8 hr.

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Figure 6.14 Simulation results for Cell-V. (A) Electrolyte potential distribution. (B) Electrolyte concentration in different time levels. (C) Concentration of oxygen in gas phase. (D) Concentration of oxygen in electrolyte phase. (E) Overpotential distribution. (F) Solid phase potential.

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materials are completely charged. If there were not any side reaction in the real case, it would be the real behavior of the system. However, when the cell voltage reaches about 2.4 V, side reactions become dominant and drop the cell voltage as discussed before. The share of different reactions are plotted in Figs. 6.13C to 6.13F for different time levels. It should be noted that the applied current is the only driving source for all side reactions. It means that the integral of current density J over each electrode on all the reactions is equal to the applied current. From the figures we can see that at the beginning of charge process (t = 0 hr), side reactions can be neglected since all the applied current is dedicated to main reactions. From another perspective the cell voltage at the beginning of the charge is about 1.9 V, and as we know, water dissociation is very slow at that voltage. By increasing the cell voltage to 2.3 V, or t  8 hr, the share of the main reaction becomes less, and side reactions start showing up. The process continues until t  10 hr, where most of the current goes to side reactions rather than to the main reaction. Finally, at t = 15 hr, it is clear that the share of the main reaction is almost negligible due to the fact that almost all active materials are charged; hence all the applied current drives the side reactions. It is clear that the oxygen side reaction is almost dominant, and the kinetics of hydrogen side reaction is very poor, meaning that we can neglect the hydrogen side reaction in their simulation. In Fig. 6.14, other simulation parameters are shown. For instance, Fig. 6.14A shows the variation of electrolyte phase potential. As it can be seen, the electrolyte phase potential increases in charge process and reaches a maximum value at about φe  2.21 V, meaning that the cell is not charged anymore. Fig. 6.14B shows the variation of electrolyte concentration at different cell regions and different time levels. Again, the electrolyte concentration reaches a maximum, which is another indicator for a fully charged cell. A very good result is shown in Figs. 6.14C and 6.14D. In these graphs the concentrations of oxygen in gas and electrolyte are respectively plotted. As it is clearly seen, at the beginning of charge process, the oxygen concentration is not considerable; however, by continuing the charging process the oxygen concentration becomes higher and higher due to the fact that the share of side reactions becomes important in benefit of oxygen reaction. This fact can be further investigated if we plot the overpotentials of the main and oxygen reactions as we did in Figs. 6.14E and 6.14F. It is evident that as the charging process continues, the share of the main reaction

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becomes more and more negligible, whereas the share of oxygen reaction becomes more and more dominant. Comparison of Figs. 6.14C and 6.14D from another viewpoint indicates that the oxygen in the gas phase has flatter distribution than that in the electrolyte phase. This is due to the fact that the diffusion coefficient of oxygen in the gas phase has a higher value than that in the electrolyte phase. In other words, oxygen in the gas phase becomes homogeneous much faster than in the electrolyte phase. The results of the figures prove this fact.

6.10.3 Numerical simulation of electrolyte stratification using two-dimensional modeling In previous sections, we studied one-dimensional modeling of lead–acid batteries. Although the one-dimensional model is very accurate and much useful information can be obtained, in some cases, at least two-dimensional modeling should be performed. Simulation of electrolyte stratification is an example of such cases. In this phenomenon, natural convection takes place inside the battery cell due to the electrolyte concentration gradient. Since during charge or discharge, electrolyte concentration takes place (as was discussed in Fig. 6.12B), a more concentrated electrolyte becomes heavier and sinks, whereas a less concentrated electrolyte rises due to gravity force making a natural convection movement. The induced natural convection causes the electrolyte to become stratified, which in turn results in nonuniform usage of electrodes. To numerically capture the phenomenon, the Navier–Stokes equations should be coupled with governing the electrochemical system of equations. In this case, at least a two-dimensional space should be modeled because electrolyte movement has no meaning in one dimension. Electrolyte movement occurs due to the following reasons: 1. In portable devices like cars, the case of the battery moves, and so does the electrolyte. 2. Released gases inside the battery cause the electrolyte to move. 3. As mentioned before, electrolyte concentration is a major source of electrolyte movement. 4. The temperature gradient in a battery may be a driving force for electrolyte movement. Independently of the mechanism responsible for electrolyte movement, the Navier–Stokes equations should be coupled with battery governing equations to simulate the electrolyte movement. In the lead–acid batteries, electrolyte moves in porous media such as electrodes and separators.

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Normally, the porosity of the regions exerts a lot of force on electrolyte and renders the acid movement; however, in rib regions of separators the electrolyte has enough space for circulation and natural convection. Since stratification occurs in porous media, the Navier–Stokes equations should be written in the form such that the effect of porosity would be included in the equations. The proper form is given in Eq. (6.14). Electrolyte stratification is studied in discharge process under constant temperature condition. In this case the side reactions are dropped out of the governing system of equations, and the Navier–Stokes equations are added to the system. The simplified system of equations is as follows: ∇ · (σ eff ∇φs ) − Aj = 0, ∇ · (k ∇φl ) + ∇ eff

· (keff D ∇(ln c )) + Aj = 0,

∂(ε c ) Aj + v · ∇ c = ∇ · (Deff ∇ c ) + a2 , ∂t 2F ∂ρv μ + v · ∇(ρv) = −∇ p + v · (μ∇v) + ρ g[1 + β(c − c◦ )] + (εv), ∂t K ∂ρ + ∇ · (ρv) = 0. ∂t

(6.92) (6.93) (6.94) (6.95) (6.96)

The existence of Navier–Stokes equations and the continuity equation require special attention for numerical solution. Patankar [45] was one of the pioneers of FVM and gave a proper algorithm called SIMPLE to solve such systems. The details of the method are given in Appendix E, and more can be found in CFD textbooks such as [45,68]. To show the numerical simulation of acid stratification, we chose CellIV from Appendix A. All the necessary parameters such as geometrical dimensions and electrochemical characteristics are given in the same appendix. Alavyoon et al. [50] was the first who used the cell to investigate the effect of electrolyte stratification. They used the holographic laser interferometry method for measuring electrolyte concentration and the laser Doppler velocimetry (LDV) for measuring the flow field. The cell consists of three regions, namely a positive electrode, a free space for electrolyte, and a negative electrode. The electrodes and the free space have 2-mm thickness, and the charging current is very low, about 9.434 mA cm−3 . Since the charging current is low, the temperature of the cell does not vary too much during the test, and we can assume an isothermal model at T = 25◦ C. Alavyoon et al. [50] proposed a system of equations for simulation of electrolyte stratification, in which instead of solving the full Navier–Stokes

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equations, they used a creeping flow and reduced the momentum equation. Moreover, they made many simplifying assumptions: 1. The kinetic rates of reactions were assumed to be constant in the direction of cell thickness. 2. Electrolyte diffusion was assumed to be constant. In reality the diffusion coefficient is a function of both concentration and electrode porosity. 3. It was also assumed that the porosity of electrodes is constant, which is not an accurate assumption. They solved the resulting system of equations using FDM and compared their results with experimental test. It is good to notice that before putting the battery cell under test, Alavyoon et al. made a preparation routine: 1. After preparation of the setup, the cell was filled with 5 M sulfuric acid, and the cell was discharged with I = 9.34 mA cm−2 until the cell reaches the cut-off voltage Vcut = 1.5 V. 2. Then the cell was filled with 2 M sulfuric acid and kept for 48 hours so that the electrolyte becomes uniform all across the cell. On the other hand, Gu et al. [37] investigated this problem once again using full Navier–Stokes equations. In this case, the model proposed by Gu was more accurate than that of Alavyoon. The only thing that was not considered in their simulation was the preparation process. They did not model the preparation process, and as we will see, the process makes changes in initial conditions. We show that the preparation process can be modeled using a one-dimensional model, and as we will see, it affects the results. Here the preparation process is simulated using a one-dimensional model, and the results are shipped to a two-dimensional model. Figs. 6.15 and 6.16 show the results of a one-dimensional simulation. Fig. 6.15A shows the cell voltage variation. It shows that the cell requires about 5.5 hr for full discharge. The shares of current density in solid and electrolyte phases are plotted in Fig. 6.15B. In the same graph the sums of both current densities are also plotted. It is quite evident that the sum of both current densities is constant and equal to I = −9.34 V, which is the result of electroneutrality. The electrolyte concentration variation is shown in Fig. 6.15C, and as it can be seen in this cell, the electrolyte concentration reaches zero almost in all regions except about 0.4 M in the negative electrode, which is not considerable. Fig. 6.15C illustrates the variation of porosity during discharge. As can be seen, the preparation process results in a nonuniform

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Figure 6.15 Simulation of discharge phase of preparation process for Cell-IV. (A) Cell potential during discharge. (B) Share of current density. (C) Electrolyte concentration. (D) Porosity change. (E) Active material distribution. (F) Distribution of SoC.

porosity distribution. This result also can be seen in the active material distribution shown in Fig. 6.15E and state of charge in Fig. 6.15F.

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Figure 6.16 Simulation of discharge phase of preparation process for Cell-IV. (A) Cell potential during discharge. (B) Share of current density. (C) Electrolyte concentration. (D) Porosity change. (E) Active material distribution. (F) Distribution of SoC.

Fig. 6.16 shows the same results for the rest process, where the cell is put in rest for 48 hours. The voltage of the cell remains constant (Fig. 6.16A),

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Figure 6.17 Cell-IV model and numerical grid. (A) Cell model. (B) Numerical grid.

and as it can be seen in Fig. 6.16B, both solid and electrolyte current densities are zero. The only parameter that changes during the rest is the electrolyte concentration because the cell is filled with 2 M sulfuric acid, and from Fig. 6.16C it is clear that it takes 48 hours for the electrolyte to become uniform. From Figs. 6.16D to 6.16F we can see that the porosity, active area, and SoC do not change during the rest period. Therefore the initial values for stratification simulation should be taken from these figures. The fluid flow is simulated using a SIMPLE algorithm given in Appendix E. The simulated domain is shown in Fig. 6.17A and the numerical grid is shown in Fig. 6.17B. As can be seen, a non–uniform mesh is used for simulation. Also, note that for giving a proper visualization, the x and y axes are independently scaled. The results of the simulation are shown if Figs. 6.18 and 6.19 for time levels t = 15 and t = 30 min, respectively. Figs. 6.18A and 6.19A show the velocity vectors at electrolyte region. It is clear that the electrolyte tends to move downward near the electrodes because, during the charging process, acid is produced inside the electrodes according to the electrochemical reaction of electrodes. But it is evident that the electrolyte near the positive electrode is denser than the negative electrode because of stoichiometric coefficients of lead–acid main reactions. Figs. 6.18B and 6.19B show the natural convection that takes place inside the electrolyte region. Some vortexes are seen near the top of the cell because of electrolyte movement. The result of electrolyte movement is translated into electrolyte stratification as is plotted in Figs. 6.18C and

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Figure 6.18 Results of simulation at t = 15 min. (A) Velocity vectors. (B) Velocity field. (C) Electrolyte contours.

Figure 6.19 Results of simulation at t = 30 min. (A) Velocity vectors. (B) Velocity field. (C) Electrolyte contours.

6.19C. The movement of electrolyte causes the denser electrolyte to sink and the lighter ones to rise. Therefore, along the vertical cross-sections of the cell, we see an acid gradient, also known as acid stratification. If we do not couple the Navier–Stokes equation with the other governing equations, then electrolyte stratification could not be captured. To show this argument, we draw the same results of Fig. 6.19 in the absence

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Figure 6.20 Results of simulation at t = 30 min without fluid flow. (A) Velocity vectors. (B) Velocity field. (C) Electrolyte contours.

Figure 6.21 Comparison of electrolyte concentration between with and without electrolyte movement at section A − A.

of electrolyte movement in Fig. 6.20. As can be seen, since we do not have any velocity field (comparing Figs. 6.20A and 6.20B), the electrolyte does not show any gradient in a vertical direction. The vertical contour lines in Fig. 6.20C support this argument. Fig. 6.21 shows the electrolyte concentration gradient at midheight cross-section of the battery cell. The figure shows that including elec-

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Figure 6.22 Comparison of vertical velocity component at section A − A.

trolyte movement and neglecting it make a significant difference in final results. Hence if in a battery, free electrolyte exists, then the simulation of electrolyte movement is crucial even though the electrolyte movement is slow and creepy. The vertical component of the velocity field plotted in Fig. 6.22 at the same height supports this argument. The velocity at most reaches about 0.1 mm s−1 , which is a slow movement. Finally, electrolyte concentration gradients in the vertical direction at the center of the electrolyte region in different time levels are plotted in Fig. 6.23. It is clear that as time passes, the gradient becomes more significant.

6.10.4 Simulation of thermal behavior of lead–acid batteries Lead–acid batteries, just like any other battery technology, faces with thermal problems. During charge or discharge, according to electrochemical reactions and Joule heating, the battery gets warm and may undergo thermal runaway, as was discussed with more details in Section 6.5. Studying thermal rise (TR) or thermal runaway (TRA) requires an insight into understanding the involved physical phenomena. This study can be done by numerical simulation if a proper system of governing equations is modeled and solved. The purpose of the present section is investigating the thermal behavior of lead–acid batteries by solving the governing equations. Governing equations of lead–acid batteries are fully described in Section 6.5. For the simulation of thermal behavior of lead–acid batteries,

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Figure 6.23 Comparison of vertical velocity component at section A − A.

Eqs. (6.16)–(6.32) should be solved. Since the oxygen recombination and consequently thermal-runaway problem occur in the sealed or gelled lead– acid batteries, electrolyte movement can be neglected; hence the conservation of momentum or Navier–Stokes equations are not going to be modeled. The heat generation in the cell is obtained using Eq. (6.44) with details given in Section 6.5. It should be noted that heat dissipation should also be simulated because dissipation of energy plays an important role in thermal behavior. In this study the resistance model described in Section 6.5 is used for simulation. To numerically illustrate the present modeling, Cell-V from Appendix A is chosen and simulated. This cell was introduced by Srinivasan [39] for simulation of pulse charging. Since all the necessary data for this cell are available, it is used for the present thermal modeling as a test case. The chosen cell is a VRLA battery in which the side reactions accompany the main reactions according to Eqs. (6.8)–(6.13). The rates of reactions are determined by the Butler–Volmer equations (6.19)–(6.21). All the required data and parameters of the battery are given in Appendix A. Fig. 6.24A shows a physical illustration of the cell. As can be seen in this figure, the ambient temperature is constant (Tambient = 25◦ C), and heat is dissipated via convection. Since these electrodes are located in the middle of an electrode set inside the middle battery cell, with a good approximation,

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Figure 6.24 A typical valve-regulated lead–acid cell model.

we can consider that the left and right electrodes are at the same temperature, and hence no heat transport exists between the electrodes. This means that the left and right boundary conditions are of adiabatic type and heat is dissipated into ambient only from the top and bottom of the battery case. The proper numerical boundary conditions are illustrated in Fig. 6.24B. To numerically solve the governing equations, they should be discretized through a computational domain. In this study, we use the semiimplicit method for pressure-linked equation (SIMPLE) introduced by Patankar [45]. A nonuniform computational grid (Fig. 6.24B) is generated to reduce the computational cost and improve the accuracy where the gradients of variables are high. A comprehensive grid study has been performed to insure that the results are grid independent. Fig. 6.25 illustrates the parameter sensitivity study results. Fig. 6.25A shows that when the time step is equal to t = 5 seconds, changing the grid size in j-direction affects the final results (in this case the positive electrode terminal temperature). However, if the time step is reduced to t = 1 second, them the results would be independent of grid size in j-direction (Fig. 6.25B). Reducing the time step does not alter the accuracy of the simulation, as can be seen in Fig. 6.25C. Finally, increasing the grid size in i-direction does not give any more accurate

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Figure 6.25 Grid study procedure. (A) Sensitivity to mesh size (t = 5). (B) Sensitivity to mesh size (t = 1). (C) Sensitivity to time step. (D) Sensitivity to mesh in x-direction.

results (Fig. 6.25D). As a consequence, using a grid size of the order of Im × Jm = 15 × 35 and choosing the time step equal to t = 1 second guarantees that the result would be grid independent. This mesh size and time step are used for the rest of the simulation. The governing equations contain highly nonlinear sources, which are linearized by Newton method to obtain a stable solution. The system of equations is solved with a segregated method, and iterations are performed until the relative error is less than 10−8 . The charging voltage of a battery is very important in TRA. Culpin [51] has carried out a set of experimental tests and indicated that TRA appears at float charging about 2.63 V per cell, whereas the batteries at 2.40 V per cell showed a much lower and more uncertain trend to TRA. In this study the battery cell is simulated at three different floating voltages 2.26, 2.36, and 2.46 V. The cell is overcharged from a fully charged state (SoC = 1) to ensure that the main reactions do not contribute to heat

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Figure 6.26 Time history of simulated results for float charging V = 2.26 V. (A) Voltage and current. (B) Cell Temperature.

generation. Therefore all the input current is spent on driving the side reactions, and the effect of side reactions can be better understood. Figs. 6.26–6.28 show the results of simulation at floating voltage of V = 2.26 V. Fig. 6.26A shows the applied voltage of the cell and the resulting current density. Since the cell is being charged from SoC = 1, all the input current goes into the generation and consumption of oxygen and hydrogen. At the beginning of charge the generation of oxygen at the positive electrode and its consumption at the negative electrode are not equal. Hence the battery requires a proper current to drive the reactions. When the charging continues, these rates balance each other, and the applied current reaches a limit, which can be seen in Fig. 6.26A. The current drop follows an exponential trend since it is governed by the Butler–Volmer equation (6.20) and is known as the natural behavior of lead–acid batteries. Fig. 6.26B shows the variation of terminal temperature versus time. This figure shows that the terminal temperature reaches an equilibrium after a while and the cell temperature does not vary after that. The temperature plateau forms because the heat generation and dissipation come to an equilibrium. Fig. 6.27 shows the variation of different parameters at the cell halfheight y = 12 H at different time levels. Figs. 6.27A and 6.27B show the current densities of the main and side reactions. It can be seen that the current of the main reaction is zero since the battery is fully charged and no current goes into the main reaction. On the other hand, the oxygen current has a moderate value, which is decreasing until it reaches a specific

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Figure 6.27 Variation of different properties at the cell half-height vs. for V = 2.26 V. (A) Jmain and JO2 . (B) JH2 . (C) Concentration of oxygen in liquid phase. (D) Concentration of oxygen in gas phase.

equilibrium amount. Moreover, we can see that the current that drives the hydrogen evolution reaction is small, which shows that the hydrogen reaction has a slow kinematic rate and we can even neglect the hydrogen evolution reaction, as can be clearly seen from Fig. 6.27B. As a conclusion, in this cell, all the charging current enters the oxygen generation and consumption reactions. In Figs. 6.27C and 6.27D the oxygen concentrations in liquid and gas phases are shown, respectively. From these figures we can see that at the beginning of the charge, oxygen evolves very fast and its rate decreases as charging continues.

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Figure 6.28 Contribution of reversible and irreversible heat of different reactions. (A) Heat of main reaction. (B) Heat of hydrogen reaction. (C) Heat of oxygen reaction. (D) Joule heating. (E) Total heat source inside the cell. (F) Variation of cell heat capacity.

To have a better understanding of the heat sources and sinks in a lead– acid battery, the generated heats of different reactions and heat dissipation

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are plotted in Fig. 6.28. As expected, according to Fig. 6.27A, the generated heat of the main reactions is zero. The same argument is true for hydrogen reaction, as we can see from Fig. 6.28B. The main source of heat inside the battery is the heat generation of the side reaction shown in Fig. 6.28C. In this figure the reversible (Eq. (6.37)) and irreversible (Eq. (6.38)) heats of the reaction are plotted for different cell regions (i.e., positive electrode, separator, and negative electrode). The total reversible and irreversible heats are also plotted in the same figure. We can see that the net reversible heat in all the regions is zero. Moreover, from the figure we can see that only the irreversible heat is responsible for heat generation inside the battery cell. Fig. 6.28D shows the Joule heating in different cell regions. As it was stated before, the Joule heating is small since the battery is in a fully charged state, and hence its resistance is negligible. If the charging process begins at SoC < 1, the Joule heating will have a large value. The other point about the Joule heating is that it exists even in the separator region because the potential field exists inside the separator. The heat dissipation from the upper and lower walls are plotted in Fig. 6.28E. The total heat generation is also shown in this figure and compared with heat dissipation. This figure shows that at the beginning of the charging process, the total heat generation inside the battery is positive resulting in battery thermal rise. When the charging continues, the battery cell temperature increases. As a result, the temperature difference between the battery and ambient becomes larger. The larger the temperature difference, the larger the heat dissipation. After about 10 hr, the heat generation and dissipation balance each other out, and the battery temperature stays constant (see Fig. 6.26B). Finally, the variation of internal heat capacity of the cell at the middle height y = 12 H is shown in Fig. 6.28F. The heat capacity of the cell does not change significantly since the initial composition of the cell does not change. Although the oxygen generation and acid concentration variation affect this value, we can see from the figure that the variation is negligible. Figs. 6.29 and 6.30 show the same results for float charging V = 2.36 V. From Fig. 6.29B we can see that increasing the charging voltage increases the battery temperature; but at this voltage the cell temperature rise will not exceed the limit (i.e., 60◦ C), hence thermal runaway would not be expected. The higher temperature rise happens since the oxygen generation rate becomes faster and more current passes through the cell. The increase in cell current means that the irreversibility becomes larger and more heat is

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Figure 6.29 Time history of simulated results for float charging V = 2.36 V. (A) Voltage and current. (B) Temperature profile.

Figure 6.30 Contribution of reversible and irreversible heat of different reactions. (A) Heat of oxygen reaction. (B) Total heat source inside the cell.

generated, as clearly shown in Fig. 6.30A. It is worth noting that other heat sources can be neglected, as discussed for the case of float charging 2.26 V. Fig. 6.30B shows the sum of all heat sources and dissipations inside the cell. Comparing with Fig. 6.30A, it is clear that the net reversible heat generation is zero (as in the case of float charging 2.26 V) and only irreversible part remains for heating the cell. We can also see in the figure that the generation and dissipation rates will balance each other, which results in a stable cell temperature after about 5 hr (comparing with Fig. 6.29B).

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Figure 6.31 Time history of simulated results for float charging V = 2.46 V. (A) Voltage and current. (B) Temperature profile.

Figure 6.32 Contribution of reversible and irreversible heat of different reactions. (A) Heat of oxygen reaction. (B) Total heat source inside the cell.

Figs. 6.31 and 6.32 show the same results for float charging V = 2.46 V. In this voltage the heat generation becomes too strong, and according to Fig. 6.31B, the cell temperature exceeds the limit in less than 1 hr, that is, TRA occurs. From Fig. 6.32A we can see that the irreversibility becomes much larger than the previous charging. Therefore, the heat generation is much more than the previous cases. It should be restated that in all the simulations the total reversible heat is zero and only the irreversible heat is responsible for the temperature rise.

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Figure 6.33 Temperature contours at different time levels for V = 2.46 V. (A) At time level t = 0 min. (B) At time level t = 20 min. (C) At time level t = 40 min. (D) At time level t = 60 min.

