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Energy storage : a new approach [2 edition]
 9781119083597, 1119083591

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Energy Storage 2nd Edition

Scrivener Publishing 100 Cummings Center, Suite 541J Beverly, MA Publishers at Scrivener Martin Scrivener ([email protected]) Phillip Carmical ([email protected])

Energy Storage 2nd Edition A New Approach

Ralph Zito and Haleh Ardebili

This edition first published 2019 by John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA and Scrivener Publishing LLC, 100 Cummings Center, Suite 541J, Beverly, MA 01915, USA © 2019 Scrivener Publishing LLC For more information about Scrivener publications please visit 1st edition (2010), 2nd edition (2019) All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at Wiley Global Headquarters 111 River Street, Hoboken, NJ 07030, USA For details of our global editorial offices, customer services, and more information about Wiley products visit us at Limit of Liability/Disclaimer of Warranty While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials, or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Library of Congress Cataloging-in-Publication Data ISBN 978-1-119-08359-7 Cover image: Wind turbine, Rachwal | ∙ Solar Power Plant, Tangencial | Cover design by Kris Hackerott Set in size of 11pt and Minion Pro by Manila Typesetting Company, Makati, Philippines Printed in the USA 10 9 8 7 6 5 4 3 2 1

Contents Preface to the First Edition


Preface to the Second Edition


Acknowledgements to First Edition


Acknowledgements to Second Edition



Introduction 1.1 The Energy Problem 1.1.1 Increasing Population and Energy Consumption 1.1.2 The Greenhouse Effect 1.1.3 Energy Portability 1.2 The Purposes of Energy Storage 1.3 Types of Energy Storage 1.4 Sources of Energy 1.5 Overview of this Book

2 Fundamentals of Energy 2.1 Classical Mechanics and Mechanical Energy 2.1.1 The Concept of Energy 2.1.2 Kinetic Energy 2.1.3 Gravitational Potential Energy 2.1.4 Elastic Potential Energy 2.2 Electrical Energy 2.3 Chemical Energy 2.3.1 Nucleosynthesis and the Origin of Elements 2.3.2 Breaking and Forming the Chemical Bonds 2.3.3 Chemical vs. Electrochemical Reactions 2.3.4 Hydrogen 2.4 Thermal Energy 2.4.1 Temperature 2.4.2 Thermal Energy Storage Types 2.4.3 Phase Change Materials

1 1 2 3 4 5 6 10 12 15 15 15 19 26 27 28 31 31 35 36 37 39 39 40 42 v



3 Conversion and Storage 3.1 Availability of Solar Energy 3.2 Conversion Processes 3.2.1 Photovoltaic Conversion Process 3.2.2 Thermoelectric Effects: Seebeck and Peltier 3.2.3 Multiple P-N Cell Structure Shown with Heat 3.2.4 Early Examples of Thermoelectric Generators 3.2.5 Thermionic Converter 3.2.6 Thermogalvanic Conversion 3.3 Storage Processes 3.3.1 Redox Full-Flow Electrolyte Systems 3.3.2 Full Flow and Static Electrolyte System Comparisons

43 46 48 49 49 50 50 51 51 54 54 55

4 Practical Purposes of Energy Storage 4.1 The Need for Storage 4.2 The Need for Secondary Energy Systems 4.2.1 Comparisons and Background Information 4.3 Sizing Power Requirements of Familiar Activities 4.3.1 Examples of Directly Available Human Manual Power Mechanically Unaided Arm Throwing Vehicle Propulsion by Human Powered Leg Muscles Mechanical Storage: Archer’s Bow and Arrow 4.4 On-the-Road Vehicles 4.4.1 Land Vehicle Propulsion Requirements Summary 4.5 Rocket Propulsion Energy Needs Comparison

59 59 62 63 64


71 72 75 77 78 83 85

Competing Storage Methods 5.1 Problems with Batteries 5.2 Hydrocarbon Fuel: Energy Density Data 5.3 Electrochemical Cells 5.4 Metal-Halogen and Half-Redox Couples 5.5 Full Redox Couples 5.6 Possible Applications

6 The Concentration Cell 6.1 Colligative Properties of Matter 6.2 Electrochemical Application of Colligative Properties 6.2.1 Compressed Gas 6.2.2 Osmosis

66 66 66 67 69 69 70

89 89 91 93 94

6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19 6.20 6.21 6.22 6.23 6.24 6.25 6.26 6.27 6.28




6.2.3 Electrostatic Capacitor 6.2.4 Concentration Cells: CIR (Common Ion Redox) Further Discussions on Fundamental Issues Adsorption and Diffusion Rate Balance Storage by Adsorption and Solids Precipitation Some Interesting Aspects of Concentration Cells Concentration Cell Storage Mechanisms that Employ Sulfur Species Balance Electrode Surface Potentials Further Examination of Concentration Ratios Empirical Results with Small Laboratory Cells Iron/Iron Concentration Cell Properties The Mechanisms of Energy Storage Cells Operational Models of Sulfide Based Cells Storage Solely in Bulk Electrolyte More on Storage of Reagents in Adsorbed State Energy Density Observations Regarding Electrical Behavior Concluding Comments Typical Performance Characteristics Sulfide/Sulfur Half Cell Balance General Cell Attributes Electrolyte Information Concentration Cell Mechanism and Associated Mathematics Calculated Performance Data Another S/S−2 Cell Balance Analysis Method A Different Example of a Concentration Cell, Fe+2/Fe+3 Performance Calculations Based on Nernst Potentials 6.28.1 Constant Current Discharge 6.28.2 Constant Power Discharge Empirical Data

95 96 101 107 109 113 116 118 119 120 122 126 127 132 134 137 140 141 143 145 145 146 146 149 150 153 155 156 157 158 160

7 Thermodynamics of Concentration Cells 7.1 Thermodynamic Background 7.2 The CIR Cell

163 163 166

8 Polysulfide – Diffusion Analysis 8.1 Polarization Voltages and Thermodynamics 8.2 Diffusion and Transport Processes at the (−) Electrode Surface 8.3 Electrode Surface Properties, Holes, and Pores 8.4 Electric (Ionic) Current Density Estimates

175 176 177 179 183


Contents 8.5 Diffusion and Supply of Reagents 8.6 Cell Dynamics 8.6.1 Electrode Processes Analyses 8.6.2 Polymeric Number Change 8.7 Further Analysis of Electrode Behavior 8.7.1 Flat Electrode with Some Storage Properties 8.8 Assessing the Values of Reagent Concentrations 8.9 Solving the Differential Equations 8.10 Cell and Negative Electrode Performance Analysis 8.11 General Comments

184 186 186 186 198 198 206 207 219 225

9 Design Considerations 9.1 Examination of Diffusion and Reaction Rates and Cell Design 9.2 Electrodes 9.3 Physical Spacing in Cell Designs 9.3.1 Electrode Structures 9.4 Carbon-Polymer Composite Electrodes 9.4.1 Particle Shapes and Sizes 9.4.2 Metal to Carbon Resistance 9.4.3 Cell Spacing 9.5 Resistance Measurements in Test Cells 9.6 Electrolytes and Membranes 9.7 Energy and Power Density Compromises 9.8 Overcharging Effects on Cells 9.9 Imbalance Considerations


10 Electrolytes, Separators, and Membranes 10.1 Electrolyte Classifications 10.2 Ionic Conductivity 10.2.1 Measurement Techniques 10.2.2 Nyquist Plot Circuit Fitting 10.3 Ion Conduction Theory 10.3.1 Ion Conduction in Liquid Electrolytes 10.3.2 Ion Conduction in Polymer Electrolytes 10.3.3 Ion Conduction in Ceramic Electrolytes 10.4 Factors Affecting Ion Conductivity 10.5 Transference Number 10.6 Electrolytes for Lithium Ion Batteries 10.6.1 Liquid Electrolytes Non-Aqueous Electrolytes Aqueous Electrolytes

245 246 247 247 249 251 252 256 260 262 263 264 264 264 268

227 228 229 229 233 235 235 236 237 239 240 244 244

Contents 10.6.2 Solid and Quasi-Solid Electrolytes Polymer Electrolytes Ceramic Electrolytes 10.7 Electrolytes for Supercapacitors 10.8 Electrolytes for Fuel Cells 10.9 Fillers and Additives

ix 270 270 272 272 276 282

11 Single Cell Empirical Data 283 11.1 Design and Construction of Cells and the Materials Employed 283 11.2 Experimental Data 287 12 Conclusions and Future Trends 12.1 Future of Energy Storage 12.2 Flexible and Stretchable Energy Storage Devices 12.3 Self-Charging Energy Storage Devices 12.4 Recovering Wasted Energy 12.5 Recycling Energy Storage Devices 12.6 New Chemistry for Electrochemical Cells 12.7 Non-Electrochemical Energy Storage 12.8 Concentration Cells 12.8.1 Pros and Cons of Concentration Cells 12.8.2 Future Performance and Limitations

289 289 290 294 295 298 300 301 302 303 304

Appendix 1


Appendix 2






Preface to the First Edition The main purpose of this book is to present a different phenomenological approach to practical energy storage. Throughout the book, a main thread of the problems associated with the generation, transmission, conversion, and storage of energy is proposed, and various technologies are addressed primarily from a phenomenological viewpoint. The exploration of new processes is a major part of the continual search for improved means of the generation and storage of energy. Although some mathematical developments are presented, this book is not intended as a text on thermodynamics or electrochemistry. Certainly, thermodynamics is employed and generous use is made of mathematical tools, but this is not a text directed toward the development or teaching of such principles. It is assumed that the reader possesses some knowledge of elementary classical physics and mathematics and differential calculus in order to easily understand some of the details of thermodynamics and diffusion processes upon which the mechanisms of concentration cell operations are based. This book presents a broad review of energy technology only in summary fashion in order to provide a background so that the concentration cell approach will be viewed in context with other available means of energy storage. It is necessary to cover a reasonable portion of these subjects in order to make this narrative understandable as an alternative presentation. The rationale behind the concentration cell approach should become apparent as the reader moves through the arguments and reviews. The primary aim is to suggest and describe an alternative approach to energy storage other than the ones that have been pursued in the past, especially so vigorously within the past thirty to forty years. The recognized need for improved and practical ways to store large quantities of energy for later use, such as in load leveling for the electric power utilities, has resulted in innumerable programs sponsored by the Atomic Energy Commission, the Energy Research and Development Administration, and the present Department of Energy. xi


Preface to the First Edition

However, many of the concepts regarding the processes at electrode surfaces evolved during the prolonged writing of this book, and the mathematics was developed subsequent to the experiments with laboratory cells described here. The work is hardly complete, and much still needs to be explained. Our astonishment was significant when we first observed the large voltages produced in symmetrical cells with electrodes of large micro-areas. Hopefully, this book will engender interest in acquiring a basic understanding and stimulate further explorations into other, alternative methods of storing energy that may have been largely overlooked, such as the class of phenomena generally referred to in electrochemistry as “concentration cells.” There is much opportunity to store energy in an efficient and easily reversible fashion. Since this type of cell makes use of the colligative properties of substances, many different combinations of materials can be employed in such cells. The first five parts are devoted to a general discussion of energy issues and the presentation of tutorial information. The author gathered most of the data and operation and construction details during the various development projects undertaken in redox and concentration types of cell and battery development. Much of the technical information, performance data, and design parameters were obtained while the author was with the Westinghouse Electric Co. Research Center in Pittsburgh, the General Electric Co., the Research and Development Center in Schenectady, NY, and the Technology Research Laboratories, Inc. in Durham, NC. The practical application aspects of a category of phenomena in physics and electrochemistry have been largely overlooked. That category is the source of electric potentials that can be produced from concentration differences of the same chemical ionic species opposite electrode surfaces. The only practical answer to all-around energy storage needs may lie in the application of the class of phenomena referred to as electrochemical concentration cells. The electrical potential results from structured cells with intense differences in concentration of the same elemental (chemical) specie at two different oxidation states. Storage continues to be a main concern in the whole spectrum of energy related issues. There is no question regarding the efficacy or scientific soundness of the principles employed for storing energy in this fashion. The approach is well beyond such serious concerns. The questions that remain are those regarding the ultimate practicality and competitiveness with respect to other methods of reversible energy storage in such matters as its ultimately achievable efficiency, cost, and energy density.

Preface to the First Edition xiii The information presented in this book is the result of forty years of search and experimental researching for a method of storing energy in a reversible, dependable, and life-long manner. Research has largely been centered on the electrochemistry of what has in recent years become known as redox systems. Both static and full flow electrolyte systems have been explored. The resultant system that satisfied most, if not all, of the imposed practical requirements and appeared to offer the least limited performance with future development was the concentration cell – a significant departure from the normal path of such studies. This mechanism for storage is based upon Nernst’s equation for the chemical potential that can be derived from the ratio of concentrations of the same ionic substance at opposing electrodes in an electrochemical cell. There are two further volumes planned in this series on energy storage, the first of which, tentatively titled, Concentration Cells: Fabrication Methods and Materials, is due to be published by Wiley-Scrivener in September 2011. The aim of this second volume will be to provide the engineer and scientist with the most comprehensive coverage of the concentration cell yet written, with a view toward employing the concentration cell in the storage of energy on a large scale. Ralph Zito

Preface to the Second Edition The purpose of this new edition is to present an expanded and updated description of the methods of energy storage including emerging and future trends. There is no doubt that energy storage will remain a critical component of our society’s sustainability efforts for the foreseeable future. Thus, an expanded and modern version of this book is deemed necessary. To resolve and manage the unique energy challenges of the 21st century, much effort has been focused on identifying the most efficient, effective and durable energy storage solutions that can adequately meet our multifaceted life requirements. In this book, we address a wide range of energy storage methods including electrochemical, electrical, mechanical, thermal, and chemical. In this new edition, the emphasis is on basic mechanisms and we explore, in more depth, various energy storage techniques. A special focus is given to alternative electrochemical storage, including concentration cells.  This book intentionally presents a broad review of energy technology so that contemporary developments may be viewed in context and a comparison may be drawn between the various available means of energy storage. This new edition is the result of the second author’s doctoral education at the University of Maryland, College Park, three years of experience as a scientist at the General Electric Co., Research and Development, in Schenectady, NY, postdoctoral experience at Rice University, and an academic career of more than fifteen years at the University of Houston. In this new edition, an expanded discussion of energy fundamentals has been added. More examples and details have been provided on energy storage methods. A new chapter about electrolytes and membranes has been added that discusses battery, supercapacitor and fuel cell electrolytes with more emphasis on the fundamentals of ion transport phenomena. Several new sections highlighting the future trends in energy storage have also been added to the final chapter of this book. The sections include new trends in flexible and stretchable energy storage devices, self-charging energy storage, recovering wasted energy, methods of recycling energy xv


Preface to the Second Edition

storage devices, new chemistry for electrochemical cells, and trends in non-electrochemical energy storage. This book is intended for upper level engineering and science students with some background in material science, physics and chemistry. This book is also suitable for engineers, scientists, and entrepreneurs with technical background who have an interest and curiosity about energy storage. Haleh Ardebili September 20, 2019

Acknowledgements to First Edition I am taking advantage of this written opportunity to express my deep appreciation and thanks to some of the many people who have made writing this book possible. With their participation and contributions over the years in research and development programs at TRL, Inc. and GEL, Inc., they have made many contributions to accumulating more experience with these energy technologies. It is not practical to even attempt to list all those individuals who, over many years, have designed and constructed experimental hardware, from which we have learned how to cope with a multitude of related problems, and who have made writing this book possible. In particular, I would like to acknowledge the following for their direct influences on the developments. Special thanks go to Prof. Richard S. Morse and Prof. David Ragone at MIT, who both launched this whole area of redox systems in Cambridge, MA in 1968. Also, I would like to thank Keith Poulin for his administrative and managerial skills, Graham Wilson, whose fertile imagination, combined with his engineering capabilities, resulted in most of our very early hardware display devices, Joseph O. Dixon for his engineering skills, faith, and confidence, D. Morris for his problem solving, tenacity, and positive attitude, and Prof. Charles Harman from Duke University for his technical and personal support and patience. There are so many people, including my dedicated secretaries over the years, Phoebe Rasmussen, Sarah Tortora, and Patricia Pearson, who need to be commended for staying with us through difficult situations. Others include the many shop and laboratory people who coped with rather messy conditions in handling such materials as graphite and carbon black and unfriendly electrolytes in large quantities. I would like to thank one individual in particular, Sir John Samuel, who has steadfastly maintained a confidence in our technical accomplishments and has been responsible for obtaining much needed support for our programs. This book reflects only a tiny portion of all the work from so many others who preceded its writing and made writing it possible. Ralph Zito xvii

Acknowledgements to Second Edition This book would not be possible without a lifetime support from many people and organizations. I would like to first acknowledge all funding agencies that have supported my research: University of Houston (UH), U.S. National Science Foundation (NSF), Department of Defense (DoD), Texas Center for Superconductivity at UH (TcSUH), Subsea Systems Institute (SSI), and Texas Space Grant Consortium (TSGC). I would like to express my sincere gratitude to all the students who have diligently worked in my lab at the University of Houston, and who have contributed to advancing science and engineering, specifically, in the area of energy storage. In particular, I would like to mention my PhD students, Qin Li, Mejdi Kammoun, Mengying Yuan, Taylor Kelly, William Walker, Sean Berg, Bahar Moradi, Anh Vu, Sarah Aderyani, and Qiang Fu. A special acknowledgement is extended to Dr. Mejdi Kammoun for his dedication to research in my lab. I would like to offer my special thanks to all my colleagues and friends at the University of Houston, and other universities, national labs, and industry, who have collaborated and interacted in research and educational activities. I am humbled to be a part of this distinguished scientific community. The support and dedication of the administrators and the staff of the Department of Mechanical Engineering, the Cullen College of Engineering, and the University of Houston, are also greatly appreciated. Finally, I would like to thank my family for their love, patience and support, especially, my mother, Parichehr, my son, Arvin, and my husband, Pradeep. I dedicate this book to the loving memory of my father, Bijan Ardebili (1944–2015) who was the most inspiring and positive supporter in my life. Haleh Ardebili

1 Introduction Billions of years ago, during the Big Bang nucleosynthesis, chemical energy was stored in the chemical elements. We now store electrochemical energy in our modern-day batteries. Over the estimated 14 billion years life of our universe, energy storage has been and will continue to be an essential part of all things in existence. In the last few decades, we have become dramatically more dependent on reliable and long-lasting energy storage. Several reasons have contributed to this increased reliance, including the widespread use of modern portable or wearable devices, significant growth of human population, and the rising demands of the 21st century lifestyle. Meanwhile, there are other driving factors at play. Our conventional sources of energy like nonrenewable fossil fuels continue to decline, and the environmental, economic and political concerns surrounding the generation and use of energy are growing, leading to a wide range of energy challenges and concerns. To resolve and manage the unique energy challenges of the 21st century, much effort has been directed toward identifying the most efficient, effective and durable energy storage solutions that can adequately meet our multifaceted life requirements. In this book, we will address a wide range of energy storage methods with emphasis on their basic mechanisms. In this chapter, we will first discuss the “energy problem” and the motivation underpinning the development of efficient approaches to store energy. We’ll then classify, compare and discuss various methods of energy storage. In the final section, an overview of this book will be provided.

1.1 The Energy Problem All the discussions and dire announcements in technical literature during recent years have certainly made everyone aware of the “energy problem.” There is not much doubt that we are confronted with a real problem of domestic and international importance. The critical issues concerning the Ralph Zito and Haleh Ardebili. Energy Storage 2nd Edition: A New Approach, (1–14) © 2019 Scrivener Publishing LLC



Energy Storage 2nd Edition

availability of energy sources and their efficient use are rapidly becoming vitally important. Increasing population, in conjunction with the greaterthan-ever energy and materials demands that people are making in order to increase their comfort, travel, and other lifestyle choices is indeed causing greater stress. All of these require not only an increased availability of energy but also more effective ways of utilizing what is available.

1.1.1 Increasing Population and Energy Consumption

9 8 7

Figure 1.1 World population increase.


35 20 40













6 20

World Population (Billions)

Human population is growing at a rapid rate. The world population is projected to reach about 9 billion by 2040, as depicted in Figure 1.1. Consequently, energy consumption has risen significantly due to the increasing human dependency on various technologies and the power and energy they require. According to the 2016 International Energy Outlook (IEO) by U.S. Energy Information Administration (EIA), the total world energy consumption is projected to surpass 900 quadrillion Btu by 2040. British thermal unit (Btu) is defined as approximately 1,055 Joules. The electrical power and industrial sectors are projected to each increase over 300 quadrillion Btu by 2040. The measured and projected total delivered energy consumption per sector is shown in Figure 1.2. Clearly, we must seek sustainable and global energy solutions to combat the significantly large energy consumption demands of the 21st century. The main efforts of research and development have been directed toward the development of new alternatives or finding more primary sources of energy. For the present, and until the discovery of a new class of phenomena, we have a fairly good idea of what can be accomplished. We know what alternative sources are possible – alternative presumably to



World Energy Consumption (quadrillion Btu)

1000 Industrial Sector


Residential Sector


Transportation Sector 400 Commercial Sector 200

Electric Power Sector




40 20












Figure 1.2 World energy consumption by sector type.

petroleum products. Yet none of them are nearly as attractive for portable or motive power unless we significantly lower our criteria. It would appear that sources of energy are plentiful on planet Earth. However, they are often locally unavailable, too bulky, too unpredictable (solar and wind), and/or too dangerous to be portable. An effective method for storing energy would greatly reduce the problem and would provide low-cost energy for everyone. It seems that not nearly as much attention or support has been directed toward the problem of storage as that which has been directed toward generation. Perhaps this difference is due to the absence of many promising approaches to accomplishing the latter. This book presents a different approach and aims to stimulate additional efforts toward the search and development of better storage.

1.1.2 The Greenhouse Effect The Earth is believed to be approximately 5 billion years old. For a relatively long time, life could not be sustained outside the ocean. This is mainly due to the absence of a properly formed atmosphere that could protect life from lethal radiations. Several billions of years later (about 600 million years ago) the Earth’s atmosphere, capable of protecting life, was formed. For millions of years, the atmosphere and the surface of Earth have effectively reflected the solar radiation to Space, and the consequent infrared (IR) radiation has passed through the atmosphere back to Space as depicted in Figure 1.3. However, it is believed that more recently, some of the IR radiation has been absorbed by the greenhouse gases below the


Energy Storage 2nd Edition


Sun 2 1

3 4

Earth’s Atmosphere

Earth Greenhouse Gasses


Earth’s Surface


Figure 1.3 (1) Solar rays radiated toward Earth, (2) some reflected by Earth’s atmosphere, (3) some reflected by Earth’s surface, (4) infrared radiation passes through the atmosphere and out to Space, and (5) some of the infrared radiation is absorbed and re-emitted back to Earth by the greenhouse gases.

Earth’s atmosphere and the absorbed IR radiation is re-emitted back to Earth. This phenomenon is commonly referred to as the “Greenhouse Effect”. It can be mainly attributed to the use of fossil fuels and the human industrial evolution. The greenhouse effect presents yet another major reason for the global community to actively move toward environmentally friendly and renewable energy harvesting, conversion and storage.