Moreover, after about 1 hr, the cell temperature exceeds the limit, and the battery enters the unstable situation. Fig. 6.32B shows that in this period the total irreversible part is not balanced by dissipation. Meanwhile, the net reversible part in the positive and negative electrode has been canceled. It worth noting that in the present study the cell temperature is not considered as a bulk temperature and the distribution of temperature inside the battery is also modeled. Fig. 6.33 shows the temperature contours inside the cell at different time levels for float charging of V = 2.46 V. Since

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the chosen cell is slender and its aspect ratio (hight over width) is large, the temperature contours cannot be clearly displayed. Therefore, for better presentation, the x and y axes are scaled differently. From the figure we can see that the temperature inside the battery is not uniform, and there exists a temperature gradient inside the battery. This means that using a bulk temperature for modeling of lead–acid batteries is not quite accurate. From these results we obtain important information. For example, as mentioned before, there are different theories about TRA, all of which can be categorized into two distinct paradigms that conflict with each other. The first paradigm [55,54,56], which is most accepted by many researchers, states that the heat generation due to oxygen reduction at the negative electrode is the main source of temperature buildup, which can result in TRA. This paradigm further states that the Joule heating not only contributes to temperature buildup but also accelerates the internal oxygen reduction cycle. The second paradigm [57,58] states that according to the Hess law, the closed oxygen cycle cannot produce any heat since it is a closed cycle. This paradigm further discusses that the only heat generation mechanism in a VRLA battery is the Joule heating. From the present study and numerical simulation another paradigm can be deduced, which links both mentioned paradigms. It has been shown that heat generation of each electrode reaction can be split into reversible and irreversible parts. The reversible part can be found by considering the enthalpy of each reaction and changes sign during charge and discharge. On the other hand, for each reaction, the irreversible part defined by Eq. (6.38) should also be considered. This term can be found by considering the local current density and the open-circuit voltage of the reaction noting that it is always positive. Besides the heat generated due to the electrochemical and chemical reactions, the Joule heating is also a source of heat in a battery. The Joule heating can be found by solving Eq. (6.43). A closer examination of the Joule heating and the irreversible part of heat generation due to the electrochemical reactions shows that these parts have the same nature. To be more specific, the Joule heating can be found by multiplication of current and voltage: qJoule = IV . On the other hand, the irreversible part of heat generation of each reaction is qirr. = JU. Comparing these equations, we conclude that these parts have the same nature or qirr. is of Joule heating type in which I is replaced with the local current density J, and V with local open-circuit voltage U. From Eq. (6.38) we can obtain the irreversible part of heat generation of each reaction: qirrev = −Jmain U main − JO2 U O2 − JH2 U H2 .

(6.97)

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According to these results, a new notion is proposed, called the generalized Joule heating, which is the sum of classical Joule heating and the irreversible part of the generated heat due to the chemical and electrochemical reactions: qGJH = |φ · ik | + qirrev = qJoule + qirrev .

(6.98)

By defining this notion we can conclude that instead of the classical Joule heating, the generalized Joule heating qGJH is the only responsible mechanism for TR or TRA in batteries or specifically VRLA batteries. In addition to the heat sources, heat sinks also play an important role in the thermal behavior of a battery. Heat sinks have also been introduced to the governing equations with a resistance model developed in Section 6.5. The results of the numerical simulation give detailed information about the internal physical phenomena. Firstly, they show that the temperature distribution inside the battery is significant and the assumption of a bulk temperature, which is commonly used in VRLA battery simulations, is not quite proper. The temperature distribution depends on the boundary conditions and can be worse in different operational conditions. However, we can obtain this temperature gradient using the proposed model. Secondly, the results show that the total reversible heat of the reactions at the positive and negative electrodes cancel each other out, and hence the net reversible heat generation is zero inside the battery. Therefore we can conclude that in contrast to the first paradigm, we should not consider the heat generation due to the reactions at only one electrode. If we add up the heat sink at the other electrode, then the total amount of heat generation would be zero. Finally, the results show that besides the Joule heating, the irreversible part of heat generation due to the chemical and electrochemical reactions contributes to temperature rise. Hence, in contrast to the second paradigm, the Joule heating is not the only responsible mechanism for TRA.

6.11 Summary In this chapter, we introduced lead–acid batteries in more detail. Different aspects of batteries are discussed. Active materials, electrochemistry, main and side reactions, and other issues related to this type of batteries were explained. Finally, applications of lead–acid batteries were introduced. The governing equation of lead–acid batteries are obtained out of the general governing equations of batteries. The properties are also presented

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via tables, equations, and relations. This information was used to simulate lead–acid batteries under different operational conditions. Finally, different cells were simulated and studied using numerical simulation of lead–acid batteries governing equations derived in this chapter. The first study showed the capability of numerical methods in predicting the behavior of a lead–acid battery under discharge condition. Since it that case, side reactions were not important, a simplified one-dimensional system of equations was simulated using the Keller–Box method. After that, the problem of charging in which side reactions are important was considered. Again the problem was simulated using a one-dimensional model. The second problem was solved using the finite volume method instead of the Keller–Box method. It is quite evident that the problem could be solved also by using the Keller–Box method. Selection of the solution method is a choice of interest. The problem of electrolyte stratification was considered, in which electrolyte movement was important. Therefore the problem could not be solved in a one-dimensional space; thus a two-dimensional model was considered for simulation. Again finite volume method was chosen, and SIMPLE method introduced by Patankar was used for simulation of flow field. The last sample was chosen for studying thermal behavior of lead– acid battery in overcharge process. In this type of batteries, side reactions play an important role in estimation of battery behavior. It was shown that without simulation of side reactions, we cannot have a good estimation of cell voltage and other physical phenomena.

6.12 Problems 1. What are the main advantages of lead–acid batteries? 2. Although lead–acid batteries do not have the highest energy density and specific energy among other battery technologies, they still are the most manufactured types. Investigate and compare the total number of produced lead–acid batteries with other technologies. 3. Investigate difference between gelled and AGM batteries. 4. In the case of constant properties in one-dimensional model, simplify the governing equations (6.49)–(6.54). 5. Between different heat dissipation mechanisms, which one is not considerable and can be neglected in lead–acid batteries? Why? 6. Calculate the efficiency of lead–acid cells under standard conditions. 7. Plot the diffusion coefficient of sulfuric acid versus temperature in the range 25–55◦ C.

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8. Plot the diffusion coefficient of dissolved oxygen in electrolyte versus temperature in the range 25–55◦ C. 9. The finite element method is very popular in numerical simulations. Try solving Cell–II using FEM. 10. In case of one-dimensional simulation including side reactions, obtain an appropriate system of equations suitable for the Keller–Box method. 11. Regarding Fig. 6.12B, if the battery is put under rest after t = 105 s until the electrolyte become uniform all over the cell: (a) Give an estimation for electrolyte concentration. (b) Using the Bode relation, give an estimation for battery voltage. 12. Studying Fig. 6.12, how can we check if the proper boundary conditions are satisfied? 13. Comparing Figs. 6.14C and 6.14E, how can you explain the difference between the shape of oxygen gradients in gas phase, which is very flat, and in liquid phase, which has a stepwise shape? 14. Explain why in Fig. 6.33 the upper part of the cell is hotter that the lower parts. 15. Estimate an average heat dissipation for Figs. 6.33A–D. Compare your results with numerical values shown in this chapter.

CHAPTER 7

Zinc–silver oxide batteries Contents 7.1. Zinc–silver oxide battery components 7.1.1 Zinc electrode 7.1.2 Silver oxide electrode 7.1.3 Separator 7.1.4 Electrolyte 7.2. Types and applications 7.2.1 Button cells 7.2.2 High-power primary prismatic cells 7.2.3 High-power secondary prismatic cells 7.3. Electrochemical reactions 7.4. Governing equations of zinc–silver oxide batteries 7.5. Mathematical model 7.5.1 Conservation of electrical charge in solid and electrolyte phases 7.5.2 Conservation of species 7.5.3 Balance of energy 7.5.4 Conservation of mass 7.5.5 Electroactive area 7.5.6 Reaction rate 7.6. Heat sources and sinks 7.6.1 Heat of electrochemical reactions 7.6.2 Joule heating 7.6.3 Heat dissipation or sinks 7.7. Physico-chemical characteristics 7.8. Numerical simulation of zinc–silveroxide batteries 7.8.1 One-dimensional simulation 7.8.2 Two-dimensional isothermal model 7.8.3 Simulation of thermal behavior of zinc–silver oxide batteries 7.8.4 Two-dimensional simulation with water cycle 7.9. Summary 7.10. Problems

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Zinc–silver oxide batteries (also known as silver–zinc batteries) are very mature and primarily developed for space and aircraft applications. In these applications, the specific energy and energy density are more important than cost. The first Soviet Sputnik satellites, as well as US Saturn launch vehicles and the Apollo Lunar Module, were powered by rechargeable silver–zinc batteries. Since the positive plates are made of silver mono and Simulation of Battery Systems https://doi.org/10.1016/B978-0-12-816212-5.00011-8

Copyright © 2020 Elsevier Inc. All rights reserved.

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divalent oxides, they become very expensive comparing to other technologies. The good news is that the silver can be entirely recycled and reused again and again. The technology offers both primary (nonrechargeable) and secondary (rechargeable) versions; however, silver–zinc batteries are most known for their huge application in primary button cells. In primary button cells the amount of active materials is minimal and not very significant regarding the cost of the cell. Silver-oxide primary batteries account for over 20% of all primary battery sales, which shows the importance of this technology over the other types. Another advantage of silver–zinc batteries is their long life, which is important in many applications such as watches. A tiny button cell will keep a watch running 24 hours per day for 3 to 5 years. As mentioned, the main application of silver–zinc oxide batteries are button cells for small applications and military or space applications with large modules. However, new experiments show that silver–zinc technology may provide up to 40% more run time than lithium-ion batteries. Also, another advantage of silver–zinc batteries over lithium ones is that in silver– zinc batteries the electrolyte is water-based, which makes them safer and also more tolerant of thermal runaway and flammability problems. International Electrotechnical Commission (IEC) has designated letter S to indicate silver–zinc batteries in which zinc acts as the negative electrode and silver oxide acts as the positive counter electrode. The electrolyte of such batteries is a binary solution of an alkali metal hydroxide, usually KOH or in fewer applications, NaOH. The nominal voltage of coin type S-batteries is about 1.5 V with maximum open-circuit voltage at about 1.65 V. The open-circuit voltage of silver–zinc batteries may reach 1.95 V depending on the composition of silver electrode since the open circuit voltage of divalent silver oxide is about 1.95 V, whereas the open circuit of monovalent type is about 1.65 V. Silver–zinc technology has many advantages over present commercial technologies. The main characteristics of silver–zinc batteries are: • The technology has very high energy density and specific energy. They had the highest energy density before lithium technologies. • They also have very high power density and specific power, which made them the best choice in applications where high power peak is required. For example, in the lunar command module the main source of energy was a hydrogen/oxygen fuel cell due to its higher energy content. However, fuel cells have problems with delivering high peak power;

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Table 7.1 Main characteristics of zinc–silver oxide batteries [69]. Parameter Comment

Specific energy Approximate energy density Specific power Discharge efficiency Energy/consumer-price Self-discharge rate Cycle durability Nominal cell voltage Cutoff voltage Temperature range

approximately 132 Wh kg−1 approximately 505 Wh L−1 high, but widely variable 40–90% (highly dependent on load) 0.1 Wh $−1 0.7%/month (average) about one hundred for secondary types 1.55 V typical (1.6 V noload, fresh) 1.2 V 0 to +60◦ C typical (storage to −40◦ C)

therefore the command module was equipped with silver–zinc batteries to overcome the problem. • One of the most exciting characteristics of silver–zinc batteries is their flat voltage curve during constant current discharge. Other battery technologies usually show decay in voltage during constant current discharge. However, the voltage of these batteries remains constant until all the active material is finished and then the voltage suddenly drops. • The technology has a low self-discharge, meaning that it can be put in a shelf for a long time. Their shelf life is claimed to be better than zinc–air batteries. • Usually silver–zinc batteries have better performance at low temperature than zinc–air and lithium–based batteries. The main characteristics of batteries are summarized in Table 7.1.

7.1 Zinc–silver oxide battery components In preparation of a zinc silver–oxide cell, many different materials are used. Different cell types have different requirements. Here we briefly discuss typical materials.

7.1.1 Zinc electrode The negative electrode is made of pure zinc. It is well known that zinc shows high half-cell potential, low polarization, and high limiting current density [70]. Zinc is unstable in aqueous alkali and reduces water into hydrogen. Therefore the electrodes are made of alloys with cadmium, aluminum, lead, or sometimes with mercury.

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7.1.2 Silver oxide electrode The positive electrode of the cells are made of silver oxide. There are three oxidation states namely monovalent Ag2 O, divalent AgO, and trivalent Ag2 O3 . Among these, trivalent is very unstable and cannot be used in batteries. Monovalent oxide is the most stable one and is used in button cells. However, the divalent form contains more energy than monovalent version. Silver oxides are not good conductors; hence they show high polarization, which is not acceptable for a battery. To increase its conductance, the oxide is mixed with graphite powder. The mixture increases the electrical conductance but decreases the energy content. Therefore a proper percentage should be obtained for each application.

7.1.3 Separator Separators are required for preventing internal short–cuts. Since the cell is filled with an alkali solution, the separators should meet the following characteristics [70]: 1. Permeable to water and hydroxyl ions. 2. Stable in strong alkaline solutions. 3. Not oxidized by the solid silver oxide or dissolved silver ions. 4. Retards the migration of dissolved silver ions to the anode. There are many different materials for making separators such as cellophane, polyvinyl alcohol (PVA), or polyethylene. Separators are typically made of fibrous woven or nonwoven polymers. The fibrous nature gives them strength and durability.

7.1.4 Electrolyte The electrolyte is made of potassium hydroxide KOH or sodium hydroxide NaOH. Potassium hydroxide has higher electrical conductivity. Hence it is a better choice than sodium hydroxide. The conductivity of alkaline electrolytes depends on both its concentration and temperature. Therefore special attention should be paid on selecting a proper concentration. Normally, the electrolyte concentration is selected between 20 to 40 percent on a weight basis.

7.2 Types and applications Since zinc–silver oxide batteries are expensive comparing to other highenergy content batteries like lithium–based technologies, their applications

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become limited. However, the number of produced cells are not limited, but they are made vastly in different companies. As discussed before, they are manufactured in two major types; namely button cells and prismatic high–power types. Secondary types are also available and are becoming more and more popular in some applications.

7.2.1 Button cells Button cells are the most common and well-known type that are produced in huge number. Almost all sizes are available in the market for different purposes. Comparing to the other technologies, silver–zinc types provide more energy in the same size. Hence they are attractive for any application. Since button cells require a limited amount of silver, the cost is not very significant. Thus the manufacturers have no financial concerns about the cost. Other characteristics of technology have helped it to penetrate the market. For example, the technology has a low self-discharge, which means that its shelf life is very long and also has a long service life. This property is attractive for the end users. Almost all button cells are made of monovalent silver oxide or Ag2 O and can provide from 5 to 25 mAh [70]. Since they are made of monovalent silver oxide, their open-circuit voltage is about 1.55 V.

7.2.2 High-power primary prismatic cells The ability to provide very high-power and energy make them very attractive for space applications. The power and energy content are not the only characteristics of the technology. Zinc–silver oxide batteries provide a very flat discharge curve, which is very important for electronic devices. Moreover, the electrolyte of the cell is aqueous that is very tolerant of thermal runaway and flame. Therefore these batteries are safer than other highpower technologies such as lithium-based batteries. These characteristics make primary zinc–silver oxide batteries the first choice for torpedoes and other military uses.

7.2.3 High-power secondary prismatic cells Zinc–silver oxide secondary batteries exhibit the highest specific energy and energy density comparing any other commercially available aqueous secondary batteries. They have a low internal resistance, which makes them attractive for high-power applications where the current rate may reach 20 times the nominal capacity or 20C rate. For this reason, in applications

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Figure 7.1 Electron and ion movement during the discharging process.

where performance outrages the cost, secondary zinc–silver oxide batteries are superior to other technologies. Airplanes, submarines, space crafts are examples of such markets. The only drawback of rechargeable form is that cyclic life is not high enough. Normally secondary batteries can be recharged between 10 up to 250 times. Fortunately, the battery can be recycled and reproduced again and again.

7.3 Electrochemical reactions To understand and analyze the function of a battery, knowing and assessing the reactions occurring inside the cell is the first step. In a zinc–silver oxide battery, during discharge, electrons are produced at the negative electrode according to the following reaction: Discharge

− −−  Zn + 2 OH−  −− −− −− −− − − ZnO + H2 O + 2 e .

(7.1)

Charge

As a result of these reactions, ions move from the positive electrode toward the negative electrode through the membrane and the produced electrons by crossing the external circuit (see Fig. 7.1) to generate power. The concentration of ions in the electrolyte depends on their mobility and induced electrical field inside the electrolyte. On the positive electrode,

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electrons are consumed according to the following electrochemical reactions: Discharge

− −−  2 AgO + H2 O + 2 e−  −− −− −− −− − − Ag2 O + 2 OH ,

(7.2)

Charge

Discharge

− −  Ag2 O + H2 O + 2 e− −  −− −− −− −− − − 2 Ag + 2 OH .

(7.3)

Charge

These are the main reactions of a zinc–silver oxide battery. The overall cell reaction then is Discharge

−  Zn + AgO −  −− −− −− −− − − Ag + ZnO.

(7.4)

Charge

The main reactions are accompanied by several side reactions where the most famous ones are Discharge

2− −−  Zn + 4 OH−  −− −− −− −− − − Zn(OH)4 ,

(7.5)

Charge

Discharge

− −  Zn(OH)42− −  −− −− −− −− − − ZnO + 2 OH + H2 O,

(7.6)

Charge

Discharge

− −−  2 OH−  −− −− −− −− − − 12 O2 + H2 O + 2 e

(7.7)

Charge

at the negative electrode and 1 2 O2

Discharge

− −−  + H2 O + 2 e−  −− −− −− −− − − 2 OH

(7.8)

Charge

at the positive electrode. Reactions (7.5) and (7.6) describe the zincate cycle that is responsible for electrode shape change and is very important when cyclic life is investigated. Reactions (7.7) and (7.8) constitute the internal water cycle during which water is dissociated at the positive electrode and recombines at the negative counter electrode. This reaction is dominant in charge process for secondary zinc–silver oxide configurations when the cell is nearly charged.

7.4 Governing equations of zinc–silver oxide batteries Ever since Wang et al. [29] and Gu et al. [35] proposed their single-domain model, many research studies have applied to various types of batteries. The

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principal privilege of their presented model is its capability to simulate the behavior of any type of battery systems. In other words, since every type of battery has its own specific governing equations, properties, and boundary conditions, the model has been built in such a flexible format to adapt itself to any variations regarding its input data and information. Regarding zinc–silver oxide batteries, the most modelings are either based on the experimental tests [71–74] or similar to Dirkse’s detailed survey of the electrochemical behavior of silver electrode [75]; they only include analyzing the behavior of electrodes and not the of entire cell. As a result, there are only a few zonal or single–domain approach models available for these kinds of batteries. Glancing at previous works, the first modeling based on governing equations, which was valid only on a single positive electrode, was presented by Blanton et al. [76]. Later, Venkatraman and Zee [77] presented a mathematical model during high rates of discharge based on porous electrode theory [78,46] by using Blanton’s isothermal model. [76]. But most importantly, a set of governing equations with a single-domain approach, as the first and only complete model has been presented so far, was introduced by Torabi and Aliakbar [43]. They used a reduced one-dimensional isothermal model for validation, and to ensure that thermal effects had been neglected correctly, they carried on an experimental test with a low current density trying to keep the temperature of the cell constant. Interestingly enough, in all the open literature, modelings are only valid for discharge in isothermal conditions. Hence the literature lacks the following items: 1. The present models are only valid for discharge, and no model is capable of simulating the charging process. 2. Side reactions are not considered in any previously presented models. Although side reactions are important in the charging process, they were not considered before. 3. All the previously presented models are isothermal. In all the previously mentioned papers, the temperature is entirely assumed to be constant because the simulation is done under room temperature. Consequently, the energy equation is not taken into account, and thermal effects are not considered. Therefore the results of their modeling may be way far from reality, especially in high discharge currents. The isothermal model may be great for button cell types that are used to operate under very low current, but it fails to give good simulation when working with high-power cells.

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Figure 7.2 A typical zinc–silver oxide cell.

The above literature survey shows that despite all the research mentioned, in general, there is no evidence of an accurate model for zinc–silver oxide batteries. In this chapter, we present a general system of governing equations of zinc–silver oxide batteries. The system is coupled with an energy equation to give a complete set of equations that can predict the thermal behavior of the cell as well. The advantages of the present model are as follows: 1. The charge process is also modeled, and the effect of porosity variation is added. 2. From different side reactions discussed before, the water cycle that is important in charging is modeled. It is worth mentioning that the zincate cycle is not considered yet, and at the time it is a topic of research. 3. An energy equation is added to the governing equations, meaning that the model is capable of predicting the thermal behavior of the system. Hence, the present model has many advantages over the other models in the open literature. In this chapter, we discuss the model and present simulation results.

7.5 Mathematical model A cross-section of typical zinc–silver oxide battery cell is shown in Fig. 7.2, where porous electrodes and separator are filled by a binary alkaline electrolyte, usually KOH or NaHO. The main and side reactions are defined by Eqs. (7.1)–(7.8). Here we neglect the zincate cycle side reactions since the cyclic life is not the concern of the present book. Hence the governing equations of zinc–silver oxide batteries are:

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• At the positive electrode: Discharge

− −−  Ag2 O + H2 O + 2 e−  −− −− −− −− − − 2 Ag + 2 OH ,

(7.9)

Charge

Discharge

− −  2 AgO + H2 O + 2 e− −  −− −− −− −− − − Ag2 O + 2 OH ,

(7.10)

Charge

1 2 O2

Discharge

− −−  + H2 O + 2 e−  −− −− −− −− − − 2 OH .

(7.11)

Charge

• At the negative electrode: Discharge

− −−  Zn + 2 OH−  −− −− −− −− − − ZnO + H2 O + 2 e ,

(7.12)

Charge

Discharge

− −−  2 OH−  −− −− −− −− − − 12 O2 + H2 O + 2 e .

(7.13)

Charge

As always in general, we can cast all the reactions into a general formulation as 

zj

νj Mj = nj e− ,

(7.14)

j

where νj is the stoichiometric coefficient, Mj is a general chemical formula for each species, zj is the charge number, and nj is the number of transferred electrons. According to our discussions, to attain and characterize the cell behavior, the following transport equations must be simultaneously solved: 1. Conservation of electrical charge in the solid phase to obtain the solid phase potential φs . 2. Conservation of electrical charge in the electrolyte phase to obtain the electrolyte phase potential φe . 3. Conservation of species to obtain active material concentration. Active materials such as zinc or silver oxides can be calculated using their stoichiometric relations. Hence any separate transport equation is unnecessary, and in the present study the only active material involved in the equations is the concentration of electrolyte c OH . 4. Conservation of species to obtain oxygen concentration in gas and electrolyte phases. 5. The balance of energy to obtain the thermal behavior of the cell. Therefore, for simulation, we have to obtain a system of governing equations for zinc–silver oxide batteries.

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7.5.1 Conservation of electrical charge in solid and electrolyte phases Conservation of electrical charge φs in the solid phase in its volumeaveraged form suitable for simulation is ∇.(σ eff ∇φs ) − jOH = 0.

(7.15)

This equation represents the Ohm’s law in solid media, where jOH is the kinetic rate of production (or consumption) of the electron governed by the Butler–Volmer relation (discussed later). In these equations, since we are dealing with a porous medium, effective properties must be used. As it is well established, in any porous media the effective values are obtained by modifying the normal values using the Bruggeman relation [29]. For instance, the effective solid conductivity σ eff is corrected as σ eff =



σk m1.5 ,

(7.16)

k

where m is the mass fraction of each species in the solid phase (and not the porosity). The potential distribution φe in the electrolyte phase is governed by modified Ohm’s law, in which the effect of concentration is also included. Considering this effect, the following transport equation governs the electrolyte phase potential: ∇.(κ eff ∇φe ) + ∇.(κDeff ∇ lnc OH ) + jOH = 0.

(7.17)

In the same manner as discussed in the solid-phase case, the effective electrolyte conductivity κ eff is calculated using the equation [76] κ eff =

εF 2

RT

(DK+ + DOH− )c OH ,

(7.18)

where ε is the porosity of the medium, and c OH is the electrolyte concentration. The electrolyte concentration is one of the key parameters, the distribution of which should be determined by solving the conservation of species as is discussed in the next section; DK+ and DOH− are the diffusivities of potassium and hydroxide ions, respectively. Comparing Eqs. (7.15) and (7.17), the source terms jOH in both equations are the same with different signs. It means that all the current that exits the solid electrode will enter the electrolyte and vice versa.

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It is clear that electrolyte phase potential is continuous through the cell, but the solid phase potential is only valid inside solid electrodes. Therefore, even though the solid phase potential is not valid in the separator region, it is common to solve Eq. (7.15) in the separator region as well. From the physical point of view, we know that its values are meaningless, but the reason we do in this way is that the single-domain formulation is not violated. In other words, we do not need to solve the solid phase potential by another specified algorithm. It is clear that the solid phase potential in the separator does not contribute to any physical property, and therefore its calculation does not generate any error.

7.5.2 Conservation of species The micromacroscopic model for conservation of species for electrolyte concentration of OH ions is expressed by the equation  ◦  ◦  t− + 1 t ∂(ε c OH ) OH OH = ∇ · (Deff ∇ c ) − +∇ − jOH . ∂t F zF

(7.19)

Since the electrolyte is confined to porous media (porous electrodes and separator), the effective diffusion coefficient appeared in Eq. (7.19) is related to its bulk value with Bruggeman [29] relation OH = ε 1.5 DOH , Deff

(7.20)

where DOH is the diffusion coefficient of hydroxide ions in the electrolyte. According to Eqs. (7.9)–(7.12), the number of transferred electrons is z = 2. Finally, t−◦ is the transference number of hydroxide ions. Although t−◦ depends on electrolyte concentration and temperature, due to the lack of data, it is usually assumed to be constant. Consequently, its gradient or the last term in Eq. (7.19) becomes zero. It is worth mentioning that since practical zinc–silver oxide batteries are made of compact sandwiched cells, where there is no free electrolyte between the electrodes, the convective terms are not considered in Eq. (7.19). It is because the porosity of the medium almost renders the electrolyte movement causing a negligible velocity field. In other words, mass transfer due to convection is not significant in the field leaving diffusion and migration as the major dominant mass transfer mechanisms. The source term jOH of Eq. (7.19) is the sum of all the reactions that contribute to generation or consumption of hydroxide ions, namely reactions (7.9)–(7.11) at the positive electrode and reactions (7.12) and (7.13) at the negative electrode.

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229

A similar expression is valid for the concentration of oxygen in the liquid phase. Eq. (7.21) defines the relation ∂(ε c O2 ) 1 O2 O = ∇ · (Deff2 ∇ c O2 ) − j . ∂t 4F

(7.21)

Again the diffusion coefficient of dissolved oxygen should be corrected to account for the porosity of the region using the Brugmann relation [29] O

Deff2 = ε1.5 DO2 .

(7.22)

Also, the source term jO2 of Eq. (7.21) comes from the reactions that contribute to generation or consumption of oxygen, namely reaction (7.11) at the positive electrode and reactions (7.12) and (7.13) at the negative electrode.

7.5.3 Balance of energy Thermal analysis of batteries requires the energy equation to be taken into account in addition to the equations mentioned. Since all of the introduced governing equations are strongly coupled to the temperature change due to the Butler–Volmer equation and physico-chemical properties, it is obvious that the isothermal condition is rarely the case, especially in high-power applications. Consequently, in high-power applications, the assumption of constant temperature during discharge cannot be that reasonable. However, for low-power applications, such as button cells used in watches, toys control systems, and other similar systems, the constant temperature assumption is fairly the case. If the thermal effect is to be considered, all thermal-electrochemical governing equations including energy balance equation should be solved simultaneously to achieve fulfilling results that can model the whole essential parameters of a battery cell under real operating conditions and various initial inputs. Besides the temperature distribution, which is the objective of the solution of the energy equation, all physical and chemical parameters, such as thermal and electrical conductivity, diffusion coefficient, density, and so on, should be modified according to the local temperature. The energy equation in the general form is represented by the equation ρ cp (

∂T + υ × ∇ T ) = ∇ × λ∇ T + q, ∂t

(7.23)

where the first term on the left–hand side of the equation demonstrates the transient form of heat. In the transient term, T and t represent the tem-

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perature and time, respectively, whereas the density ρ and specific heat cp vary for different battery parts such as electrodes, electrolyte, and separator. Even for a single electrode consisting of Ag, AgO, and Ag2 O with different weight percentages, the above parameters differ because each species has its density and specific heat. Therefore, for each specific part of the electrode or battery, an averaged property should be obtained by applying the following equation, where s and e show the solid and electrolyte phases: (ρ cp )avg = cp,e ρe ε + (1 − ε)



cp,s ρs ,

(7.24)

s

where the sum over s is carried on all different available solid species. The second term in Eq. (7.23) indicates heat transfer caused by the displacement of electrolyte. Indeed, the velocity item υ is related to convection heat transfer and is in association with electrolyte displacement. Since in zinc–silver oxide batteries the electrolyte is confined either in the electrodes or in the separator, the velocity of electrolyte is usually small, and we can neglect its movement. Consequently, the second term on the left–hand side of Eq. (7.23) is negligible. The first term on the right–hand side of Eq. (7.23) accounts for conduction heat transfer or Fourier heat transfer inside different parts of the cell. Again, the heat conductivity λ should be accounted for all different species and should be averaged by means of the equation λavg = λe [εωe ]1.5 +



λs [(1 − ε)ωs ]1.5 .

(7.25)

s

Finally, the last and most important item of the energy equation is dedicated to net heat generation or q inside the battery cell, which is considered to be internal volumetric net heat production in a zinc–silver oxide battery. This term requires special attention and is discussed in more detail in the following sections. By applying the approximation and operational assumption of zinc– silver oxide batteries the energy balance equation can be rewritten as a modified form of the following equation, where all battery parameters are clearly initialized and are available: ρ cp

∂T = ∇ × λ∇ T + q. ∂t

(7.26)

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231

7.5.4 Conservation of mass Since silver and its different oxides have different densities, the porosity of the positive electrode changes according to electrochemical reactions. A mass balance [29] can be used to determine the porosity change of the silver electrode: ∂εs 1 =− ∂t 2F −

1 2F

 

MW Ag2 O ρ Ag2 O



2MW AgO ρ AgO



2MW Ag ρ Ag

MW Ag2



aAg in,1

O

ρ Ag2 O

aAg in,2 ,

(7.27)

where MW denotes the molecular weight, and a is the electroactive surface of the positive electrode. The first term on the right-hand side of this equation indicates the volumetric change of the electrode due to the conversion of Ag2 O to Ag according to reaction (7.9), whereas the second term indicates the volumetric change due to reaction (7.10). A similar equation can be obtained for porosity change of the zinc electrode: ∂εs 1 = ∂t 2F



MW Zn ρ Zn



MW ZnO ρ ZnO



aZn in,4 .

(7.28)

Eq. (7.28) shows the volume change due to reaction (7.12).

7.5.5 Electroactive area The electroactive area is a very effective parameter in battery simulation. It should be noted that the active area differs in charge and discharge since charge and discharge have opposite electrochemical directions; hence different materials act as the active material. For both electrodes, the active area is a function of the state of charge or SoC. When a cell is at its fully charged condition or SoC = 1, the active areas for both electrodes are at their maximums for discharge and are zero for charge. But at SoC = 0, the active area is at its maximum for charge and is zero for discharge. Mathematically, the active area is obtained from SoC according to the following relations: A = Amax SoC ξ

(7.29)

A = Amax (1 − SoC ξ )

(7.30)

for discharging and

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for charging process. In these equations, Amax is the maximum available active area in a fully charged state, and ξ is an exponent that shows the dependency of active material to tortuosity of the electrodes.

7.5.6 Reaction rate The rate of generation of species at electrode/electrolyte interface (indicated by subscript se) can be determined as rse = −

  νj j

nj F



inj ,

(7.31)

where the rate of generation or consumption is determined by the Butler– Volmer equation. For zinc–silver oxide batteries, the Butler–Volmer equation for reaction j can be written as [77] 

αaj nj F Ci pi Mk pk ) k ( ) exp( (η)) Ci,ref Mk,ref RT  −αcj nj F Ci qi Mk qk ) k ( ) exp( (η)) , − i ( Ci,ref Mk,ref RT

inj = i◦j i (

(7.32)

where i◦j is the exchange current density, C and Cref are the concentrations of involved specie at electrode/electrolyte interface and reference state, respectively, and the involved species are hydroxide ion OH− for main reactions and oxygen O2 for side reactions. Also, Mk is the molar density of species k. Other parameters appeared in this equation are the Faraday constant F, the universal gas constant R, and the cell temperature T. The anodic and cathodic charge transfer coefficients αaj and αcj are usually obtained from experimental tests with the following condition: αaj + αcj = 1.

(7.33)

Finally, the overpotential η is defined as η = φs − φe − Uj,ref ,

(7.34)

where φs and φe are the potentials at the solid electrode and electrolyte, respectively, and Uj,ref is the equilibrium potential of each reaction. It should be noted that since all reactions occur at the electrode/electrolyte interface, all parameters should be considered at the same interface. The total current density jOH is the sum of all reactions at any electrode/electrolyte interface that contributes to production or consumption

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233

of hydroxide ions. Therefore, at the positive electrode/electrolyte interface, it becomes [43] jOH = aAg (in1 + in2 + in3 ),

(7.35)

where in1 to in3 are the currents produced/consumed by reactions (7.9)– (7.11), respectively, and aAg is the electroactive surface area of the positive electrode. Similarly, at the negative electrode/electrolyte interface, jOH can be found from the relation jOH = aZn (in4 + in5 ),

(7.36)

where in4 and in5 are the currents produced/consumed by reactions (7.12) and (7.13), respectively, and aZn is the electroactive surface area of the negative electrode. The current density jO2 for oxygen reactions is the current responsible for production or consumption of oxygen. Hence it is jO2 = aAg in3

(7.37)

jO2 = aZn in5

(7.38)

for the positive electrode and

for the negative counter electrode.

7.6 Heat sources and sinks According to Eq. (7.26), the net heat source term is the only unknown part. To evaluate this item, we may need to have a comprehensive outlook of all thermal-electrochemical reactions and mechanisms inside the battery cell. Moreover, due to the complexity of assessing the heat source term in a zinc–silver oxide battery, a clear overview of heat generation and dissipation mechanisms inside the battery cell is highly in demand. To obtain an appropriate formulation for the net heat generation, cognition of the nature of mechanisms is of crucial importance. Internal heat generation is inevitable due to the nature of a battery cell. In other words, there are some mechanisms inside a battery cell leading to producing a considerable amount of heat called heat sources. On the other hand, heat sinks also play a vital role in dissipating the produced heat to the ambient resulting in a lower temperature. Also, heat dissipation mechanisms are highly in

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demand owing to their roles in maintaining the internal thermal balance of the cell. In the present section, we first explain the heat generation due to electrochemical reactions. Then we discuss the Joule heating and finally detail heat dissipation mechanisms. These mechanisms are responsible in temperature rise or thermal runaway in zinc–silver oxide batteries.

7.6.1 Heat of electrochemical reactions One of the main factors contributing to producing heat during discharge is assumed to be the released heat throughout electrochemical reactions. Indeed, the exothermic nature of reactions in zinc–silver oxide battery during battery operation can be considered as an internal heat source. According to Eqs. (7.9)–(7.13), main and side reactions are present at each electrode. These reactions produce and consume energy during charge and discharge. Regarding these reactions, at the positive electrode, Eqs. (7.9)–(7.11), and at the negative electrode, only Eqs. (7.12) and (7.13) are responsible for heat generation [43]. Based on the research carried on by Torabi and Esfahanian [31,32], the amount of generated heat throughout electrochemical reactions can mainly be divided into reversible and irreversible parts. The reversible part is the one that can be either positive or negative regarding the current direction during charge or discharge. However, irreversible heat of reactions is the part of electrochemical reactions and is always positive regardless of the direction of reactions; therefore, it always causes an increase in the cell temperature. Due to different nature of these parts, their value can be calculated by applying various formulas. The reversible part of heat generation for reaction i can be obtained from the following equation, where i is the Peltier coefficient for reaction i and can be calculated by applying Eq. (7.40) for each reaction [32]: qi,rev = ji i .

(7.39)

In Eq. (7.39), ji is the total current of generation and consumption of each species. For hydroxide ions and oxygen, ji can be obtained from Eqs. (7.35)–(7.38) at the positive and negative electrodes. The Peltier coefficient for reaction i is also defined as i =

T si . ni F

(7.40)

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235

In Eq. (7.40), T, n, and F represent the temperature, the number of transferred electrons in each reaction, and Faraday’s constant, respectively. Also, changes in the amount of entropy of electrochemical reaction i can be evaluated by the method introduced in [31,32] for dealing with the entropy of an electron. According to the electrochemical reaction i during charge or discharge, the sign of si similarly changes. Unlike the reversible part, the irreversible part of the heat of electrochemical reactions experiences changes in the sign neither during charge nor discharge and always remains positive and leads the battery to thermal rise. This heat can be calculated by applying the equation qi,rev = ji ηi .

(7.41)

In this equation, ηi is called the overpotential and illustrates how much the battery cell is away from its open-circuit conditions and can be obtained from the following equation, where the term Ui,ref represents the potential of reaction i in the equilibrium: η = φs − φe − Ui,ref .

(7.42)

7.6.2 Joule heating The Joule heating is the inevitable part of any electrical system. It can be defined as the produced heat when current flow passes through a nonconductive material. The amount of Joule heating can be obtained by the equation qJoule =|



ik .∇φk | .

(7.43)

k

Regarding the Joule heating, the amount of the produced heat is always positive; therefore the Joule heating is always responsible for increasing the temperature of the battery cell. According to Eq. (7.43), this heat source is in a close relationship with the changes in the gradients of potentials in solid and electrolyte phases. Furthermore, the higher the electrical resistance is, the more Joule heating is generated. As a result, the amount of Joule heating produced in each part of the cell is a strong function of the existence of nonconductive materials. Moreover, there is a striking resemblance between the nature of Joule heating and irreversible part of the heat of electrochemical reactions because they both are positive and cause an increase in the temperature of the cell.

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Simulation of Battery Systems

Figure 7.3 Illustration of thermal resistances in the modeled battery.

7.6.3 Heat dissipation or sinks Zinc–silver oxide batteries are normally used in moderate temperature around 30◦ C. However, in high-power applications, its temperature may rise to about 100◦ C, which is still not too much for considering radiation heat transfer. Moreover, this battery technology is always a sealed battery in which no mass flow exists out of the battery shell. These assumptions show that the equivalent circuit model for heat dissipation of Fig. 7.3 is quite perfect for zinc–silver oxide battery simulation.

7.7 Physico-chemical characteristics A zinc–silver oxide battery is a multicomponent medium like any other technology. During charge and discharge, the active materials convert to other materials with different physical properties. Consequently, for an accurate modeling or simulation, we have to obtain accurate properties. As stated before, each property should be obtained according to the mole fraction of involved elements. The formula for obtaining physical properties is  eff =



k m1.5 ,

(7.44)

k

where k is the desired property of the kth component in the mixture, and m is its mole fraction. Table 7.2 presents the properties of different chemical components that participate in zinc–silver oxide battery reactions.

Zinc–silver oxide batteries

Table 7.2 Properties [77]. Material ρ Cp g cm−3 J g−1 K−1 Ag 10.49 0.234 AgO 7.48 0.343 Ag2 O 7.14 0.343 Zn 7.13 0.389 ZnO 5.61 2.910 Separator 1 1.340 Electrolyte 2.12 2.520 H2 O (l) 1 4.184 OH− − −

λ

σ

237

s◦

W cm−1 K−1

−1 cm−1

J mol−1 K−1

4.30 0.207 0.207 1.20 0.207 0.0024 0.002 0.0058

6.28 × 105 8.33 × 102 1 × 10−8 1.83 × 105 1 × 10−2 1 × 10−4

42.6 117 121.3 41.6 43.6



− − −

− − 69.9 −10.8

Potassium hydroxide KOH is very common in zinc–silver oxide batteries. The properties of KOH strongly depend on its concentration, some of which are tabulated in Table 7.3.

7.8 Numerical simulation of zinc–silveroxide batteries Like any other battery types, zinc–silver oxide batteries are made of some cells connected in series. In each cell a consecutive set of positive and negative plates with a proper separator in between forms the structure. As a result of this arrangement, each positive (negative) plate, except plates locating at the ends, simultaneously reacts with two neighboring negative (positive) electrodes at both sides. By considering this symmetrical order, as shown in Fig. 7.4A for a zinc–silver oxide battery, we can assume that our model consists of half of a positive plate and half of one of its neighboring negative plates with a full-length separator in between. Since this supposed model replicates in the entire cell, technically, a battery cell can be assumed as an array of several independent cells locating next to each other.

7.8.1 One-dimensional simulation For numerical simulation, verification, and fundamental study, Cell-Ag from Appendix B is selected. The cell is operated with low current density where the temperature rise was observed to be about 1◦ C. The temperature rise is so low that the assumption of isothermal model is applicable. Fig. 7.5 shows the variation of simulated voltage compared with experimental test. The figure shows a good agreement between the simulation and experiment even in capturing sudden changes in input current. As can

Table 7.3 Properties of potassium hydroxide [70]. Conductivity Specific % KOH Specific at 18◦ C, heat at gravity 18◦ C, −1 cm−1 at 15.6◦ C cal/g ◦ C 0 1.0000 0.999 5 1.0452 0.17 0.928 10 1.0918 0.31 0.861 15 1.1396 0.42 0.801 20 1.1884 0.5 0.768 25 1.2387 0.545 0.742 30 1.2905 0.505 0.723 35 1.344 0.45 0.707 38 1.3769 0.415 0.699 40 1.3991 0.395 0.694 45 1.4558 0.34 0.678 50 1.5143 0.285 0.66

Freezing point, ◦C

0 −3 −8 −14 −23 −6 −58 −48 −40 −36 −31 +6

Boiling point, ◦ C

Vapor pressure mm Hg

Viscosity vP.

at 760 mm Hg

at 100 mm Hg

at 20◦ C

at 80◦ C

at 20◦ C

at 40◦ C

100 101 102 104 106 109 113 118 122 124 134 145

52 52.5 53 54 56 59 62 66 69 71 80 89

17.5 17 16.1 15.1 13.8 11.9 10.1 8.2 7 6.2 4.5 2.6

355 342 327 306 280 250 215 178 156 140 106 70

1 1.1 1.23 1.4 1.63 1.95 2.42 3.09 3.7 4.16 5.84 8.67

0.66 0.74 0.83 0.95 1.1 1.31 1.61 1.99 2.35 2.59 3.49 4.85

Zinc–silver oxide batteries

239

Figure 7.4 A typical cell sandwich. (A) Description of a cell sandwich. (B) Electron and ion movement.

Figure 7.5 Comparison of present simulation and experimental test.

be seen, the simulation results favorably follow the voltage step of the experimental test. This type of verification is very important from the point that usually the simulation results are used to be verified with a single-step experimental test. For example, a single-step constant current (or voltage) discharge of a battery is carried on by an experimental test, and the numerical code tries to simulate the experiment. However, they may be unable to accurately predict the current interrupt tests such as shown here. Another point in studying Fig. 7.5 is its compatible behavior at the end of the discharging process with the experimental results. After t = 300 s, the voltage profile sharply drops, which is attributed to mass limitation effects. It is known that the active material of either the positive or the negative electrodes is finished; however, it is not known which of them is actually

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Figure 7.6 Variation of species on the zinc electrode.

Figure 7.7 Variation of species on the silver electrode.

responsible. In practice, we have to use a reference electrode to become aware of this issue, but here we can use the numerical results to find out which electrode is the limiter. Figs. 7.6 and 7.7 show the active materials of the negative and positive electrodes, respectively. From Fig. 7.6 we can deduce that at the beginning of the test, the plate consists of pure zinc, and after t = 300 s, almost all the negative active material is converted to ZnO. However, Fig. 7.7 shows that the positive electrode is initially composed of AgO; however, after t = 300 s, only about 50% of the active material is converted to Ag. This fact indicates that in this test, the limiter is the

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241

Figure 7.8 Variation of electrolyte concentration.

negative electrode, or in other words, the sharp decrease in the voltage curve is attributed to the mass limitation of the negative electrode. The variation of electrolyte concentration is shown in Fig. 7.8. Although in zinc–silver oxide batteries, according to Eq. (7.4), the electrolyte is not considered as an active material, but the half-cell reactions, that is, Eqs. (7.1)–(7.3), indicate that the electrolyte actually plays as an active material (note that the side reactions are not simulated here). During the discharging process, the electrolyte is generated at the positive electrode and is consumed at the negative counter electrode with the same rate. Therefore, as it is clear in Fig. 7.8, electrolyte concentration increases in the positive electrode and decreases in the other one. The integral of each graph would be one since the graphs are normalized by initial electrolyte concentration.

7.8.2 Two-dimensional isothermal model Again Cell-Ag from Appendix B is selected for numerical simulation, verification, and the fundamental study. As stated before, the cell is operated with low current density, and the temperature rise was observed to be about 1◦ C, which is an indicator of a good isothermal situation. In the two-dimensional case, where the temperature gradient is not a case (like button cells), the governing equations are (7.15), (7.17), and (7.19). Since in these applications the cell is of primary type, and the rate of discharge is very low, side reactions have a very negligible effect and are dropped out.

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Simulation of Battery Systems

Figure 7.9 Domain of interest.

The obtained system should be numerically solved using a proper scheme. The Keller–Box method is a powerful tool for simulation of battery systems, although the method is not suitable for two-dimensional cases. Therefore another scheme, such as finite volume, finite element, or finite difference, should be selected. Here we use the finite volume method [45, 68] for solving the system of equations. Note that the system of governing equations is coupled and also that they are nonlinear and contain nonlinear source terms. Consequently, the resulting system of equations is highly stiff and very sensitive to initial guess. To be able to solve the system of equations, first, the source terms should be linearized. Then due to the stiffness of the resulting system of algebraic equations, iterative methods with successive underrelaxation method were used. A nonuniform grid was generated as shown in Fig. 7.9 to obtain good accuracy with minimal computational cost. It should be noted that since the cell sandwich is very slender with an aspect ratio of about 40, a uniform grid would result in a large number of computational volumes. Using the mentioned nonuniform grid eliminates the problem, and a sensible mesh can be used to solve the problem with acceptable accuracy. Fig. 7.10 shows the simulation results validated by the experimental test. The accuracy of simulation can be seen from the figure since the model can accurately predict the three successive sudden changes in applied current.

Zinc–silver oxide batteries

243

Figure 7.10 Verification of the present simulation.