1.1.3 Energy Portability Energy portability is another major challenge. We cannot carry windmills around – they are huge and dangerous. A waterfall, due to topographical considerations, is not available everywhere, and its size is immense for the intermittent power and energy produced. Solar cells can be designed to be portable but their usage is confined to the available sunny days and limited daytime hours. There are really not too many attractive choices for portable energy harvesting and storage. Batteries and supercapacitors are the least obtrusive and the most predictable limited secondary sources, but they are not practical as large-scale primary or secondary sources. Windmills and photovoltaic cells are almost useless without either storage or the assistance of an electric utility power grid, which operates on nuclear power or coal fuel. It would appear that the energy source trap has merely changed shape.



Ideally, high energy and power density “batteries” of some sort that are charged by nuclear or fossil fuel would be a good solution for smoothing the irregularities in the distribution and availability for the planet’s population. The term “batteries” used here only refers to some mechanism for practical storage. So far, the most promising is probably an electrochemical method. Compressed air, metal springs, flywheels, etc., all have very serious drawbacks. Most generating facilities are not portable, nor would most people wish to live with them in their midst.

1.2 The Purposes of Energy Storage Storing energy in its many forms in nature is a vital part of all processes as well as life itself on Earth. As we explore these processes and their importance to us, we can gradually make observations that lead to some revealing conclusions. One of our purposes is to examine the general area of energy storage and to identify the key mechanisms that have significant roles in nature and civilization. Then we will develop a description and a reasonably detailed understanding of why we need to store energy and how the various mechanisms we employ work to satisfy these needs. Most of this book is devoted to electrochemical processes and, in particular, full flow electrolyte cells that are frequently referred to as redox batteries. In a very general sense, there are only three purposes for the storage of energy: to make an energy supply portable from essentially non-portable sources, to store from an ongoing source for use at a later time, and to change the ratio of power-to-energy, as accomplished by flywheels, capacitors, etc. All applications of energy storage can be put into one or more of these categories. Certainly, if we wish to power a portable power tool or an electric automobile, a hydroelectric plant is hardly practical. However, if we use the energy produced by the hydroelectric station and store part of what is not immediately needed in an electric battery, it becomes a pragmatic concern for the electric vehicle. Nuclear energy sources are hardly portable on a small scale. But in similar fashion, as for nonportable hydroelectric stations, storing portions of the generated energy in some sort of device such as a battery could become useful in mobile electric vehicles. In the second instance, we might have the need to store solar energy during the daylight hours for use after sunset to power lights, etc. There are many cases where the convenience factor is not met – where the generation


Energy Storage 2nd Edition

or the source of energy is occurring at a time that does not coincide with the need or application time. It must be noted here that when we refer to energy we are essentially referring to the “capacity to do work” in the classic sense of Force × Distance. This capacity is being transferred from one source to a device that is capable of storing that capacity for work to be used at a later time. The storage of energy in one form or another for later use has been extremely important not only to mankind but also to virtually every form of life. Storing energy in the form of chemical structures, such as carbohydrates, enables life forms to survive for periods of time between food intake activities. Obviously, all activity and motion are manifestations of exchanges of energy from one level to another. The exchange of potential energy for kinetic energy of motion for falling objects is a simple example. Few energyexpending actions make use of prompt sources. It can be argued that even the processes that produce solar radiation make use of energy that was stored at some earlier stages of cosmic evolution. An energy bank deposit must be made at an earlier time for energy to be in an available and usable form at the time of demand. This stored, but temporarily unavailable energy is usually not in a form that is needed for the specific operations at the demand time. Hence, there is usually a conversion mechanism accompanying the stored energy device in order for the energy reserve to be useful. There are many examples that can illustrate this simple, but basic, premise. They are presented in a later section of this book. In biochemical processes, the stored carbohydrates, perhaps in the form of sugars, are oxidized in order to produce the heat necessary to sustain life.

1.3 Types of Energy Storage We can classify energy storage in many different ways. Storage can be classified based on the energy type, portability, cost, energy density, power density or other. All classifications are valid and serve specific purposes associated with the selection and application of a suitable energy storage method. From a phenomenological perspective, energy storage can be classified based on the energy type and mechanism. Here, we classify energy storage into six main types: (i) electrochemical, (ii) electrical, (iii) mechanical, (iv) thermal, (v) chemical, and (vi) biological, as shown in Figure 1.4. Commonly known batteries and fuel cells are considered electrochemical energy storage devices in which the active materials at the



Energy Storage


Electrochemical Electrical

Chemical Thermal Fossil Fuels



Capacitors Ice Storage

Fuel Cells

Biological Starch



Nanocapacitors Flywheels

Concentration Cells

Molten Salt

Superconducting Magnetic Energy Storage (SMES)

Oxygen Biofuels

Hydraulic Accumulator

Solar Pond

Supercapacitors/ Ultracapacitors

Figure 1.4 Classification of energy storage types.

electrodes undergo chemical reactions and the chemical energy is converted into electrical energy. On the other hand, the traditional capacitor stores purely electrical charge in the electrodes and is therefore considered an electrical energy storage device. Similarly, in supercapacitors, electrical charge is stored in the electrodes where generally, no chemical reaction takes place; however, the electrolyte can undergo salt dissociation where ions play a major role associated with the double-layer effect (further description will be given in a later chapter). In general, supercapacitors can be considered as electrical or electrochemical energy storage devices. Mechanical energy storage is one of the earliest human used energy storage; examples include bows and arrows, springs and flywheels. Another relatively old storage type is thermal energy storage, and includes ice, molten salt, and solar pond. An example of thermal energy storage using molten salt (phase change material) is depicted in Figure 1.5. In a sense, chemical energy is stored everywhere and in all chemical elements and compounds in existence. From the perspective of human consumption, fossil fuels (i.e., gasoline, coal, etc.), hydrogen (used for hydrogen fuel cells), and oxygen (used for solid oxide fuel cells) are some of the most commonly used chemical energy storage materials. Biological energy storage is also another critical storage method inherent in all biological organisms; examples include starch and glycogen. Biofuels are

Energy Storage 2nd Edition


Thermal Energy Storage Tower Hot Solar Rays


Steam Generator

Electric Turbine

Molten Salt Cold

Solar Mirrors

Figure 1.5 Thermal energy storage using a molten salt phase change material.

being used as a replacement or an alternative to fossil fuels for a variety of applications including transportation. Devices that store energy can be compared based on their energy density and power density in a famously known plot called the Ragone plot as shown in Figure 1.6. The vertical axis of this plot represents the energy density in Watt. hour/kilogram (W.h/kg) and the horizontal axis represents the power density in W/kg. In some Ragone plots, the energy and power


10 Hrs

1 Hr

Fuel Cells

Energy Density (Wh/kg)

100 NICd Lead-Acid Battery

0.1 Hr Lithium Battery

36 sec




3.6 sec

Double-Layer Capacitors


0.36 sec

36 msec AluminumElectrolytic Capacitors


0.01 10

100 Power Density (W/kg) 1000

Source US Defence Logistics Agency

Figure 1.6 Ragone Plot (Courtesy of U.S. Defense Logistics Agency).




density axes may be switched. The diagonal time lines represent the discharge time, which is the time to release the energy stored based on the specific energy and power densities of the device. As shown, a fuel cell can provide relatively high energy density; however, its power density is lower compared to other devices like capacitors, or even batteries. On the other hand, capacitors can provide very high power density compared to other devices. Energy and power densities can be expressed in terms of gravimetric (per kilogram) or volumetric (per cubic meter) densities. The choice of the unit representation is often governed by the specific applications under consideration. For example, in certain applications, volumetric density may be a higher priority in consideration due to the dimensional and shape constraints of the application system, such as in cell phones or laptop computers. In other applications with weight constraints, lighter devices that deliver more energy or power (i.e., higher gravimetric densities) may have a higher priority such as the case of electrical automobiles. For the majority of applications, both high gravimetric and high volumetric densities are desirable, and both should be evaluated when a device enhancement is made, including improvements in the materials or the design. We should note that realistically, as we improve the energy density of an energy storage device, we may inadvertently reduce its power density due to the inherent physical and chemical constraints and tradeoffs in the mechanism of energy storage, active and passive material properties, and design parameters. This tradeoff can be observed from the Ragone plot (Figure 1.6) where fuel cells, on the top left corner, exhibit high energy and low power densities, and conversely, the capacitors, located on the bottom right corner of the plot, show high power and low energy densities. Batteries and supercapacitors provide a good compromise, where relatively moderate or high energy and power densities, can be attained for a wide range of common applications. In addition to the energy and power densities, there are several other important application requirements that often must be considered, to investigate a suitable energy storage device. These additional requirements may include cost, reliability, safety, cycle life, and flexibility, which are not included in a typical Ragone plot. In other words, before we can assess the suitability of an energy storage device for a certain application, we may need to make a comprehensive comparison of various energy storage types and devices, with respect to multiple pertinent parameters.


Energy Storage 2nd Edition

1.4 Sources of Energy In most scientific or socioeconomic discussions about the sources of energy on Earth, often a common list of sources are mentioned that generally include • • • • • • •

Fossil fuels Nuclear energy Hydropower Biomass Solar energy Geothermal energy Wind and tidal energy

Although these are valid sources of energy (i.e., fossil, nuclear and renewables), perhaps we can delve a little deeper into the “original” or “primary” sources of energy. There are essentially four “primary” energy sources of which we are presently aware. There is only one source that we know of from which all others have derived, and that source is nuclear. This can be a philosophical issue, bordering on the question of primal causes. That question is perhaps unanswerable, and it has no relevance to our subject matter in this book, except to dispel some of the vague popular literature and general conversations on the problem of finding or developing alternative energy in order to meet the increasing demands worldwide. In actuality, we know little of the causes and the effects, such as whether electromagnetic radiation is the result of nuclear (sub-atomic) phenomena or the reverse. Regardless, the overall macroscopic benefits to us on Earth are sunlight and all its attendant benefits and nuclear reactor systems. However, despite all the delightful speculations, the subject of available energy sources on Earth reduces to the following list: • Nuclear (in the form of radiation from the Sun) • Nuclear (in the form of terrestrial based reactors to produce steam power) • Gravitational (tides due to the motion of our Moon about the Earth) Various other substances and processes are referred to as power or energy, such as coal, petroleum, wood, wind, hydroelectric, etc. Yet if we simply trace their sources back to the origin, then it becomes obvious



that they are all different manifestations, perhaps in delayed fashion of the three sources identified above. For example, hydroelectric power and wind power are, in essence, different manifestations of solar energy. Water evaporating from bodies of water at lower elevations, as a result of solar heating, and transporting the vapor to areas of higher elevation to become rainwater for reservoirs can provide electric power. Similarly, solar heating of various ground regions causes air turbulence, which can be quickly transformed into horizontally moving air masses as wind and can be utilized for windmill operations. Burning coal or petroleum to run heat engines of various sorts merely makes use of the solar energy that was incident on the Earth’s surface long ago and over a long period of time. These resources will eventually become extinct, especially at the increasing rate with which people everywhere are depleting them. The advertised searches for new or alternative energy sources need to be identified in clearer terms. If what is meant by “finding and developing alternative energy” is to actually find some source other than solar in one form or another, or nuclear, then that declaration is misleading, if not actually false. All we know scientifically is clearly evident. Looking for an unidentified supply of usable energy is almost like looking for a new color that no one has yet seen – a rather impossible task. To look for this new, unidentified source implies discovering a new phenomenon in nature. Such discoveries do happen, but rarely by design or systematic searching. The development of new technologies comes only after a discovery has been made. Near the end of the 19th century, Henri Becquerel accidentally discovered radioactivity when his photographic plates placed close to uranium ore showed signs of exposure. In a series of steps over a rather short period of time other investigators such as Madame Curie, with her separation of radium from masses of pitch blend, and almost concurrently with Roentgen, who discovered what he called X-rays that emitted from a cathode ray tube whose anode was bombarded by high speed electrons of X-rays, all suggested a far more complex structure of matter. Then, others such as Bohr and Rutherford began developing hypotheses and models to explain these new events in the laboratory. Without these discoveries we probably would not have any knowledge of nuclear processes or how to make practical use of the immense energy available from within the atom. Until there is empirical evidence of some new phenomenon, or at least a workable theory, there is little purpose to searching aimlessly for something that may not exist, and if it does exist, we have no idea what to look for. The idea of developing new energy sources may be pointless, unless the intention is to engineer and make those sources, of which we are already aware, more readily available


Energy Storage 2nd Edition

for application. Meanwhile, perhaps something will come along before the earth runs out of suitable radioisotopes and fossil fuels. It appears that until, and only if a truly new source appears, we will have to utilize the innumerable variations of presently known energy source systems.

1.5 Overview of this Book This book focuses on the phenomenological aspects of energy storage and explores various approaches and techniques of storing energy. A particular attention is focused on electrochemical and concentration cells. Chapter 2 provides an overview of the fundamental concepts of energy. This includes basic concepts behind classical mechanics and mechanical energy, electrical energy, chemical energy, and thermal energy. Chapter 3 discusses conversion and storage of energy and presents various forms of energy storage that are familiar to all of us. The major purpose of this book is to emphasize the importance of this general approach to energy storage as a rather new technical viewpoint wherein the physical principles employed are basic to all matter. Perhaps the best example of this is the mechanical and thermal behavior of gas compression. This book focuses on a single and particular class of artificial means of energy storage, a general class of energy conversion known as electrochemical processes. More specifically, this book concentrates on aqueous systems that are largely compatible with ambient conditions of temperature, pressure, and a chemical environment. Some mention will be made of these other systems and devices, for example, non-aqueous systems, and solid electrolyte based electrochemical cells, to retain a better perspective regarding the electrochemical systems of prime interest here. The reasons for this selection will become clear as the argument develops and as some background information associated with major application criteria is provided. In the chapters to follow, we outline the requirements of our energy needs and establish a background of information upon which we can more clearly assess the various storage options available to us. However, the primary purpose is the description of a new approach. The science is not new, but it has received scant attention as a possibility for reducing our energy problems. The first three sections describe the background and rationale of electrochemical storage and, more specifically, redox types of systems, based on first-hand knowledge in developing earlier attempts at large-scale energy



storage for load leveling and standby power. Justifications for redox systems for large-scale, bulk energy, stationary storage applications will also be identified. The first six chapters of this book present not only a general background of energy requirements, competitive methods for storage, and a description of the basics of concentration cell operations, they also provide sufficient information to comprehend this different approach to storage along with some justifications for taking that technical direction. Chapter 7 focuses on the thermodynamics of cells. However, in order to fully appreciate the important details of cell operational mechanisms, it is necessary to delve a bit into the various key processes of diffusion and transport processes. This entails rather lengthy and tedious mathematical analyses and the major portion of such activities is presented in Chapters 8 and 9. Chapter 10 presents discussion of various types of electrolytes used in electrochemical cells including liquid, gel, and solid electrolytes. Chapter 11 provides some empirical data on a single cell. Finally, Chapter 12 presents emerging and future trends in energy storage and a few concluding remarks. A significant amount of space is devoted to explaining the thermodynamics of the concentration cell, and the explanations of the operating principles are numerous. Moreover, keeping track of the materials transport mechanisms and the source of electric potential can be confusing. However, in order to contend with the many aspects of present-day energy technology, it is important to have a working knowledge and comprehension of the basic underlying physics. This book is directed to those readers who would like to have an appreciation of the wide vistas of energy technology and wish to acquire enough understanding to make independent evaluations of modern trends in such matters as conservation, fossil fuels versus solar energy, and so on. The purpose of this book is to stimulate interest in the subject of energy and the dynamics of the physical world around us. Some mathematics is employed where it is necessary in order to establish quantitative energy relationships in solving problems or in the design of devices. The purpose of the mathematics is to provide those who have the interest and skills with a more in-depth understanding of the factors and limitations of the physical world as we now know it. There are also many excellent texts written on energy, theoretical mechanics, and the mathematics of theoretical physics. Some of these are listed in the accompanying bibliography.

2 Fundamentals of Energy The concept of energy is elusive and mysterious. It is constantly being reexamined for greater understanding. It’s an idea or concept about which nearly everyone thinks they have some understanding. It is interesting to note how we refer to the idea by such phrases as “burning excess energy,” “using a lot of energy,” or someone “has a lot of energy,” as if it were a fuel of some sort. The repetition of the word, as with most concepts, conveys a vague feeling of comfort with the notion, that we indeed have a grasp of it. In this chapter, we discuss the concept of energy. We review the fundamentals of various types of energy including mechanical, electrical and chemical, and interject some historical perspectives of energy and its usage.

2.1 Classical Mechanics and Mechanical Energy Frequently, in ordinary conversation or in more popular literature, the term energy is confused with or substituted for force. This confusion dates back many centuries beginning with human attempts to comprehend physical phenomena and the world around them. Even after the beginning of what is known as the scientific method, attributed to Galileo, much about the subject has confounded our comprehension. In the next pages we will address the issue of what energy is.

2.1.1 The Concept of Energy We may never be actually able to observe the quantity named “energy,” but its effects are certainly and easily observable. Perhaps this also adds to the mystery of energy. Lindsay and Margenau presented a timeless review of the history of physical concepts in their book, Foundations of Physics (1936). Though dated, this text presents a very comprehensive treatment of basic concepts both in classical and quantum mechanics. Since we have

Ralph Zito and Haleh Ardebili. Energy Storage 2nd Edition: A New Aprroach, (15–42) © 2019 Scrivener Publishing LLC



Energy Storage 2nd Edition

seen little to compare with this and we don’t wish to compete with their treatment of mechanics, we quote their pointed and well-said statements: All the problems of classical mechanics can be solved without reference to it (energy). The question at once arises: why then should it have been introduced at all? This is what we wish to discuss. We must first remark, however, that the idea if energy is historically much older than the name. Without doubt it goes back at least to Galileo and his observation with respect to machines that “what is gained in power is lost in speed,” referring to the fact that the force required to lift a weight (by means of a pulley system) multiplied by the distance through which the force has to be applied remains constant, though either factor in itself may vary. The concept of work is involved here. Its importance was, however, overshadowed by Galileo’s epoch-making discoveries of the laws of motion, and it was not until the time of Huyghens that it again became prominent in the concept of “vis viva” or “living force”, i.e., a quantity varying as the mass multiplied by the square of the velocity. The attribution of the term energy to the concept of “vis viva” did not come until the 19th century. (1936)

The origin of the word energy is Greek, and it means active, or the capacity to do work. In more recent times, the idea has taken hold that energy is some ethereal form of substance. In quantum and relativistic mechanics, energy is interchangeable with mass (matter) as indicated, for example, in the celebrated equation from Einstein’s Relativity Theory, E = mc2. Kinetic energy remains kinetic energy when, for example, a moving mass slows down by frictional losses. The collective kinetic energy of the mass is disbursed as kinetic energy (motion) of many more, smaller particles (molecules) and measured as a temperature rise. However, kinetic energy can be converted or swapped for potential energy when, for example, a mass loses velocity while slowing down as it moves against an attractive field of force such as gravity. In more recent times, the idea has taken hold that energy is a substance. For those who wish to acquire a detailed knowledge of the history of the concept of energy and force, including the evolution of these concepts over the centuries, the books Concepts of Mass and Concepts of Force by Max Jammer would be helpful. As far as we know, consistent with our observations, energy cannot be created or destroyed. We will ignore, in this book, the processes wherein energy and matter are equivalent, i.e., that matter can be transformed into energy and vice versa because all the phenomena associated with our lives do not occur at the atomic nuclear level. The availability of energy to do work can and does change. We can only transform energy from one of its

Fundamentals of Energy


two forms into the other, when possible, to suit our practical purposes, such as from kinetic energy to potential energy and back to kinetic. Ultimately, all these transformations resolve in an increase of entropy (disorder) in the universe and, therefore, an increase in world temperature. When all things are finally at the same temperature, there is no more useful energy. Another penetrating and early analysis of the subjects of energy (kinetic and potential), mass, and force was put forth by James Clerk Maxwell, author of the electromagnetic theory of radiation, among other profound and basic contributions to physics. In his preface to Matter and Motion, he states: Physical science, which up to the end of the eighteenth century had been fully occupied in forming a conception of natural phenomena as the result of forces acting between one body and another, has now fairly entered on the next stage of progress – that in which the energy of a material system is conceived as determined by the configuration and motion of that system, and in which the ideas of configuration, motion, and force are generalized to the utmost extent warranted by their physical definitions. (1877)

What is energy? It is not observable in the classical sense. Theoretical physicists have interminably debated its ability to be observed and have variously defined the concept over many years. First, it is necessary to identify or define what we mean by an “observable quantity.” Interestingly, there are few directly observable. The rest of these concepts, such as energy, momentum, temperature, etc., are inferred by various experiments and indirect observations. What we call momentum is calculated and seen as the force necessary to change the momentum of a body. We are using the word “observable” in a rather sophisticated sense. It is necessary to divorce our ideas of what we directly experience from those of theories intended to explain the process or sensation. Force, a push or pull, is a directly observable magnitude, as are the velocity of a body in motion, pressure, volume, length, temperature, and even quantity of heat. In reality, the subject of what is legitimately observable is highly debatable. Not only are the semantics of the subject in question, but we must also identify the level of primitiveness or experience of the individual making the observation. However, such quantities as entropy and enthalpy in thermodynamics are not directly observable. Implicitly, we think of energy as being some sort of substance or quality that enables one to overcome an opposing force and move an object to a different position spatially in opposing this force. Examples of certain non-observables are electric and magnetic fields. We can observe their effects, but we are not able to see or observe the field


Energy Storage 2nd Edition

directly. In the case of a magnetic field, the attraction force of a magnet for iron particles can be seen or experienced directly, but not the field itself. In fact, the magnetic field is an invention by physicists to structure a logical and working explanation of that observation. Similarly, we cannot experience an electric field directly, but surely the consequence of such a field is seen or felt when a statically electrified insulator, such as a rod of glass or plastic, gathers bits of fur or paper. The dilemma we tend to fall into is the result of familiarity with the concepts and objects that we think we understand but unfortunately do not. The idea of energy in any form is very complex and sophisticated. Sometimes energy is described as having many different forms – chemical and mechanical energy and perhaps thermal energy as temperature. In actuality these are superficial manifestations of the same thing. We need to distinguish basic ideas and concepts from those involving superficial property assessments. In deference to undue involvement in philosophic issues, let us take a deeper look into the matter of observables, dimensions, and measurable quantities in the physical sciences. As always, along with the benefits reaped from greater numbers of tools and concept evolution in physics, the more likely we become separated from basic contact with the outside, physical world. The tendency is to become too involved with word pictures and symbols, and then it becomes increasingly difficult to separate the reality that we “directly” perceive from the physical models generated in our minds and placed there by myriad conversations and teachings in physical science. Returning to the subject of energy, for example, when we see a large mass such as an automobile moving at speed of, say, 60 mph, we might remark upon the large amount of kinetic energy the automobile has, especially if it collides with a rigid object and suffers severe damage. In this case, as in all others, we do not really make a direct observation of its “energy.” In fact, it is not possible to see or experience energy as such, only the consequences of its transference. In order to make sense out of our observations, we simply invent concepts and weave a fabric of theories, hypotheses, and other explanatory structures. In this manner, we are attempting to not only acquire a better “understanding” of the world around us but also to be able to predict the results of our actions, and perhaps even control their outcome by making use of these concepts. Now, let’s take a critical (analytical) look at the main subject of this book – energy. We cannot make direct measurements of this rather elusive substance. Only the effects or results of energy can be observed and measured. We cannot hold energy in our hands as we can grasp material

Fundamentals of Energy


substances such as a stone or stick of wood, nor can we see it as directly as we see wood, that same stone, or a source of colored light. Energy makes no sound that we can perceive. In fact, energy is more of a concept in classical physics than a “substance,” as in a material object. The idea of energy existing is mostly to explain motions and activities in life. If someone is very active and has great strength, we tend to say that the individual has a lot of energy, implying that he or she is in possession of a larger quantity of some substance that enables him to perform tasks greater or faster than normal. When an explosion takes place we think that a great amount of energy is released. Exactly what do we mean? Large forces are manifest when an explosion takes place, and we see solid objects torn apart and many particles of matter thrown away from the center of the explosion with great velocities. Knowing that large forces operating over very short periods of time are required to do this, the conclusion is that this large amount of stored (potential) energy is released in the form of much action and motion.