As stated in one-dimensional modeling, this type of verification is very important. Although both one- and two-dimensional models are accurate, the prediction of two-dimensional method is seen to be more consistent with the experimental test. The accuracy of the one-dimensional model comes from the fact that the applied current density is very low (of order 0.1 A cm−2 ). This makes a nearly uniform potential distribution over the plates and also causes a slow reaction rate all over their surface. Hence a near-zero gradients of parameters take place across the cell width, and the situation resembles a one-dimensional case. However, the consistency of the twodimensional model is due to the fact that the physical interpretation of the two-dimensional model is more consistent with the real prototype (this fact is fully explained in Chapter 5). Thus the physical properties and involved phenomena are better simulated. To better investigate the difference between one- and two-dimensional simulations, here the same cell is simulated under a higher input current density; that is, the cell is discharged with i = 0.3 A cm−2 . The variation of cell voltage versus time is plotted in Fig. 7.11. In this case the higher current density causes the reaction rates to become nonuniform all over the pales, which in turn results in a formation of a nonuniform electrolyte concentration, potential distribution, reaction rate, and any other parameter as well. Moreover, in that case the temperature gradient will be developed inside the cell, which makes the cell reactions more and more nonuniform. However, for the present investigation, we have neglected the heat generation phenomenon since the model is isothermal. The thermal effect

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Simulation of Battery Systems

Figure 7.11 Comparison between the results of one- and two-dimensional simulations.

Figure 7.12 Electrolyte concentration contours at t = 30 s. (A) One-dimensional model. (B) Two-dimensional model.

itself requires a lot of specific attention and requires a different investigation. The results show that even neglecting the thermal effects, there would be some differences between one- and two-dimensional simulations when the current density increases. Other simulation results are summarized in Figs. 7.12–7.14. Fig. 7.12 shows the electrolyte concentration contours at t = 30 s. As it can be seen, at this time the electrolyte concentration distribution differs in both simulations. For the one-dimensional model, the electrolyte concentration is

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245

Figure 7.13 Electrolyte concentration contours at t = 60 s. (A) One-dimensional model. (B) Two-dimensional model.

only a function of cell thickness, whereas in the two-dimensional model, it is a function of both cell width and cell height. In the figure, it is also clear that the electrolyte is more concentrated near the positive electrolyte lug. This phenomenon takes place because the reaction rates are faster at the lug and slower at the bottom of the cell because a potential drop exists from top to bottom, resulting in lower overpotential and consequently lower reaction rate. However, in practice, it is well known [50,79] that the electrolyte movement due to the natural convection inside the battery results in electrolyte stratification, which leads to accumulation of more concentrated electrolyte at the bottom of the cell. Since we do not model the electrolyte movement in the present study, this effect is not seen in the result. Electrolyte stratification may be the subject of future investigation. Fig. 7.13 shows the same results for t = 60 s. The results show that the electrolyte concentration contours differ in both simulations; however, they are not as different as it was at t = 30 s. In other words, as the time moves on, the electrolyte distribution takes a two-dimensional form, and later on, it becomes uniform again at the end of discharge. This phenomenon happens since, at the end of discharge, the cell potential drops (see Fig. 7.11); hence a lower reaction rate happens. Since the reaction rates become slower, diffusion becomes dominant and makes the electrolyte concentration to become

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Simulation of Battery Systems

Figure 7.14 Electrolyte concentration contours. (A) Electrolyte concentration. (B) Potential in electrolyte.

more uniform. As a result, the cell potential reaches the same value in both simulations. This explanation can be better seen from Fig. 7.14, which shows the electrolyte concentration across the cell width at the middle height of the cell. It can be seen that at t = 30 s the difference is more obvious than at t = 60 s. Moreover, as shown in Fig. 7.14B, the difference between the simulations can be seen in electrolyte potential at the same cross-section. It should be noted that although from the figure it is shown that the electrolyte potential distribution difference deviates more at t = 60 s, as it was explained before, the overall overpotential (see Eq. (7.34)) reaches the same value at that time.

7.8.3 Simulation of thermal behavior of zinc–silver oxide batteries In the previous section, we carried an isothermal simulation on Cell-Ag from Appendix B. We mentioned that two-dimensional simulation is vital in some cases, especially if thermal effects are considered. In this section, we also add the thermal effects to the same two-dimensional simulation. For thermal effect, besides the governing equations that were used for isothermal modeling, we should also add the energy balance (7.26) to the governing equations. Adding the energy equation, all the necessary auxiliary equations that were described in Section 7.4, including source terms and dissipation mechanisms, should be incorporated into the simulation codes.

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247

The first step to model the cell is generating a two-dimensional grid by applying some necessary assumptions. Fig. 7.9 shows such a generated mesh used for the previous isothermal study. The generated mesh is also suitable for thermal simulation because, the generated grid should be capable of covering the important nodes with the great rate of gradients and changes in the number of vital parameters. In a zinc–silver oxide battery cell the nodes located in the edges of each zone have the potentials of rapid changes in the variables. As a result, the nodes near the edges of each zone required to be smaller in size to cover the effective changes in vital areas by more computational grids. It results in a nonuniform mesh, facilitating to observe even tiny gradients at all the zonal interfaces. Again finite volume was chosen as the numerical tool, and since there are different regions (electrodes and separator) with different properties, a multizone or multiblock scheme was used for simulation. With this scheme, it is very easy to deal with the internal zonal boundaries or interfaces. All the interior interfaces are treated as internal boundaries without needing any special treatment. All the boundary conditions should be applied only on the exterior boundaries. The nonlinear source terms of the GEs require linearization, for which we have used Newton linearization. The resulting nonlinear stiff system of equations was solved by iterative techniques using proper underrelaxation parameters. Also, the developed solver was a segregated scheme, in which each equation is solved independently. After modeling the battery system, it is vital to confirm the validity of the results. To do that, the results of the present simulation are compared with those resulted from experimental tests. Unfortunately, the only available experiment data was carried on in isothermal condition. Consequently, to validate our model, the boundary conditions were chosen such that the temperature of the cell remains constant throughout the simulation. Fig. 7.15 illustrates the comparison of the simulated current density with current density obtained from the experimental test. As it is shown in the figure, the simulation results are in a very good agreement with experimental tests. It was mentioned that heat generation inside a battery cell is mainly due to electrochemical reactions and Joule heating. Separating these two parts from the experimental test is very hard; however, one of the benefits of the numerical simulation is that these parts can be easily separated and studied. The first result of the simulation is dedicated to the amount of Joule heating produced in different parts of the cell during modeling. To have a

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Simulation of Battery Systems

Figure 7.15 Validation with the experimental tests.

Figure 7.16 Joule heating.

better understanding, the Joule heating for different regions (i.e., electrodes and separator) are calculated and plotted in Fig. 7.16. We can see from the figure that at the beginning of discharge the amount of Joule heating generated in the cathode is greater than that in other parts due to the existence of nonconductive materials. However, as time goes on, silver–oxide reduces to metallic silver at the cathode, and hence its conductivity increases, which in turn results in decreasing Joule heating at the positive electrode. On the other hand, a very different phenomenon occurs at the negative electrode. At the beginning of discharge, the negative electrode is mainly composed of metallic zinc with a very high electrical conductivity. Therefore, the Joule heating is very low compared with the positive electrode. During the discharge of the cell, the negative electrode converts from metallic zinc to zinc–oxide; consequently, its conductivity decreases, and the Joule heating increases accordingly. Summing up, during the first stage of the discharge, the Joule heating is mainly due to the heat generated at the positive elec-

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Figure 7.17 Heat of reactions. (A) Reversible part. (B) Irreversible part.

trode, and at the final stages of the discharge, the negative electrode is the main source of Joule heating. From the figure it is clear that during the last moments of simulation, significant Joule heating generation in the anode affects the total Joule heating production in the whole cell and results in a dramatic increase in the amount of Joule heating, which may have subsequent effects on the stability of the cell. As mentioned before, another part of heat generation is due to the electrochemical reactions. The generated heat due to the electrochemical reactions can be divided into the reversible and irreversible parts as fully discussed in previous sections. Fig. 7.17 illustrates the reversible and irreversible parts of heat generation due to the electrochemical reactions. To have a better understanding, these parts are separated and plotted in different graphs. From the figure we can deduce that both reversible and irreversible parts have more value at the beginning of the discharge, and as the time goes on, electrochemical heat generation tends to zero. This trend makes sense in the way that at the beginning of discharge, electrochemical reactions are faster and produce more energy, and as the time passes, the active material of the electrode finishes, and hence the electrochemical reaction rates become lower and lower. Moreover, the reversible part of the heat generation is higher than the irreversible part, but at the same time, the irreversible part is not negligible. Not only the irreversible part is negligible, but also it is very high compared to the Joule heating. This fact can be deduced by comparing Figs. 7.16 and 7.17.

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Figure 7.18 Heat sources.

Figure 7.19 Heat sinks.

Fig. 7.18 illustrates a good comparison of the contributions of different parts of heat generation during the discharge process in the observed zinc– silver oxide battery cell. We clearly see that the reversible part of the heat of electrochemical reactions is responsible for heat generation during the first moments of modeling. However, the trend experiences a decline as time goes on. On the other hand, the Joule heating starts increasing during the last moments of simulation and even overtakes other heat production lines. As mentioned before, in addition to heat generation, heat dissipation plays an important role in thermal behavior of a battery. To study the effect of heat dissipation to the surrounding, the amounts of heat released from the simulated cell from its upper and lower walls are calculated and illustrated in Fig. 7.19. The figure shows that the heat dissipation increases as the cell temperature increases. This effect is a very natural process since as the cell temperature increases, the convection term becomes greater in magnitude.

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Figure 7.20 Heat generation.

The figure also shows that the heat dissipated from the upper part is lower in magnitude than the dissipated heat from the lower part. This difference happens because the lower part has less thermal resistance than the upper part. In fact, the headspace above the battery cell, consisting of an air gap, plays as an insulator to the heat dissipation mechanism, resulting in lower heat dissipation from the top wall. The total heat dissipation, which is the sum of dissipation from the upper and the lower walls, is shown in the figure with a thick solid line. As shown in the figure, the dissipation rates from both sides reach a stable condition at the end of the discharge process. It means that the cell is unable to dissipate more energy than the shown value. This phenomenon is very important for determining the final temperature of the battery. The whole heat source (generation plus dissipation) for each individual electrode is plotted in Fig. 7.20. The figure indicates that the net heat source is higher in the positive electrode at the early stages of the discharge, but gradually, the negative electrode becomes dominant. This is the consequence of the fact that at the early stages, the positive electrode is a semiconductive material with very low conductivity (see Table 7.2) and gradually becomes metallic silver with a very high conductivity. Therefore the amount of Joule heating decreases during the process. Moreover, as time passes, electrochemical reactions decreases, and the heat generation declines due to the electrochemical reactions. On the other hand, the inverse process that occurs at the negative electrode makes it the main source of heat generation at the end stage of the discharge process. It is clear that the heat generation due to the electrochemical reactions declines as in the positive electrode. However, the Joule heating becomes dominant so that at the end

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Figure 7.21 Net heat sources and sinks.

of the process, it not only out ranges the negative electrode but also changes the decreasing trend of the overall heat generation (indicated by the thick solid line). As can be seen, at the end of the process the total heat generation starts increasing. This increase in the total heat generation is the direct effect of Joule heating since the Joule heating starts an exponential increase in the negative electrode as shown in Fig. 7.16. Fig. 7.21 demonstrates the same results in another point of view. In this figure the net generation and dissipation in the whole cell are plotted. This graph would be useful in bulk simulation and estimation of cell thermal behavior. In other words, what is plotted in this figure is used as the source term q in Eq. (7.26), which is the algebraic sum of heat generation and dissipation in the cell. As discussed in detail before and according to Fig. 7.21, the net heat generation curve experiences a decrease during simulation, whereas during the last moments, it again becomes upward and may lead the cell to instability. It is clear that the main objective of the above study is to determine the variation of the cell temperature during the process. Fig. 7.22 shows the temperature distribution throughout different parts of the cell at t = 240 s and t = 680 s, which is near the end of the discharge process. Fig. 7.22A shows that at the middle period of discharge the hottest portion of the electrode is located near the bottom of the cell. Moreover, during the last stage of the discharge, the hottest spot moves toward the top of the cell. This phenomenon is because at the early stage of discharge, the lower part of the cell at the positive electrode has the highest electrical resistance since they are the furthest path to the electrode lug (located at the top of the electrode). Hence the electrical overpotential has the highest value in this region, resulting in higher irreversible heat generation. At the end of

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Figure 7.22 Temperature distribution inside the cell. (A) t = 240 sec. (B) t = 680 sec.

discharge the irreversible heat becomes negligible as discussed before, and hence irreversible heating does not contribute to temperature rise. On the other hand, the upper wall has more heat resistance and does not allow the heat to escape from the upper wall. Consequently, the upper parts become warmer than the lower parts. In all cases the whole battery cell becomes warmer as can be clearly seen from the figure. It is also worth mentioning that the main temperature gradient inside the cell is in a vertical direction, whereas in horizontal direction, there is a very slight and negligible temperature gradient. It is translated to the fact that the cell experiences a zero gradient condition at the left and right sides resembling an adiabatic wall. According to the discussion, the internal temperature of the cell always experiences an increase during the discharging process. To have a better view of the process, the averaged temperature of the cell is plotted in Fig. 7.23. From the figure it seems that the cell temperature is going to become stable at a moderate value. However, the temperature temporal slope curve shows that although the temperature slope declines with a steep slope from the beginning of the simulation, over the last stage of discharge, it changes its direction and starts increasing. This means that the temperature curve also experiences a sharp upward movement simultaneously from that moment on. This sharp increase in temperature can be interpreted as

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Figure 7.23 Temporal variation of the cell temperature.

the result of an increase in Joule heating in the negative electrode. If the cell discharge continues, then it may lead the cell to become out of control and cause further damages to the system. In other words, a thermal runaway may occur at this battery cell.

7.8.4 Two-dimensional simulation with water cycle In secondary or rechargeable zinc–silver oxide batteries, water cycle takes place mainly during charge where the cell voltage exceeds 1.7 V. Water cycle starts with water dissociation at the positive electrode, at which water dissociates into hydroxide ions according to reaction (7.11). The hydroxide ions travel to the negative electrode and recombine to generate water according to reaction (7.13). When the cell is fully charged and still is under overcharging, the main cell reactions stop due to the lack of active material. However, it is observed that at this period the current density flowing to the cell is not zero and has a moderate value. Theoretically, this current should be consumed by side reactions. As mentioned in Section 7.4, the current drives both zincate and water side reactions. In the present section the role of the water cycle is considered. For the current study, Cell-Ag-W from Appendix B is selected for simulation. All the physical and electrochemical parameters are available in the appendix. It was discussed that the present modeling has the advantage that accounts the effect of variation of the state of charge and active area of each electrode. The proper relations are described by Eqs. (7.27)–(7.30).

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Figure 7.24 Validation and comparison of present modeling including the effect of SoC.

Figure 7.25 Variation of active area versus time.

The cell is simulated using finite volume method as was done for the previous example. A two-dimensional grid, like shown in Fig. 7.9, is generated to obtain best performance, that is, the minimum computational cost and maximum accuracy. A grid study has been conducted to become confident that the results are independent of grid size. Fig. 7.24 shows the validation of the present simulation. The figure shows that the present simulation can properly simulate the cell during discharge. Moreover, from the figure it is quite clear that modeling state of charge gives better agreement with the experimental test. Fig. 7.25 shows the variation of active area versus time. We can see that in both electrodes the active area decreases; however, the active area of the positive electrode decreases faster than that of the negative electrode. This can be attributed to more active material in the negative electrode. It is also interesting that for both electrodes, active materials decrease linearly versus time.

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Figure 7.26 Variation of chemical species with respect to time.

Variation of different chemical species is illustrated in Fig. 7.26. The figure shows that divalent silver oxide reduces to metallic silver during discharge at the positive electrode. Meanwhile, metallic zinc is oxidized into ZnO at the negative counter electrode. One conclusion of this figure is that as the cell discharges, the internal resistance of the positive electrode decreases, and it becomes more conductive, whereas we have the opposite situation at the negative electrode. It means that the negative electrode becomes more and more nonconductive. This simulation is completely consistent with the thermal study in the previous section. Fig. 7.26 shows the bulk values for different electrodes, and the contour plots of active materials are shown in Figs. 7.27–7.30. The figures show that a nonuniform concentration exists for all different species. Moreover, the gradient of species varies as the cell continues to discharge. As stated before, side reactions or more specifically, water cycle, are important during the charging stage where the cell potential exceeds 1.7 V. Simulation of the charging process starts after the cell is discharged for 100 s. As discussed before, the cell is discharged with constant current until it reaches a specific state shown in Figs. 7.25–7.30. Now the cell is charged with V = 2.3 V, and different parameters are studied. During a constant voltage charging, the current reaches an asymptotic value in real test. However, in theory, if we do not consider the side reactions, then the current should reach zero because of after the cell is fully charged, the active area goes to zero, and no current should be imposed. Fig. 7.31 illustrates two completely different behaviors if side reactions are modeled or not. Without considering side reactions, the current exponentially reaches zero since there is not active material for further charging.

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Figure 7.27 Contours of Ag active material at the positive electrode. (A) t = 50. (B) t = 100.

Figure 7.28 Contours of AgO active material at the positive electrode. (A) t = 50. (B) t = 100.

However, considering side reactions results in a moderate current passing through the cell that is more consistent with experimental data. Moreover, we see that the current reaches a specific limit and never become zero. In other words, if we continue charging, then water cycle continues its operation, and the required energy is supplied from the charger. As shown in Fig. 7.31, when the side reactions are modeled, the current reaches an asymptotic value. In Fig. 7.32, this case is analyzed in more detail by separating the share of the currents of the main and side reactions. From the figure we see that at the first stage of charge, since the cell is not fully charged, almost all the energy input drives the main reaction, and the

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Figure 7.29 Contours of Zn active material at the positive electrode. (A) t = 50. (B) t = 100.

Figure 7.30 Contours of ZnO active material at the positive electrode. (A) t = 50. (B) t = 100.

side reactions are negligible. However, as the charging process continues, the battery becomes fully charged, which in turn results in activation of side reactions. In the range of 50 < t < 200 s, both the main and side reactions have moderate values, but after t > 200, the battery becomes almost fully charged, and the main reactions become negligible. As we see and stated before, the current density of side reactions asymptotically reaches a moderate value. According to electrochemical reactions of zinc–silver oxide batteries, during the charging process, hydroxide ions are consumed in the positive electrode and generated at the negative electrode. In Fig. 7.33, hydroxide

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Figure 7.31 Current under constant-voltage charging.

Figure 7.32 Variation of cell current versus time.

concentration profile is plotted in different time levels across a cross-section located at the middle height of the cell. Since the charging process is started immediately after the discharging process, a nonuniform concentration gradient exists at t = 0. The figure shows that the concentration of hydroxide ions decreases at the positive electrode and increases at the negative counter electrode. One interesting result obtained form this curve is that as the charging process continues, the hydroxide concentration continues to increase at the negative electrode even more than its initial concentration that is, 8.9 mol l−1 . This fact never can be obtained without modeling the water cycle because by neglecting the side reactions, during the charging process, all the concentrations reach their initial values if the cell is fully charged. Since a two-dimensional modeling is studied here, the cross-section results do not give a complete perspective of the whole process. To have a better demonstration, contour plots at time levels t = 0, t = 50, t = 100,

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Figure 7.33 Hydroxide ion concentration at middle height in different time levels.

and t = 200 seconds are plotted in Fig. 7.34. The contour plots show that hydroxide concentration does not have a uniform pattern all across the electrode and has different values at different locations. Also, it has a strong time dependency, whereas its value and pattern differ in different time levels. The contour plot of dissolved oxygen concentration is plotted in Fig. 7.35. The figure shows that the contour levels are almost uniform across the cell. In fact, during the charging process, oxygen is generated and consumed at the positive and negative electrodes, respectively. The electrolyte generation and consumption should make a concentration gradient as was the case for hydroxide ions. However, since the diffusion coefficient of oxygen is very large, it becomes uniform across the cell due to diffusion. This is why the contour plots of oxygen are much more uniform than the hydroxide ions.

7.9 Summary In this chapter, we studied zinc–silver oxide batteries and their applications, advantages, and disadvantages were discussed. Then we explained the governing equations of the battery and gathered the physico-chemical properties from the open literature. After that, to show the capability of the numerical simulation in studying these types of battery, we numerically investigated different zinc–silver oxide cell models. First of all, we discussed a one-dimensional model in which only the main reactions existed. After that, we introduced a more accurate model in two dimensions and studied different parameters. We then studied the effect of thermal model and carried on many different studies. Finally, we studied the effect of side reactions in more detail.

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Figure 7.34 Hydroxide concentration contours. (A) t = 0. (B) t = 50. (C) t = 100. (D) t = 200.

7.10 Problems 1. If the energy equation is not considered, how do we impose the environment temperature to the model? 2. What is the difference between imposing environmental temperature to the model and considering the energy equation? 3. In some cases, we can consider the energy equation for studying the thermal behavior of the cell but model the cell temperature as a bulk value. Rewrite the energy equation for considering this effect. 4. How can we adapt the energy equation for one-dimensional formulation? 5. Describe the effect of side reactions in charging process.

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Figure 7.35 Oxygen concentration contours. (A) t = 0. (B) t = 200.

6. If we neglect the side reactions, then what would be the final situation of curve 7.34? 7. Integrate the hydroxide concentration shown in Fig. 7.34 for different time levels. What do you obtain? Explain the reason.

CHAPTER 8

Lithium-based batteries Contents 8.1. Lithium-based battery components 8.1.1 Positive electrode 8.1.2 Negative electrode 8.1.3 Separator 8.1.4 Electrolyte 8.2. Types and applications 8.2.1 Lithium–cobalt oxide, LCO (LiCoO2 ) 8.2.2 Lithium–manganese oxide, LMO (LiMn2 O4 ) 8.2.3 Lithium–nickel–manganese–cobalt oxide, NMC (LiNiMnCoO2 ) 8.2.4 Lithium–iron phosphate, LFP (LiFePO4 ) 8.2.5 Lithium–nickel–cobalt–aluminum oxide, NCA (LiNiCoAlO2 ) 8.2.6 Lithium–titanate, LTO (Li4 Ti5 O12 ) 8.3. Electrochemical reactions 8.4. Governing equations of lithium-based batteries 8.4.1 Conservation of mass and momentum 8.4.2 Conservation of electrical charge 8.4.3 Conservation of chemical species Conservation of lithium ions in electrolyte Conservation of lithium ions in solid phase 8.4.4 State of charge 8.4.5 Conservation of energy 8.5. Boundary and initial conditions 8.6. Physical characteristics 8.6.1 Conductivity of electrolyte 8.6.2 Solid conductivity 8.6.3 Open-circuit voltage 8.7. One-dimensional model of lithium-based batteries 8.7.1 Conservation of electrical charge in solid phase 8.7.2 Conservation of electrical charge in electrolyte phase 8.7.3 Conservation of chemical species 8.7.4 Kinetic rate or Butler–Volmer equation 8.8. Simulation of lithium-based batteries 8.8.1 Cell voltage 8.8.2 Concentration of lithium ions in electrolyte 8.8.3 Concentration of lithium ions in solid particles and state of charge 8.8.4 Simulation result 8.9. Summary 8.10. Problems

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Figure 8.1 Schematic illustration of intercalation process in lithium-based batteries.

Lithium-based batteries are very famous due to their high energy content. Because of their high energy density and specific energy and power, they are made in huge number, and with the increase of electrical vehicles, the demand is becoming higher and higher. These batteries have a different chemistry than other conventional ones, because in this technology the electrochemical reactions occur through a process known as insertion or intercalation rather than through classical electrochemical reactions. In the intercalation process the lithium ions are inserted between layers of a host material. In both electrodes as shown in Fig. 8.1, intercalation takes place; hence a similar phenomenon exists during charge and discharge. In a lithium battery, all the three components, the positive and negative electrodes and the electrolyte, are considered as active materials. During discharge, intercalated Li+ ions are moving out of their host material (in most cases, carbon) and travel to the positive electrode and intercalate inside the positive material. During charge, a reverse process takes place, and the inserted ions move back to the negative electrode through nonaqueous electrolyte and separator. Usually, the voltage of a single lithium cell is higher than 3.6 V, which is way higher than the water dissociation voltage. This fact limits the electrolyte choices to nonaqueous ones.