2.1.2 Kinetic Energy The quantity we call kinetic energy implies motion (kinetic from the Greek signifying movement). It is the energy associated with physical bodies in motion. Potential energy is the latent ability to do work. Work and energy are essentially synonymous as employed in mechanics. However, doing work implies action. Let us get a bit more involved in the details, units, and ways of measuring energy, always keeping in mind that we quantify on the basis of the results of transformations of what we refer to as energy. Pushing on a body to move it against frictional resistance forces and the force of gravity are too familiar to all of us in our everyday experiences. The force we must exert through our leg muscles to climb a flight of stairs is frequently experienced and readily comprehensible. The term “work” may have originated in times past because of the idea that effort had to be exerted to do the common tasks necessary to life. The amount of energy involved in lifting weights to certain heights is also defined as work. One can appreciate how we probably acquired the terminology and the idea that energy, or work, is defined as force times distance. In other words, the amount of energy required or produced is the product of the distance through which a force is acting times the force. This is simply expressed as

Force × distance = F × d = Energy (Work)



Energy Storage 2nd Edition

The above equation assumes that the force is constant throughout the distance it is acting. Before getting into the more quantitative aspects of energy, we should examine the basic units of measurement that are employed. That subject is usually referred to as dimension theory or dimensional analysis. Despite the seemingly abundant, and perhaps unlimited, number of phenomena and effects in nature ranging from electric, mechanical, thermal, magnetic to hydraulic, gravitational, and more, there is only a limited number of basic units that we must use in analyzing, measuring, and calculating these different processes. Most processes can be described mathematically in terms of four basic parameters. Along with the most commonly used units, they include the following: Dimension – Units 1. 2. 3. 4.

Mass (M) – grams, slugs Length (L) – meters, feet Time (t) – seconds Electric charge (q) – coulombs

Let’s quickly explore the meaning of what has just been stated. Later in this book, we will return to the subject of dimensions when we pursue the subjects of energy conversion and use. All physical (mechanical) processes and properties can be described in terms of mass, length, and time. If we include the electric field, magnetic field, properties of conductance, permeability, electric current, etc., in our catalog of phenomena, then we need to add the fourth dimension, electric charge, to the list. Units of measurement are necessary to quantify these parameters. There are many forms of the units in which energy is measured. Depending on whether the English or metric system is employed, the most common units are as follows. In Equation (2.1) above we defined energy or work as the product of force multiplied by distance. The units of force (F) are found, from its definition, as the product of mass times acceleration, or

F = M × a,


where M is mass, and a is acceleration, as illustrated in Figure 2.1. Acceleration is the rate of change in velocity of a body (mass). Since velocity, v, is expressed as the rate of change in distance versus time, its units are v = L/t. The rate of change in velocity with time, similarly, is v/t = L/t2.

Fundamentals of Energy


Inertial mass

Push force

Mass M

Pull force

Figure 2.1 Acceleration of a mass by a constant force.

We experience numerous types of forces. Among them are the forces due to gravitational attraction, attraction and repulsion of electric charges in an electric field, and magnetic attraction and repulsion forces. Somewhere in the development of these observations and concepts, the decision was made that all these forces are dimensionally the same. In order to remain consistent in the logic of mathematical representation, all types of forces have the dimensions of ML/t2. How did this occur? Mechanical force as mass times acceleration was probably first proposed by Newton in his Principia Mathematica as part of his Laws of Motion. Maintaining that a body in motion will remain in motion and that a body at rest will remain at rest unless acted upon by an external force seemed a reasonable and almost intuitive proposition after its statement. Unfortunately, the statement itself provides no method for measuring this force, and its utility is limited. In order for any postulate regarding physical interactions to be useful, there must be some way to quantify the results and to accurately predict future results given enough facts or data. It seems that Newton may have felt that the idea of force was intuitively known and did not need further explanation. Even though most of us proceed through life without being bothered by much thought of the concept of force, that sentiment does not pervade all of theoretical physics. The ideas of force and energy are still subjects of great conjecture and debate. The history of the development of physical concepts is not the prime concern here, but some knowledge of their evolution does serve to bring more closely to our attention and scrutiny a better appreciation of terms that we employ daily. Sometimes it is necessary to begin understanding or developing a body of knowledge in order to make certain basic assumptions on an entirely intuitive basis. As scientifically unsatisfying as that may be, it is unavoidable at times. One could draw a weak comparison to plane geometry (Euclid) with regard to its various axioms and the declaration that parallel lines never meet. Even the concept of straight lines is rather intuitive in nature.


Energy Storage 2nd Edition

Perhaps the best definition is that a force is required to change the motion of a body. Many problems arise in finding acceptable definitions for the basic parameters of physical science, namely, the abstract concepts of mass, time, force, and energy. However, we must learn to be satisfied with definitions that leave something to be desired in order to move on toward generating a working body of mechanics that enables us to design and build practical devices that serve our purposes. An interesting definition of energy comes from the Grolier Encyclopedia, which states: Energy can be measured in terms of mechanical work, but because not all forms of energy can be converted into useful work, it is more precise to say that the energy of a system changes by an amount equal to the net work done on the system… In classical physics, energy, like work, is considered a scalar quantity; the units of energy are the same as those of work. These units may be ergs, joules, watt-hours, foot-pounds, or foot-poundals, depending on the system of units being used. In modern science, energy and the three components of linear momentum are thought of as different aspects of a single fourdimensional vector quantity, much as time is considered to be one aspect of the four-dimensional space-time continuum… Energy exists in many different forms. The form that bodies in motion possess is called kinetic energy. Energy may be stored in the form of potential energy, as it is in a compressed spring. Chemical systems possess internal energy, which can be converted by various devices into useful work; for example, a fuel such as gasoline can be burned in an engine to propel a vehicle. Heat energy may be absorbed or released when the internal energy of a system changes while work is done on or by the system. (1993)

The force of gravity on ponderable bodies that have the quality or property of mass is given as

Force = Mass gravitational acceleration constant = M

g (2.3)

From experience we learned that the larger (more massive) the body the greater the force needed to accelerate that body to the same velocity on a smooth, low friction surface. The rationale of defining the mass and force is rather circuitous because we employ the same phenomena to assess each parameter. In determining the mass of an object, its weight is used. Hence, the more a body weighs the proportionately greater is its mass. The property of a body’s mass to resist being moved or accelerated is known as inertia. Thus, a mass that is acted upon by a gravitational force has a

Fundamentals of Energy


weight directly proportional to its mass, as illustrated in Figure 2.2. Since all bodies fall in a constant gravitational field with the same acceleration (Galileo), the gravitational and inertial masses are declared to be one and the same. This principle of equivalence explains why all bodies, when acted upon by gravity and permitted to fall freely with no opposing forces, will experience the same acceleration. Hence, the velocities they attain over the same vertical distance will be equal. The force acting upon a mass with a gravitational mass, Mg is

Fg = Mg g


The inertial mass, Mi, which is accelerated by some force, F, described in Equation 2.2 is expressed as

F = Mi a


More properly, force is defined as directly proportional to the rate of change in momentum of a body, or


d mv dt



Suspended mass, M

Gravitational mass

Force of gravity

Figure 2.2 Weight of a mass due to gravitational attraction.


Energy Storage 2nd Edition

With this assumption of equivalence, or Mi = Mg, we arrive at a means of measuring force in a quantitative manner. A mass of 1 kg is thought to have a force of 1 kg times 9.80 m/sec2, or 1 Newton. In the “c-g-s” system (centimeter-gram-sec), 1 gram of mass has a force of 1 dyne exerted upon it by gravity at the Earth’s surface. The acceleration constant of gravity in the metric system of measurements is 980 cm/sec2. In the English system of units, the acceleration, g, is 32 ft/sec2, and in the metric system g = 980 cm/sec2. By inspection we see that the unit of force, as defined above, is

F = M L t–2


Now, is that true for all types of forces, including electrical and magnetic? The answer apparently is yes. As the need arises in the development of our discussion of energy, we will explore each of these types of forces. Returning again to the main topic, we will see how the idea of energy is explored and how a useable and quantitative definition is generated. The formulation of relationships in physics involves a lot of mathematical trickery. Some of these manipulations might even appear at times to employ the practice of self-deception in order to arrive at the desired answers to posed questions. The derivation of the very familiar expression for kinetic energy, 1/2 mv2, is interesting. How is it that the energy of a moving body with mass, m, is the product of its momentum, mv? Plus, where does the factor ½ come from? Look again at how we have defined the idea of energy, or work, in Equation (2.1) as the scalar product of force times the distance. Without questioning further at this point how we have justified this leap of confidence, the next step is to quantify the idea. If force can be defined as the product of the mass of a body times the acceleration produced by that force acting upon the body, as in equation (2.2), then an increment, dE, in energy, E, in moving a small distance, dx, can be represented as

dE F . dx

m a  .  dx


d2x  .  dx dt 2


where acceleration, a, is represented as the second derivative of distance with respect to time. So far so good, if we accept the force times distance premise. It is now necessary to integrate distance in order to obtain an expression for a finite

Fundamentals of Energy


amount of energy produced by a force, constant or otherwise, moving through a distance, x. Thus,


F  . d m

d2x . dx dt 2


Since acceleration is the first derivative of velocity, v, we substitute dv/dt for acceleration in equation (2.8) and obtain

E m

dv . dx  dt


If we make the dt term the denominator under dx, we obtain another velocity term under the integral sign and have the following upon integration:

E m dv .

dx dt

m v . dv

1  mv 2 2


which is the well-accepted formula for the instantaneous kinetic energy of a moving body. An example of a device that stores kinetic energy is the flywheel (Figure 2.3). We can think of a flywheel as a kinetic battery. Flywheels offer

Composite Rim

Magnetic Bearing Vacuum Chamber




Source: Beacon Power, LLC



Figure 2.3 (a) Typical design of a flywheel as an energy storage device (courtesy of Beacon Power, LLC), (b) NASA Flywheel (Courtesy of


Energy Storage 2nd Edition

advantages of high reliability and excellent cycle life, but are not suitable for small portable applications. The modern flywheel rotors are often made of composite materials for optimum strength and density properties, and the bearings are superconducting and electromagnetic to minimize friction. Flywheels store rotational kinetic energy (Urot) expressed as


1  I  2



where I is the moment of inertia (kg m2), and is the rotational velocity (rad/s). For a hollow cylinder or disc, the moment of inertia is equal to the mass times the radius r squared (I=mr2). For a solid cylinder, the moment of inertia is I=(1/2) mr2. The speed of a modern high-speed flywheel with electromagnetic bearing can reach as high as 100,000 rpm in vacuum.

2.1.3 Gravitational Potential Energy The problem of defining energy has been attacked in a rather pragmatic fashion. Perhaps, we can approach this concept from a different perspective. For many thousands of years, we have known that lifting weights requires doing what we generally now call “work”, and that weight lifting can be converted into other useful forms of work, such as turning a paddle-wheel, grinding grain, and powering woodworking tools. The energy stored after lifting a weight is referred to as gravitational potential energy. Many thousands of years ago, early humans harnessed this energy by placing heavy stones on high altitudes to be later released for survival and defense. The modern examples of the potential gravitational energy usage include the clock pendulum, water behind a dam, swings, and rollercoasters. To quantify the gravitational potential energy, we must identify relevant parameters including height and mass of the object. We calculate the gravitational potential energy (U) stored in an object, located at a certain position above the ground, by multiplying the mass of the object (m) by gravity (g) times the height (h) expressed as

U = m.g. h where U is in Joules, m is in Kg, g is in m/s2, and h is in meters.


Fundamentals of Energy


2.1.4 Elastic Potential Energy Humans have been using uncoiled mechanical springs in bows and arrows for thousands of years, for hunting and survival. When using bows and arrows, we are in fact harnessing the elastic potential energy that was stored in the unoiled springs. As time passed, upon the emergence of coiled springs, the usage of mechanical springs expanded to many common applications including watches. Imagine, as you twist the knob on a watch, you are causing the spring to coil, and thus, you are powering the watch with elastic potential energy. The watch would then uncoil, gradually in time, during the operation and thus, release the stored energy. Figure 2.4 shows the schematics of the coiled springs in an Elgin watch that was commonly used from 1864 to 1968. To quantify the potential energy stored in elastic coiled structures such as a spring, we express the elastic energy U as

1 U      k   x 2 2


where k is the spring constant, and x is the displacement of the spring. For deformable bodies that can undergo deformation d or strain (the derivative of displacement with respect to position), the linear elastic strain energy (U) and strain energy density (u) (i.e. energy per volume), can be calculated as


1 F  d 2

1   k  d 2 2


Figure 2.4 Illustration of the coiled mainsprings used in Elgin pocket watch (Courtesy of Elgin National Watch Co.)


Energy Storage 2nd Edition

1   2


1 E  2



where, F is the force on the body, k is the material stiffness (or spring constant), is the stress in the body, and E is the elastic modulus. The strain energy density in one dimension can be converted to 2D or 3D based on the actual state of stresses and strains in the body. For example, if a body is subjected to plane stress (i.e., the normal and shear stresses in a plane are non-zero), the strain energy density (u) is expressed as


1 ( 2

x x

y y






where x and y refer to the directions of the normal stresses and strains, and xy is the xy are the shear stress and strain, respectively, in the xy plane.

2.2 Electrical Energy Imagine a charged particle at a certain voltage. The electrical or electrostatic energy (or work) associated with this particle can be calculated by multiplying its charge (in coulomb) by the voltage (in volt):

Electrostatic Energy of a Particle Charge = Voltage Charge (2.18) Now, imagine two particles with respective charges q1 and q2 (coulomb). The electrostatic potential energy U (joules) between the two bodies depends not only on their charges, but also the distance between the bodies r (meters) expressed as

U  r

1 q .q    1 2 4. . o r


where o is the permittivity in vacuum equal to 8.854 187 817 . . . ×10−12 F m−1. Now, instead of a charged particle, let us take a look at a charged plate, as in the case of a capacitor. A classical capacitor consists of an insulator or

Fundamentals of Energy


dielectric material that is placed between two charged parallel plates. The capacitance of a classical parallel plate capacitor is expressed as


o r

A d


where C is the capacitance (Farad or F), o is the permittivity in space equal to 8.854 × 10–12 F/m, r is the relative permittivity of the material between the plates, also known as the dielectric constant of the material, A is the area of the plates (m2), and d is the distance between the plates (m). To determine the energy stored in a capacitor, we will first start with the basic relation between the capacitance, charge and voltage. In an ideal capacitor, the capacitance is assumed to be constant and is expressed as following:




where Q is the charge on the capacitor plates (or electrodes), and V is the voltage across the plates. A strategy to compute the total energy stored in a capacitor is to first calculate the energy of an infinitesimal amount of charge added to the capacitor plate. Then, we can integrate the term over the total charge on the plate and find the total energy stored in a capacitor. Let’s define the differential energy, dU that is associated with an infinitesimal amount of charge, dq. This small charge dq is added to the capacitor plate, at a certain voltage, V(q), as depicted in Figure 2.5, and expressed in the following equation as

dU = V (q). dq


We can now calculate the total energy in a capacitor by integrating Equation 2.22. The total stored energy is the integration of the differential energy elements dU over the total charge Q: Q



dU 0


V q . dq 0


q . dq C



Energy Storage 2nd Edition Voltage, V Slope = 1/Capacitance dU



Charge, Q

Figure 2.5 Differential energy dU of an infinitesimal charge dq on a capacitor plate.

Using Equations 2.23 and 2.21 (Charge=Capacitance Voltage), we obtain:


1 2 Q /C 2

2 1 CV /C 2


Therefore, the electrical energy stored in a capacitor can be expressed as

Energy in a Capacitor

1 C  V 2 2


We can also geometrically ascertain the total stored energy by calculating the area under the line in a voltage-charge plot. In the case of a capacitor, the relation between voltage and charge is linear as shown in Figure 2.5, and the area under the line is that of a right triangle, equal to ½ Q V. After substituting Q = C .V, the same expression for stored energy is obtained as in Equation (2.25). Now, let us discuss energy storage in batteries. In the case of batteries, the electric charge stored in the active chemical compounds of the electrodes is referred to as the capacity:

Capacity = Current



Note that capacity (Amp.Hour) and capacitance (F) are different parameters. The general formula to calculate the electrical energy stored in a battery is expressed as

Energy = Nominal Voltage Capacity


Fundamentals of Energy


Common units for battery parameters are Watt.Hour or W.h for energy, Volt or V for the voltage, Amp.hour or A.h for capacity, Amp or A for current, and hour or h for time. Specific capacity (gravimetric) is commonly expressed in the units of Ah/kg (or mAh/g), and area capacity (mAh/cm2). For energy, Watt.hour can be converted to Joules or J using the relations: Watt = Joules/Second and hour = 3600 seconds. The energy calculated for batteries, capacitors and other storage devices are often normalized with respect to either mass (g), volume (cm3), or footprint area (cm2) of the device (e.g. battery or supercapacitor) or the active material inside the device (e.g. cathode material in the battery). This allows for a more precise and uniform comparison and evaluation of the performance of various devices (and associated materials). Normalized energy calculations can be better utilized when we are deciding on an optimum energy storage device for a specific application that may have design constraints in weight, volume or footprint area.

2.3 Chemical Energy Chemical energy can be defined as the energy stored in the chemical bonds of the atoms, molecules and chemical compounds. By this definition, every material that contains chemical bonds is storing chemical energy. However, to power our vehicles, electronics, and other devices, we utilize specific types of materials where chemical energy can be more effectively and efficiently released and converted to electrical, mechanical, or thermal energies. Fossil fuels, bio-fuels, hydrogen, and battery electrodes are several examples of materials that can store chemical energy, and be used for our daily energy consumption. Let us begin our discussion of the chemical energy with a brief overview of its history and the origin of chemical elements.

2.3.1 Nucleosynthesis and the Origin of Elements When the first atoms were formed during the Big Bang, energy was stored in the chemical elements that are now prevalent in our modern stars, planets and cosmic bodies. Thus, chemical energy storage is as old as our universe. The age of the universe is estimated to be about 13.8 billion years based on the mapping of the universe conducted by Planck European Space Agency mission. The age of the universe and the major events in the universe are illustrated in Figure 2.6. Our universe is believed to have started (t=0 s) with the Big Bang event that


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was followed immediately by a rapid expansion referred to as the inflation (t=10-32 s). Next, the protons began to form (t=10-6 s) followed by nuclear fusion (t=10-2 s to t=180 s). The first atoms that formed during the Big Bang nucleosynthesis are known as the “light elements” consisting of Deuterium (2H), Tritium (3H), Helium-3 (3He), Helium-4 (4He), Lithium-7 (7Li) and Baryon-7 (7Be). Some of the primary nucleosynthesis reactions during the Big Bang are the following:

p + n 2H + H + 2H 3H + p 4 He + 3H 7Li + 3 He + 4He 7Be +


H + 2H 3He + n H + 2H 4He + n 7 Li + p 4He + 4He 7 Be + n 7Li + p (2.28)



fo et rm ra at l hy io d n ro g


fu ar Nu cle

n Pr ot o






fla tio


fo r








where p is for proton, n is for neutron, and is the photon gamma rays. The number on the top left of the element represents the number of neutrons in the isotope. For example, 2H is Deuterium or a hydrogen isotope with 2 neutrons, and 3H is Tritium or a hydrogen isotope with 3 neutrons. The formation of Helium-4 and Lithium-7 are also depicted in Figure 2.7. The explosion of a star, known as Supernova, has led to the creation of several other relatively heavier elements. These elements include carbon, nitrogen, oxygen, neon, and sulfur, which are heavier than the “light elements” such as helium and lithium. Supernova is the last stage of a star’s

Present Universe


10–32 s 1 μs

Figure 2.6 The age of the universe.

0.01 s to 3 min

380,000 years

13.8 Billions years

Fundamentals of Energy 2H



n 4 He


γ 7 Li

Figure 2.7 Formation of Helium-4 and Lithium-7 during the Big Bang nucleosynthesis.

life. The layered structure of a red giant star that is close to supernova event is shown in Figure 2.8. Each layer in the exploding star, contains specific elements, and the outer layers have relatively lighter elements compared to the inner layers of the star. The chemical elements found in our bodies and on our planet can be all traced back to the stars and the Big Bang. It may be interesting to think of humans as star-based beings, composed of elements that are billions of years old and came from the stars. With this new perspective, let us take a look at the specific sources for different elements.

Nonburning Hydrogen Hydrogen, Helium Helium, Neon Helium, Carbon, Neon Oxygen, Carbon Oxygen, Neon, Magnesium Silicon and Sulfur Iron and Nickel Core

Figure 2.8 Layered structure of red giant star close to supernova.


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Jennifer Johnson, an observation astronomer at Ohio State University, has provided updated sources of chemical elements for our solar system. Figure 2.9 summarizes these sources based on Johnson’s work, and the previous investigation by researchers at Meteorite Laboratory at North Arizona University. The chemical elements in our solar system can be traced back specifically to the Big Bang fusion, cosmic rays fission, merging neutron stars, dying small (low mass) stars, exploding large (high mass) stars, and exploding white dwarfs. The human-made elements are also included. As astronomers collect more data and expand our understanding of the stellar events, more specific sources may be identified. The investigation of the rate of expansion of the universe has led to some very interesting findings. To put it in simple terms, it appears that gravity alone cannot fully explain our observations of the universe. Therefore, scientists have proposed the presence of a matter and energy, known as the “Dark Matter” and “Dark Energy” that may have contributed to the rather puzzling behavior of the galaxy and the universe. The dark matter is believed to be not directly observable, and essentially, it cannot emit light or interact with the electromagnetic radiation, hence, the reference to the term “dark”. Although not observable, the dark matter and dark energy are believed to cover most of the modern universe, more than 95%, as illustrated in Figure 2.10 (a). The composition in the

C Cosmic ray N M Human-made fision BSL N Merging neutron L Exploding large B C N O F Ne stars stars C SL SL L L L Exploding S Dying small stars D Al Si P S Cl Ar white dwarfs L LD LD LD LD LD B Big Bang fusion

H B Li Be BCS C Na Mg L L

K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge LD LD LD LD LD LD LD LD LD LD LD LD L L Rb L


Cs Ba NS NS Fr N

Ra N


As L

Se L

Br L

Kr L


Hf Ta W Re Os Ir Pt Au Hg NS NS NS NS NS NS NS NS NS La Ce Pr Nd Pm Sm Eu NS NS NS NS NS NS NS Ac Th Pa U Np Pu Am N N N M M M N


Bi N

Po N

Po Rn N N

Gd Tb Dy Ho Er Tm Yb Lu NS NS NS NS NS NS NS NS Cm Bk Cf Es Fm Md No Lr M M M M M M M M

Figure 2.9 Periodic table of elements and their origins [Sources: J. Johnson, OSU; Meteorite Laboratory, NAU].

Fundamentals of Energy Atoms, 4.60%

Dark Matter, 24.00%

Photons, 15.00%

Neutrino, 10.00%


Atoms, 12.00%

Dark Matter, 63.00%

Dark Energy, 71.40%



Figure 2.10 (a) Distribution of atoms, dark matter and other in (a) modern universe (today), (b) more than 13.7 billion years ago (Source: NASA).

pie chart is based on the results obtained by the Wilkinson Microwave Anisotropy Probe (WMAP) at NASA. “Dark energy” can be considered as the source of anti-gravity, responsible for accelerating the expansion of the universe.

2.3.2 Breaking and Forming the Chemical Bonds During a chemical reaction, the chemical bonds can be either broken or formed. Depending on the nature of the chemical reaction, and the specific reactants and products involved, energy may be either released or absorbed in the reaction. If the energy is released during a reaction, it is known as an exothermic (heat is released) reaction, and if the reaction absorbs energy, it is an endothermic (heat is absorbed) reaction. The energy to break or form a chemical bond can be estimated as shown in Table 2.1 for selected types of bonds. Table 2.1 A selection of average bond energies (KJ/mol). Bond type

Bond energy (KJ/mol)

Bond type

Bond energy (KJ/mol)

Bond type

Bond energy (KJ/mol)















































Energy Storage 2nd Edition

To determine whether a reaction absorbs or releases energy (endothermic or exothermic), we can use the formula of enthalpy of reaction as

Enthalpy of Reaction (ΔH) = Sum of Bond Energies of the reactants – Sum of Bond Energies of the products (2.29) A positive enthalpy indicates that the energy is absorbed (endothermic). A negative enthalpy indicates released energy (exothermic). Another aspect of chemical reaction that can be determined is whether it is spontaneous or not. Through the formula for Gibbs free energy (GE), we can ascertain spontaneity of the chemical reaction. If GE is negative (ΔG0), then the reaction will not take place spontaneously, and it will require energy. If ΔG = 0, the reaction will neither go forward or reverse, and it is said to be in a state of equilibrium. Imagine two chemical compounds A and B forming a by-product AB. The driving energy for this reaction (ΔGro) is the difference in the values of the standard Gibbs free energy (G energy) of the products (in this case AB) and the standard Gibbs free energy of the reactants, A and B:

Gibbs Free Energy of Reaction = Sum of Gibbs Energy of Products – Sum of Gibbs Energy of Reactants OR ΔGro

ΔG of ( prod )

ΔG of (reac )


If A and B are simple elements, this is called a formation reaction, and since the G energy of formation of elements is zero then:

Gr o = Gf o (AB) 2.3.3 Chemical vs. Electrochemical Reactions One way to visualize the differences between a chemical and electrochemical reaction is by considering the transport pathways for the ions and electrons as illustrated in Figure 2.11. In a typical chemical reaction between two chemical compounds or elements A and B, a by-product AB can be

Fundamentals of Energy Chemical

Electrochemical (Primary)



Electrochemical (Secondary/Rechargeable)

Load B










Load B


Electronic Path e–





Electrolyte AB




Rx A



Charging e–











R(1-x) A





Electrolyte R(1-x) BC R+ ions


Discharging Load e– A


BC e–

Electrolyte R+ ions




Figure 2.11 Graphical comparison of chemical and electrochemical reactions.

formed directly at the site of A and B. The exchange of electrons and ions are made at the reaction site, and the reaction may continue until all A and B are consumed or converted to the product AB. In an electrochemical reaction, such as in a battery, the elements or compounds A and B (i.e., electrodes) are not in contact, and are separated by the solid electrolyte or separator. Therefore, the pathways for the transport of ions and electrons are effectively separated. In other words, in an electrochemical cell, ions move through the electrolyte where no electron transport is allowed. At the same time, the electrons are sent to the external circuit, to power a device (current is supplied in the opposite direction).

2.3.4 Hydrogen Hydrogen is another material commonly used to store chemical energy, to be later converted to other forms of energy like electrical energy. For example, in proton exchange membrane fuel cells (PEMFCs), hydrogen is converted into protons (hydrogen ion or H+) and electrons (e-). The protons transport through the fuel cell membrane, and they combine with the oxygen at the other electrode to form water (H2O). The electrons generated from the hydrogen ionization (oxidation), exit the electrode and enter the external circuit to power the load. Hydrogen can be stored using two main approaches: physical storage or  chemical storage (Figure 2.12). The physical storage of hydrogen


Energy Storage 2nd Edition Hydrogen Storage

Physical Storage

Liquid Hydrogen

Chemical Storage

Metal Hydrides

Compressed (Gaseous) Cryocompressed


Carbohydrates Ammonia

Figure 2.12 Two main approaches to store hydrogen: physical and chemical storage.

includes liquid hydrogen, compressed (gaseous) hydrogen, and cryocompressed hydrogen. A carefully designed, reliable and robust container is generally required to safely store and transport the hydrogen in its various physical forms, to prevent unwanted chemical reactions and catastrophic explosions. The chemical storage of hydrogen involves chemical compounds that contain the hydrogen element and can be retrieved at a later time through a chemical process. These compounds include hydrides, carbohydrates, hydrocarbons, and ammonia and they are used as a medium to store the hydrogen. A potential advantage of chemical storage is that some of the chemical compounds may be in the form of solid pellets that are relatively stable and easy to transport. On the other hand, releasing hydrogen from the chemical compounds may require special processing conditions including higher temperatures. For example, to release hydrogen from metal hydrides, a relatively high temperature around 120 °C (248 °F) to 200 °C (392 °F), may be needed, and that is mainly due to the very strong bonding of hydrogen to these compounds. One way to assess various hydrogen storage types is by comparison of their gravimetric and volumetric energy densities. For example, the metal hydrides may exhibit a relatively good volumetric energy density; however, their gravimetric energy density can be lower than that of hydrocarbon fuels.

Fundamentals of Energy


2.4 Thermal Energy Thermal energy is another common and important type of energy that can be stored, converted, transferred, and utilized for various applications. Let us start with some basic concepts. When water freezes into ice, the heat flows out, and consequently, its thermal energy is transferred. As the ice melts, the heat flows back into it, and the thermal energy is stored inside the water. A commonly known relation between thermal energy transfer and temperature change, is the following:




where Q is the thermal energy (or heat) transferred to or from an object, m is the mass of the object, T is the temperature change, and c is the specific heat capacity of the material in the object. Let us explore further the concept of temperature, types of thermal energy storage methods, and phase change materials in the next sections.

2.4.1 Temperature Temperature is another term that we use almost on a daily basis, like, “the temperature is hot or cold”, but its concept can be somewhat vague. What does a temperature really represent? Before we answer this question, let us consider the relevance to energy. Can the temperature play a role in how we identify whether thermal energy has transferred into or out of an object? The answer is yes. If we observe a temperature change in the object, then a transfer of thermal energy to or from an object has occurred. But, now we return to the question: “What is temperature?” The temperature of an object can represent how fast the molecules in the object are vibrating. In other words, temperature is related to the kinetic energy of the molecules. Higher the temperature, faster the molecules vibrate, and more kinetic energy they possess. To find the exact relation between temperature and kinetic energy, various derivations and approaches including the kinetic theory model, and statistical thermodynamics approach have been employed, and are readily available in the literature. Here, we will not delve into any of those derivations, but we’ll discuss the main conceptual associations. Let us start with the famous equation for an ideal gas:

PV = nRT



Energy Storage 2nd Edition

where P is pressure, V is volume, n is number of moles, R is the gas constant, and T is temperature. We can relate pressure P to the force exerted by the gas atoms inside a container. These forces can be eventually related (omitting the derivations) to the kinetic energy of the molecules as

P  V

2 N  .   KEavg 3


where N is the number of molecules, and KEavg is the average kinetic energy of the molecules (or particles). As discussed in previous sections, kinetic energy is also related to mass and velocity, and therefore, we can obtain an expression between temperature and the mean square average velocity of the molecules as

1 3 m  v 2     k  T 2 2



where m is the mass of the molecule (particle), v is the velocity of the molecules, k is the Boltzmann constant, and T is the absolute temperature. Rearranging the terms, we obtain the expression for temperature as


1 m  v 2 3k


Here, we see that temperature is directly related to the mean square average of the velocity of the molecules or particles, and faster this is, higher is the temperature.

2.4.2 Thermal Energy Storage Types Let us take a look at how thermal energy is stored. In general, thermal energy storage is classified into three main categories: (i) sensible heat storage, (ii) latent heat storage, and (iii) thermochemical energy storage, as depicted in Figure 2.13. The concepts of sensible and latent heats are illustrated in Figure 2.14. As the material is heated, the temperature also begins to increase. This type of heat is referred to as the sensible heat, where both heat input and temperature are altered and the phase of the material remains the same (i.e., liquid or solid). In the case of a phase change material, as the

Fundamentals of Energy


Thermal Energy Storage


Sensible Heat


Latent Heat

Heat Pump Heat of Reaction


Liquids Solids

Thermal Chemical Pipeline

Liquid-Gaseous Solid-Solid

Figure 2.13 Thermal energy storage types.


PCM changes from solid to liquid, during heating and absorbs/stores thermal energy Liquid

Phase Change Temperature TPC

Latent Heat Sensible Heat Solid

Sensible Heat

PCM changes from liquid to solid, during cooling, and releases thermal energy

Input Heat

Figure 2.14 Temperature vs. energy storage during sensible and latent heating in phase change materials.

temperature reaches a certain temperature referred to as the phase change temperature Tpc, the material undergoes a phase-change, (e.g., solid to liquid) and thermal energy is stored in the material. This stored thermal energy at a relatively constant temperature, is referred to as latent heat. Similarly, when cooling from high temperatures, as the temperature decreases, the heat is released. When the temperature reaches Tpc, the


Energy Storage 2nd Edition Phase Change Materials


Paraffin Compounds Non-Paraffin Compounds


Salt Hydrate




InorganicInorganic InorganicOrganic

Figure 2.15 Classification of phase change materials.

material returns to its original phase (e.g., liquid to solid), and releases the stored thermal energy.

2.4.3 Phase Change Materials Phase change materials can be classified into three categories: Organic, inorganic, and eutectic as shown in Figure 2.15. The most commonly known organic phase change materials are paraffin compounds, for example, used as candles. Organic PHMs can also be non-paraffin based. Examples of inorganic PHMs are salt hydrates and metallic materials. A eutectic is a material composed of two or more components that has a minimum melting composition, where each individual component melts and freezes simultaneously. This leads to a mixture of the two components crystals. Therefore, a eutectic generally freezes or melts without undergoing any segregation. Eutectics can be composed of organic-organic components, inorganic-inorganic components or inorganic-organic components. More information on thermal storage can be obtained from several books and journal review articles, including a comprehensive review by Sharma, A., et al., 2009.

3 Conversion and Storage Energy storage is a growing issue in our society. Fossil fuel resources such as petroleum are becoming not only more scarce in some areas but also increasingly inaccessible and costly. Most nations are now addressing the possibilities of providing energy in as many forms possible from sources other than fossil fuel. Petroleum products have largely been responsible for the immense progress made in Western society. Since the beginning of the 20th century, petroleum products have enabled the development of railroads, aircraft propulsion, and large ocean-traversing vessels, as well as the automobile. As far as we know now, there is no other equivalent source of energy in the form of combustible fuel, except perhaps for alcohol (ethanol). However, we are rapidly realizing that alcohol is not a practical solution to the greatly increasing demands for portable world energy. Here, we will briefly review the role and merits of many competitive mechanisms available for secondary energy sources, ranging from compressed air and fly-wheels to electrochemical cells. Electrochemical cells still appear to be the most promising method of storing energy, which is probably the principal reason for its attractiveness. There are many practical considerations involved in selecting one means or another to store energy for use over time. Among these factors, perhaps cost is the most critical. Considerations such as safety, availability, life, dependability, cost, etc., would follow next. Unfortunately, nature doesn’t provide unlimited choices to achieve an inexpensive reliable, safe, and readily available means of storing useful energy. To focus our attention, we should establish two general categories of energy sources and identify them as primary or secondary. In reality, there is only one source in the strictest sense of the origin of energy. Secondary sources are actually intermediary places or devices where energy, from one of the few primary sources, is stored until needed at a later time. We will describe and compare the various mechanisms offered by nature and made practical by present-day technology to store energy, followed by Ralph Zito and Haleh Ardebili. Energy Storage 2nd Edition: A New Approach, (43–58) © 2019 Scrivener Publishing LLC



Energy Storage 2nd Edition

a section on electrochemical systems that concentrate on the redox type of devices that now seem to offer the most promise for solving long-term storage problems. Subsequently, details of performance, electrochemistry, and methods of constructing and testing these systems are presented. First, in order to establish a firmer base, a physics background has been presented, along with the development of an argument for redox cells as practical devices. This book describes a particular class of approaches to large-scale energy storage. Large-scale is defined here as amounts of energy exceeding 1 kWh that is stored in a single unit or group of energy storing units. The class of main interest here is that of reversible electrochemical cells. Since most cells or batteries with which we are familiar in our daily lives, such as the dry cell, alkaline cells, lithium ion batteries, and the ubiquitous lead acid battery, are not likely candidates, they will not be covered here. Only the approaches that have a chance to succeed as secondary batteries for solar, wind, load leveling, and emergency power applications will be discussed. Storing energy in its many forms in nature is a vital part of all processes and life itself on Earth. As we explore these processes (Table 3.1) and their importance to us, we can gradually make some observations that lead to revealing and important conclusions. Unfortunately, nature is not very cooperative when it comes to providing a multitude of materials with the totality of desirable and needed properties, such as low-cost, safety, availability, electrochemically well-behaved in aqueous solutions, and a large enough energy and power density potential to make simple, practical storage systems. All applications or manifestations of energy storage can be put into one or more of these categories. Certainly, if we wish to power a portable power tool or an electric automobile, a hydroelectric plant is hardly useable. However, if we use the energy produced by the station for storage in an electric battery, it becomes a practical situation. That’s an example of the portability or mobility reason for storage. In the second instance we might have the need to store solar energy during the daylight hours for use after sunset to power light, etc. There are many cases where the convenience factor is not met – where the generation of energy is occurring at a time that coincides with the need or application time. It must be pointed out here that when we refer to energy we are essentially referring to the “capacity to do work” in the classic sense of Force × Distance. It is this capacity that is being transferred from one source to another device that is capable of storing this capacity for work.

Conversion and Storage


Table 3.1 Energy sources accumulated by “natural” processes over recent and remote past. Energy source

Process of formation


Form of delayed solar, non-uniform heating of earth


Form of delayed solar evaporation and condensation, rain

Incident solar

Promptly available energy when converted to thermal or Photovoltaic


Gravity of moon and relative motion of Earth & moon


Residual thermal energy (compression) of earth formation


Remnants of initial matter formation processes

Fossil fuels

Accumulated conversion of organic matter deposits (solar)

Organic fuels

Wood, hydrocarbon gasses, (delayed solar)

Metal stressing

Steel springs

Elastic deformation

Rubber bands

Elevated weights

Impact devices, clocks

Compressed gases

CO2 or compressed air powered devices

Chemical reactions

Gun powder and explosive mixtures

Masses in motion

Fly wheels, rotating masses and linear, high-speed impact


Batteries and fuel cells

Storage of energy in one form or another to be used at a later time has been extremely important not only to mankind, but also to every form of life. Storage of energy in the form of chemical structures such as carbohydrates enables life forms to survive for periods of time between food intake activities. Other, simpler forms of energy storage that are familiar to all of us are listed below. These forms of energy are categorized for the purpose of distinguishing basic differences in the origin and the physical processes involved.


Energy Storage 2nd Edition

Since all of the above are commonly known and familiar to the reader, there is little need to give many examples of each of these processes or mechanisms. It is interesting to note here that energy sources that are derived from “natural” sources provide, by far, the largest amount of the energy consumed by industry and commerce. Especially in the case of fossil fuels, the available energy density is much greater than any other competing man-made source for most common applications – nuclear power not included. Returning to a less esoteric domain, the outline of the presentation is as follows, listing the primary topics that will be covered: 1. Primary energy sources – On a grand cosmic scale, primary sources are limited to nuclear and gravitational. All subsequent forms of energy are consequences, as far as we know at present, of these two principal sources. These are sources over which humans have no control. 2. Secondary sources of energy – It is necessary to define what we mean by “secondary.” Usually the definition involves some process of storing, and is associated with those resultant processes over which we do have control. 3. Conversion processes – Those devices that enable us to transform one form of energy into another more.

3.1 Availability of Solar Energy Indicative to this explanation is a comparison between the solar energy that is available from photovoltaic conversion mechanisms and the solar energy that is available via plants’ conversion into combustible fuels, such as methanol. The total available energy via combustion from 1 gallon of gasoline in an internal combustion engine is about 36kWh. The conversion efficiency of an automobile engine to mechanical output is at most about 20%. Thus, if we had very efficient electric motors instead, we would need to have at least 1/5 of the 36kWh × 500 million gallons per day. That number is 3,500 million kilowatt-hours of energy from sunlight per day. Again, if the conversion efficiency were 100%, dividing the numbers would reveal that we need about 5,000 square miles of area to provide the equivalent of gasoline. Unfortunately, we must multiply that figure by 10 to account for conversion losses, giving us a total of about 50,000 square miles.

Conversion and Storage


However, the following problem remains: How does one use solar energy to operate cars without storage? The practicality and cost of solar energy for consumer use to power industry and residential homes has become a widespread concern that stems from questioning energy sources and their limits. A simple estimate of available solar energy per unit land area, especially if it is to be collected and transformed directly into electric power, reveals that immense land areas are required to provide even modest amounts of energy for commercial use as illustrated below. The following calculations present a rather dismal prospect for widespread use of solar energy as a substitute for the more “conventional” sources presently in use. Regardless, the problem of providing energy to meet the increasing demands of the future is very serious. Other than oil and coal, there are no realistic answers. Nuclear, along with cheap and reliable energy storage (such as batteries), could be an answer to welldesigned, plug-in hybrid vehicles, giving a range of 200 or more miles on a single charge. As an example, consider a middle region of the North American Continent and use simple approximations. At high noon (normal incidence) on a cloudless day at the Equator, incident solar power surface density is in the range of 0.12 watts per square cm. Since sunlight is available only half the day (about 10 hours of useful daylight), the energy density is reduced to approximately 1.2 watt-hours per square cm per day. And, due to Earth’s rotation on its axis, the sun’s radiation makes a changing angle to the Earth’s surface perpendicular. Therefore, we can simply approximate that by another 50% factor. Hence, the energy density over a 24-hour period is further reduced to 0.6 watt-hours per square cm. The above energy per unit area per day becomes 0.6Wh/cm2 = 3.6 Wh/ 2 in , or 3.6 Wh/in2 = 500 Wh/ft2. Since there are about 5,300square feet per square mile, there are 5300×5300×500 = 28 million × 500WH from 1 square mile, or 15,000 megawatt hours per square mile in any one clear day at the equator. At latitudes of the mid-United States region, we might need to divide that number by two, making the total sunlight energy equal to about 7,000 megawatt-hours per day per square mile of area. Now let’s take a brief look at what the order of magnitude of the needs are for domestic energy per day or per year in the United States. According to the website Globilis, the amount of energy produced and consumed per year in 2002 was 4 million kilowatt-hours. That number reduces to 10,000 million kWh per day consumption.


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So, if we want to produce (assuming 100% conversion efficiency of electromagnetic energy from the Sun) that amount of energy, we would need an area of about 1400 square miles. Present-day efficiencies of solar photovoltaic cells can be in the order of 10% to 20%. Using these lower efficiency figures, we need 7,000 to 14,000 square miles of accessible area to generate the electrical energy presently being used by the United States. Imagine the maintenance and access roads required for such a solar collector field. Simply the problem of keeping the surfaces of semi-conductors and collectors clean and repaired would be monumental. Now, consider the gasoline situation. The United States used 21 million barrels of oil per day in 2007. The yield at the refineries was about 20 gallons of gasoline per barrel of oil, which means the nation actually used over 400 million gallons of gasoline per day – an astronomical figure. The problem remains unsolved. In addition, we must also have available storage to make the energy portable and useable whatever peak demands arise. Now, consider corn as a source of methanol. The data from the US Department of Agriculture shows that the average yield of corn per growing period of at least six months is about 150 bushels per acre. A bushel of corn will yield 2.5 to 3 gallons of ethanol. Ethanol has less than half the energy content per gallon than gasoline. Thus, we would obtain about 200 gallons of gasoline equivalent per acre of cornfield. Returning to the gasoline consumption rate above, if the United States consumes about 400 million gallons per day multiplied by 300 days per year, then the country consumes about 12,000 million gallons per year. After dividing the numbers for corn yield, close to 100 million acres of farmland would be required. A square mile is equal to 700 acres. After dividing again, it appears that about 150,000 square miles of farmland is needed under ideal circumstances – no provision is made here for roads, buildings, fertilizer, machinery, etc., – which is not very encouraging.

3.2 Conversion Processes The following are non-mechanical methods for converting from one form of energy to another. A simple mechanical example would be the generation of electric energy by mechanically moving an electric conductor through a magnetic field as is done in a dynamo. This is intended only as a brief review of a few well-known means employed to convert energy from one form directly into an electrical output.

Conversion and Storage hν




n-type p-type

Figure 3.1 Photovoltaic semi-conductor cross-section.

3.2.1 Photovoltaic Conversion Process The photovoltaic process (Figure 3.1) converts direct conversion from light energy (photons) to electricity (volts and amps) via p-n solid-state devices. Electrons are raised to the necessary high levels of kinetic energy and cross over the junctions with a corresponding voltage. Efficiencies are generally above 20% and thin film device can be less expensive, but at lower efficiencies.