8.1 Lithium-based battery components In lithium-based batteries, four main important components play a vital role in cell behavior. These components are positive and negative electrodes, separator and electrolyte [80]. Each of these parts has its own research field and has a strong effect on the performance of the cell. When we speak

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about the performance, we have to consider all different aspects such as higher capacity, longer cyclic life, lower self-discharge, better thermal behavior, and lower cost.

8.1.1 Positive electrode The positive electrode can be made of different materials such as LiMnO2 , LiCoO2 , and LiNiO2 . To provide a large capacity, cathodes should contain a large amount of lithium. On the other hand, the cathode material should exhibit a reversible reaction in charge or discharge so that its structure does not change dramatically. Otherwise, it will have a low cyclic life. Moreover, the electronic conductivity on ionic mobility of the materials should be very high to minimize the internal resistance. In addition to the these properties, the material should be stable in its environment, especially in contact with the electrolyte. Finally, they should be very cost effective since usually the anode is made of carbonaceous materials, which are not so expensive, and the main material cost is due to the cathode.

8.1.2 Negative electrode In practice, most of negative electrodes are made of graphite or other carbon-based materials. Many researchers are working on graphene, carbon nanotubes, carbon nanowires, and so on to improve the charge acceptance level of the cells. Besides the carbon-based materials, different noncarbonaceous materials are working with and under consideration. For example, amorphous Sn composite oxide (ATCO) was first introduced by Fuji Photo Film Co, Ltd. in 1995. ATCO has some advantages over graphite because it can accept twice the gravimetric capacity, but it has a very short cyclic life, which makes it unattractive. Some transition metal oxides including CoO, NiO, CuO, and FeO also provide higher capacity but just like ATCO show a short cyclic life. By contrast, LiTiO exhibits a large cyclic life (about hundreds of thousands of cycles) but has a lower capacity in comparison with carbonaceous materials.

8.1.3 Separator Like any other battery type, the separator has two main functionalities: 1) preventing from internal short circuit and 2) providing a medium for ionic transport. Lithium battery separators are generally divided into microporous membrane and nonwoven cloth. The former is made of both

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natural and synthetic polymers including polyolefins, polyvinylidene fluoride, nylon, and so forth.

8.1.4 Electrolyte The electrolyte of lithium batteries should be selected from nonaqueous materials since the cell voltage is very high so that water is dissociated. Therefore, the electrolyte is typically made of a lithium-ion-rich mixture of organic carbonates such as ethylene carbonate or diethyl carbonate. A lithium-based salt is mixed with a nonaqueous electrolyte including lithium hexafluorophosphate (LiPf6 ), lithium hexafluoroarsenate monohydrate (LiAsF6 ), lithium perchlorate (LiClO4 ), lithium tetrafluoroborate (LiBF4 ), and lithium triflate (LiCF3 SO3 ).

8.2 Types and applications Although in lithium-based batteries all the four components are important, normally they are categorized according to their cathode (the positive electrode) material and are named after their cathode composition. Numerous materials are considered and studied in many research groups and universities. However, only a few of them have become commercialized and mass produced. Here we introduce and briefly discuss some of the most famous lithium batteries. As mentioned before, lithium battery types are usually named after their chemical material used in their positive electrode, but it is not true in all cases. For simplicity, they are given an abbreviation name that is more simple to remember. For example, lithium–cobalt oxide has the chemical symbol LiCoO2 and is abbreviated as LCO. Similar abbreviations exist for other lithium batteries.

8.2.1 Lithium–cobalt oxide, LCO (LiCoO2 ) Li–cobalt oxide is a very popular battery due to its high energy content. Most mobile phones or laptops are equipped with LCO batteries. The negative electrode is made of graphite, but other materials such as nickel, manganese, and/or aluminum are under research. Because cobalt is an expensive material, LCO is usually expensive. Other than its cost, a short cyclic life and low thermal stability are considered as the main drawbacks of these cells. For thermal reasons, the cells should not be charged by currents over their C-ratings. In practice, the

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Figure 8.2 Kiviat diagram of LCO (modified from [82]).

cells are recommended to be charged by 0.8C. This means that LCO is not appropriate for fast charging. In Fig. 8.2 a Kiviat or spider plot of LCO cells is shown. Other than parameters shown on the figure, toxicity, fast-charge capabilities, the selfdischarge rate, and the shelf life are important parameters to be analyzed. LCO batteries loosed their interest comparing with NMC and NCA because of its high cost and performance.

8.2.2 Lithium–manganese oxide, LMO (LiMn2 O4 ) LMO was first introduced in 1983 and commercialized in 1996 by Moil Energy. Comparing with LCO, these cells have a lower internal resistance, that is, a higher specific power. In a 18650 package, LMO can provide currents up to 20–30 A with moderate heat buildup. Consequently, LMO is a good candidate for high-power applications such as electric vehicles. The main drawbacks of LMO cells are their low specific energy and life span, as shown in Fig. 8.3. It is good to know that new designs have improved the specific power, safety, and life span. These improvements have occurred by blending cobalt with other metals such as nickel, manganese, or aluminum.

8.2.3 Lithium–nickel–manganese–cobalt oxide, NMC (LiNiMnCoO2 ) Similar to LMO, nickel–manganese–cobalt cells are used in power applications. A 18650 cell can deliver up to 2800 mAh and be discharged with 4 to 5 amperes. If customized, NMC can be discharged up to 20 A but with lower cyclic life and lower capacity. Silicon can be added to the negative

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Figure 8.3 Kiviat diagram of LMO (modified from [82]).

Figure 8.4 Kiviat diagram of NMC (modified from [82]).

electrode to increase its capacity up to 4000 mAh in a 18650 cell, but the mechanically unstable nature of silicon in charge and discharge makes it difficult to commercialization. The cathode combination is usually one-third nickel, one-third manganese, and one-third cobalt. The combination makes the cell to be less expensive than LCO or LMO, resulting in a better market. Since nickel has a lower cost, battery manufacturers tend to remove the cobalt and use more nickel in the positive electrodes. Besides the cost, nickel can accept more energy, and hence the NMC cells have more specific energy than LCO. Fig. 8.4 shows the Kiviat graph of NMC cells.

8.2.4 Lithium–iron phosphate, LFP (LiFePO4 ) Lithium–iron phosphate was discovered in University of Texas in 1996 with other contributors. Comparing other lithium batteries, LFP shows a good thermal resistance and higher cyclic life and can provide very high power.

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Figure 8.5 Kiviat diagram of LFP (modified from [82]).

LFP is more tolerant to full charge conditions, and hence its safety is superior over other types. Unfortunately, LFP suffers from energy content, and its open-circuit voltage is 3.2 V. Moreover, the LFP cells have higher self-discharge. Also just like other lithium types, LFP is very sensitive to cold temperature. Even storage in cold temperature shortens its service life. In many applications, LFP is used for replacing lead–acid starting batteries. By making a pack of four cells in series, a battery with 12.8 V is obtained whose open-circuit voltage is very similar to 12-V lead–acid starting batteries. Fig. 8.5 shows the Kiviat plot for LFP cells. As can be seen, its low specific energy is the main issue, whereas other characteristics show reasonable values. Again, it should be emphasized that the figure does not include other important characteristics.

8.2.5 Lithium–nickel–cobalt–aluminum oxide, NCA (LiNiCoAlO2 ) Lithium–nickel–cobalt–aluminum oxide batteries have been produced since 1999 for specific applications. The cells have many similar behavior like NMC including high specific power, high specific energy, and reasonable life span. However, they have cost and safety issues. The specific characteristics of NCA are plotted in Fig. 8.6.

8.2.6 Lithium–titanate, LTO (Li4 Ti5 O12 ) In contrast to other types, LTO cells are named after the composition of the negative electrode. The negative electrode of LTO is made of Li4 Ti5 O12

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Figure 8.6 Kiviat diagram of NCA (modified from [82]).

Figure 8.7 Kiviat diagram of LTO (modified from [82]).

instead of graphite. The positive electrode is similar to that of LMO, LFP, or NMC. The open-circuit voltage of LTO is about 2.4 V and can be fast charged. The main advantage of LTO is its specific power as a specific cell can be discharged by 10C or 10 times its rated capacity. Other advantages of LTO are cyclic life and its performance at low temperature. Fig. 8.7 shows the Kiviat plot for LTO, and as it can be seen, these cells have desirable characteristics regarding specific power, safety, performance, and life span. However, they are costly, and their specific energy is not very high. Table 8.1 compares the mentioned technologies. Note that the data are given for an average cell of the named technology. Different manufacturers may provide different data according to their own production line characteristics.

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Table 8.1 Comparison of different lithium battery technologies [82]. Property

LCO

LMO

NMC

LFP

NCA

LTO

Since Open circuit voltage, V Typical Operating voltage, V Specific energy, Wh kg−1 Charge, C-rate Discharge, C-rate Cyclic life Max. temperature, ◦ C Cost, $ kWh−1

1991 3.6

1996 3.7

2008 3.6

1996 3.2

1999 3.6

2008 2.4

3.0–4.2

3.0–4.2

3.4–4.2

2.5–3.65

3.0–4.2

1.8–2.85

150–200

100–150

150–220

200–260

90–120

50–80

0.7C–1C 1C

0.7C–1C 1C–10C

0.7C–1C 1C–2C

0.7C 1C

1C, 5C max 10C

500–1000 150

300–700 250

1000–2000 210

1C 1C, 25C on some cells 1000–2000 270

500 150

3000–7000 Very safe

N/A

N/A

∼ 420

∼ 580

∼ 350

∼ 1000

8.3 Electrochemical reactions In lithium-based batteries, charge and discharge take place through intercalation process. In an intercalation process, lithium ions are reversibly removed or inserted into a host. The structure of the host should not change significantly during intercalation or deintercalation. For most technologies, the positive electrode is made of a metal oxide that can provide enough space for accepting lithium ions. Providing enough space happens if the metal oxide has a layered structure just like graphite. The negative electrode is usually made of graphite, which itself has a layered structure and can provide the necessary space for lithium insertion. When a Li-ion cell is charged, as expected, the positive material is oxidized, and the negative material is reduced. In this process, lithium ions are deintercalated from the positive electrode and move toward the negative electrode through the electrolyte; then they are inserted into the negative electrode. If the negative electrode is made of graphite, then the intercalated electrode will be Lix C6 , where 0 < x < 1. Since in a lithium-ion battery, metallic lithium is not present, they are chemically less active and hence safer. Intercalation process, just like any other electrochemical reaction, is made of two half-cell reactions. In the positive electrode the following half-cell reaction takes place: charge

+ − −−  LiM  −− −− −− − − Li1−x M + x Li + xe ,

discharge

(8.1)

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where M indicates any metal oxide. The intercalation process of the negative electrode is charge

−−  C + xLi+ + xe−  −− −− −− − − Lix C.

(8.2)

discharge

Consequently, the overall cell reaction is charge

−  LiM + C −  −− −− −− − − Li1−x M + Lix C.

(8.3)

discharge

Eqs. (8.1)–(8.3) can be applied on any available lithium battery technology. Example 8.1. Apply Eqs. (8.1)–(8.3) to LCO and obtain the electrochemical reactions of LCO cells. Answer. In LCO cells the positive electrode is made of lithium cobalt oxide or LiCoO2 . Therefore, according to Eqs. (8.1)–(8.3), the half-cell reactions are charge

+ − −−  LiCoO2  −− −− −− − − Li1−x CoO2 + x Li + xe

discharge

at the positive electrode and charge

−−  C + xLi+ + xe−  −− −− −− − − Lix C discharge

at the negative electrode. Then the overall reaction is charge

−−  LiCoO2 + C  −− −− −− − − Li1−x CoO2 + Lix C. discharge

8.4 Governing equations of lithium-based batteries When speaking about lithium batteries, it should be kept in mind that there are many different available configurations both in industrial and research scales. The vast variety of chemistry leads to many different mathematical models available in technical reports and papers. A quick literature review reveals that for simulation of lithium batteries, both zonal and single-zone formulations are used by different researchers. In many battery systems, dilute solution theory can be applied for modeling the electrolyte. However, in the case of insertion batteries, where

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Figure 8.8 Schematic of a Li-ion cell.

the insertion process is diffusion-limited [83], dilute solution theory cannot provide a correct model, and we should use the theory of concentrated solution. In the book, we consider this model for modeling the electrolyte. Giving a general model for all types of lithium-based batteries is important, and it should be noted that the model must be adopted for any specific technology. Doyle et al. [83] gave a general model for insertion batteries which is general enough to include a full range of materials used in making lithium cells. Since the model is general and is presented in a general vector form, it can be easily applied to any system. However, when experimental data are available for a specific type, we can modify the model to include the real data. Fig. 8.8 shows a schematic model of the Li–on battery. The cell sandwich consists of a Cu collector with negative active material or electrode, an Al collector on which positive active material is pasted, and a separator in between. The positive and negative electrodes consist of porous solid matrices whose pores are flooded by a proper electrolyte. Neglecting side reactions, during charge and discharge, the following electrochemical reactions occur at the positive and negative electrodes, respectively: discharge

−  Li −p + s + e− −  −− −− −− − − Li −s + p ,

(8.4)

charge

discharge

+ − −−  Li + p  −− −− −− − − Li −p + e .

(8.5)

charge

Both equations can be cast into the single equation discharge

+ −  Li+ + e− + θs −  −− −− −− − − Li − θs ,

charge

(8.6)

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where θs stands for any solid insertion site. This site can be the negative or positive electrode. In a general form, in an intercalation process a fraction of inserted lithium ions contribute to charge and discharge. Thus for generality, Eq. (8.6) can be rewritten as discharge

−  x Li+ + x e− + θs −  −− −− −− − − Lix −θs .

(8.7)

charge

Example 8.2. For LMO, write the negative and cathodic reactions and compare the reactions with Eq. (8.7). Answer. The electrochemical reactions of LMO are given by [30] as follows: Positive electrode discharge

−−  Liy−x Mn2 O4 + x Li + + x e −  −− −− −− − − Liy Mn2 O4 . charge

Comparing the reaction with Eq. (8.7), we see that θs = Liy−x Mn2 O4 , which is the positive material in which intercalation occurs. Negative electrode discharge

+ − −−  Lix C6  −− −− −− − − Li0 C6 + x Li + x e .

charge

Again by comparison with Eq. (8.7) we see that θs = C6 or a graphite negative electrode. Similar expressions can be written for other lithium-ion technologies including LCO (Liy-x CoO2 ), LTO (Liy-x TiO2 ), and so on. The governing equations of Li-ion batteries have been studied and presented [84] with the following assumptions: • There is no gas phase. • Electrolyte is assumed to be concentrated binary. • Side reactions are ignored. • Charge transfer kinetics follows the Buttler–Volmer equation. • Volume changes are neglected, so porosity is assumed to be constant. Just like other battery technologies previously discussed in this book, the electrochemical behavior of lithium cells can also be described by the conservation laws of chemical species and electrical charge.

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8.4.1 Conservation of mass and momentum Lithium-ion batteries are usually available with a solid electrolyte. Therefore electrolyte movement has no meaning and hence is not the case. Consequently, convective terms in all other governing equations are dropped whenever needed.

8.4.2 Conservation of electrical charge In an electrochemical battery cell, the electrical current in an external circuit is carried by electrons, which is governed by Ohm’s law. The volumeaveraged equation for solid potential distribution is given by Eq. (3.73) and in case of lithium batteries is reduced to ∇ · (σ eff ∇φs ) − jLi = 0,

(8.8)

where φs means the potential distribution in solid, and σ eff is the effective electrical conductivity. It is worth mentioning that in writing Eq. (8.8), only the main reactions are considered, and side reactions are neglected. When side reactions exist, we need to modify the source term jLi so that the current generation or consumption of all reactions is taken into account. The current source term jLi in Eq. (8.8) is the reaction current resulting from production and consumption of Li + species in cell, introduced as jLi = As ij ,

(8.9)

where As is electroactive or specific interfacial area of the positive and negative solid electrodes, and ij is the current density of each reaction defined by Eq. (3.10). It is quite evident that in the separator region, both As and ij are zero, and hence there is no source term at the right-hand side of Eq. (8.8). For conventional electrodes made of spherical particles, the specific interfacial area is calculated from the equation As =

Ase , Vs

(8.10)

where Ase is the total surface area of spheres in the volume, and Vs is the volume of a sphere. Thus Eq. (8.10) becomes As =

Ase 4π r 2 3εs = εs 4 s3 = , Vs rs 3 π rs

(8.11)

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where rs is the average radius of electrode particles. Note that εs is the solid phase porosity related to the electrode porosity εe , polymer matrix fraction εp , and conductive filler fraction εf by the relation εs + εe + εp + εf = 1.

(8.12)

As mentioned before, the current density resulting from each reaction is governed by the Butler–Volmer equation (3.10). In lithium-ion batteries, the same argument is true, and since in the present model, side reactions are neglected, the current density reduces to 

ij = i◦j exp



αaj F

RT

 ηj

  αcj F ηj , − exp −

RT

(8.13)

where αaj and αcj are the anodic and cathodic charge transfer coefficients for electrode reactions, respectively, F is Faraday’s constant, and R is the universal gas constant. By definition, i◦j is the current density of an electrode reaction and for the insertion process both in the positive and negative electrodes is calculated by i◦j = kce αaj (cs,max − c se )αaj (c se )αcj ,

(8.14)

where k is the kinetic transfer rate constant. According to this equation, the exchange current density depends on both lithium concentrations ce and cs in electrolyte and solid phases. Also, it is a function of area-averaged solid state lithium-ion concentration c se at the electrode/electrolyte interface and the maximum lithium concentration cs,max in solid phase. The full discussion about different definitions of concentration is given in Section 8.4.3. The last parameter of Eq. (8.13) that requires further discussion is the electrode overpotential ηj . According to discussions of this parameter in Section 3.2, the overpotential is defined as ηj = φs − φe − Uj ,

(8.15)

where Uj is the open-circuit potential of the positive and negative electrodes. Since in a lithium-based battery, we are dealing with different materials for the positive electrode in different technologies, and even there are different negative electrodes, we should pay special attention to select a proper value for equilibrium of open-circuit potential for both electrodes. In this chapter, we give the open-circuit voltages of some famous electrodes. There are more data available in the open literature for different battery types. The readers are advised to be very careful in using these data.

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Figure 8.9 Plate nomenclature.

In any case the open-circuit voltage of both electrodes is a function of the state of charge and temperature. For a full discussion about the issue, see Section 8.6.3. In some cases where the current density is very low, we can assume that a uniform current density is formed all over the electrodes. Hence, instead of using the Butler–Volmer equation to find the current source, jLi can be simply found by the relation jLi =

Iapp . Ad

(8.16)

In Eq. (8.16), A is the electrode projected area as shown by hatched pattern in Fig. 8.9 or, mathematically, A = h × w.

(8.17)

Analogous to the solid phase potential, the potential distribution in the electrolyte phase is obtained using volume-averaged equations (3.74). Inside the electrolyte the applied current is carried by charged ions, for which the modified form of Ohm’s law yields ∇ · (κ eff ∇φe ) + ∇ · (κDeff ∇ ln(ce )) + jLi = 0,

(8.18)

where φe is the electrolyte phase potential, κ eff is the effective ionic conductivity, κDeff is the diffusion conductivity of the electrolyte, and jLi is the interfacial current density at the solid-electrolyte interface. The argument about the source term of Eq. (8.8) is also true for the source term in Eq. (8.18). In other words, the source term is actually the

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sum of all current generation or consumption sources or reactions. Also, note that the two sources in the above two equations are the same with negative sign. This means that the current generated in the electrodes should go into the electrolyte phase and vice versa.

8.4.3 Conservation of chemical species In an intercalation or insertion process, lithium ions exist both in the electrolyte and electrodes. Therefore we need to drive the necessary equations to obtain its concentration in both media. In the electrolyte phase, diffusion, migration, and generation exist and contribute to the concentration gradient. However, in solid particles the only mechanism is diffusion. Here we discuss these effects in more detail.

Conservation of lithium ions in electrolyte From the general mass balance equation, the conservation of lithium ions in electrolyte yields to 1 − t+◦ Li ie · ∇ t+◦ ∂(εe ce ) = ∇ · (Deeff ∇ ce ) + , j − ∂t F F

(8.19)

where ce is the electrolyte concentration, which is the same as the concentration of lithium ions in the electrolyte phase. Since the electrodes are porous, the electrolyte porosity εe should be considered in the transient term. The porosity of the medium implies that the effective properties should be used. For instance, the effective diffusion coefficient Deeff , which is a function of electrolyte concentration and temperature, should be corrected according to the Bruggeman relation [29] Deeff = De εe1.5 .

(8.20)

It is worth mentioning that the Bruggeman relation is widely used in different battery technologies, and in almost all of them the porosity exponent is selected as 1.5, but it is well known that 1.5 is not the only value. The exponent should be obtained experimentally. Doyle et al. [66] discussed the matter in more detail and had found different values for different parameters. Consequently, it is better to write the Bruggeman relation as follows: Deeff = De εeζ ,

(8.21)

where ζ is a general parameter that should be obtained experimentally. Also, this parameter can be used as an adjusting parameter in different simulations.

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Figure 8.10 Illustration of solid concentration in an electrode solid particle. (A) During insertion. (B) During de–insertion.

In Eq. (8.19), t+◦ is the transfer number of the Li + , which is measured relative to the solvent velocity. If the transfer number is assumed to have a constant value, as is the case in many simulations, the last term of Eq. (8.19) vanishes. Although the transfer number is not actually a constant parameter, due to the lack of experimental results, it is considered as a constant.

Conservation of lithium ions in solid phase In the construction of solid electrodes, graphite and positive material powder are used with an average radius of rs . As illustrated in Fig. 8.10, lithium ion concentration increases at the outer surface of the particle during insertion and decreases during de–insertion. The rate of generation at the surface obviously depends on the current density and is determined by the Butler–Volmer equation. In a solid particle, mass transport takes place mainly due to diffusion. Therefore the concentration gradient is governed by the equation  2  ∂ cs 2 ∂ cs ∂ cs = Ds + . ∂t ∂ r2 r ∂r

(8.22)

Eq. (8.22) is expressed in spherical coordinate with r being the main axis and should be solved alongside with other equations. Moreover, Ds is the diffusion coefficient of lithium ions in the solid material of both electrodes. The solution of Eq. (8.22) requires two boundary conditions at r = 0 or the center of the particle and r = rs or the surface of any spherical particle. For a symmetric condition, the concentration is symmetric at r = 0, and hence at the center of any particle (for both positive and negative materials) we have ∂ cs = 0. (8.23) ∂r

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Simulation of Battery Systems

At the outer surface or r = rs , the rate of lithium ion consumption is equal to the rate of insertion or deinsertion. Thus we have: jLi = −Ds

∂ cs , ∂r

(8.24)

where jLi is the current density described by the Butler–Volmer equation. In a simpler case, Gu and Wang [84] proposed another model by assuming a bulk value for ion concentration in a particle. By this assumption Eq. (8.22) reduces to ∂(εs cs ) jLi = . ∂t F

(8.25)

As explained before, εs is the solid phase porosity defined as the fraction of volume of solid phase over the whole volume. An area averaging over a spherical particle of either the positive or the negative electrode gives Ds jLi (¯cse − cs ) = , lse As F

(8.26)

where lse is the microscopic diffusion length explained by Wang et al. [29] (see Section 3.4.2). In the case of solid spherical geometry, the characteristic length is related to its radius by the relation rs lse = . 5

(8.27)

From Eq. (8.26) we can obtain the area-averaged lithium concentration at the surface of particles or c¯se at the electrode/electrolyte interface. This parameter is used to determine the current density according to the Butler–Volmer equation. Example 8.3. For lithium batteries with low current density such as coin cells, give an approximation equation for solid phase concentration change of lithium ions for a moderate time duration. Answer. When a cell is subjected to low current density, we can neglect the porosity change and concentration gradient. Hence Eq. (8.25) can describe the model. Moreover, the current density can be assumed as constant according to Eq. (8.25). Thus for small changes, Eq. (8.25) is simplified as cs jLi = . t εs F

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This equation gives the proper relation for concentration change: cs =

jLi t. εs F

The simple relation obtained here may be used in many applications, especially in low current density types. The lower the current density, the more accurate the result.