3.2.2 Thermoelectric Effects: Seebeck and Peltier The search continues for materials with lower thermal conductivity, higher electrical conductivity, and higher Seebeck coefficients. Some materials that are used as a thermoelectric heat pump (cooling device), such as bismuth telluride, would give higher conversion efficiencies if they could be operated at higher sustained temperatures. Unfortunately, electrical contact deterioration, materials diffusion within the structure, and melting points exclude them from such application. The amount of heat loss due to heat conduction along the paths from the hot to the cooler junctions significantly contributes to conversion efficiency degradation. Joule heating, due to ohmic resistance of these paths, adds to the inefficiencies. The attractive- ness of no mechanically moving parts does not quite make these devices practical, except in special applications where inertness and long life are important. In addition,


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these types of devices are subject to the ever-present Carnot efficiency limitations encountered by all heat engines. Heat to electrical efficiency is 12 to 18%. Thermal energy, Q, pumped into or out of a thermoelectric junction per unit time is simply expressed as

iπ = dQ/dt = current × Peltier voltage


The Seebeck effect is utilized in many ways, but it is most commonly used in the well-known thermocouple used to measure temperature. The materials (usually metal alloys) are selected for this purpose on the basis of their thermoelectric coefficients and stability at high temperatures. The alumel-chromel, or iron- constantan materials couples, is very common. The Seebeck coefficient is very dependent on both operating temperature ranges and materials choice.

3.2.3 Multiple P-N Cell Structure Shown with Heat It is necessary to conduct heat to and from the hot and cold junctions, respectively, in parallel for an array of junctions while still providing electrical conduction in series (Figure 3.2). At times this can be a somewhat formidable engineering and materials application problem. In general, semiconductor materials with large Peltier potentials, low thermal conductivity, and high electronic conductivity are desirable. Unfortunately, these properties can become mutually exclusive because of the physics of the processes. For example, at 300°C temperature differential, output is about 0.05 volts per junction.

3.2.4 Early Examples of Thermoelectric Generators Thermoelectric energy conversion systems were devised in the mid-19th century and employed for limited practical purposes to operate communications equipment. There are records of Edison experimenting with very large arrays of metal alloy wire junctions wrapped around Franklin stoves to generate sensible electric power. In more recent times, numerous similar devices were constructed in Russia and utilized to broadcast news and propaganda to remote areas in that country that had no other sources of electrical power to operate radios.

Conversion and Storage

Thermal conductor Electrical insulator


Thermal conductor Electrical conductor

Figure 3.2 Multiple themoelectric junctions – thermopile. From: Introduction to Energy Technology, Ann Arbor Science, 1976.

3.2.5 Thermionic Converter For a thermionic converter, theoretical efficiency is about 75%, practical efficiency can be about 15%, and the hot electrode comes to about 1600–2000°C. These devices, with a perhaps misleading name, convert heat energy into electrical output primarily by “boiling” conduction band electrons from metallic surfaces placed in a vacuum (Figure 3.3). The drawbacks to this approach seem to be largely associated with their operating life, particularly with the emitter.

3.2.6 Thermogalvanic Conversion Thermogalvanic conversion is a process going directly from a temperature differential to an electrochemical potential. An example of a symmetrical cell employing the phenomenon described by the Gibbs-Helmholtz relationship is shown in Figure 3.4. For thermogalvanic conversion, dE/dT ~ 0.01 volts per deg C, efficiency can be about 15%, and temperature range is from 140° on the cold side to 350° on the hot side.


Energy Storage 2nd Edition Collector

Emitter – Heat source

– Thermal energy in

– –

– –


– – – – – – – –



i RL Electrical energy out

Figure 3.3 Illustration of an operating thermionic converter.

Thermogalvanic conversion Silver deposited

Silver removed

Heat flow


Solid AgI electrolyte Ag+

Load Limitations: Low efficiencies, about 15 to 20% Limited operating life before polarity reversal is necessary Irregular accumulation of silver causes short cycle life Advantage over semiconductor thermoelectrics is much higher couple voltage dE/dT ~ 10–2 volts/deg – C

Figure 3.4 Thermogalvanic cell depiction employing silver iodide electrolyte.

Conversion and Storage


There is another mechanism for transforming heat energy directly into electrical form without the necessity of mechanically moving parts or any other such complexities. As in the case of thermoelectric effects, there is an analogous one in chemistry. It is known as the thermogalvanic effect, as expressed by the Gibbs-Helmholtz equation. At first glance, this could be mistaken as another colligate property of matter. However, upon closer examination we see that the electric potential/temperature behavior of substances is quantitatively dependent upon the material property itself, namely, the free energy of formation as a function of temperature. The equation relating the enthalpy, H, and the free energy change ΔF, to the rate of change of free energy with temperature is



(ΔF ) T


where pressure is assumed to be constant. The equivalent of free energy (in watt-second) can be put into terms of voltages and the quantity of electric charge, e.g.,

ΔF = nFf E


where E is in volts, Ff is Faraday’s number of 96,500 coulombs per equivalent, and n is the electric charge on the ion. Thus, we then have the following equation as a net expression for the change in electrochemical voltage as a function of temperature:

nFf T


ΔH nFf E




ΔH nFf T



By applying this last relationship to a symmetrical “thermo-galvanic cell” that employs the silver/silver chloride electrodes, as shown in Figure 3.4, the value of ∂E/∂T at an average temperature of 300°C can be estimated from enthalpy data that is readily obtained from sources such as the


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Handbook of Chemistry and Physics, published by The Chemical Rubber Publishing Co. Solid silver iodide, or AgI, was explored some years ago as a possible electrolyte for a galvanic version of thermoelectricity. This compound has properties that lend themselves to such application because its electrical ionic resistivity as an electrolyte is a usable range, and the gradient of voltage versus temperature has unusually high values. Some of this data is presented in Figure 3.4.

3.3 Storage Processes 3.3.1 Redox Full-Flow Electrolyte Systems The storage of electrical energy in chemical form by reversible electrochemical processes is in widespread use. The main limitations to all present electrochemical couples (batteries) are their shelf and cycle (operational) life as well as their energy density and power delivery capabilities. For stationary applications, energy and power densities are not of primary importance. Their energy turnaround efficiencies, cost, and operational life are more significant. This technical exhibit describes an electrochemical cell, which stores energy on the basis of concentration differences at opposite electrodes and between the same chemical species. The low electric potentials at which these cells operate eliminate the possibilities of water electrolysis and the formation of hydrogen or oxygen gas at the electrodes during charging. Thus, a sealed system can be constructed that requires no maintenance. Since the cell is chemically symmetrical, its cycle life is indefinitely great. One of the few basic electrochemical processes that can be employed in an energy cell and also has little deleterious effects upon either electrode or cell structure materials is

S + 2e



This energy storage system is described and compared with other methods as candidates for numerous space applications. This technical exhibit is about simple energy storage for applications where reliability and life are among the primary considerations. In those cases where the primary energy is generated from solar, wind, tides, and even I.C. engines, there is a need for storage for some period

Conversion and Storage


of time. This energy is then used at a later time when the demand requires it. The electrochemical system that is proposed here and identified for reference convenience as a symmetric cell appears to be an attractive process. A brief review of a number of more likely competitive systems in terms of performance is also presented. Cell electrochemistry, transport processes, and cell performance are also presented here.

3.3.2 Full Flow and Static Electrolyte System Comparisons Redox cells with two electrolytes and all liquid reagents can be designed and operated as either static electrolyte systems or full flow electrolytes. In the first case the electrolytes remain in their respective compartments (negative and positive sides of a cell with a barrier or membrane separating the two compartments) as in conventional batteries. The drawing in Figure 3.5 shows such a basic design. The static electrolyte version of the redox cell offers the advantages of simplicity in design, no mechanically moving parts, and it is easily sealed. However, the energy and power densities of any particular cell design are a compromise because inter electrode cell spacing will affect both coulombic capacity as well as internal cell resistance and reagent availability for discharge. Also, charge retention time is significantly smaller because of ionic and molecular diffusion across the membrane separator.

Static redox cell in discharge mode


Anolyte reagent



Catholyte reagent

Separator, Membrane

Figure 3.5 Fixed electrolyte redox cell.


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In the full flow configuration, the two electrolytes are circulated from reservoirs into and out of their respective cell compartments through appropriate manifolds and pumps as shown in the Figures 3.6 and 3.7. The following are the advantages of this design approach: • • • •

Indefinitely long charge retention times No cell imbalance problems with large cell arrays System can be electrically shut off by electrolyte draining Separation of energy capacity from power delivery parameters in system designs • Ability to be recharged chemically by replacing electrolytes • Electrolytes may be electrically recharged in an external device In some applications the advantages or salient characteristics of full flow redox may outweigh the necessary additional complexity and mechanisms. In striving for the realization of a very long-life secondary battery, one of the developmental paths that can be followed is the use of reagents that remain in solution at all times. In other words, there is no deposition, removal, or change of composition or change in structure of solid reagents at electrode surfaces in the energy storing process. There are very few choices of chemical components that have all the properties necessary to make it a practical electrochemical process. Those that are candidates are listed in Chapter 5. Virtually all of the materials combinations have the singular drawback of having dissimilar materials on opposite electrolytes. Furthermore, since these reagents are in solution, there is the Redox electrochemical energy system

Anolyte tank

Catholyte tank



Circulation pumps

Figure 3.6 Full flow electrolyte redox battery system.


Conversion and Storage

Anolyte tank

Catholyte tank





Circulation pumps

Separation of power output from energy storage system parts Example shown below of internal combustion engine and the fuel storage

Horse power hours Horse power Gasolene tank


Mechanical power output I.C. Engine

Oxidizer, Air

Figure 3.7 Comparison of power/energy separation to I.C. engine system.

inexorable transport of catholyte materials into the anolyte region and vice versa. In most cases, there is no direct method of returning these unwanted components from one electrolyte to their origin. An example of such a redox cell is the chromium/iron couple where chromium and iron chlorides are in the negative and positive sides of a two-compartment cell. Since the energy process participants all have positive charges, once any chromium diffuses to the iron side of such a cell it is lost permanently. This type of transfer of ions results in a gradual deterioration of the electrolyte, and the cell will cease functioning until a new electrolyte is introduced from an external source. Another example of a redox system employs the sulfur/bromine couple and was first developed by TRL, Inc., for National Power, PLC. This is also a materials asymmetric couple whose cycle life is limited by unwanted diffusion of sulfide ions from the negative cell side into the positive, bromine side. Cation membranes are employed (usually NAFION) to maintain effective electrolyte separation, and cycle life can be large, but the


Energy Storage 2nd Edition Power station


Users Discharging electricity during peak usage hours

Storing electricity at night

AC/DC invertor +

V5+ V5+/V4+ Electrolyte tanks



H+ V4+

V2+/V3+ Electrolyte tanks

e– V3+ Cell





Electrolyte storage tanks for positive and negative electrolytes, energy conversion cell, pump for circulating electrolytes and pipes. (This battery delivers or stores electric power by changing the oxidation states V(2), V(3), V(4), V(5) of vanadium ions.)

Figure 3.8 Schematic of a vanadium redox cell.

electrolytes must be chemically processed periodically in order to sustain performance. A redox system that employs the same chemical species for energy storage currently in development (University of New South Wales, Australia) is the vanadium redox. In order to recognize an operating cell that meets the conditions of all liquid electrolytes and has the same chemical reagents on both sides, it is necessary to have at least three oxidation states of an anion or cation that are soluble in a polar solvent such as water. Vanadium is the only element with those properties that is reasonably well behaved. The circulating electrolyte design being pursued for load leveling application is illustrated in the drawing of Figure 3.8. In principle the electrolytes should have an indefinitely long life because vanadium ions are on both sides of the cell, and any unwanted diffusion can be corrected electrically by the transport during the recharging mode. Some limitations, however, are (1) high cost, (2) electrolyte maintenance, (3) hydrogen evolution and an ineffective seal, and (4) energy density limited by materials solubility in water.

4 Practical Purposes of Energy Storage 4.1 The Need for Storage Despite the continuing challenges with using electrochemical batteries for long-term, bulk, or large vehicular propulsion power, they are still favored in many applications. These applications can range from small vehicles to computer power, because of their inherent simplicity in configuration and because they provide direct output power in electrical form. Immense attention and support has been expended on energy matters in recent decades, mostly with regard to its sources or production, i.e., wind, solar, methanol fuels, etc. Relatively little resources have been devoted to the storage of energy. In fact, a significant number of the problems with which we are faced could be resolved with more effective storage. Energy is frequently available to the user, but often not at the needed time or location. Usually, until a technology shows imminent promise as practical, industrial and/or commercial products, they are not supported by commercial enterprises, which must realize a return on investment in the predictable future. A recent exception to this is the lithium-ion or polymer-based technology, which has found widespread use in portable devices such as computers, navigation equipment, cameras, etc., and use as auxiliary power for electric hybrid cars. Despite its limited life and rather high cost, it presents an opportunity for making some progress in areas in which other batteries cannot be applied. Let us take a brief look at the overall energy situation about which there is so much scientific discussion. The broad view certainly encompasses not only the issue of storage but also the matter of primary energy sources. In actuality, we have very few options. From a very practical standpoint there are only two primary sources, and they are either solar or nuclear energy. More will be discussed in later chapters. The position that seems to predominate in any discussion these days about the environment and energy favors alternative energy sources. What Ralph Zito and Haleh Ardebili. Energy Storage 2nd Edition: A New Approach, (59–70) © 2019 Scrivener Publishing LLC



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are these alternatives? Unfortunately, there are precious few alternatives that we are, at present, aware of to “solve the world’s energy problems.” In Chapter 5 we will examine these options more closely. For the present, the options can be summed up as being either fossil fuel (petroleum products) processes or nuclear processes. All others, such as wind, solar photovoltaic, geo-thermal, lunar, etc., are at best only supplementary sources when one considers the immense amount of energy that we use and will increasingly continue to use on a global scale. We are no longer at the level of one manpower to one horsepower per person that was predominant well before 1900 in Western civilizations. Now we are at the tens to hundreds of horsepower machinery at the disposal of each person. To sustain the level of accommodations in travel, living comforts, and convenience that modern societies enjoy will require increasing amounts of readily available energy. Meanwhile, our oil reserves are surely dwindling because they are finite (no new oil deposits are being created on Earth at this rate of usage), and the demand is increasing. At present, wind and solar, other than solar thermal, provide small amounts of intermittent and usually unpredictable power to the user. There are no large-scale, practical, and economic means available as yet to store vast quantities of energy to make their extensive use practical. In most instances, windmills and solar collector field outputs are connected to a nearby electric power utility to “level” the power in a continuous fashion. We certainly don’t want to be in a situation where our electrical appliances work well only when the sun is shining brightly or only when there is enough wind. Hence, we must either store this energy when the source is producing more than we need at that moment, or we sell our excess wind-generated power at windy times to a power grid, and then buy some back when the local moving air mass (wind) is insufficient to power our home, office, or factory. That situation ties everything inexorably to available power lines. That restriction reduces the places where alternative sources such as these can be employed effectively. Hence, no remote, stand-alone wind or solar power is practical without storage. All of the above are fine as auxiliary power for perhaps reducing the use of and our dependence upon fossil fuels. However, unless we drastically change our way of life in the more civilized and prosperous parts of the world, these measures will not solve our overall energy needs for the future. So far, the only two sources with universal applicability are fossil fuels and nuclear. Nuclear along with hydro can run virtually all of our stationary power needs. However, portable nuclear power is very limited to large vehicles such as ocean-going ships. There has been much speculation

Practical Purposes of Energy Storage


about developing nuclear-powered aircraft and large land-going vehicles. Unfortunately, the issues of fissionable mass criticality, controls, safety, radiation, and shielding greatly limit its applicability to moving conveyances. Certainly, we cannot look forward to street buses and private cars being powered this way. That leaves us with the situation of how to power all the various vehicles and industrial equipment even if we have universal nuclear power for power-generating plants. Should we use gasoline, diesel fuel, or propane? Perhaps, but there may be a time in the not so distant future when science will have found either a new source of portable energy or an entirely new approach (mechanism) for the creation of available energy, in much the same fashion that, prior to 1900, nuclear energy was inconceivable. Knowledge of the atomic nucleus, with all of its associated subatomic particle physics, gamma radiation, relativity theory, and quantum mechanics, didn’t exist. Thus, there was no way to even think constructively about such matters before that time. Similarly, there may indeed be a whole way of attacking the energy question if we could be armed with new knowledge of the nature of things in the Universe. Then again, there may not be any other alternatives beyond what we now know about matter, energy, space, time, etc. If there is some qualitatively new approach that utilizes new principles of physics in generating usable energy to satisfy our needs, and if we find it in time, the limited availability of fossil fuels may just buy us that needed time and we will not have to change our lifestyles too drastically. However, storing energy effectively would make all primary energy sources available for many applications, including load leveling. It would also enable us to physically transport available energy from one place to another without wires. In addition, electric vehicle propulsion could become practical. Storing energy in a directly useable, electrical form, as distinguished from compressed gas, would be ideal. Batteries and some forms of regenerable fuel cells offer that opportunity. They are able to store energy provided electrically from a source (e.g., generators, solar photo-voltaic) by changing oxidation states of ionic materials. Then they are able to deliver the major portion of that energy at a later time directly in the form of electrical output. Virtually all of the complexities and inefficiencies of converting from one form of energy to another is avoided. Converting, for example, from mechanically moving components, such as expanding gas or rotating flywheels, requires a generator and regulation. An electrochemical device


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requires little ancillary equipment. The basic simplicity of such systems make them quite attractive, even though in most instances there is the necessity for electronic circuitry to manage the power and probably convert from ac to dc and back again.

4.2 The Need for Secondary Energy Systems The ability to store energy from a primary source for later use is important in many situations, especially when that primary source of energy is from an uncontrollable and variable source, such as solar radiation through the atmosphere at the Earth’s surface. Even in those instances where the primary source is controllable, such as in petroleum-fueled gas turbines, there may be a need for peak bursts of power at random or scheduled times. Also, the ability to store energy would provide uninterrupted power delivery to the electrical load in those instances where a temporary breakdown or malfunction is encountered at the primary source. The diagram of Figure 4.1 illustrates the idea of the multiple functions that storage might provide in a total energy system. The storage stage shown can serve the following purposes: • Provide uninterrupted power • Briefly provide peak power exceeding that of the primary source • Smoothing function in those instances where the power from the source is not constant • Standby, emergency source.

Primary energy source

Storage, buffer stage secondary battery

Electronic control system

Figure 4.1 Function of storage in a total energy system.

Electric load

Practical Purposes of Energy Storage


4.2.1 Comparisons and Background Information There are a number of mechanisms that have been employed, and currently still are, for storing energy to either make it portable from one place to another or use it for peaking and standby purposes. Among the more common methods are the following, listed in Table 4.1 along with their respective energy densities and general areas of application. Where the primary energy source is electrical, such as in photovoltaic collectors, the secondary electrochemical cell has proven most applicable and practical. There are those instances where the primary source is in the form of mechanical energy, and the flywheel is the practical solution to smoother operating characteristics, but its energy density is quite low. Electrical capacitors also have low energy densities, and their problems with use are compounded by the steep discharge curve and the necessity to operate at very high voltages in order to obtain even modest capacities. There are a few more options for storage such as superconducting magnets and full flow redox, but they all seem to be complex in construction as well as in operation. To date, these options seem not to offer significant improvements in storage capabilities over more conventional systems. Even though electrochemical cells offer the simplest form of storage, batteries have their own limitations and problems. Their relatively short life in conjunction with high costs and very limited energy and power densities creates a rather discouraging picture for widespread applications where large amounts of energy must be stored. To date, it appears that the lithium-ion cell and the vanadium redox system are in the forefront for application, respectively for motive power, and for load leveling applications. Table 4.1 Secondary sources, energy storage methods. Process

ED, wh/lb

User or application



Suspensions, triggers



Peak power

Compressed gas

1 to 4

Propulsion and starting


4 to 7

Peak and interim power


10 to 40

General purpose


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If we are to develop alternative energy sources such as solar and wind power, it is mandatory that some form of practical storage be available. These alternative (non-fossil fuel) sources are mostly unpredictable and very dependent upon climatic conditions and the time of day necessitating either storage or connection into a very large power grid where loads and sources can be shared and programmed. Without some form of storage, stand-alone solar and wind systems would not be practical. Fossil fuels have been providing us with a primary source, and it represents the basis upon which most of our society functions. Petroleum derivatives provide more than an order of magnitude greater energy and power densities than any other source of portable power. In the case of hydrogen/oxygen fuel cells, energy is stored by generating hydrogen from some primary energy source as fuel for use at a later time in an electrochemical cell. Hydrogen gas is generated by employing one primary energy source or the other, e.g., hydro-electric, nuclear, solar, etc., and it represents a portable means of storing energy for use at a later time by being oxidized in a heat engine or an electrochemical fuel cell. It is important to emphasize that redox types of electrochemical systems offer the greatest promise to date for a solution to storing large quantities of energy in a stationary situation. The promise of success is largely due to the higher degree of control available in full flow cells compared to fixed electrolyte and reagent cells. Redox offers independence from cell imbalance problems and the possibility of very long life since reagents are fluids and there are no solid deposits on electrodes. The only possible competition at present seems to be the possibility of the “liquefaction” of coal as a fuel for the internal combustion engine. Speculating further, there is always the possibility that combustible fuels, resembling the organic hydrocarbons, can be artificially produced by some process at large centers of primary energy sources such as the nuclear reactor. However, all of these appear to be a ways off in the future, if ever, in a practical sense.

4.3 Sizing Power Requirements of Familiar Activities Examining the energy needed to perform a number of familiar activities may prove informative. Some of these activities are common, everyday actions. The illustrations that follow emphasize the familiar in order to gain some perspective on the range of energy and power required to perform various common tasks.

Practical Purposes of Energy Storage


One of the most common activities with which many can identify is the act of throwing a ball. A hardball baseball weighs approximately 1⁄2  lb, and when thrown by an adult with some practice, it achieves velocities in excess of 50 to 60 miles per hour. Speeds of over 80 mph have been reported. Consider the case of a 60 mph ball speed. That corresponds to 88 feet per second. To calculate the energy of motion, we use the formula 1/2mv2, with m being in units of slugs in the English system. The exercise in estimating the energy and power associated with throwing the ball is as follows:

Kinetic Energy

1 mv 2 2 1 0.5  lbs 2 32  ft/sec 2


88  ft/sec  ft. lb 60 ft. lb


Remember, the mass units have been converted to slugs by dividing the weight by the acceleration of gravity. That amount of energy would be equal to, for instance, the impact energy of dropping a 1 lb weight from a height of 60 feet. Now consider the power involved. It will be recalled that power is the rate of doing work, or expending or transferring energy. A simple estimate of the power that a human arm can develop for a brief period of time is found by taking the above information and making a few realistic assumptions. The swing or arc of the pitcher’s arm from the beginning of the throw to the moment the ball is released will be approximated as 180°, or half of a full circle. The radius of the pitcher’s arm is about 2.5 feet. Hence, the total distance through which the ball is accelerated to its release velocity is

Arc Distance = ½ (2 R ) = 2.5

8 ft


Next, if we assume that the ball is linearly accelerated through the throw, an average speed of the ball through the arc while still in the hand of the pitcher is about 44 ft/sec. The time then spent in accelerating the ball is 8ft/44ft/sec = 0.2 sec. During this short time, about 60 ft-lb of energy was imparted on the ball, corresponding to 300 ft-lb per second as the average rate of doing


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work, or power output of the pitcher’s arm. Converting this number to other, more familiar units, we get

300 ft-lb/sec = 300 ft-lb/sec/550 ft-lb/sec-hp = 0.54 hp, (4.3) or

0.54 hp = 0.54 hp × 746 watts/hp = 400 watts.