8.4.4 State of charge The local state of charge for each electrode is used in may equations and is a key parameter for cell modeling. This parameter is evaluated by the fraction of solid-electrolyte interface concentration and maximum solid concentration and should be evaluated for each electrode separately: SoC =

c¯s,e cs,max

,

(8.28)

where ¯cse is the area-averaged lithium ion concentration at electrode/electrolyte interface, and cs,max is the maximum concentration of lithium in the solid phase.

8.4.5 Conservation of energy Neglecting electrolyte movement in lithium cells, the general energy balance or Eq. (3.23) is reduced to ∂(ρ cp T ) = ∇ · λ∇ T + q. ∂t

(8.29)

The heat capacity of the battery, like other battery technologies, is a combination of heat capacity of all the involved species. Moreover, the thermal conductivity λ is also a function of different species and porosity of the medium. These parameters can be obtained from the following equations: ρ cp =



εk ρk cpk ,

(8.30)

ε k λk .

(8.31)

k

λ=

 k

The heat source of Eq. (8.29) can be evaluated using Eq. (4.23). The result is

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Simulation of Battery Systems





Uj q = As ij φs − φe + T + σ eff ∇φs · ∇φs + keff ∇φe · ∇φe + keff D ∇ ln ce · ∇φe . T (8.32) In this equation, all the parameters are previously explained, and the energy balance is also completely discussed in Chapter 4.

8.5 Boundary and initial conditions Boundary and initial conditions are quite analogous to those of lead–acid and zinc–silver oxide batteries. In other words, for initial values, we choose the uniform state as follows: T = T◦ , ce = ce,◦ , cs = cs,◦ .

(8.33) (8.34) (8.35)

Moreover, the initial values for φs and φe are obtained by solving Eqs. (8.8) and (8.18) with constant concentration and temperature values. We again emphasize that from the mathematical point of view, we do not need to specify initial values for φs and φe since their governing equations are steady state. However, for numerical simulation, we need some initial guesses, and an improper guess will cause numerical instability. For the concentration of chemical species, zero flux is the physical boundary condition: ∂ ce ∂ cs = = 0. (8.36) ∂n ∂n This argument is true for potential in the electrolyte since no ionic current is passed through the solid walls or current collectors: ∂φe = 0. ∂n

(8.37)

The boundary condition for solid potential is zero flux on all the solid walls except for the lugs. Hence the following equation is appropriate for all the solid walls except the lugs: ∂φs = 0. ∂n

(8.38)

At lugs, we put φs = 0 for the negative electrode since we need to have a reference point for potential and φs = Vs at the positive lug if we have a

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283

constant voltage condition and −σ eff

∂φs = Iapp ∂n

(8.39)

for a prescribed applied current.

8.6 Physical characteristics As discussed in the previous sections, there a lot of choices for different parts of lithium batteries. Many different materials are tested and analyzed every day to obtain better performance and higher energy content. Consequently, gathering and summarizing a single document for the material property is almost impossible. In this section the most famous materials that are widely used in different lithium battery technologies are discussed. More information is available and updated in the open literature.

8.6.1 Conductivity of electrolyte LiPF6 is frequently used as the electrolyte in lithium-based batteries. The salt is solved in a nonaqueous mixture of ethylene carbonate (EC) and dimethyl carbonate (DMC). The conductivity of the electrolyte depends on the composition of the electrolyte. If a 2:1 v/v mixture of EC/DMC is used, then at 25◦ C, the following relation is fitted by Doyle et al. [66] to experimental data: k0 =4.1253 × 10−4 + 5.007 × 10−3 c − 4.7212 × 10−3 c 2 + 1.5094 × 10−3 c 3 − 1.6018 × 10−4 c 4 , (8.40) where c is the electrolyte concentration in mol dm−3 . Alternatively, many cells use LiPF6 in a 1:2 v/v mixture of EC/DMC. Again Doyle et al. [66] gave the following fitted curve: k0 =1.0793 × 10−4 + 6.7461 × 10−3 c − 5.2245 × 10−3 c 2 + 1.3605 × 10−3 c 3 − 1.1724 × 10−4 c 4 . (8.41) The above data are valid over the range of 0.1 to 4.0 M, and the unit of conductivity is S cm−1 . It should be noted that when applying Eqs. (8.40) and (8.41) for porous electrodes, the effect of porosity should also be considered. This effect can be applied by the Brugmann relation as follows: k=

ε k0 , τ

(8.42)

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Simulation of Battery Systems

where τ is the tortuosity correction and in many cases is related to ε by τ = ε −0.5 .

(8.43)

However, for better modeling of lithium cells, we can use Eq. (8.42) and choose τ as a calibration parameter. In other words, the conductivity in porous electrodes can be written as k = ε p k0

(8.44)

and determine p by adjusting the simulated results with experimental data.

8.6.2 Solid conductivity Other than the electrolyte, the solid phase conductivity σ is also important. For graphite electrode, Doyle et al. [66] suggested: σLix C6 = 1

S cm−1

(8.45)

In the open literature, there are many different values for conductivity of Liy Mn2 O4 . For instance, again Doyle et al. [66] suggested: S cm−1

σLiy Mn2 O4 = 0.038

(8.46)

In another research, Fuller et al. [83] gave: σLiy Mn2 O4 = 1

S cm−1

(8.47)

The value strongly depends on the composition and additives of the electrode.

8.6.3 Open-circuit voltage The open-circuit voltage or the equilibrium voltage of the electrodes strongly depends on the used materials. As discussed before, there are many types of lithium cells available both in industrial and experimental scales. Since the range of materials is very vast, we need to pay special attention to selecting a proper value for the open-circuit voltage. In any case the opencircuit voltage of materials strongly depends on the cell temperature with a nearly linear dependency. Thus we have: Uj = Uj,ref + (T − Tref )

∂ Uj , ∂T

(8.48)

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Figure 8.11 Open-circuit voltage of ULix C6 [66].

where Uj,ref is the open-circuit voltage of the electrode at reference tem∂ Uj perature Tref , and shows the dependency of the open-circuit voltage ∂T ∂ Uj depend on the state of the charge on the temperature. Both Uj,ref and ∂T of the electrode. For most practical cells, graphite is the chosen material in the negative electrode. Based on the experimental data, Doyle et al. [66] fitted the following relation for open-circuit voltage of Lix C6 electrode: ULix C6 = −0.16 + 1.32 exp(−3.0SoC ) + 10.0 exp(−2000.0SoC ),

(8.49)

where SoC is the local state of charge in terms of solid-state lithium concentration at the electrode/electrolyte interface determined by the relation cse SoC = . (8.50) cs,max Eq. (8.49) is graphically shown in Fig. 8.11. In the same reference the temperature dependency of the open-circuit voltage is given as dULix C6 =344.1347148 dT

exp(−32.9633287SoC + 8.316711484) 1 + 749.0756003 exp(−34.79099646SoC + 8.887143624) − 0.8520278805SoC + 0.362299229SoC 2 + 0.2698001697. (8.51)

×

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Simulation of Battery Systems

Figure 8.12 Open-circuit voltage of ULiMn2 O4 [66].

For LMO cells, Doyle et al. gave the following relation for LiMn2 O4 electrode: ULiMn2 O4 =4.19829 + 0.0565661 tanh[−14.5546SoC + 8.60942]   1 − 0.0275479 − 1 . 90111 (0.998432 − SoC )0.492465 − 0.157123 exp(−0.04738SoC 8 ) + 0.810239 exp[−40(SoC − 0.133875)].

(8.52)

Fig. 8.12 graphically shows the open circuit voltage for LiMn2 O4 . For manganese spinel, the temperature dependency of open-circuit voltage is given by dULiMn2 O4 =4.31274309 exp(0.571536523SoC ) dT + 1.281681122 sin(−4.9916739SoC ) − 0.090453431 sin(−20.9669665SoC + 12.5788250) − 0.0313472974 sin(31.7663338SoC − 22.4295664) − 4.14532933 + 8.147113434SoC − 26.064581SoC 2 + 12.7660158SoC 3   SoC − 0.5169435168 2 − 0.184274863 exp − . (8.53) 0.04628266783

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8.7 One-dimensional model of lithium-based batteries In Sections 6.10 and 7.8, we explained in detail simulations of lead–acid and zinc–silver oxide batteries. For simulation, different numerical methods, including the Keller–Box and finite volume methods, were used. It is quite evident that the same methods can be applied for the case of lithium-ion batteries. In fact, many researchers have applied these methods (especially the finite volume method) for simulation of governing equations of lithium-based batteries including but not limited to [66,83,85–88,36]. Although the governing equations can be solved using any of the previously used methods, here we try to solve the equations using some semianalytical methods. The analytical solution can be obtained in some circumstances by making some reasonable assumptions. The main advantage of the present modeling is its fast response, which makes it suitable for online purposes such as monitoring or control. The readers are encouraged to solve the governing equations with numerical methods and compare the results with the results obtained by the present modeling. For developing the model, an LMO (lithium, manganese dioxide) cell is selected. The following reactions are considered as the main reactions of the cell: discharge

−−  Liy−x Mn2 O4 + xLi + + xe −  −− −− −− − − Liy Mn2 O4

(8.54)

charge

at the positive electrode and discharge

+ − −  Lix C6 −  −− −− −− − − Li0 C6 + xLi + xe

(8.55)

charge

at the negative electrode. In this study, we neglect the side reactions. The model used for simulation is the same as shown in Fig. 8.8, where the cell sandwich consists of a Cu collector with an Lix C6 negative electrode, an Al collector on which an Liy Mn2 O4 is pasted, and a separator in between. The positive and negative electrodes consist of porous solid matrices whose pores are flooded by LiPF6 . The governing equations of Li-ion batteries have been studied previously in a general form. The presented governing equations have no analytical solution; hence we have to make the following assumptions to obtain the answer: • The gas phase is ignored. • Concentrated binary electrolyte is assumed to be the case. • Side reactions are ignored as stated before.

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Simulation of Battery Systems

• Although charge transfer kinetics follows the Buttler–Volmer equation,

• • • •

we can assume that a uniform current density is distributed all over the electrodes. Volume changes are neglected, so the porosity of electrodes is assumed to be constant. The current direction is normal to electrode plates, so two- and threedimensional effects are neglected. The Biot number is assumed to be very small; hence an isothermal assumption is used. The diffusion coefficient and transference number are assumed to be constant.

8.7.1 Conservation of electrical charge in solid phase In an electrochemical battery cell the electrical current in an external circuit is carried by electrons and is governed by Ohm’s law: σ eff

d2 φs Li − j = 0, dx2

(8.56)

where φs is the solid phase potential, and σ eff is the effective electrical conductivity of the electrode defined as σ eff = σ ε 1.5 .

(8.57)

Eq. (8.56) is the one-dimensional form of Eq. (8.8) with constant σ eff and jLi .

8.7.2 Conservation of electrical charge in electrolyte phase In the electrolyte phase, charged ions carry the applied current, for which the modified form of Ohm’s law should be applied. Eq. (8.18) in onedimensional form with constant property assumption is reduced to κ eff

2 d2 φe eff d + κ (ln(ce )) + jLi = 0, D dx2 dx2

(8.58)

where φe is the electrolyte phase potential, κ eff is the effective ionic conductivity, and κDeff is the diffusional conductivity of the electrolyte. The effective properties, as usual, should be corrected according to the Brugmann relation κ eff = κε 1.5 .

(8.59)

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The conductivity of electrode highly depends on the composition of electrolyte. For the assumed electrolyte composition consisting of LiPF6 in a 2:1 V/V mixture of ethylene carbonate (EC) and dimethyl carbonate (DMC), κ is determined by Eq. (8.40).

8.7.3 Conservation of chemical species Conservation of lithium-ion follows Eq. (8.19), which according to the assumption of one-dimensional space and constant properties is simplified as ◦ ∂ ce ∂ 2 ce 1 − t+ εe = Deff 2 + (8.60) jLi . ∂t ∂x F In this equation, ce is the concentration of lithium ions in the electrolyte phase, ε is the electrolyte porosity, and F is Faraday’s constant. Moreover, the transference number of lithium ions t+◦ is assumed to be constant, and hence the last term in Eq. (8.19) is dropped out. By the assumption of constant properties, we take the diffusion coefficient Deff out of the differential operator. Note that, as usual, Deff should be corrected according to the Bruggeman relation Deff = Dε1.5 .

(8.61)

The current density distribution jLi can be assumed constant all over the electrodes according to the previously made discussions; thus Eq. (8.16) is used instead of the Butler–Volmer equations. The conservation cs of lithium ions in solid phase is obtained using Eq. (8.22), which by the constant property assumption is simplified to εs

∂ cs jLi = . ∂t F

(8.62)

8.7.4 Kinetic rate or Butler–Volmer equation The rate of reaction is determined by the Butler–Volmer equation or Eq. (8.13), which we repeat for better reference:      αaj F αcj F ij = i◦j exp ηj − exp − ηj

RT

RT

(8.63)

The equation requires the values for exchange current density as expressed by Eq. (8.14): i◦j = kce αaj (cs,max − c se )αaj (c se )αcj

(8.64)

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Simulation of Battery Systems

Figure 8.13 Illustration of a one-dimensional cell.

with overpotential defined as ηj = φs − φe − Uj .

(8.65)

8.8 Simulation of lithium-based batteries In this section, we present an analytical simulation of Li-ion batteries. In some cases, some parameters should be obtained analytically. Hence the solution is better to be called semianalytical instead of analytical. Therefore the aim of the present simulation is to give a semianalytical simulation in which for some equations, we propose an analytical expression, and for some others, we give a numerical method. Such a combination, as we will show later, gives an accurate enough result and is very fast. Fig. 8.13 shows a one-dimensional model of a lithium-ion cell. The cell consists of a positive electrode, a separator, and a negative electrode. The boundary conditions are applied on exterior boundaries drawn by larger circles. The left boundary is located at x = 0, and the right boundary at x = xp . The interior boundaries located at x = xn and x = xs are respectively located at the positive electrode/separator and separator/negative electrode interfaces.

8.8.1 Cell voltage The cell voltage can be obtained from Ecell = φs (xp ) − φs (0).

(8.66)

Since the voltage is a potential parameter, we can choose an arbitrary reference point. For convenience, the voltage of the negative electrode is chosen as the reference point, that is, φs (0) = 0.

(8.67)

Ecell = φs (xp ).

(8.68)

Hence Eq. (8.66) becomes

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The solid phase potential can be obtained by simply integrating Eq. (8.56). Since the parameter and the source term are constant, the integration yields: jLi 2 x + C1 x + C2 . (8.69) 2σ eff By proper boundary conditions we can obtain the integration constants C1 . Eq. (8.69) not only gives the cell voltage φs (xp ) but also can be used to determine the solid phase potential distribution along the positive and negative electrodes. For the negative electrode, the following boundary conditions should be applied: φs (x) =

φs (0) = 0, ∂φs (x)  = 0.  ∂ x x=xn

(8.70) (8.71)

Eq. (8.71) comes from the fact that at the negative electrode/separator interface, no current exists in the solid phase. In other words, all the current is carried by charged ions in the electrolyte phase. On the other hand, Ohm’s ∂φs (x) law indicates that the current density is defined as i = −σ . ∂x Substituting Eqs. (8.70) and (8.71) into Eq. (8.69), we obtain the solid phase potential in the negative electrode: φs (x) =

jLi  2 x − 2xn x , 2σ eff

0 ≤ x ≤ xn .

(8.72)

For the positive electrode, depending on the operating mode, different boundary conditions can be applied. For the constant voltage mode, the boundary conditions are: φs (xp ) = Ecell , ∂φs (x)  = 0,  ∂ x x=xs

(8.73) (8.74)

where Ecell is a predefined value. In this case, we can obtain an exact solution by applying these boundary conditions in Eq. (8.69). The result is φs (x) =

jLi 2 jLi ( x − 2x x ) + E − xp (xp − 2xs ), s cell 2σ eff 2σ eff

xs ≤ x ≤ xp . (8.75)

Eq. (8.75) gives the potential distribution along the positive electrode in constant voltage mode. However, if the cell is under a constant current

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test, Eq. (8.73) should be replaced by −σ eff

∂φs (x)  = −i .  ∂ x x=xp

(8.76)

The negative sign on the right-hand side is because the current density has a different direction in the positive and negative electrodes. Applying Eq. (8.74) to Eq. (8.69) gives C1 = −

jLi σ eff

xs ,

(8.77)

and applying Eq. (8.76) to Eq. (8.69) gives C1 = − Note that i =

xp xs

i σ eff

.

(8.78)

jLi dx, which with constant jLi results in i = jLi (xp − xs ).

(8.79)

We see that both boundary conditions (8.74) and (8.76) give the same result for C1 but does not give any value for C2 . Mathematically, in a constant current mode, we have many answers for solid phase potential along the positive electrode. From physical interpretation we know that only one of them has a physical meaning, and now we have to find the proper answer. To find the correct answer, we have to solve all the governing equations, which requires numerical algorithms and is not our goal here. As another way, we can use physical and electrochemical concepts to choose the correct answer out of all possible answers. We know that the cell potential is also determined by the relation Ecell = Eocv ± [(ηct )a + (ηc )a ] ± [(ηct )c + (ηc )c ] − iRi ,

(8.80)

where Eocv is the open-circuit voltage, (ηct )a and (ηct )c are activation polarizations or charge-transfer overvoltage at anode and cathode, respectively, (ηc )c and (ηc )a are concentration polarizations at anode and cathode. Also, Ri is the internal cell resistance, and i is the current density passing through the cell. In this equation, the signs − and + represent discharge and charge, respectively. Eq. (8.80) is the key to choose the proper solution when it is compared with Eq. (8.68). In other words, out of infinitely many solutions of Eq. (8.69), the correct one satisfies Eq. (8.68). However, we have to find a

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method for finding an accurate value for Ecell . To do this, we will discuss each term in Eq. (8.80). The first term on the right-hand side of Eq. (8.80) is the open-circuit voltage of the cell, which can be found from the relation Eocv = Up − Un ,

(8.81)

where Up and Un are open-circuit voltages of the positive and negative electrodes versus a reference electrode, respectively. The open-circuit voltage of the negative electrode is given by Eq. (8.49), and its temperature dependency is defined by Eq. (8.51). For the present sample, which is an LMO cell, the open-circuit voltage of the positive electrode is given by Eq. (8.52), and its dependency on temperature is also given by Eq. (8.53). The overpotential ηct at the positive and negative electrodes can be evaluated from Eqs. (8.63)–(8.65). In these equations, αa and αc are anodic and cathodic transfer coefficients for electrode reactions, respectively. These parameters can be used as the calibration parameters to make sure that the numerical results are fitted to the experimental data. The same argument is true for k in Eq. (8.64), which is the kinetic transfer rate constant. Determination of overpotential requires electrolyte concentration in electrolyte and solid phases. This requires that the electrolyte and solid phase concentration differential equations (i.e. Eq. (8.60) and (8.62)) to be solved. The solution of these equations is postponed to Sections 8.8.2 and 8.8.3. Although concentration polarization ηc is significant, we neglect these terms because the present modeling deals with low current density (or low power) applications. In this situation the mass transfer mechanisms such as diffusion have enough time to deliver active materials to reaction sites. The last term in Eq. (8.80) includes the internal resistance, which should be calculated in some manner. Esfahanian et al. [89] gave a physical model for the calculation of internal cell resistance. Fig. 8.14 shows the proposed model, in which the resistance of the electrodes are modeled using two resistors in series to account for internal resistances of the solid and electrolyte phases. The solid and electrolyte resistances are indicated by Rs and Re , respectively, with subscripts + and − indicating the positive and negative electrodes. In separator region the solid phase resistance Rss is usually very large, and we can neglect it, but the electrolyte resistance Res can be calculated using some physical relations. This is why the resistance model in the electrolyte is calculated using a parallel model instead of a series one. The solid resistance of any region (i.e., the electrodes and separator) can be calculated by the general known formula of the resistance:

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Figure 8.14 Schematic of resistance model.

Rs =

d σ eff A

,

(8.82)

where d and A are the thickness and cross-section area of the region, respectively. In a one-dimensional model, we are dealing with a cross-section with a unit area A = 1 cm2 . The effective solid conductivity includes the effect of electrode or separator porosity into account. For the electrolyte resistance, the same formula can be used, but with the effective ionic conductivity instead of the electronic conductivity. Therefore the ionic resistances are calculated from Re =

d κ eff A

.

(8.83)

By determining the cell resistance we have all the parameters required for calculating Ecell from Eq. (8.80), and, as discussed before, by equating it with Eq. (8.73) we can determined the exact solution of Eq. (8.69). By this algorithm, there is no need to solve differential equation (8.58) to obtain potential distribution in electrolyte because the electrolyte potential distribution can be easily found by Eq. (8.65) in both electrodes. In separator region the electrolyte potential is almost linear, and therefore we can find it by connecting the electrolyte potential values at electrodes/separator interfaces by a single line.

8.8.2 Concentration of lithium ions in electrolyte The lithium ion concentration in the electrolyte phase is required for evaluation of the Butler–Volmer equation, which in turn is necessary for evaluation of activation polarization. To obtain the value, Eq. (8.60) should be solved. Hopefully, by present assumptions the equation can be analytically solved, and the exact solution can be found. The equation can be rearranged as

Lithium-based batteries

A

∂ 2 ce A ∂ ce + J= , 2 eff ∂x D ∂t

295

(8.84)

where for simplicity the following parameters are defined: ⎧ Deff ⎪ ⎨ A = , ε 0 ⎪ ⎩ J = 1 − t jLi .

(8.85)

F

Note that in each region, A , Deff , and J are constant but have different values from one region to another. Eq. (8.84) is a nonhomogeneous partial differential equation with homogeneous boundary conditions at both ends: ∂ ce (0, t) = 0. ∂x ∂ ce (xp , t) = 0. ∂x

(8.86) (8.87)

The initial condition is assumed to be constant: ce (x, 0) = ce0 .

(8.88)

Although the boundary conditions are homogeneous, we cannot solve Eq. (8.84) because, as discussed before, A , Deff , and J have different values in different regions. To overcome this difficulty, we have to split the domain into three different regions over each of which, these parameters are constant. Therefore, we have to simultaneously solve three partial differential equations to obtain the concentration profile over the three different regions. To distinguish these profiles, we rewrite Eq. (8.85) for all three regions as follows: Ai

∂ Ci ∂ 2 Ci Ai + Ji = , ∂ x2 Di ∂t

i = n, s, p,

(8.89)

where Ci stands for concentration profile in each region with i = n, s, p referring to the negative electrode, the separator, and the positive electrode, respectively. Fig. 8.15 illustrates a typical concentration profile during discharge, and as we can see, the concentration profile is almost linear along with the separator but has a curvature along the electrodes. Moreover, the concentration profile is continuous at all regional interfaces, but its slope suddenly changes. Mathematically speaking, Eq. (8.89) makes a system of three dif-

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Figure 8.15 A typical concentration profile during discharge.

ferential equations. For each equation, we need two boundary conditions and one initial condition. Fig. 8.15 depicts the following boundary conditions for each region: Negative electrode The electrolyte has zero slopes at its left boundary and is continuous at electrode/separator interface. Separator The electrolyte concentration is continuous at its both ends. Positive electrode The electrolyte has zero slopes at its right boundary and is continuous at electrode/separator interface. Interior boundaries From physical understandings, the current density at the interior boundaries x = xn and x = xs is continuous. The above physical assumptions yield six required boundary conditions as follows: 1. At x = 0 the zero flux exists, that is, ∂ Cn  = 0.  ∂ x x=0

(8.90)

2. At x = xn the concentration is continuous: Cn (xn ) = Cs (xn ).

(8.91)

3. At x = xn the current density is equal in both the negative electrode and the separator: Dn

∂ Cn  ∂ Cs  = Ds .   ∂ x x=xn ∂ x x=xn

(8.92)

4. At x = xs the concentration is continuous: Cs (xs ) = Cp (xs ).

(8.93)

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5. At x = xs the current density is equal in both the positive electrode and the separator: Ds

∂ Cp  ∂ Cs  = Dp .   ∂ x x=xs ∂ x x=xs

(8.94)

6. At x = xp the zero flux exists, that is, ∂ Cp  = 0.  ∂ x x=xp

(8.95)

These boundary conditions are sufficient for obtaining a closed-form solution for concentration profile. The obtained governing equation with proper boundary conditions as described by Eqs. (8.90)–(8.95) can be solved analytically. Here we can use the method of Green functions to obtain the solution. According to this method, for a multiregion system with different properties in a onedimensional space, the solution of Eq. (8.89) is [90] Ci (x, t) =

M   L

j=1

  Gij (x, t|x , τ ) 





τ =0

c0 dx +



t

τ =0



A Gij (x, t|x , τ ) i Ji dx dτ , Di L 

(8.96) where Gij (x, t|x , τ ) is the associate Green function of Eq. (8.89) with boundary conditions (8.90)–(8.95). The first integral on the right-hand side of Eq. (8.96) takes into account the effect of initial conditions, and the second integral gives the effect of the source term in concentration generation. For the present study, Eq. (8.96) can be expanded as 

xn

Ci (x, t) =  +

0 xp

xs  t

  Gi1 (x, t|x , τ ) 

 

Gi3 (x, t|x , τ ) 



τ =0

τ =0

c0 dx +



xs

xn

xn

τ =0

c0 dx

c0 dx

An Jn dx dτ Dn τ =0 0  t  xp Ap Gi3 (x, t|x , τ ) Jp dx dτ. + Dp τ =0 xs

+

 

Gi2 (x, t|x , τ )

Gi1 (x, t|x , τ )

(8.97)

As is clear form Eq. (8.97), the analytical solution is available if the Green function is known. In fact, obtaining proper Green functions is the key in

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obtaining the solution. Let us now have some discussion about Green function and its role on finding the solution of a partial differential equation. Later we discuss the method of finding the proper Green function. The Green function is a two-variable function with two parameters. The variables of the function are space and time. In the one-dimensional space, as in our case, the space parameter is shown by x and time by t. The parameters x and τ of the Green function are separated by a vertical line to emphasize that they are parameters and not variables. The Green function represents the effect of a unit source, which is released spontaneously at time t = τ at x = x , where the domain initial condition is zero with zero boundary conditions. Therefore, for any position x at t < τ , the Green function is zero, but for t > τ , the whole domain is influenced by the unit source. Consequently, when the Green function is evaluated at τ = 0, there is a source term in the domain released at t = 0. In view of Eq. (8.97), it is clear that for the first three integrals, the Green function is evaluated at τ = 0; hence these three terms are responsible for taking the effect of initial condition into account. Whereas the first three integrals are evaluated at τ = 0, the other two integrals are evaluated from τ = 0 to τ = t over the whole domain. The effect of these integrals is the effect of sources in the final solution. It is worth mentioning that since the boundary conditions of the problem (i.e., Eqs. (8.90) and (8.95)) are zero, the above five integrals are sufficient, but if the boundary conditions were not homogeneous, the other two integrals should be added to Eq. (8.97). In solving differential equations using Green functions, the main problem is obtaining the Green function itself because the Green function of any problem is different from other problems depending on the differential equation and the boundary conditions. For partial differential equation such as Eq. (8.89), Özisik [91] and Hahn [90] gave the following equation for the Green function: 

Gij (x, t|x , τ ) =

∞ 

k=1



Dj exp(−βk2 (t − τ ))  Aj



1 ψin (x)ψjn (x ) , Nk

(8.98)

where βk is the kth eigenvalue corresponding to the kth eigenfunction, ψin . Also, Nk is the norm of the kth eigenfunction defined as Nk =

M =3  j=1

=

Dn An





Dj Aj

0

xn



xi+1

xi

 ψjk2 (x )dx

2 ψ1k (x )dx +

Ds As



xs

xn

2 ψ2k (x )dx +

Dp Ap



xp

xs

2 ψ3k (x )dx . (8.99)

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299

Finally, the eigenfunction ψik for Eq. (8.89) in the Cartesian coordinate system are given by Hahn [90] as    βk βk ψik = Aik sin   x + Bik cos   x , 

Ai

Ai

(8.100)

where βk is the eigenvalue of the problem and is discussed later. For the three regions, we have i = n, s, p; hence there are six unknowns associated with each βk . If βk are known, then Aik and Bik in Eq. (8.100) can be found using boundary conditions, that is, Eqs. (8.90)–(8.95). It should be noted that eigenfunctions are not unique, and any multiplication of them by a constant makes another eigenfunction. For this reason, we can select one set of eigenfunctions without any difference in the final result. Therefore, in applying boundary conditions, it is customary to set one of the nonvanishing coefficients to unity. In general, the boundary conditions of the Green function are of the same type as the original problem but are homogeneous. In our case where the boundary conditions are homogeneous themselves, the same conditions are applied on the Green function. Hence applying the boundary conditions gives: ∂ψ1k  = 0, which yields:  ∂ x x=0     0 0 A1k cos γ1 − B1k sin γ1 = 0,

1. Eq. (8.90) results in

xn

xn

(8.101)

where γ1 is defined as βk xn γ1 =  .

(8.102)

An

Eq. (8.101) states that A1k = 0, and as stated before, we can choose B1k = 1 without any error in the final result. Therefore the first eigenfunction is   x ψ1k = cos γ1 .

(8.103)

xn

2. Eq. (8.91) results in ψ1k (xn ) = ψ2k (xn ), which can be simplified as 







xn xn cos γ1 = A2k sin γ2 + B2k cos γ2 , xs xs

(8.104)

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where analogous to Eq. (8.102), we have βk xs γ2 =  .

(8.105)

As

3. Eq. (8.92) results in Dn

∂ψ1k  ∂ψ2k   = Ds  . Consequently, we have ∂ x xn ∂ x xn 







xn xn −K1 sin γ1 = A2k cos γ2 − B2k sin γ2 , xs xs where Dn K1 = Ds



(8.106)

As . An

(8.107)

4. Eq. (8.93) results in ψ2k (xs ) = ψ3k (xs ), which gives 





A2k sin γ2 + B2k cos γ2 = A3k sin γ3



xs xs + B3k cos γ3 , xp xp

(8.108)

where again we have βk xp γ3 =  .

(8.109)

Ap

5. Eq. (8.94) results in Ds

∂ψ2k  ∂ψ3k   = Dp  , which yields ∂ x xs ∂ x xs



K2 A2k cos γ2 − B2k sin γ2



    xs xs = A3k cos γ3 − B3k sin γ3 ,

xp

xp

(8.110) where



Ds K2 = Dp 6. Finally, Eq. (8.95) results in

Ap . As

(8.111)

∂ψ3k   = 0, which gives ∂ x xp

A3k cos γ3 − B3k sin γ3 = 0.

(8.112)

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301

These boundary conditions can be cast into a matrix form as ⎡ cos γ1

⎢ ⎢ ⎢−K1 sin γ1 ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎣ 0

⎤⎡

  − sin γ2 xxns   − cos γ2 xxns

  − cos γ2 xxns   sin γ2 xxns

sin γ2

cos γ2

K2 cos γ2

−K2 sin γ2

  − sin γ3 xxps   − cos γ3 xxps

0

0

cos γ3

0 0

⎡ ⎤



0

1

0

− sin γ3

B3k

0

⎥⎢ ⎥ ⎢ ⎥ ⎥⎢ ⎥ ⎢ ⎥ ⎥ ⎢A2k ⎥ ⎢0⎥ 0 ⎥⎢ ⎥ ⎢ ⎥   ⎢ ⎥ ⎢ ⎥ xs ⎥ − cos γ3 xp ⎥ ⎢ B2k ⎥ = ⎢0⎥ . ⎢ ⎥ ⎥ ⎢ ⎥  ⎥ ⎢ ⎥ ⎢ ⎥ ⎢0⎥ + sin γ3 xxps ⎥ ⎢A3k ⎥ ⎦⎣ ⎦ ⎣ ⎦

(8.113) Since the right-hand side of Eq. (8.113) is homogeneous, for the system of equations to have a nontrivial solution, the determinant of the coefficient matrix should become zero:    cos γ1   −K sin γ 1 1     0    0     0

− sin









  − cos γ2 xxns

− cos   sin γ2 xxns

sin γ2

cos γ2

K2 cos γ2

−K2 sin γ2

0

0

γ2 xxns

γ2 xxns

   0 0    0 0      − sin γ3 xxps − cos γ3 xxps  = 0.     − cos γ3 xxps + sin γ3 xxps    cos γ3 − sin γ3 

(8.114) Eq. (8.114) has infinitely many solutions for βk , or in other words, it has infinitely many eigenvalues. Example 8.4. For a sample shown in the following table, draw the lefthand side of Eq. (8.114) versus β and discuss the result. Negative electrode

Separator

Positive electrode

0.0104 9.7 × 10−8

0.0056 9.0 × 10−7

0.0179 2.2 × 10−7

0.232

0.357

0.380

Length (cm) Diffusion Coefficient, Di (cm2 s−1 ) Porosity, εi (-)

Answer. From the given data we find the following values: xn = 0.0104,

xs = 0.0160,

xp = 0.0339

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Figure 8.16 Determinant versus β .

and 9.7 × 10−8 = 4.18 × 10−8 , εn 0.232 Ds 9.0 × 10−7 As = = = 2.52 × 10−6 , εs 0.357 Dp 2.2 × 10−7 Ap = = = 5.79 × 10−7 . εp 0.380

An =

Dn

=

By having these values, γ s and Ks can be found using the corresponding equations. Then the determinant of Eq. (8.114) can be evaluated and plotted versus β . Fig. 8.16 shows the variation of matrix determinant versus β . As we can see, the graph repeatedly cuts the x axis, or in other words, it has an infinite number of zeros. Example 8.5. Find the first positive roots of problem 8.4. Answer. Here we only discuss the first three roots, and the other ones can be found similarly. Inspecting Fig. 8.16 indicates that the first root is β0 = 0, the first positive root is near β = 0.08, and the second one is about β = 0.13. Different numerical schemes can be used to obtain the roots. For example, the bisection method can be easily applied to finding the roots of the problem. In this method, we have to find two consecutive points; for one point, the sign of the matrix determinant is positive, and for the other one, it is negative. For the first root, we see that for β = 0.0798, the value of the determinant is −0.00153 and the corresponding value for β = 0.0799 is

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303

0.00143. Hence the root exists between these two values. By the successive bisection method we obtain the root β1 = 0.0798516701. To find the second nonzero positive root, we see that the matrix determinant is 0.0023 at β = 0.1305 and is equal to −0.00089 at β = 0.1306. Again by applying the bisection method, the root is found to be β2 = 0.1305721526. The method can be used to obtain all roots of the matrix determinant. In the following table the first seven roots are shown. β0

β1

β2

β3

β4

β5

β6

0

0.0798516

0.1305721

0.2176830

0.2954248

0.3471612

0.4564114

It should be noted that β s are all positive. Once β s are known, for each β , Eq. (8.113) is solved to obtain the coefficients of its corresponding eigenfunction. As we can see, the equation contains five equations, whereas there are only four unknowns. To solve the system of equations, we can choose any four equations out of the five. The solution gives A2k , B2k , A3k , and B3k . Consequently, the eigenfunctions are known. Example 8.6. Obtain the coefficients of eigenfunctions corresponding to β s obtained in Example 8.5.

Answer. Having a β , obtaining the coefficients is as easy as solving Eq. (8.113). Just it should be noted that Eq. (8.113) is a system of five equations with only four unknowns. Therefore we can select any four equations that we prefer to solve the system. For example, for the first nonzero root β1 = 0.0798516, we choose the first four equations and write the system of equations as ⎡ ⎤⎡ ⎤ ⎡ ⎤ 0 . 499511 0 . 866307 0 0 A 0 . 282568 ⎢ ⎥ ⎢ 2k ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎢0.866307 −0.499511 ⎥ ⎥ ⎥ 0 0 ⎢ ⎥ ⎢ B2k ⎥ ⎢−0.253866⎥ ⎢ ⎥⎢ ⎥=⎢ ⎥. ⎢ ⎢0.720601 ⎥ ⎢ ⎥ 0.69335 −0.994138 0.108122⎥ 0 ⎢ ⎥ ⎢A3k ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦ 0.108122 0.994138 B3k 1.35927 −1.41269 0

The result is 

A2k

B2k

A3k

B3k

T

T  = −0.0787806 0.3715994 0.2680391 0.6066140 .

We apply the same procedure for any other β . The results for the first six roots are tabulated in the following table.

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Simulation of Battery Systems

βk

0 0.0798516701 0.1305721526 0.2176830508 0.2954248345 0.3471612530 0.4564114399

A2k 0 −0.0787806520 −0.5309579660 −0.9127678081 −0.0575968086 +0.4735179905 +0.3022688508

B2k 1 +0.3715994927 −0.1588446164 −0.2273640902 −0.2610575223 −0.6247490265 −0.4502927277

A3k 0 +0.2680391599 −0.2493468963 +0.4141284024 +0.1771614456 −0.3252240573 +0.9728475249

B3k 1 +0.6066140563 +0.4960696893 +1.4749514236 +0.2613105608 +1.3245414986 +0.0835810605

The last step in finding the Green function is calculating the norm of the eigenfunctions after obtaining the coefficients of eigenfunctions using Eq. (8.98). Inspecting the equation shows that all the parameters are known and the norm Nk can be calculated for each βk . Example 8.7. Having the eigenfunctions of Example 8.5, calculate the norm for each β . Answer. When the eigenfunctions are known, the calculation of the norm is easy. The corresponding norm for each β is simply calculated using Eq. (8.98) as follows: Nk =

Dn An

 0

xn

2 ψ1k (x )dx +

Ds As



xs

xn

2 ψ2k (x )dx +

Dp Ap



xp

xs

2 ψ3k (x )dx .

For example, for β1 , we can write 





9.7 × 10−8 0.0104 2 0.07985 N1 = sin √ x dx 4.18 × 10−8 0 4.18 × 10−8     9.0 × 10−7 0.016 0.07985  + x −0.078 sin √ 2.52 × 10−6 0.0104 2.52 × 10−6  2 0.07985  x dx + 0.371 cos √ 2.52 × 10−6     2.2 × 10−7 0.0339 0.07985  + 0.268 sin √ x 5.79 × 10−7 0.016 5.79 × 10−7  2 0.07985 x dx + 0.606 cos √ − 7 5.79 × 10 = 0.0028464817. The same procedure should be done to find other norms corresponding to each β . For the first six β s, the norms are tabulated as follows.

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Lithium-based batteries

k

0

1

2

3

4

5

6

βk

0

0.0798516

0.1305721

0.2176830

0.2954248

0.3471612

0.4564114

Nk 0.011214 0.0028464817 0.0025618140 0.0100729783 0.0016168154 0.0081471600 0.0047429873

After finding β s, Ak , Bk , and Nk , we obtain the Green function according to Eq. (8.98), and the concentration profile can be obtained using Eq. (8.97). Summing up, the procedure is as follows: 1. Finding βk using Eq. (8.114). 2. Finding Ak and Bk corresponding to each β using Eq. (8.113). 3. Calculating Nk using Eq. (8.99). 4. Finding C (x, t) using Eq. (8.97) with the Green function defined by Eq. (8.113).

8.8.3 Concentration of lithium ions in solid particles and state of charge Concentration in solid particles is essential to calculate the state of charge of the electrodes, which in turn is important to calculate the open-circuit voltage. Therefore we have to solve Eq. (8.22) with proper boundary conditions. In case of constant current density, the solution is discussed in Example 8.3 and hence is not repeated here. After obtaining the concentration of lithium ions in the solid phase, the local state of charge of a battery or cell, SoC, is calculated using Eq. (8.28). In addition to evaluation of the open-circuit voltage, SoC is an important parameter in many applications, including monitoring devices and online simulators.

8.8.4 Simulation result To show the results of the present simulation, we simulated the lithium cell from Appendix C. The cell was simulated numerically by Doyle et al. [66], and the details can be found in the reference. The cell properties are tabulated in Appendix C. The same values were used for explaining Examples 8.4–8.7. Therefore the way the concentration profile is obtained can be traced through the same examples. The simulated voltage of the cell is shown in Fig. 8.17. As we can see, the present simulation can give accurate results when the applied current is low. For the present case, the simulated voltage accurately follows the experimental test and numerical simulation when the discharge current is 1C (i.e., i = 1.75 mA cm−2 ) or lower. However, for the case of 4C or

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Figure 8.17 Simulated cell profile.

Figure 8.18 Concentration profile at t = 5 min.

i = 7 mA cm−2 , the error arisen from the present model becomes larger. It is because, at high current density situations, the constant parameter assumption of the present model fails. In such situations the concentration polarization plays an important role in determining the battery performance. Concentration polarization results in nonhomogeneous conditions inside the cell hence, and thus all the properties and other parameters such as reaction rate cannot be considered constant. Therefore in high current operations the present assumptions fail to give accurate results; however, investigation of results shows that the error of such assumption is not very high. For example, Fig. 8.17 indicates that even in 4C operation, the present simulation can follow the experiment and CFD results with reasonable accuracy. Figs. 8.18–8.20 show the variation of electrolyte concentration across the cell at different time steps. The figures show that the analytical solution of the governing equations provides very good results with a maximum er-

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Figure 8.19 Concentration profile at t = 20 min.

Figure 8.20 Concentration profile at t = 59.85 min.

ror less than 10%. The importance of the analytical simulation is its runtime speed. In fact, the present modeling requires some milliseconds to obtain a specific profile, whereas the CFD methods, although very accurate, require couples of minutes to perform a full simulation.

8.9 Summary There are many lithium-ion technologies available both in industrial and lab scales. In the present chapter, we briefly introduced the main industrial versions and discussed the involved electrochemical reactions. Each technology has its own pros and cons. Since the quality of different manufacturers differ, we cannot give exact characteristics for different cell types. Therefore the technologies are qualitatively compared. The data shown in tables and graphs are typical values for industrial cells, and if referred to each individual battery catalog, we may find many diverse data.

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The general governing equations of lithium-ion batteries are presented in vector form. For different purposes, we may use a one-, two-, or threedimensional model, which can be easily derived from the vector form. Moreover, since Li-ion batteries have many different technologies, the governing equations are expressed in general form. For each technology, the equation can be adapted from the general form. In contrast to previous chapters, the governing equations of lithiumion cells are solved analytically for the one-dimensional case. The purpose of this type of simulation is to obtain a very fast simulator with accurate enough results. The present model is very useful in applications where the simulation speed is crucial, such as monitoring and control systems. To obtain such an analytical solution, many simplifying assumptions have been made. The results showed that in low current density applications, the proposed assumptions are accurate enough and the model can perfectly predict the cell behavior. However, in high current density applications, the model deviates form experimental test results. It should be noted that the model can be improved so that the effect of parameter and source term variations be included. If done, the model would be capable to accurately predict the cell behavior even in high current density conditions. No need to remention that the governing equations may be simulated without any simplifying assumption, using numerical schemes explained for simulation of other battery types. All the numerical methods such as Keller–Box, FVM, FDM, or FEM can be used to simulate the governing equations of lithium-based batteries in the same manner as was done for the other batteries.

8.10 Problems 1. Plot the electrolyte conductivity at 25◦ C for EC/DMC with 2:1 v/v mixture (i.e., Eq. (8.40)) versus electrolyte concentration. 2. Plot the electrolyte conductivity at 25◦ C for EC/DMC with 1:2 v/v mixture (i.e., Eq. (8.41)) versus electrolyte concentration. 3. Plot the temperature dependency profile of graphite (i.e., Eq. (8.51)) versus SoC. 4. Plot the temperature dependency profile of LMO (i.e., Eq. (8.53)) versus SoC. 5. For different Li-ion technologies (i.e., LFP, LTO, NMC, LCO, and NCA), obtain the electrochemical reactions.

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6. For the one-dimensional model, obtain the proper system of governing equations for Li-ion batteries. 7. Repeat problem 1 for the two-dimensional case. 8. For the case of constant current density, simplify the one-dimensional model obtained in problem 1. 9. Solve the governing equations of Li-ion batteries in the onedimensional form using the Keller–Box method. 10. Solve the previous problem using the finite volume method. 11. In case of nonisothermal form, obtain the exact solution for energy equation in the one-dimensional space. 12. By including the energy equation simulate the cell from Appendix C and plot the cell temperature with time. 13. The governing equations of LMO can be solved in the twodimensional space using the same techniques explained for other battery types. Using FVM, solve the system of governing equations of LMO batteries in the two-dimensional space and compare the results with the one-dimensional situation.

CHAPTER 9

Techno-economic assessment of battery systems Contents 9.1. Introduction 9.1.1 Battery technology 9.2. Environmental effects of different types of batteries 9.2.1 Environmental impacts of Photovoltaic battery systems 9.2.2 Environmental effects of electric vehicle battery systems 9.3. Summary 9.4. Problems

312 314 314 315 327 350 351

Batteries play an essential part in our lives. In our daily activities, we mostly use batteries in various applications such as our cellphones, remote control, laptops, and so on. Energy storage batteries, often seen as the key to broader deployment of cleaner renewable energy, are not usually the subject of much environmental debate. They can be used to power electric cars and store output from renewable electricity generation such as wind turbines and solar panels. Batteries also require large quantities of assorted metals and minerals in their manufacturing process. These materials for the production are often mined, produced, and transported with significant consumption of money and energy, leaving behind a significant environmental footprint. Use and consumption of these resources also produce waste, contributing to the environmental impact. An increasing greenhouse gas effect causes an increase in the average temperature of the Earth’s surface. The manufacture and transportation of batteries emit exhaust and other pollutants into the atmosphere, thereby contributing to the greenhouse effect. In this chapter, we try to carry out a comprehensive study on the environmental effects of different kind of batteries. In the first part of this chapter, we introduce batteries used in the energy systems. In this chapter we use various types of batteries including lithium-ion, lead–acid and nickel–cadmium. We compare the battery energy storage systems and present the environmental effects of using them. Finally, we review many research outcomes related to the environmental impacts of battery energy systems. To improve the understanding of this Simulation of Battery Systems https://doi.org/10.1016/B978-0-12-816212-5.00013-1

Copyright © 2020 Elsevier Inc. All rights reserved.

311

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chapter, we focus on some groups of studies. In the first part, we look at the research studies on the environmental impacts of various types of batteries. In the second part, we focus on emission-related concerns of batteries for renewable electricity production and batteries used in electric vehicles.