So, it seems that a human can generate significant amounts of power, but only for very brief periods of time.

4.3.1 Examples of Directly Available Human Manual Power Mechanically Unaided Arm Throwing An experienced baseball player can easily pitch a ball at 60 mph. Taking the data available for a typical ball pitching, the following information is generated: • • • •

Ball weight ~ 8 oz Propelled velocity ~ 88 ft/sec Kinetic energy ~ 60 ft-lbs Power Out for 0.2 sec ~ 1/2hp ~ 400 watts. Vehicle Propulsion by Human Powered Leg Muscles As an addendum to the above, consider the amount of energy a human can generate on a sustained, but limited, basis with leg muscles only. This is illustrated below in the form of bicycling uphill. Employing the same mathematics as before in ball pitching, we can see how much power is required to pedal at different speeds up hills with different inclinations. A 180 lb man with a 20 lb bicycle, pedaling at the speed of 3 mph, moves vertically at the rate of grade fraction × speed. In the steepest case shown in Table 4.2, a speed of 5 mph is 88×5/60 = 7 ft/sec. At a 5% grade, that would be 0.05 × 7 = 0.35 ft/sec in the vertical direction. Lifting 200 lbs at a rate of 0.35 ft/sec requires about 70 ft-lb/sec of power. Assuming a 100% efficiency of the bicycle mechanisms and no friction loss to the road surface, converting to hp and watts results in almost 100 watts.

Practical Purposes of Energy Storage


Table 4.2 Hill climbing: 180 lb person + 20 lb bicycle. 3% grade at 3 mph

~50 watts

5% grade at 3 mph

~100 watts

10% grade at 5 mph

~200 watts

Using similar arithmetic, the numbers for a bicyclist climbing different hills at different rates are shown below. As we know from experience, it is possible to sustain such effort for only brief periods, perhaps limited to minutes rather than hours of such power outputs. Before leaving the realm of human muscle power capabilities in transforming one form of energy into another, we will make an estimate of the sort of performance that can be achieved with the use of mechanical aids that enable us to store some energy over a short time to be released in an even more brief interval of time. The use of spring-like materials, such as resilient wood and steel, have enabled us to hurl objects much further and with greater speeds than are possible by efforts of directly throwing. A classic example of this is afforded by the bow-and-arrow of antiquity and modern times. The limit to the storable energy is, of course, the arm and shoulder strength of the archer. As a sample calculation, let us assume that the archer is capable of exerting a 100 lb pull on a bow string, and that the string is pulled back a full 2 feet from its rest position. In addition, if we assume the arrow weighs about 4 oz (while these figures may not be completely accurate, they are sufficient for illustrative purposes), then the computations below follow. There is a constant force term, k, that one can assume for the bow that expresses the force as a function of the tension or stretch of the bowstring. That distance, x, at the center of the bow is the difference from the zero point of the string when there is no pulling force to the position of the string center at maximum extension, or pull. The incremental force, dF, is then

dF = k dx

(4.5) Mechanical Storage: Archer’s Bow and Arrow The energy, E, stored in the bow string upon extending it a distance, x, from the relaxed position and normal to the string is

E = ∫ k x dx



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where k is the “spring constant” of the bow. If we assume that k is constant over the string extension, then E = (1/2)kx2. For a bow and arrow with the properties given below, we obtain the results shown for energy and speed of the projectile: • • • • • •

Full Bow Pull ~ 100 lb Total string displacement ~ 2 feet Arrow weight ~ 4 oz Energy imparted to arrow ~ 100 ft-lb Exit velocity ~ 112 ft/sec ~ 76 mph Power out for 0.018 sec ~ 1 hp.

The energy with which the arrow leaves the archer’s bow is quite high, and the speed exceeds that which throwing can attain. This was a formidable distance weapon in its time, and it still is employed for silent or stealth operations. The exit energies of other projectiles, such as bullets propelled by gunpowder, is mentioned elsewhere in this book. Among the more commonly found forms of portable energy sources for general use are the myriad electrochemical cells and batteries that are available everywhere and are used for almost every imaginable purpose. These applications range from the flashlight to large batteries that are used to start internal combustion engines and to power electric vehicles. These are all secondary devices in the sense that the capacity for releasing or making energy available to the user has been provided by some primary source at an earlier time. In the case of a primary battery, its energy deposit is made at the time of manufacture. In the case of a rechargeable battery, its ability to repeatedly store energy upon electrical recharging has also been provided at the time of its manufacture. At the other extreme is the hydroelectric generating station or dam. Here also, the system is secondary because the water at a useable elevation in the reservoir was provided by the sun as accumulated rainfall. These two secondary sources have the energy characteristics given below: • Flash light with 2 D-cells ~ 0.5 to 1 watt over 5 to 10 hours intermittent • Hydro-electric Generation (water fall) at 40 gallons per minute from 100 foot elevation ~ 1 hp (746 watts). There is a huge difference in the economics, the life, and the practicality of all the various means of storing and making available energy for practical purposes. A quick look at the costs of the energy from these two sources is pertinent in our overall views on the subject.

Practical Purposes of Energy Storage


Let’s assume that the price of a D-cell alkaline battery with a useable capacity of around 6 watt-hours is $1. That would make the cost of energy on a larger scale about $160/kwh. Compare that to the figure of about $0.10 per kWh for energy from an electric power utility. This should help explain why we don’t use primary batteries to power electric cars or our homes. However, the convenience factor of having a small amount of portable energy and power when needed for lights, computers, and other electronic apparatus makes it well worth the cost – as long as we are not using large amounts of energy at that cost rate. The single shortcoming of an electrochemical battery is its inability to deliver high, short-term bursts of power that one can obtain from chemical reactions like explosions or mechanical contraptions like catapults and large springs.

4.4 On-the-Road Vehicles Now, we will turn our attention from human powered efforts to devices that are capable of doing greater amounts of work, such as the internal combustion engine in passenger cars. Consider the amount of energy and power required to propel an automobile in its different modes of operation. The least amount of energy per unit time is needed to maintain cruise speeds, but much more power is required in accelerating and hill climbing. And, of course, passing on an upgrade hill is the most demanding on power from the engine. Most modern cars are designed with engines that will deliver the type of performance that motorists demand under the worst conditions. That means that the purchaser is buying a vehicle with a power plant significantly larger than what is needed to more slowly climb hills and that perhaps doesn’t accelerate from 0 to 60 mph in less than 10 seconds.

4.4.1 Land Vehicle Propulsion Requirements Summary A typical or test vehicle that weighs about 3,000 lb when loaded with passengers is used here for illustrative purposes. The calculations below show a few of the energies and power levels required to accomplish the cited performances. At 60 mph the kinetic energy of the vehicle is

½ mv2 = ½ (3000 lbs/32 ft/sec2) × (88 ft/sec)2 = 363,000 ft-lb (4.7)


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If the vehicle were to accelerate from full stop to 60 mph in 10 seconds, the power requirement of the engine would be

Hp = (3.6×105 ft-lb/10sec)/(550 ft-lb/sec/hp) = 65 hp


In order to maintain a speed of 25 to 35 mph, about 25 to 35 hp is required to overcome road friction, drive train losses, and wind resistance. These calculations indicate that almost twice the engine power is required to climb hills and accelerate at the rates shown than what is needed to propel the car along a level road to overcome the frictional forces, etc., encountered in machinery and wind and road contact. It should be noted here that cruise power must be added to the hill climbing power because those cruise losses still pertain.

4.5 Rocket Propulsion Energy Needs Comparison For rocket propulsion, the equation for thrust force versus mass and energy of the propellant is


d(Mv) dt

ΔM 2 m M Δt


where ϕ = fuel energy density m = mass of fuel M = total mass of expelled material. Gasoline as fuel, with liquid oxygen, gives 12,000 pounds of thrust per gallon burned per second. All the above applications of portable power sources have both power and energy density requirements far outside the current or future battery capabilities.

5 Competing Storage Methods Electrochemical secondary batteries are still the predominant means of energy storage in everyday situations. They take the form of either the ubiquitous dry cells (button cells included) or the familiar lead storage battery. Uses have a very wide range, from small electronic devices to power systems for diesel railway locomotives and emergency standby power sources. Despite their undeniable utility and the fact that they have made possible a vast variety of consumer and industrial products, they each suffer from some debilitating problems. Unfortunately, the life of most batteries is very limited either in terms of the number of charge/discharge cycles or simply their age. The aging and deterioration of batteries, as compared to mechanical or electronic devices, are caused by many inherent factors over which we have little control. These inherent factors have mostly to do with the very nature of chemical changes and their limited reversibility. Frequently, this irreversibility and cumulative deterioration are not only due to chemical changes but also the physical changes brought about by the chemical processes, such as lowering the mechanical strength of materials resulting from chemical changes. Why and how do batteries fail? What have we done to circumvent or improve upon the limitations? It is not our intention here to analyze in any detail the many mechanisms that contribute to battery failure. However, a brief review of some of the more familiar electrochemical cells might contribute to our understanding of why the search continues for improved electrochemical processes that might be applied to the purpose of energy storage. Some typical, common secondary cell reactions include the following: • NiCad Cd + 2Ni(OH)3 = CdO + H2O + Ni(OH)2 (from Junger) • Zinc/Iron cell Zn + Fe2O3 = 2FeO + ZnO

Ralph Zito and Haleh Ardebili. Energy Storage 2nd Edition: A New Approach, (71–88) © 2019 Scrivener Publishing LLC



Energy Storage 2nd Edition • Lead-Acid Pb + PbO2 + H2SO4 = 2PbSO4 + 2H2O (from Plante) • Nickel-Iron Fe + 2Ni(OH)3 = FeO + 2Ni(OH)2 + H2O (from Edison) • Ni-MH Charge to the right, discharge to the left: • Ni(OH)2 + M = NiOOH + MH, where the metal MH is usually a charged hydrogen absorbing alloy such as LaNi4.7Al0.3(H6) • Lithium Ion (−) anode: C + Li+ + xe− = LixC • (+) cathode: LiMO2 = Li1−xMO2 + xLi+ + xe−

An anode is where oxidation takes place, and cathodes are reduction sites. There are other lithium cells that employ solid and/or polymer electrolytes such as the Li-ion SPE cells that use LiCoO2 cathodes and graphite anodes. The Li/SOCl2, lithium thionyl chloride, cell has energy producing reactions at the electrodes during discharge – anode is Li = Li+ + e−, and cathode is 2SOCl2 + 4e− = 4Cl− + SO2 + S.

5.1 Problems with Batteries Despite their convenience, batteries have a number of limitations and serious disadvantages. The cost per unit of stored energy is hardly important when the application calls for very small amounts of energy such as in watches, cell phones, and other small electronic communication devices. However, it becomes a major consideration in motive power or bulk energy storage cases. When one considers large-sized energy facilities, other matters in addition to initial capital equipment (battery) costs become important. Safety, cycle, standing life, and dependability become issues to address. In virtually all electrochemical batteries, solid materials that are part of the electrode structures participate in the energy storing processes. Electrochemical reactions in which solid materials undergo changes in structure or composition are not completely reversible. Hence, there are inexorable and unwanted changes that occur with each cycle that eventually result in the demise of the battery. The lead-acid cell is an example of this situation. The principle energy storing reaction is essentially

PbO2 + Pb + 2H2SO4 = 2PbSO4 + 2H2O


Competing Storage Methods


The inherent failure mechanisms of batteries include the following: • • • • •

Shrinkage of the negative plate sponge lead Sulfation of plates, exposed plates, or long discharged times Changes in crystalline structure Spallation of active materials Grid corrosion by long overcharging.

These are all consequences of chemical reactions involving solids at or part of the electrode structures. Other batteries, such as the Edison cell, nickel-cadmium, lithium-ion, etc., involve changing the chemical composition of solid materials. All of the energy cells listed above are fairly common and presently available. They all have solid reagents at both electrodes. The continual cycling of cells directly involves or necessitates the restructuring of these solids as the cell is charged and discharged. Unfortunately, such physical (chemical) processes are not entirely reversible. Each time the electrode materials chemically change, a certain degree of irreversibility is encountered. Materials are redeposited with the same uniformity, some of the solids become physically detached from the electrode, and some loss of continuity or capacity results. The operating life of the electrochemical cells is limited by these mechanisms. In addition to the limitations cited above, there are also concerns with the passivation of electrode materials, or memory effects in which cell capacity or electrical performance is adversely changed as a result of continued partial charge cycling. Because of these factors and the need to develop energy systems with longer lives and more useable operational characteristics, a search has been continuing for electrochemical cells with energy storing (producing) reagents in either a liquid or gas phase. Such systems are usually referred to as either fuel cells or redox batteries. In the former gas phase, reagents such as oxygen and hydrogen are introduced into the reaction chamber, or electrode compartment, and oxidation of the fuel (hydrogen) and reduction of the oxidizing agent (oxygen) takes place with the release of energy in the form of electric current to an external load. In redox batteries, essentially the same process takes place but with different reagents and the singular difference that the reaction products can be regenerated externally to be reused in the cell an unlimited number of times. In actuality, the water produced in a fuel cell can also be “reused” by dissociation into H2 and O2. However, in the redox system the intent is to reuse the reagents rather than discard them.


Energy Storage 2nd Edition

Energy density of an electrochemical couple is obviously one of the many important properties to be considered when evaluating it as a potentially viable means of storing energy. Because of the large difference in available energy per unit weight of electrochemical processes as compared to other currently employed sources of energy, a different level of standards must be adopted in their assessment. A comparison between batteries and fossil fuels shows a vast difference in energy levels. This situation is illustrated in Table 5.1, which shows the performance numbers for fossil fuels. The remarkable aspects of these sources of energy are that they are in convenient liquid form, easy to handle, and they make use of the surrounding oxygen in the atmosphere as oxidant. Because of these features, petroleum products have risen rapidly to their prominent position as universal fuels for most mobile and remotely powered machines ranging from electric power generators to automobiles and aircraft. In applications where power, energy capacities, safety, and “portability” are critical, there is no rival. Only in very specialized situations such as in naval vessels, where the ability to remain at sea for great distances and for long periods of time without refueling, has nuclear power competed. Also, in many instances where air pollution is a concern, or where fossil fuels and hydropower are not options, nuclear power is competitive. A quick comparison of some of the better-known energy sources for doing our work is presented below. Relevant details about gasoline as our universal fuel are also illustrated below. Table 5.1 Heats of combustion for some hydrocarbons. Fuel type (Phase)




Ethane (G)




n-Pentane (L)




n-Hexane (L)




n-Heptane (L)




n-Octane (L)




Methanol (L)




Ethanol (L)




Competing Storage Methods


5.2 Hydrocarbon Fuel: Energy Density Data The following is an outline of the energy densities of various common hydrocarbon fuels that have been employed for the propulsion of vehicles via internal combustion engines. The most common among these fuels are the petroleum derivative polymers, heptane and octane. Table 5.1 shows the total energy of combustion for some of these fuels when combustion is complete and when it takes place at 20°C. These hydrocarbons may be represented as having the general molecular form of CxHyO2. When complete combustion takes place, assuming no formation of nitrogen compounds, the general form of the reaction is

nO2 +CxHyOz




n x

y 4

z 2


Such complete combustion is an idealized situation since some of the products are CO and the process is invariably not isothermal. However, these data do give representative maximum values for their useful available energy. Octane rating of gasoline is usually the percentage of octane to heptane in the fuel mixture. A 100-octane fuel might then be all octane. Ethanol fuel’s energy density of over 8 kWh/Kg is a very high value, especially when compared to most other processes for storing energy in a controllable and usable form. However, when one examines the other factors associated with its actual application to the conditions necessary to its use and the characteristics of engines, we see this number significantly diminished. A look at the amount of energy available from a gallon of octane for the mechanical energy necessary to propel a vehicle gives the following figures. Maximum energy conversion efficiency of a tuned, constant speed I.C. engine is in the order of 20%. Hence, at a specific gravity of about 6 pounds per gallon, only 1/5 of the 35.7 kWh per gallon is actually available, or about a 7 kWh/gallon maximum.


Energy Storage 2nd Edition

Burning 1 gallon, 6 pounds, of octane completely requires 24 pounds of oxygen for the reactions. That amount of O2 has a volume at STP of 260 ft3. Since only 20% of the composition of air is oxygen, a minimum volume of air of 5 × 260 = 1300 ft3 is required to “burn” 1 gallon of octane. Under ideal conditions, a vehicle with a mileage performance of 30 mpg traveling along a roadway at 60 mph would be consuming 2 gallons of octane per hour. That vehicle engine would be taking in a minimum of 2,600 ft3 of 20°C air per hour for complete combustion to occur. The air cleaner and fuel system must be able to handle airflow rates at least in the range of 3,000ft3/hour, or 50 ft3/minute. After over a century of development, we have learned how to accomplish these feats with what have become extremely reliable machines. However, two very important facts should be borne in mind: 1. Only a small fraction of the stored energy is utilized in the current internal combustion engines. 2. A much larger amount of oxygen in both weight and volume must be available from the surrounding atmosphere for the fossil-fueled I.C. engine to function. An interesting fact should be noted here about energy densities. The extremely attractive energy density of petroleum products is made possible only because it is not necessary to carry the oxidizer on board the vehicle. The oxygen is supplied by the immediately surrounding environment and is available when needed at any demand rate. If it were necessary to carry the oxygen on board the vehicle, the number would be quite different. An amount of oxygen about four times the weight of fuel is needed. Thus, if the practical 20% efficiency of conversion is taken into account, 1.2 kWh/lb of fuel would be realized, and if we add the four times weight of oxygen, the figure becomes 240 Wh/lb. If now we also include the weight of nitrogen in the air, we obtain the low value of 40 Wh/lb for the energy density of a power source system that not only had to carry the fuel but also its own air comprised of only 20% O2. All of the proceeding may seem a bit unfair in assessing the “true” energy density of a hydrocarbon fuel, but in actuality this is the situation with which any contending energy system must compete. The “unfairness” lies mainly in the fact that these fossil fuel systems breathe air and do not have to carry the weight of the oxidant along. By virtue of the same fact, the combustion products are exhausted into the atmosphere, thus increasing pollution problems in some instances.

Competing Storage Methods


To eliminate air pollution entirely, it is necessary to devise energy producing systems that do not involve combustion products of any sort, other than perhaps water vapor, into the surrounding environment. This essentially means that a non-combustion method of producing heat or power must be substituted for the burning of fossil fuels. In fact, the only combustible materials are either carbon or compounds of carbon, all of which yield CO2 as the ultimate product of the process. The necessity of carrying not only the fuel or reducing agent on board the system but the oxidizers as well does present a severe handicap when attempting to develop a competing high energy density system.

5.3 Electrochemical Cells Electrochemical cells suffer not only from the lower energy densities associated with such processes but also from the necessity of carrying all of the chemical reagents necessary to the available energy reactions. This situation does enable the cell to remain independent of the external surroundings during its operation. These factors contribute to energy densities of electrochemical reactions, which are at least an order of magnitude lower than those of hydrocarbons. Among the most energetic of electrochemical reactions are those between metals and halogens. The highest specific energy capacity is that between lithium and fluorine because of their low atomic weights and high reactivity, or free energy of combination. The reaction of the formation of solid salt, Li + F → LiF, occurs with a free energy of 140 kcal/gm mol. wt. and at a potential of over 6 volts. This corresponds to an energy density of 3,100 Wh/Kg for the direct or electrochemical reaction of the elements. This value is quite attractive and even competitive with fossil fuels. However, there are no practical ways of handling the highly reactive components. Both the metal lithium and the element fluorine react energetically with water and most other substances associated with the construction of electrochemical cells. Hence, in order to make an operable cell employing this couple, it is necessary to find some non-aqueous, probably polar solvent in which the salt LiF is soluble and that will not react with either of the two reagents. Fused salt electrolytes are possibilities, but they involve high temperatures, difficult operating conditions, and increased hazards. There is also the problem of storing free fluorine as an available reagent for a charged cell. These problems are formidable, and no short-term, practical solutions are in sight.


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5.4 Metal-Halogen and Half-Redox Couples The term redox has been used in recent times in a manner that is not entirely clear when applied to electrochemical cells. All such processes involve oxidation and reduction. The fuel (reducing agent) is oxidized, and the oxidizing agent is reduced in the process, thus producing energy in a hopefully usable fashion. The term redox, as applied to electrochemical systems, refers to a system that employs reducing reactants and oxidizers that are passed through the cell, or reaction chamber, for the production of electric power. The reactants are subsequently regenerated at some location external to the reactor cell. In primary cells where only the discharge process takes place, the oxidation process occurs at the negative electrode, and the reduction of a chemical occurs at the positive electrode. In the case of the LeClanche (dry) cell, zinc is oxidized to zinc-oxygen compounds, and manganese dioxide is reduced. Secondary, or electrically rechargeable cells, have dual process electrodes. During charging from an external electrical source, the reduction occurs at the negative electrode and oxidation at the positive (Figure 5.1).

Enhanced concentration cell operation (Migration during charging mode) Cation membrane

Conductive, non-porous carbon electrodes (negative side)

Positive side M+

Micro-porous, activated carbon reaction and storage sites

Xj – ne = Xi

Xi + ne = XJ

Figure 5.1 Enhanced concentration cell operation.