9.1 Introduction The development of portable electronic products has increased the demand for high-performance batteries. The demand for batteries with higher energy density has led to the development and commercialization of new electrochemical systems, other than the established lead–acid and nickel– cadmium batteries. The introduction of new technological standards for wireless communication is likely to contribute to the growth of battery use in the future. To enable widespread access and large-scale diffusion of distributed electricity including energy storage in batteries, economic, technological and environmental factors need to be optimized. To give future generations the benefit of the capital of fossil fuels, nonrenewable resources can be used to create renewable substitutes. If electricity from PV battery systems is to contribute to renewable energy supply, they must give net energy yield throughout their lifetime. It should be noted that net energy generation is less important in energy systems with low energy turnover. As an example, zero energy buildings with solar systems can be used to improve the standard of living in poor environments by providing a means of moving energy from industrialized areas to rural areas. When it comes to energy storage in more specific batteries, one of the most important factors is the chemical and physical properties of some metals, which make them attractive to use as electrode materials in batteries [70]. For a few metals, batteries are the major end use. Seventy percent of the cadmium mined worldwide each year are used in NiCd batteries, and 70% of lead (virgin and recycled lead) are used in lead–acid batteries. It is challenging to design an energy system that is simultaneously affordable, able to provide a reliable supply of energy, and does not result in unacceptably high emissions of greenhouse gases. In other words, cost, emission, and security of supply can form an important triangle for the selection of materials (see Fig. 9.1). To avoid undesired effects of the use of a substance or a product, appropriate information and assessment methods are needed. If too narrow or short-term perspective is considered, then there is a risk of suboptimization. Suboptimization is experienced when the efficiency of a unit process is in-

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Figure 9.1 The energy triangle.

creased, but at the same time, the overall efficiency is decreased. To improve batteries from an environmental perspective, greater knowledge concerning the environmental impact of battery systems must be gained. Emerging battery technologies, which may be implemented on the same scale as established technologies, should be assessed before possible environmental effects occur. Important factors and constraints related to battery technologies should be identified and managed from an environmental perspective. Environmental information can be used to set up development goals and to evaluate steering effects [94]. Based on this background, the following questions are required to be answered to evaluate and manage batteries from an environmental perspective: • What are the most important environmental aspects of different battery systems? • How do different parameters influence the environmental impacts and energy flows of battery systems? • Which methods are appropriate for assessing the environmental aspects of battery systems? By identifying parameters that are relevant in describing the environmental performance of battery systems, it will be easier to conduct further assessments and comparisons of batteries. To facilitate the communication of environmental information on batteries to stakeholders throughout the battery life cycle, there is a need to summarize the environmental aspects of batteries. Identification of important parameters can be used to direct research and product improvements. From the variety of methods available for environmental systems analysis [95] appropriate methods have to be chosen to assess the environmental performance of batteries.

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9.1.1 Battery technology Rechargeable battery storage systems comprise of a wide range of technologies. They are classified based on the type of electrodes and electrolytes used in their storage system arrangements [96]. The battery system is made up of electrochemical cells that are wired in series, which generate electrical energy at a specified voltage through an electrochemical reaction. Each electrochemical cell has two electrodes (anode and cathode) and an electrolyte [97]. An electrochemical cell can convert energy from electrical to chemical energy and vice versa. At discharge, the electrochemical reactions occur at the two electrodes at the same time. Therefore electrons are provided from the anodes and are collected at the cathodes at the external circuit. During the charge state, the reverse reactions occur, and the battery is recharged through an external voltage applied to the electrodes. The popular battery technologies systems include the lead–acid, sodium–sulfur (Na-S), sodium–nickel chloride (NaNiCl2 , nickel–cadmium (Ni-Cd), lithium-ion (Li-ion), zinc–bromide (Zn–Br), polysulfide bromine (PSB) and vanadium-redox (VRFB) [96], including the new systems such as advanced valve-regulated (VRLA) lead–acid, lead–carbon, metal–air technologies, UltraBattery, battery with current collector improvement, advanced sodium-metal chloride, high performance sodium–copper chloride, and nanostructured energy materials in lithium batteries.

9.2 Environmental effects of different types of batteries There are numerous types of batteries, including lead–acid, nickel– cadmium, lithium-ion, sodium–sulfur, nickel–metal hydride, sodium– nickel chloride, redox flow batteries, and zinc–air. These vary in efficiency, energy storage capacity, the number of charging/discharging cycles they can perform, and finally cost. Sodium–sulfur (Na–S) is suitable for high-power applications with daily charge–discharge cycles (such as renewable energy systems and vehicles). These batteries are sealed, have a rapid response system, last approximately 15 years, but they are comparatively expensive. Lithium-ion (Li-ion), nickel–cadmium (NiCd) are ideal for small-size applications but are costly for multi-MW load applications, where several hours of discharge time are needed. Lead–acid batteries are widely available but can differ widely in design. Their performance at low temperature and their cycle life is below average, but can still offer storage solutions in some cases. In 2012, McManus [98] did a comprehensive

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Figure 9.2 Normalized data for battery production (to produce 100 kg).

research on the environmental consequences of the use of batteries in lowcarbon systems and the impact of battery production. His paper builds on previous research studies by providing an information base relating to the production of six battery types. The results indicated that the lithium batteries have the most significant impact on metal depletion. The primary material responsible for this is the lithium iron phosphate (LiFe-PO4), but there is also some impact on metal resource depletion from the use of the electronic component, the transistor. Fig. 9.2 shows the normalized data for battery production. The data for climate change, metal, and fossil fuel depletion and cumulative energy demand are listed in Table 9.1. This table shows that the most energy-intensive batteries in terms of their production are the lithium-ion and the nickel–metal hydride. The batteries with the lowest cumulative energy demand are the lead–acid and sodium–sulfur batteries. This pattern is repeated in the other categories, with the highest embodied CO2 and metal and fossil fuel depletion resulting from the production of the lithium- and nickel-based batteries. To understand the actual relative impacts of the production of the batteries, they must also be examined on an energy basis and are therefore shown here on a per MJ capacity basis (Table 9.2).

9.2.1 Environmental impacts of Photovoltaic battery systems In 2013, Mckenna et al. [98] studied the environmental effects of lead–acid batteries in grid-connected domestic PV systems. At the domestic level, the use of batteries in grid-connected photovoltaic (PV) systems has been proposed to minimize grid exports and improve consumer economics by

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Table 9.1 Characterized impact per kg of battery production.

Lead–acid battery Lithium-Ion (NMP solvent) Lithium-Ion (water solvent) Ni–Cd battery Ni–MH battery Sodium–sulfur

Climate change (Kg CO2 eq)

Metal depletion (Kg Fe eq)

Fossil depletion (Kg oil eq)

Calmative energy demand (MJ/kg)

0.9 12.5

0.4 20

0.3 1.6

17 90

4.4

20

1.5

88

2.1 5.3 1.2

1.5 3.2 3.2

0.7 1.6 0.4

37 90 19

exploiting retail electricity tariffs with variable pricing and increasing selfconsumption with feed-in tariffs. The use of batteries in grid-connected domestic PV systems mentioned in the previous section is investigated in their research. An economic and environmental impact analysis was presented for the use of lead–acid batteries in PV systems under current feed-in tariff arrangements in the UK. The environmental impact is considered in two areas, production impacts and in-use impacts. For both of these, the analysis compares the impact associated with adding a battery to the PV systems, compared to the same PV systems without battery. The production impact of lead–acid batteries was determined by examining the processes and materials contained within the battery. While a full life cycle assessment was not undertaken, a life cycle approach was taken following the ISO Standards. This was done using SimaPro software. Three environmental issues have been assessed: the impact on greenhouse gases (GHG), the fossil fuel depletion, and the metal depletion. These were analyzed using both IPCC data and the ReCiPe LCA methodology. The work has focused on these three areas; as previous research has shown, these are some of the major impact areas for battery use and production. In addition, GHG and fossil fuel depletion are major policy drivers within the energy arena, and the impact of metal depletion has been widely discussed as a potential area for concern associated with the use and production of batteries. The production impacts of lead–acid batteries per kg of battery weight in terms of greenhouse gases, metal depletion, and fossil fuel depletion are 0.9 kg CO2eq , 0.4 kg Feeq , and 0.3 kg oileq respectively. The contribution to greenhouse gases and fossil fuel is predominantly associated with the extraction and processing of lead and the polypropylene used in battery production. The contribution to metal depletion is dominated by

Table 9.2 Characterized data range for battery production per MJ capacity. Impact category Unit Lead–acid Lithium-ion

Climate change Ozone depletion Human toxicity Photochemical oxidant formation Particulate matter formation Ionizing radiation Agricultural land occupation Urban land occupation Natural land transformation Water depletion Metal depletion Fossil depletion

Ni–Cd

Ni–MH

Sodium–sulfur

kg CO2 eq kg CFC-11 eq kg 1, 4-DB eq kg NMVOC

5-7 (2.24-3.35)E-07 6–8 0.03–0.04

17-27 (3.34-5.23)E-07 3–5 0.03–0.05

10-15 (7.64-12)E-07 4–6 0.38–0.59

16-20 (5.44-6.85)E-07 1.7–2.1 0.12–0.25

2 1.26E-07 0.6 0.007

kg PM10eq

0.02–0.03

0.03–0.04

0.88–1.38

0.45–0.57

0.0085

kg U235 eq m2

1.1–1.5 0.12–0.17

2.7–4.1 0.15–0.23

3.4–5.3 0.21–0.32

2.3–2.9 0.13–0.16

0.3 0.04

m2

0.08–0.12

0.22–0.34

0.29–0.45

0.17–0.21

0.05

m2

0.0011–0.0015

0.0021–0.0033

0.0019–0029

0.0013–0.0016

0.00033

m3 kg Fe eq kg Oil eq

0.07–0.1 2–3 1.8–2.6

0.121–0191 28–44 2.2–3.4

0.25–0.39 7–10 3–5

0.14–0.174 9–12 5–6

0.024 4 0.5

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Table 9.3 Battery weights and production impacts. Specifications

Production impacts

Battery capacity (Ah)

Weight Per cell (kg)

Number of cells in battery

Total battery weight (kg)

Climate change (kg C02eq )

Metal depletion (kg Feeq )

Fossil fuel depletion (kg oileq )

210 420 570

38 29 37

8 24 24

304 696 888

273.6 626.4 799.2

121.6 278.4 355.2

91.2 208.8 266.4

the lead within the battery. Note that this approach assumes that a mix of virgin and recycled materials is used in battery production. The production impacts for the batteries considered in their research are shown in Table 9.3. This production impact is spread over the lifetime of the batteries in use. The in-use impact of the batteries is associated with the environmental time-varying effects of grid-electricity. From the perspective of the national grid, the effect of adding a battery to a PV system (i.e., where previously there was none) is to increase demand during the day when the battery is charging and to decrease need during the evening when the battery is discharging. These changes in need throughout the day will result in the corresponding changes in generation from fossil fuel plant. Moreover, due to losses in the battery, we can expect that the increase in a daytime generation will be greater than the corresponding decrease in generation during the evening, meaning that the battery will cause a net increase in fossil fuel generation with a resulting negative environmental impact. To calculate how the changes in demand throughout the day associated with adding the battery to the PV system can be expected to result in changes in generation from fossil fuel plant, data from the UK balancing mechanism reports were used to calculate the responsiveness of gas and coal generation to historic changes in demand for each five-minute period in 2009 to 2011. The responsiveness here refers to the change in a generation (in kWh) associated with a change in demand of 1 kWh. For some periods, the calculated responsiveness was uncharacteristically high or low due to operators switching from one plant type to another. To compensate this effect, periods were grouped into 144 sets (one for each hour of the day for every two months of the year), and the weighted average responsiveness of each plant type was calculated for each set. The average was weighted by the absolute value of the change in total generation during

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Table 9.4 Annual production and in-use impact for 430-Ah battery with 3.29-W peak PV system. Standard deviations shown in brackets. Climate change Metal depletion Fossil fuel depletion (kg oileq /year) (kg CO2eq /year) (kg Feeq /year)

Production impacts In use impacts Total In use impacts (lossless)

127.5 (26.2) 657.7 (137.3) 785.1 5.09

56.6 (11.6) 2.2 (0.5) 58.8 0.0255 (0.0188)

42.5 (8.7) 201.7 (41.9) 244.2 0.79 (0.36)

each period: n n   pgen pgen (i) )h = [( |ptotal (i)|, )(|ptotal (i)| ptotal ptotal (i) i=1

h

(

h

i=h1

p

gen where ( ptotal )h is the weighted average responsiveness of electricity generated by coal or gas plant to unit changes in total electrical demand during the time periods in set h (h1 to hn ), pgen (i) is the increase in average electrical power generation from gas or coal generating plant during the period i, and ptotal (i) is the increase in average electrical power generation for whole electrical grid during the period i. The net change in generation from coal and gas is then calculated by multiplying the weighted average responsiveness for gas and coal generation for each 5-minute time step by the net change in demand associated with adding the battery to the PV system and summing these over the entire year. The battery model determines the net change in demand:

Egen =

n  i=1

[pnet (i)t(

pgen (i) ), ptotal (i)

where Egen is the total net change in electricity generated from a type of generating plant (gas or coal) for periods 1 to n, and pnet (i) is the net change in electrical demand in time period i due to battery operation (compared to PV system without battery). We now consider the combined annual production and in-use impacts. Table 9.4 shows the combined impacts associated with adding a 430-Ah battery to a 3.29-kW peak PV system. These results are comparable to the case of adding a battery to a modern four kWpeak PV system. Not unexpectedly, metal depletion impact is dominated by battery production,

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whereas climate change and fossil fuel depletion impacts are dominated by battery use. Table 9.4 shows the information for a 430-Ah battery to a 3.3-kW peak PV system. As there are no energy losses with this battery, these values can be interpreted as the impacts associated with the lossless time-shifting of demand from the evening to the day. The difference between the in-use impacts for the lossless battery and those for the realistic battery can, therefore, be interpreted as the environmental impacts due to energy losses in the battery, which are two orders of magnitude greater than those associated with shifting demand from the evening to the day. To put these results into perspective, the total annual climate change impact for this battery has an equivalent impact in terms of kg CO2 eq/year as driving 4362 km in a “good” (180 g CO2 eq/km) UK petrol vehicle. Alternatively, using the same assumptions regarding the responsiveness of fossil fuel plant, it can also be equated to an average 2009 UK household (4460 kWh/year) increasing annual electricity consumption by 946 kWh, an increase of 21%. Substance flow analysis, energy analysis, and life cycle assessment were applied in this work to assess the potential environmental impact of battery systems. The methods were chosen to focus on different environmental aspects and geographical scales of battery applications [99]. Substance flow analysis (SFA) was used to assess whether or not a technical option could solve a problem in principle on a macrolevel. Energy analysis was applied since battery systems require energy for production and to function, and the use of energy is coupled with resource use and emissions. Life cycle assessment was used to identify significant environmental aspects of different battery technologies. Compared with other methods (e.g., material flow analysis and energy analysis), it includes potential environmental impact connected to both material and energy flows, and it can be used to determine whether decreased emissions or resource use are shifted to other environmental problems. For portable batteries, dissipative losses of toxic metals from incineration and landfills are of environmental concern. Indicators of global metal flows were used to assess the potential environmental impact of metals used in portable batteries. Metals of special concern were identified according to their lithospheric extraction indicators (LEIs), calculated for each. An LEI value greater than one shows that the anthropogenic use of metal exceeds the natural turnover and thus indicates a risk for substantially increased metal concentrations in the environment and thus increased environmental impact. The assessment of the battery market 1999 showed that cadmium

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and nickel were of concern due to their high LEIs (4.4–5.6). The ratios of battery metal flow to natural flows are 1.4 for cadmium, 0.07 for nickel, and below 0.01 for other metals, indicating higher environmental concern for cadmium than for the other metals used in portable batteries. For the projected market increase of NiMH and Li-based batteries assumed, NiMH batteries may cause a net increase in nickel use compared with 1999. However, the use of nickel in batteries has a low ratio of battery metal flows to natural flows (0.06–0.3), indicating low potential environmental impact. For the other metals studied, the LEIs are below 0.7, and the ratios of battery metal flow to natural flow are below 0.06, which suggests that these material flows have a low impact on the global level. Emerging portable battery technologies (lithium-ion and nickel–metal hydride) have a lower impact, based on indicators of anthropogenic and natural metal flows, than the NiCd technology. A growing battery market indicates that portable batteries may be an important end-use of cobalt, neodymium, lanthanum, cerium, praseodymium, and lithium. Increasing demand for these metals may result in higher metal prices, which may limit the growth of Li-ion (Co) and NiMH (AB5) technologies. Higher prices of metals used in batteries may create incentives for battery collection and recycling. Important parameters influencing the energy use throughout the life cycle of batteries were assessed through energy analysis of eight different battery technologies used in a standalone PV-battery system. Measures of the performance of the different battery technologies used in a PV-battery system were obtained by the energy return factor and the overall battery efficiency. With a battery storage capacity three times higher than the daily energy output, the energy return factor for the PV-battery system ranged from 0.64 to 12 for the different cases. This means that 8.1–156% of the energy output is required to produce the PV-battery system. If the value of the energy return factor is less than one, then the indirect energy used to produce and replace the device is greater than the energy output. In this case the device works similar to that of a nonrechargeable battery, simply moving energy from one place to another. If PV-battery systems are to contribute to renewable energy supply, it is important to improve the energy efficiency of all their components. For a PV-battery system with a service life of 30 years, the energy payback time is 2.4–46 years, depending on the battery technology and operating conditions. The energy payback time is 1.6–3.0 years for the PV array and 0.55–43 years for the battery, showing the energy-related

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significance of batteries in PV-battery systems. Some of the emerging technologies studied (e.g., Li-ion, NaS) exhibit performance suitable for use in PV-battery systems, resulting in higher energy return factors and overall battery efficiencies than for the established battery technologies. The influence of different parameters on environmental impact and energy flows of battery systems have been assessed. Important parameters influencing material flows are material requirements for battery production, battery service life, collection efficiency, and material recovery rate. Product characteristics of portable batteries that are related to losses of metals are the small size of each battery unit, a large number of battery owners, low concentration of economic value, and type of application. Portable batteries also have a short effective service life, which increases the turnover of materials. Material flows of industrial batteries are easier to control due to the limited number of owners and the large size, which reduces the risk of loss and inappropriate disposal. To decrease losses of metals to the environment, the collection of spent products is more important than the technical efficiency of recycling processes. NiCd battery recycling is energy efficient, even at very long transportation distances, at collection rates of 10–85%. Important parameters affecting energy flows in battery systems are the battery charge–discharge efficiency, the type of cycling regime, the battery service life, and the energy requirements for battery production. In cases where the focus is on the efficient use of fossil fuels, and electricity generated by solar energy can be considered as a free energy source, a high energy return factor is important. This measure may be important in the expansion phase of PV-battery systems. Sensitivity analysis showed that the charge–discharge efficiency is the battery parameter with the highest influence on the energy return factor and is most important for lithium-ion, sodium–sulfur, polysulphide–bromide, vanadium–redox, and zinc–bromine batteries. Service life, energy density, and energy requirements for battery production are of equal importance for nickel–cadmium, nickel–metal hydride, and lead–acid batteries. The overall battery efficiency provides a measure of the efficiency of a closed renewable system, where renewable energy has to be used as efficiently as possible. The battery charge–discharge efficiency has the greatest influence on the overall efficiency. Lithium-ion and sodium–sulfur are emerging battery technologies with favorable characteristics in this respect. The environmental impact of battery systems can be reduced by matching operating conditions and battery characteristics in a life cycle perspective. To decrease the environmental impact of battery systems, the development

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of battery technologies should aim at the recycling of materials, increased service lives, and higher energy densities. To decrease the environmental impact arising from the use of metals in battery systems, metals with relatively high natural occurrence should be used, and regulations implemented to decrease the need for virgin metals. To increase the overall energy efficiencies of battery systems, the development of battery technologies should aim at higher charge–discharge efficiencies and more efficient production and transport of batteries. Different methods of assessing environmental aspects of battery systems have been applied. Life cycle assessment (LCA) is a comprehensive method of describing and identifying environmental aspects of battery systems. Life cycle inventory data for material groups can be used as estimates in life cycle assessments when specific data on materials are not available. In this way, the environmental significance of a material can be identified at an early stage in the assessment. Following the identification of environmental hot spots, targeted methods may be applied. The ecotoxicity of metals is important when evaluating battery systems, and this is difficult to interpret in LCA. To thoroughly address ecotoxicity aspects, LCA can be combined with other methods, for example, environmental risk assessment (ERA) to model the exposure and response of organisms to different metals. ERA has extensive data requirements, and the focus is on assessment of the risk of actual effects that determine if measures have to be taken. Substance flow analysis can be used for a simplified assessment of potential environmental impact when few data on the battery system are available. The method can be used to assess whether or not a technical option could solve a problem in principle on a macrolevel. By relating metal flows arising from the use of batteries to natural metal flows, the potential environmental consequences of current and future battery markets can be assessed. The method used is in line with the precautionary principle since it enables assessment early in the cause-effect chain when few data on toxic effects are available. It can also be used to indicate whether environmental problems are simply shifted from one to another. Industrial batteries have a higher turnover of energy than batteries in portable applications. Consequently, analysis of energy flows is important when assessing industrial batteries. Energy analysis can be used to assess the net energy output of renewable energy systems requiring energy storage in batteries. The energy return factor and the overall battery efficiency can be useful indicators of the battery system requirements of fossil fuels and electricity from a closed renewable system, respectively. The choice of con-

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version efficiencies for different energy qualities and system boundaries are crucial for a reliable assessment. Primary energy use and emission of CO2 are most significant in battery manufacturing. Emissions and resource consumption of metals are significant in the end-of-life treatment, regardless of whether short- or long-term emissions are considered. Transportation for the collection of spent NiCd batteries has no significant environmental impact, and thus NiCd batteries can be transported long distances for recycling, and it would still be beneficial from an environmental perspective. From an environmental perspective, the optimum recycling rate for NiCd batteries tends to be close to 100%. Batteries manufactured with recycled cadmium and nickel have 16% lower primary energy requirements than if only virgin metals are used. Using recycled cadmium and nickel requires 46 and 75% less primary energy respectively, compared with extraction and refining of the virgin metal. There are considerable uncertainties associated with emissions of metals that may occur in the future. The potential cadmium and nickel emissions were 300–400 times greater than in a 100-year perspective. To avoid dissipative losses, cadmium should be used in products that will probably be collected at the end of their life [100]. Energy return factors and overall energy efficiencies are calculated for a stand-alone PV battery system at different operating conditions. Eight battery technologies are evaluated: lithium-ion (Ni), sodium–sulfur, nickel–cadmium, nickel–metal hydride, lead–acid, vanadium–redox, zinc– bromine, and polysulfide–bromide. With a battery energy storage capacity three times higher than the daily energy output, the energy return factor for the PV-battery system ranges from 0.64 to 12 for different batteries and assumptions. This means that 8.1%–156% of the energy output is required to produce a PV-battery system. If the value of the energy return factor is less than one, then the indirect energy used to produce and replace the device is larger than the direct energy output. In this case the device works as a nonrechargeable battery moving energy from one place to another. For a PV-battery system with a service life of 30 years, the energy payback time is 1.6–3.0 years for the PV array and 0.55–43 years for the battery, showing the energy-related significance of batteries and the large variation between different technologies. The overall efficiency, including energy requirements for production and transport of the charger, the battery, and the inverter, is 0.23–0.82. For some batteries, the overall battery efficiency is significantly lower than the direct energy efficiency of the charger, the

Techno-economic assessment of battery systems

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battery, and the inverter (0.50–0.85). The overall battery efficiency can be useful for identification of important parameters for improvement of the efficiency of the PV-battery systems from a life cycle perspective. The contribution of all transport to the indirect gross energy requirements is low (1.0–9.2%) with 3000 km transportation by a heavy truck. When transportation is done by plane, transport may contribute up to 74% of the gross energy requirements for batteries with low energy density (