Competing Storage Methods


Half-redox implies that the processes at one of the two electrodes in a cell involve reagents that remain mobile as liquids or gases so that they may be introduced and removed from the respective electrode region. The general form of such reactions is M+n + nX−1 → MXn. Metal-halogen cells are in the half-redox category. Halogens are liquid or gas at room temperature and can be caused to flow over the surface of an electrode as needed to sustain the reaction processes or to remove reaction products. If the product of oxidation, such as the salt, ZnBr2, in the case of the zinc/bromine couple, is soluble in the electrolyte, there are no solid products formed onto or removed from the surface of the (+) electrode during charging or discharging. Since the reactant and reaction products can, in principle, be supplied during discharge to the electrode surface and removed for regeneration at some external location, the bromine electrode qualifies as a redox type reagent. The importance of this aspect of redox behavior in this case is not primarily because of its ability to be removed from the cell for storage or reconversion elsewhere but because it reacts directly with the reducing agent, zinc, to produce the salt. Furthermore, its storage at the electrode or in a reservoir is mobile and can be made uniform. Cells with reagents that will recombine by direct union offer the possibility of an extremely long life. Another couple that has received little attention and has the attributes of half-redox cells and homogeneous cation is the Fe/Fe+3 couple. Oxidation/ reduction reaction is Fe0 + 2Fe+3 → 3Fe+2. The diagram in Figure 5.2 shows the rather interesting and unique characteristics whereby one can make use of the two oxidation states of the single element iron. The disadvantages of this system are the facts that iron is an electronically conductive solid with poor plating properties, and that hydrogen gas evolves not only upon charging but at times due to the necessarily low pH of the electrolyte. However, with further development in such matters as the use of non-aqueous electrolytes and improvements in plating quality, the ironredox system could become practical for some applications where cost is important and size is relatively unimportant. Of the four halogens (fluorine, chlorine, bromine, and iodine), bromine has been selected for serious evaluation as an oxidant because of its more favorable physical and chemical properties. For example, it is much more reactive than iodine and has lower costs, but it is not nearly as volatile and ill behaved as a storable component in a cell as either chlorine or fluorine. Chlorine will hydrolyze much more rapidly in water than


Energy Storage 2nd Edition Solid iron plating Fe0 Negative electrode

0.44 volts

Positive electrode

Fe+2 Ferrous ions 0.77 volts

Fe+3 Fe+2 + 2e Fe0 Charging

Fe+2 Fe+3 + e Charging

Figure 5.2 Energy level diagram for the iron redox cell.

bromine will, and it does not form reversible complexes as well as bromine does. Fluorine is an extremely difficult and costly material to work with in elemental form. A number of companies in recent years have produced some reversible cells and batteries on a laboratory prototype basis, in which the chlorine was stored as a clathrate (frozen hydrate) in a zinc/chlorine battery. Because of life and hazard considerations, among others, the battery has not emerged as a practical secondary source for widespread use to date. These factors contribute to the difficulties of achieving practical battery designs using higher energy density reagents such as those above. Figure 5.3 is a chart that compares the free energy of reaction per kilogram of reactants for a range of metal-bromine couples. These values were calculated from the free energies of the reaction and sum of the half-cell potentials available from the literature. The most attractive couple, strictly from an energy standpoint, is that of Li/Br. Unfortunately, there is currently no technology that allows us to either electro-deposit lithium metal out of aqueous solution or evenexist in an air/water environment. Non-aqueous solvents present problems not only of hermetic sealing but limited life, simultaneous chemical compatibility with free bromine, and low electrolytic conductivity. Fused salt electrolytes must be operated at high temperatures, and cells experience

Competing Storage Methods


Based upon electromotive potentials

Calculated energy density, Wh/lb

600 500 400 300 200

aqueous aqueous aqueous

100 0

aqueous aqueous
















Metal cation See bibliography reference 5; Latimer, Oxidation potentials

Figure 5.3 Energy densities of metal-bromine couples.

many other types of problems ranging from chemical durability, thermal expansion/contraction, mechanical strength, and sealability. A high temperature metal/halogen system seems quite impractical for any application at any time in the foreseeable future. High temperature fused salt cells, such as the Na/S system, have been constructed with some degree of success, but they do not employ free halogen as the oxidizer. All of the couples that employ aluminum and alkali metals have higher energy densities because of their high reactivity as chemical agents. This greater reactivity also results in greater problems of cell containment and reaction rate control. As we descend lower in specific energy density, the couples become increasingly easier to manage as an operating cell. However, their attractiveness as sources of energy also diminishes. One must then look for a compromise, hoping that a delightful selection can be found that is easy to mange, well-behaved, and has useful energy and power densities. To increase the likelihood of applicability of a couple, it is important to stay as much within ambient conditions as possible. A cell that is operable at near room temperature ranges is compatible with an aqueous environment and is also compatible with materials of desirable construction. Also, electrodes and separators that are chemically resistant and have reasonable costs are required.


Energy Storage 2nd Edition

Why explore and attempt to select a metal-halogen couple in the first place? The following two reasons can explain why: 1. This category of reagents as electrochemical cells offers high energy density and perhaps higher power density. 2. There is the possibility of long cycle life because of the totally reversible nature of the cell, in principle. Of all of the couples compatible with aqueous electrolyte systems and ambient conditions, the zinc-bromine has appeared most attractive. Zinc is the metal furthest from hydrogen in the electro- motive series as a reducing agent that can be plated out of solution in water. Bromine is similarly compatible in water and has a relatively low vapor pressure at 25°C. In the absence of catalysts, bromine hydrolyzes at a low rate, and equilibrium is attained at very low concentrations of HBrO in acidic solutions. The reactions at the negative and positive electrodes during discharge are Zn → Zn+2 + 2e at 0.76 volts and Br2 + 2e → 2Br−1 at 1.07 volts. The whole net reaction is simply Zn+2 + 2Br−1 → ZnBr2 at 1.82 volts. The high reaction potential that is actually realized is slightly over 1.8 volts per cell. Almost 200 watt-hours per pound of reactants are available for the reaction if it goes to completion and no other electrical losses are incurred. These figures are quite attractive for many application possibilities. Leadacid cells have a theoretical energy density for the energy storing reagents alone of 70 to 80 Wh/lb, or about 200 Wh/kg, as compared to the upper limit offered by the Zn/Br2 couple of 440 Wh/kg. The zinc/bromine system has some very fundamental problems, such as zinc plating being spongy and dendritic, zinc reacting in an acidic solution evolving hydrogen, and bromine being extremely oxidizing, difficult, and expensive to store. The second of the attributes of metal-halogen couples is their ability to return to the initial conditions of the discharged state as secondary cells. When the couples are discharged completely, the chemistry has returned to the initial conditions the cell possessed when first fabricated, and any remaining reagents will react directly if given the opportunity in cell design. Assuming no deterioration of electrodes, etc., and no loss of components from the system, the cell has no permanent memory, in principle. Obviously, if it is possible to “tame” the reaction in a manner rendering it reasonably safe, low cost, and long-life, applications such as load leveling, peaking, and even the electric vehicle are possibilities. An interesting comparison of the performances of a variety of common energy systems is provided in the Ragone type diagram in Figure 5.4.

Competing Storage Methods


Energy versus power densities Storage & generation systems 1000

1.00 Gas turbine

I.C. engine 0.10

100 Nickel Cadmium

Specific power watts/lb

Specific power hp/lb

Lead acid Fuel cells


1 1.0

10 100 Specific energy wh/lb


0.001 1000

Figure 5.4 Ragone plot for energy storage and generation systems.

We see that the performance on both an energy and power basis is far superior for heat engines such as the turbines and internal combustion engines. Fuel cells are significantly better performers on an energy basis than other batteries or electrochemical devices, but are not yet competitive on power density.

5.5 Full Redox Couples There are couples, other than metal/halogen, that are compatible with aqueous solutions and are entirely redox in character. However, the selection is not very attractive when one imposes the usual conditions of low cost, environmentally benign, well-behaved, long life, usable energy, and power densities. Table 5.2 is a partial list of the better-known reducing and oxidizing agents that could be utilized in an electrochemical cell as a source of electrical energy. Oxidizers and reducing agents (fuels) are listed as either chemical elements or ions (radicals) depending on their form in a cell at the beginning of “discharge,” or at the start of the energy-producing mode. The potentials are given in volts above or below hydrogen. Following convention,


Energy Storage 2nd Edition

Table 5.2 Redox cell reactants at room temperature. Oxidizer
































positive volts are above hydrogen, and negative volts are below hydrogen in the electromotive series. As is evident, there are relatively few choices for chemical reagents that satisfy the prerequisites of acceptable voltage, chemical stability, and being liquid in both states of oxidation, relatively safe, low-cost, and compatible with ambient conditions. Of all of the above, and more that are not listed here, the oxidizers that show most promise are still ferric ions, bromine, and vanadium in acid solutions. Of the even fewer choices available for reducing agents, it seems that sulfur (complexed) and vanadium are most likely to yield some practical systems. We have elected to pursue sulfur as the most attractive element for the reducing agent in a room temperature aqueous electrolyte. The properties that are most attractive, and quite remarkable when considered in light of its use as the reagent in the negative side of a cell, include the following: • • • • • • •

Plentiful supply Low cost High solubility as a polysulfide complex in water solutions Low equivalent weight Reasonable potential Well-behaved electrochemically Low volatility.

Competing Storage Methods


Many oxidizing agents have been tested in conjunction with sulfur as the negative electrode material. Bromine has been of most interest because of the potential for a completely reversible cell. Looking back into past decades, we see that other redox couples have been explored intensively in the past, such as that of Cr+2/Fe+3. The NASA Lewis laboratory has probably contributed most toward its research. One of the problems associated with its cycle life and coulombic efficiency is that of maintaining separation between the chromium and iron ionically in solutions. Imperfect separation is achieved by employing ion-exchange membranes between cell compartments, but cross diffusion inexorably results in deterioration of the electrolytes. Since both ions have positive electric charges, they cannot be separated during charging as in a metal-halogen couple with oppositely charged cations and anions. Anion membranes with low resistance, low cost, and very high transport number ratios for the transport of chloride ions in the NASA cell are difficult to fabricate. At present, the vanadium redox couple is receiving a fair amount of attention because of a number of interesting features. The fact that it is homogeneous, regarding the active reagent vanadium, and the reasonable solubility in water of the vanadium salts make the system attractive as a long-life energy storage mechanism. However, the negative aspects are the high cost of vanadium and its compounds, high molecular weights of the reagents, the necessity for operating electrolytes at very low pH, and the complexities of operating a system that requires full flow electrolytes. The potential market for such a system appears to be mainly as load leveling or standby emergency power sources.

5.6 Possible Applications In selecting electrochemical couples to investigate and perhaps develop into operable systems and products, it is vitally important to keep the intended application criteria constantly in mind. The application potentials impose their own needs for cost, life, reliability, and energy and power density. Some application possibilities for full flow redox systems include the following: • • • • •

Utility load leveling at generating and customer sides Peaking at user or customer side of power lines Emergency energy source in cases of power failure Portable power for convenience, or for remote locations Perhaps for electric vehicles and small boats.


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With each area of application, there are certain requirements for a secondary battery power source that determine its usefulness. For example, in the case of load leveling, battery weight or energy density and power density are not directly significant factors because they are in stationary locations. These parameters are meaningful only when converted into cost. Usually, larger physical equipment (batteries) has larger costs because of the required ground area and support structures, such as buildings. Hence, higher energy density batteries tend to be more attractive because of their associated costs. More materials of construction and a larger amount of floor space (footprint) and building needed to house the energy storage battery system increases the initial cost. However, longer cycle life systems will tend to amortize initial costs. Cost would naturally be greater as a consequence for lower energy density (ED) batteries. Storing energy output from generating equipment during low demand times to use for later peak demand periods can save in the cost of hydrocarbon fuels and generating capital equipment, but only if the economics of the battery are favorable. Storing for peak power demand periods at the user end of the power line can also save in the capital cost of transmission lines and transformers to the energy user. Cell imbalance in arrays of large numbers of battery cells can become a severe problem, especially for deep discharge cycling. Deep cycling is desirable because it reduces the amount of battery needed to store the usable energy. However, because battery cells are not identical in structure and performance, deep cycling can result not only in a decreasing usable output of stored energy but also in permanent damage to cells that have been reversed. Lead-acid batteries are particularly prone to damage by reverse charging. To compensate for this cycle life limitation, lead-acid batteries are usually not subjected to deep cycling, thus resulting in the necessity to build and purchase more energy storage capacity than will be in actual use. Safety, though always a serious consideration in the design and use of energy systems, is not as important in bulk energy storage applications because proper provisions can be made to cope with potentially hazardous situations by proper containment and trained personnel. In the case of consumer product areas such as the electric vehicle or portable power for tools and recreational purposes, safety is one of the prime concerns. In order to appreciate the directions in which battery development efforts are, or should be, headed some of the principle features or aspects of battery sources are identified below. For Bulk Storage • High energy turnaround efficiency • Flat discharge voltage versus current curve

Competing Storage Methods


• Minimum or absence of cell imbalance • Competitive costs of capital equipment amortized over cycle life • Low maintenance because of cost considerations For Portable or Electric Vehicle Use • Low capital cost • Safe operation • Highest energy density possible • Highest power density possible • Compactness in size as well as small weight • Minimum maintenance • Chemical recharge ability desirable Except for the higher sensitivity to energy efficiency in bulk energy applications, consumer applications and particularly the electric vehicle have even more difficult application requirements to meet. The latter has especially difficult criteria of safety and compactness imposed upon the battery design that exceed those for bulk energy uses. The lead-acid battery has been in use extensively for over a century, and it still retains its position of prominence in industrial and consumer applications. Even though its energy density is lower than what is desired in many instances, its familiarity, dependence over a wide range of operating conditions, and acceptable cycle life give rise to its universal character. Other contenders as large storage batteries include the Edison (nickeliron) cell and the Junger cell (nickel cadmium). Because of either cost or performance factors, their application is much more limited than the lead-acid. With a maximum specific energy density ranging between 12 and 18Wh/lb, or 26 to 40Wh/kg, the lead-acid battery leaves much to be desired in electric vehicle applications. The usual range of such Pb-Ac power vehicles is 50 to 70miles maximum. Doubling the energy density of a battery would certainly be a welcome improvement, but it would still limit a vehicle to a 100 to 150 mile range between lengthy charging times. The redox battery systems do offer the possibility of chemical “re-fueling,” thus extending the usable range of an electric vehicle indefinitely as long as sources or fueling stations of the chemical reagents are available en route. Numerous electrochemical couples making use of oxygen in the atmosphere as an oxidant have been experimented with and prototyped over recent decades. Perhaps the most common of these are the alkaline versions of metal-air cells. The more common versions include, Mg-air, Al-air, and Zn-air. Most suffer from problems of available power density and the


Energy Storage 2nd Edition

deterioration of air electrode catalysts due to poisoning contaminants. And when these devices are considered for use as chemically rechargeable cells, the complexities and inconveniences of removing and replacing many depleted metallic electrodes and handling caustic electrolytes containing the spent products are severe. Consequently, they also have not gained a position in any of the large-scale consumer/industrial application areas. To summarize, the three areas of metal-halogen, half-redox, and redox, and more specifically, the zinc/bromine, iron-redox, and sulfur/bromine have been selected for investigation over past years at TRL, and they are presented here because of certain inherent characteristics. These include the following: • Zinc/bromine couple – high energy density, well-behaved, completely reversible • Iron/ferric couple – low cost, long life, completely reversible, great safety • Sulfur/bromine couple – low cost, long life potential, no cell imbalance, chemically rechargeable. In all the cases above, reverse charging has little or no permanent, negative consequences on the cells. There are, however, some serious negative factors that have precluded their extensive use. These include such issues as safety encountered with the use of strong oxidizing agents such as bromine, difficulty with managing metal plating, the generation of hydrogen gas and the decomposition of water, rising acidity of electrolytes causing the evolution of gasses, and the compatibility of materials of construction in the presence of halogens.

6 The Concentration Cell Perhaps one of the more captivating and promising approaches to the practical storage of energy is in making use of the simple particles or billiard ball properties of matter. This view of the problem is enticing perhaps because of its direct simplicity. In other words, we might be able to make use of the straightforward “mechanical” properties of matter as described in classical molecular theory. In doing so, it appears we might be able to avoid many of the pitfalls of other energy systems that depend on specific properties of dissimilar materials (molecules), in which there are inherent mechanisms of degradation due to such issues as irreversible chemical changes, molecular diffusion from one region of the system to another, resulting in process contamination, or even just changes in molecular and physical structure that produce operational incompatibilities. This approach does seem to offer ways around such life-limiting circumstances. In general, there is only one chemical species that participates in the energy storing process, and the processes all appear to be completely reversible by means of electrical input of appropriate polarity. If one is willing to design around or endure the peculiar electrical characteristics of such devices, perhaps a series of practical, low-cost systems can be produced in order to solve some of the more pressing energy problems we face in daily life. We will proceed to outline some of the background physics and review the general behavior of concentration cells as electrochemical elements.

6.1 Colligative Properties of Matter The various properties of a large aggregate of material particles that seem to behave as though they possessed properties that can be described as due only to their common “physical” or “mechanical” attributes are frequently referred to as colligative properties.

Ralph Zito and Haleh Ardebili. Energy Storage 2nd Edition: A New Approach, (89–162) © 2019 Scrivener Publishing LLC



Energy Storage 2nd Edition

Webster’s Unabridged Dictionary defines colligative as “depending on the number of molecules or atoms rather than on their nature.” This more than implies that this class of attributes is independent of any chemical differences and is only concerned with the net effect of characterless particles regarding bulk behavior. Examples of such properties and their effects include the following: • • • • •

Lowering of vapor pressure of a liquid Lowering of the freezing temperature of a mixture Boiling point elevation Osmotic pressure of solutions Electric potential difference between same ionic solutions with different concentrations.

At low concentrations of solutes in any of the above instances, the properties of the solutions are changed in a manner directly proportional to the amount of solute particles. This is especially true when their concentrations are so low that the properties of the solvent are unaltered and the interactions between solute particles are minimum. The vapor pressure depression, po – p, of a solution where po is the vapor pressure of the pure solvent, and p is the vapor pressure of the solution is expressible as

p po po

1 Xa


Xa is the mole fraction of the solute, a, in question. Note that nowhere in these types of expressions does any property specific to the materials appear other than the vapor pressure of the solvent. However, we are concerned in these discussions only with the differences, po − p, that result from the introduction of any solute. The idea of basing an energy system on a set of colligative properties is attractive because it might become possible to make a device that is independent of specific materials and, thus, symmetrical in construction, other than such factors as pressure, concentration, and temperature differentials between the same components of the device. Hence, we might be able to avoid any degradation in performance or life of that device due to the transfer of materials that differ in composition.

The Concentration Cell


6.2 Electrochemical Application of Colligative Properties The following is an initial technical description of the principles of operation of such a concentration cell. We will also refer here to the concentration cell as a Common Ion Redox cell (CIR), as a convenient abbreviation for this class of energy storage systems. These cells are, at times, also referred to as symmetrical cells since the cells are constructed the same at each electrode side. Electrodes are physically identical, electronically conductive, and chemically inert. It makes use of the colligative properties of matter, or in this case, the large collections of iron atoms and ions in solution. In essence, the CIR cell employs the non-substance, specific properties of components and their physical behavior in an operating system. The raising and lowering of a solvent boiling point or freezing point as a consequence of the concentration of dissolved solute are colligative properties because they are virtually independent of materials composition. They are mostly strictly a result of the amount of solute molecules present in the liquid. Another example of colligative properties is the interdependence of temperature and pressure of a gas (in the ideal gas range). These parameters are independent of the specific gas that is present and only dependent on the number of molecules present. The search for a process, other than mechanical devices or electrochemical couples, that would enable the storage of energy in an inexpensive and reliable manner and that has a minimum of inherent failure mechanisms has resulted in the exploration of the characteristics of concentration cells as potentially practical devices for these applications. For the sake of brevity, we will refer to such cells as CIR devices (Common Ion Redox). These devices are all redox in nature because, as is necessarily the case in chemical transformations, reduction and oxidation of chemical species occur within the device for the transference of energy. However, the label should not imply that these devices or cells are full flow designs with provision for removing or replacing electrolytes. They may be operated as either stationary, or static, electrolyte systems. The particulars of design and physical configuration depend on the intended applications. For example, it may be desirable to employ external liquid reservoirs, pumps, etc., in a full flow system if exceedingly long charge retention times are required. In such cases, the additional system’s complexity and costs may be justified.


Energy Storage 2nd Edition

This approach very closely resembles the process of the isothermal compression of a gas to store energy for later use. However, the energy input and output in gas compression is mechanical. A pump is employed to compress the gas, such as air within a confined volume, while dissipating the heat generated to the outside world. The input is usually a mechanical piston pump. To regain the stored energy in the form of a gas at higher than ambient pressure, the gas is permitted to expand back to the outside usually via a piston or centrifugal pump, thus returning some of that energy in useful mechanical form. It is also necessary to provide a heat exchange system to not only dissipate the thermal energy to the outside during compression but to also return thermal energy to the expanding and cooling gas. In the CIR system, we are interested in a device that will operate with energy input and output directly in electrical form. In order to accomplish this end, it is obviously necessary to have electrical potential difference present, rather than a mechanical pressure, and charge transfers taking place within the device so that an electric current can flow in an associated external circuit. That circuit would provide the energy output as a current flow through an external electric load. The energy potential difference between electrodes in a concentration cell is a consequence of the electrodes being immersed in electrolytes with different concentrations of the same chemical species. If the chemical species at these electrodes exist at different oxidation states, this energy difference is manifested as a net electric voltage between the electrodes. The electric field is proportional to a function of species concentration ratios. This approach to storing energy is interesting because it appears that it is not only possible but also practical to obtain any level of electric potential in a CIR cell quite independently of the nature of the chemical agent(s), other than their solubility, electrical conductivity, and other general physical properties. Plus, the charge density is limited only by the amount of the chemical species that one can contain or compress in the locality of the electrodes. The CIR cell is not limited to the electrochemical potential difference between two chemical agents. Instead, the voltage range is a continuum limited only by the ratio of concentrations that can be achieved by physical and mechanical methods. Some cell design possibilities are suggested. Probably the most expeditious method of introducing and explaining the basic attributes and design parameters of concentration cells is to present this additional tutorial on the subject.

The Concentration Cell


6.2.1 Compressed Gas There are just a few steps in the explanation that will briefly describe some of the important, underlying fundamental principles of physics. Let us take a look at the ideal gas law,

PV = nRT


where P = gas pressure, V = volume occupied by the gas, n = number of moles (number of molecules in that volume), R = universal gas constant, and T = temperature in absolute degrees. Figure 6.1 shows a container with two compartments (equal volumes). These separate compartments can be connected by a gas pump to extract gas from one side to the other and by a valve permitting the compressed gas to flow (expand) from the high-pressure side to the other. The number of molecules initially in each side is n. If all the molecules were to be compressed into one side of the container, the compressed gas side would be at a pressure P = 2P. The amount of energy stored (isothermally) is then Liquid pump simply PV. Mechanical energy can be expressed as force times distance, or

Energy, E = F · L and, F = P· A, or, E = P· A· L

Liquid pump

Pressure vessel

Relief air valve

Figure 6.1 Closed system gas compression.





Energy Storage 2nd Edition

Such a method for storing energy in this configuration is quite ineffective in terms of volume of space. Hence, the gas (air in this case) is usually compressed from the outside infinite supply source into a confined space such as the volume, V, of one compartment. Then the limitation of how much energy can be stored in that volume, V, is limited only by the strength of the container to withstand high pressure differentials between the outside (one atmosphere pressure) and its interior. Not shown or discussed here is the considerable amount of temperature control and heat exchangers needed to maintain a near isothermal process. During the compression portion of a cycle heat, energy must be dissipated to the outside of the system, and during the expansion phase, heat must be supplied to the expanding gas.

6.2.2 Osmosis Now let us turn our attention to another complementary process, or mechanism, known as osmosis. We can now move to a different environment, from that of a free gas to that of matter in the liquid state (much closer molecular proximity). Figure 6.2 shows a simple diagram of a container with two compartments. A semi-permeable membrane separating the two compartments will more readily permit solvent (water) than the solute (salt, or other dissolved materials) to migrate through. This results in the “cell” developing Air pump

Pressure vessel


Relief air valve

Figure 6.2 Open-air osmosis compression system.


The Concentration Cell


an osmotic pressure differential, π, across that membrane. The value of π is analogous to the gas pressure in the previous relationship and found as

πv = nRT.


In this case, v is the volume of the solution. In operation the solutes remain essentially in place, and the solvent, pressured by osmosis, moves through the barrier from the dilute side to the concentrated side until the hydrostatic pressure (or some externally supplied pressure) is equalized. This is a reversible process. An externally applied pressure will move solvent through the semi-permeable separator, creating an increasingly dilute situation on the low-pressure side. The energy that went into creating a concentration differential can be reclaimed in part by permitting the osmotic pressure to reverse the situation. Unfortunately, this process is not as practical for energy storage as the compression of a gas. In both instances, the process is mechanical and would require additional processes for the transformation into useful forms of energy.

6.2.3 Electrostatic Capacitor An example of energy transfer and storage is that which is associated with the accumulation of electrical charge in a capacitor (condenser) as the result of an externally impressed electrical potential. Figure 6.3 shows such a simple arrangement in the form of a parallel plate condenser. This is a practical and widely used method for storing energy for brief periods of time in electrical circuitry, for load smoothing, and for providing large amounts of power for brief periods. The relationship between charge and voltage is

CV = Q,


where Q = electric charge in coulombs, V = electric potential in volts, and C = capacitance in farads. A simple parallel plate capacitor has a capacitance, C, that can be represented as


A , 4 d


where A = plate area, D = separation distance of plates, and ε = permittivity of the dielectric medium between the plates.


Energy Storage 2nd Edition Resistive load

+ + + + + + + + + +


– – – – – – – – – –



Voltage supply

Figure 6.3 Electrostatic capacitor.

When the plates are connected to an electric potential source, charges flow from that source to the plates until the voltage across the plates, as dictated by the above relationship, just equals the source potential. The energy stored is


1 CV 2 . 2


There is no “real” current flowing internally between the plates. However, a displacement current concept is employed to maintain continuity in the sense of a complete electric circuit. This is, again, an example of a concentration process to store energy. In this instance, the energy storing material in motion is not molecular but electric charges. Capacitors provide a very useful and convenient method of storing for later use in large power applications.

6.2.4 Concentration Cells: CIR (Common Ion Redox) Another approach to storing energy in direct electrical form, much as is accomplished in electrochemical cells that employ couplepotentials between dissimilar materials, is the electric potential obtained via concentration differentials with the same materials. Voltages can be produced within an

The Concentration Cell


electrochemical cell at the surfaces of electrodes if the concentrations of the same ionic species are different. This is expressed by the well-known Nernst equation:


nRT a log 1 , zF a2


where z = electric charge of the specific ions, F = faraday number, a1 = activities at electrode (1), and a2 = activities at electrode (2). A more convenient and a more easily calculated value for the concentration voltage is the substitution of reagent concentrations for their activities. This practice results in calculated values of electric potentials that are not as accurate as would be obtained if activities were employed but are reasonably valid if the concentrations are low. However, the ratios of the activity coefficients at different concentrations of specific ionic species do not differ much over a wide range on concentrations. For our purposes here, the agreement is close enough to provide a working basis for preliminary estimates of cell potentials. The table below shows fairly constant values for activity coefficients over a wide concentration range. This is true for weak as well as strong electrolytes. The activity, a, of an ion or ionic compound is related to the activity coefficient, γ, by the simple equation

a , c


where c is the concentration in molarity, as listed in Table 6.1, multiplied. The activities for HCl at 0.005 molarity and at 2.00 molarity multiplied by their respective concentrations are a = 0.930 × 0.005 = 0.0047 at c = 0.005 and a = 1.011 × 2.0 = 2.022 at c = 2, which are vast differences for a. Figure 6.4 is a simple representation of such a cell where the two electrode compartments are separated by a microporous or ion transfer membrane. The specific species is relatively unimportant, except for its solubility, mobility, and concentration as seen by each electrode. If the chemical species were merely a compound such as NaCl that ionizes in the solvent (water in this example), there would be no electric current flow possible even if the concentrations at the two depicted electrodes were vastly different. Only an osmotic pressure difference would be realized and no electrical potential.


Energy Storage 2nd Edition Table 6.1 Activity coefficients, γ, at 25°C. Molarity








































From: Physical chemistry by Alberty & Daniels, 1955, John Wiley & Sons.


– C1




Figure 6.4 Simple electrochemical cell.

In order to have ionic substances migrate from one electrode to another within the cell, it would necessitate an accumulation of opposite electric charge at the electrodes (as occurs in a capacitor), and there must be a closed circuit through a means external to the cell to provide an external electric current flow. The former condition is not possible because the electrolyte between the electrodes is conductive. Hence, no static electric charge can be sustained, and the latter can occur only if there is an electric potential across the cell. In order for an external current flow to be developed, there must be some mechanism at the electrodes to take on and give up electric charges (electrons). A process that can be used to complete the charge exchange cycle at electrodes is that of moving up or down the scale of state of oxidation or reduction for an element or compound. Then, a continuous path for

The Concentration Cell


electric charge flow within the electrolytic cell as well as electron migration through the external circuit can be provided. It can readily be seen that we now have a compression procedure (concentration differential of the same molecular species) that lends itself to a reversible process for the storage of energy in electrical form. The general exchange process below illustrates this for a species that we will identify as A, with two oxidation states, A+n and A+(n−1). Then, A+n + e– A+(n–1). As a very simple situation to illustrate the principle, let us assume that the element or species exists in solution as a compound, AxBy, with another species, B, that does not change oxidation state. Then, the process, as shown in Figure 6.5, would be the transport A ions across a separator from one side to the other. Furthermore, electrons would flow in the external circuit to maintain total electrical neutrality. Examining the situation quantitatively, we see that, in a cell as described in Figure 6.4, the maximum concentration of solute is limited by the solubility of that compound. In most cases, the solubility of such ionic materials is in the range of 3 to 6 molar. If A carries a single transfer charge per ion, then the maximum charge density for a given volume of 4 molar solution would be 2 × 25 AH per mole, or about 100 AH per liter per side. That figure reduces to 50 AH per liter of total volume of cell. Even with the 4 molar limitation and only one charge carrier per ion, the charge density is still quite attractive. However, one must look at the voltage at which the charges are delivered to an external load. For a simple cell in Figure 6.4, the voltage rises as the cell is “charged,” meaning that the concentration of ion A increases and

|An| |An–1|


Figure 6.5 Charges stored by surface adsorption.


Energy Storage 2nd Edition

proportionately decreases on the respective cell side. Total cell volume is 1 liter, as shown in Table 6.2 below. As one can see, the energy stored and the available voltage both rapidly diminish as the cell bulk concentration difference progressively diminishes. The storage of reagents at the electrode sites in extremely concentrated form is one of the numerous contributions or innovations. Figure 6.5 diagrammatically illustrates this effect. The reagents are stored as uncharged molecular structures within sites by “ion injection” and van der Waals-type forces. These molecular species are slowly released as ions in solution in the immediate vicinity of the electrode surfaces in accordance with the demand during discharge. The important issue in assessing or computing the performance of cells and their operating voltages is to realize that the cell potentials are the result of the electrolyte environment in the immediate region of the electrode surfaces (frequently referred to as the Helmholtz layer). Conditions in the bulk electrolyte region are mostly irrelevant with regard to electrode potentials. Only what the electrodes “see” to some limited distance away is important. To use the example above of active species A, the potential of a concentration cell is expressed essentially as


0.05 | A |n2 | A |n2 1 log . z | A |1n 1 | A |2n


Cells at TRL have been operated with as much as 1.2 to 1.4 volts open circuit. Such high voltages would entail concentration ratios of between 1020 and 1024 for the logarithm to the base ten to be high enough to achieve these potentials. It may seem that somehow the cell operation violates some fundamental law involving conservation of energy or a related issue. Even though the cells depicted in Figures 6.4 and 6.5 are almost identical, because their volumes are the same and the reagents are the same due to the electrode Table 6.2 Calculated voltages vs amp-hour input. C1


AH input

End voltage

Energy stored

3.9 M

0.1 M



~1/2(2.5) Wh











The Concentration Cell


structure and mechanism of storage, one gives immensely greater voltages. Note that the charging takes place at much higher voltages. Hence, the energy input, or work done on the charged ions in “compressing” them into storage at high molecular densities, is also great. As in compressing a metallic spring, the force needed to further compress the metal coil will increase as compression progresses, but that energy is available upon “discharge,” minus some irreversible frictional heat losses. This CIR process may be regarded as analogous to a molecular (ionic) spring. The important issues regarding the CIR approach are its very low cost, extremely long life, and abuse resistance. These are the prime factors that motivated exploring the concentration cell approach. In addition, it is an approach that departs in principle from all other batteries. This system depends solely on concentration differentials of the same materials. Hence, the problems of cell materials and structure corruption due to irreversible diffusion and non-uniform deposition of active reagents are absent. This approach to energy storage enables us to avoid irreversible materials transfer problems and presents an opportunity for exploring many different materials combinations as well as the relatively unlimited possibility of making cells with very high voltages. A brief comparison between the full flow and a static electrolyte does point out the obviously greater complexity of the former. Problems with full flow include the following: • Maintaining reasonably uniform flow through many parallel cell channels • Additional hardware of tanks and plumbing, pumps, and manifolds • Increased complexity of battery module with feeder tubes, etc. • Additional power required for pumps • Self discharge losses when sitting idle with flow • Necessity to build hardware as pressure vessels • The static cell method avoids all of these problems.

6.3 Further Discussions on Fundamental Issues The following is intended as a quick review of the origin of concentration potentials in an electrochemical cell. Sometimes these potentials are referred to as polarization voltages resulting from starvation at electrode surfaces of reactant containing electrolyte. Concentration cell behavior will be developed here in terms of familiar chemical principles.


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First, we must define the meaning of half-cell potential to understand the development of the concepts of concentration dependent voltages. The kinetics, diffusion equilibrium, sorption rates, and electric field penetration depths are not discussed at this time because they don’t contribute to the basic understanding. There are trade-offs between storage capacity and discharging ability. However, none of these issues are necessary in order to comprehend the technical approach and its configuration. The main purpose is the identification of the important issues in a stepby-step fashion so as to grasp the physical essentials of a concentration cell. Let us examine the potential between a metallic electrode and its ions in solution in its immediate vicinity. Consider the familiar configuration of a copper plate immersed in a copper sulfate solution. One may reasonably ask what the electric potential is between the copper and the solution of its own ions, as shown in Figure 6.6. Unfortunately, there is no physical way of making such a measurement without the presence of a second electrode. Such measurements have long ago been standardized by the use of either a silver/silver chloride, a calomel electrode, or a hydrogen reference electrode. In general, whenever electric potentials of the various elements are given, they refer to a hydrogen electrode, and that hydrogen reference is arbitrarily set at zero. In this case, the cupric potential is +0.337 volts with respect to hydrogen:

Cu++ + 2e → Cu + 0.337 v H2 → 2H+ + 2e 0.00 v

Pos +

Neg –

Probe electrode



CuSO4 solution

Figure 6.6 Standard reference electrode potentials.



The Concentration Cell


Since the potential of any electrode in an electrolyte is dependent on the concentration of the specific ion involved in the attendant reaction, we know that the voltage for copper is determined at a standardized electrolyte (copper ion) concentration. In most cases, this has been standardized as a concentration in which the “activity” is unity. Usually, the condition for activity = 1 at STP is in a 1 molar solution. The standard hydrogen potential is established at 1 atmosphere pressure of H2 gas phase and surrounded by hydrogen ions at unit activity. The above is important because one must establish a series of reference or standardized conditions in order to explore any of the cell properties. In the classic Daniel Cell, we have the two metals Zn and Cu in solutions of their respective salts. In Figure 6.7, a porous barrier separates the solutions. The potential between the two electrodes is as follows:

Zn → Zn++ + 2e +0.763 v Cu++ + 2e → Cu +0.337 v Net potential 1.10 v


Now, let us look more closely at the situation encountered by an electrode when it is immersed in electrolyte at a “bulk concentration” of any specific solute or ions. The diagram shown below in Figure 6.8 is a standard representation of an interpretation of ion concentration and electrode potential in view of distance from an electrode.




Copper Cu++


Porous barrier

Figure 6.7 The Daniel cell.


Energy Storage 2nd Edition Helmholtz layer Gouy diffuse layer + +

– +

+ +

– –

– + +



– – – + + + + – + – + – + + +

+ –






– +


– – +




ψ s



Figure 6.8 Electrolyte regions at electrode surfaces.

The above tells us clearly that there must be “intimate” contact between the electronic conducting electrodes and the molecular Helmholtz layer species undergoing electron exchange. This all happens within a very short distance. The usual porous electrodes are simply not useable here. Their site distances are too great. If we are not able to maintain the intimacy, then the cell will be very severely limited in performance by diffusion to the outer reaches of the bulk electrolyte. An electrode immersed in a static electrolyte (non-flowing) sees only a small distance into the surroundings. This distance is not much more than into the beginning of the Gouy diffuse layer. The layer, referred to as the Helmholtz layer, is essentially an electric double layer. The voltage, Δϕ, between the electrode and the remainder, or bulk electrolyte, is given as

Δϕ = ϕm – ϕs = ΔϕH + ψ.


The Concentration Cell


The distances involved are quite small. In most instances where electrolyte concentrations are appreciable, the diffuse layer is in the order of 10−7 cm, or a few ionic (molecular) diameters thick. Obviously, if electric current is produced at the electrode surface, or if there is significant flow in the surface region, the potential diagram above would be distorted accordingly. We can now return to the matter of electric potentials and concentration conditions. In fact, we will use one of the specific couples we have explored as a quick example. Consider the cell (see Figure 6.9) between hydrogen and an inert electrode such as Pt or carbon. One side of the cell is a hydrogen probe in an acid solution (H+ ions), and the other is a carbon electrode immersed in a solution of a mixture of ferric and ferrous ions (perhaps ferric and ferrous chloride). The anion, chloride, has little to do with the whole process. The cell reaction is

1 H 2 Fe 2


H .


Applying the Nernst equation, which says that the potential is proportional to the logarithm of the ratio of the activities, to determine voltage,


RT a H a Fe ln nF a1H/22 a Fe

Carbon electrode


Fe++ H+ Fe+++ Porous barrier

Figure 6.9 Hydrogen reference electrode cell.


Hydrogen reference electrode, Pt



Energy Storage 2nd Edition

but since aH+ = 1 and aH2 = 1,


RT a Fe ln nF a Fe



Eo is the electromotive force of an inert electrode surrounded by equal activities of ferrous and ferric ions when measured against the standard hydrogen electrode. When aFe++ and aFe+++ are equal, the last term becomes zero. These activities have direct relationships to concentrations, and in some instances (dilute solutions) it is possible to substitute concentrations for activities as a very first approximation of cell voltages. Now imagine a cell wherein both electrodes are chemically inert (carbon) and are immersed in a mixed solution of ferrous and ferric salts. The net potential between them would obviously be zero, if for no other reason than the symmetry or sameness of both sides of a cell. An oxidation/reduction process does take place but strictly to provide a mechanism for electric charge transport. In no way does this redox contribute to either the cell voltage or the energy level. As you can see from the basic thermodynamic equations resulting in the Nernst relationship, nowhere do the properties of the specific reactants appear other than their concentration. Figure 6.10 illustrates a simple concentration cell for non-porous electrodes. Both sides are the same except for the differences in concentrations of ferrous and ferric salts (not ions by themselves). Not shown is the fact that the solution is acidified and conduction is mainly by H+ ions. Cation membranes

Right side

Left side

Carbon electrodes


Fe+++ Fe++


Microporous barrier or ion exchange membrane

Figure 6.10 The iron concentration cell.

The Concentration Cell


are employed, and iron is notoriously sluggish as a charge carrier. All reagents, whether they are ferrous, ferric, or sulfide salts in polysulfide cells, are assumed to be in uniform solution in the bulk electrolyte volume on their respective sides of a cell. Cells with more practical characteristics do not have as simple a configuration as those shown above. In order to design them for more practical uses, they store reagents in a very concentrated (or very diluted) format within the electrode structures themselves. Such configurations are shown and described in considerable detail in the pages to follow.

6.4 Adsorption and Diffusion Rate Balance The following is a description of the principal aspects of a concentration cell as configured in this development and is in the form of a device where the internal charges are ions in “solution.” There are few chemical elements and compounds that lend themselves well to such processes. The active material chosen as representative of this class of device employs sulfur as both oxidizer and reducer. In addition to treating the diffusion rates through a separator into and out of the bulk storage regions, the rates of adsorption/desorption must be taken into account. As a first approximation, let us use the expression by Langmuir regarding adsorption isotherms. This approximation does not account for changes in adsorptivity as the surface sites become more occupied, nor does it account for any changes in the ratio of the coefficients αa and αd, the adsorption and desorption, in the constant relationships below.

Rate of adsorption = αa (1 – ϕ)C Rate of desorption = αdθ


C is the concentration of the species in solution being adsorbed, or the adsorbent, and θ is the fraction of the total available sites that are occupied by adsorbent at point in time. The introduction of this new term changes not only the mathematical balance equations but also the very nature of the mechanisms of storage. Now the electrode is no longer just seeing the concentration of specific ions in the surrounding bulk electrolyte, but it primarily sees the concentration of the adsorbed ions readily available at the electrode surface. Therefore, it becomes necessary to modify the model and our way of thinking about what may be happening within the cell.


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The rates are as follows: • • • •

Rg = generation rate of S= ions, always at the (−) electrode = KgI Rs = adsorption rate = α1(1−θ)C Rd = desorption rate = αdθ Rm = net diffusion rate across the membrane K m / V(2 Q1 Q o).

In more general terms, this may be expressed as a sum of differentials as follows. Figure 6.11 is a diagram of the “compartmentalized” nature of the cell, with the “Helmholtz” region being essentially what the electrode sees, and is the concentrated electrolyte in dynamic equilibrium with its solid forms on the surface of the porous electrodes. This region of electrolyte is also in dynamic equilibrium with the bulk electrolyte occupying the volume between the electrodes and the separator membrane. The bulk concentration differentials across the separator determine the diffusion rate of soluble components, e.g., sulfur complexed with sulfides and sulfide ions from one side of a cell to the opposite side.

Inner electrolyte region, CH

Conductive substrate

Bulk electrolyte region, CB

Porous carbon


Figure 6.11 Half-cell representation.

The Concentration Cell


Low solubility versus high salt solubility is an interesting issue. Also, we want salts to go into solution fast to sustain higher cell currents but to also precipitate out of solution for higher retention purposes.

dQ dt


dQ dt


dQ dt


dQ dt


dQ dt

(6.18) ppt

In equation (6.18) the terms are defined as follows: (dQ/dt)net is the net rate of increase of the specific ion. (dQ/dt)i is the rate of ion production by charging electric current. (dQ/dt)diff is the loss rate by diffusion away from the electrode. (dQ/dt)ad is the loss rate from solution by adsorption. (dQ/dt)ppt is the loss rate by precipitation of their compounds. Thus, there are the balances between the rates of desorption as well as the solubilization of the precipitated reagent, in this case, Na2S. The rates of sorption and diffusion are placed in the loss category since they represent losses from the (−) electrolyte. Furthermore, desorption and the electrical generation rates are to be considered gains, or sources of S= ions to the (−) electrolyte. In this fashion, we can handle the ensuing balance equations. Thus, the net rate, R, into the (−) electrolyte can also be represented by equation (6.19):

R = Rg + Rd – Rs – Rm,


where Rg is the production rate by electric current, Rm is diffusion rate, Rs is the sorption rate, and Rd is the desorption rate. More specifically, the above becomes






Km (2 Q1 Q 0 ) V


6.5 Storage by Adsorption and Solids Precipitation The most important parameters for optimum cell operation are (1) to maximize energy capacity by increasing the amount of charge density per unit


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area of electrode and (2) to establish high and sustainable concentration ratios of ionic components, i.e., large concentration at the (−) electrode and small concentration at the (+) electrode. One way to enhance charge capacity while reducing diffusion losses is to make use of solid precipitates. If the solubility of the sulfide compounds is exceeded, then they will precipitate out of solution and, by design, onto the surfaces of the electrodes. This provides for the additional supply of reagents and in a form that will remain within the (−) and (+) cell compartments for longer periods of time. If we stipulate a simple linear relationship between the solution and dissolution (re-solubilizing) rates for the solid sulfur and polysulfides that fall in and out of solution, we can express this additional factor as follows. Let the rates Rp1 and Rp2, the rates with which the compounds are precipitated and dissolved, be represented as

Rp1 = β1(C1 – Cs) Rp2 = Pβ2,


where β1 and β2 are constants at any given temperature, C1 is the concentration of the S= ions in solution, and P is a constant associated with the amount of solid Na2S in solid precipitate form. The term Cs is the maximum concentration that the electrolyte will tolerate prior to “salting out.” This last term is not a constant and tends to be very dependent on conditions such as temperature, presence of suspended solids, etc. The concentration tendencies of C1, being somewhat above Cs, will drive the precipitation of material out of solution. We can assume, for the sake of simplicity, that the relationships are linear. Thus, the complete rate equation takes the form





(C1 C s )

)C1 1



Km (2 Q1 Q o ) V



The net quantity of interest to us at the end of charging is the amount of S= ions available for discharge. That is found by equating the input rates and

The Concentration Cell


loss rates at dynamic balance for any charging current, I, as the maximum achievable charge, which occurs when R = 0, or when








Km (2 Q1 Q o ) (C1 C s ) V


(6.23) Our main interest in the above derivations is the evaluation of the amount of species, Qa, adsorbed within the electrode. In this case, it’s the sulfide ion in the form of the electrically neutral compound, sodium sulfide. (Ions cannot be adsorbed as such without the accumulation of an inordinately high electrical charge). It is necessary then to put θ into terms of quantity of material rather than the ratio of occupied sites to total available sites. This is easily accomplished. If we let A = the total number of available sites (per unit electrode volume), then the factor (1−θ) can be replaced in terms of A: A Qa A


Qa A



C1K a

, then the adsorption rate is

A Qa A


The only explanation for the magnitude of voltages obtained from our experimental cells is that the mechanism of “electroadsorption” or its equivalent takes place. That would necessitate a small layer of “stagnant” electrolyte at the electrode porous surfaces. This layer might be thought of as a very dense cloud layer, δ, of concentrated species (S= ions) that are about to be adsorbed. The balanced rate equation is thus modified to reflect this micro-layer assumption as








Km (2 Q1 Q o ) (C V

Cs ) 1 , (6.25)

where Cδ is the concentration of S= in the immediate neighborhood of the electrode and precipitate surfaces.


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Substituting the expression for θ in terms of Qa, the amount adsorbed, we get



Qa A





Qa C A

Km (2 Q1 Q o ) (C V

C s ) 1 . (6.26)

The time delay between adsorption and generation by electric current and charge transfer largely gives rise to this δ layer of not much more than a number of molecular diameters, or mean free paths in thickness. It is very important to the successful operation of such concentration cells, regarding their practical application, that the capacity and charge retention are not entirely, or even largely, dependant on membrane characteristics. Otherwise, we would be engaged in the continuing compromises between electrical conductivity and diffusion coefficients of such materials. Virtually everything that is done to reduce separator electrical resistance also promotes molecular diffusion in membranes. Hence, we seek mechanisms wherein molecular species can be collected to very high concentrations by some sort of bonding or retardation process, while not significantly detracting from either the cell potential or ionic mobility. The membrane serves the purpose mainly of keeping the two bulk electrolytes apart. A high effective concentration of the ionic species of interest (the S= ion in this instance) must be established and maintained throughout the charging process in order to “force” the diffusion of that ion into the carbon surfaces to be adsorbed. A gel electrolyte might very well serve that purpose. Some of our experimental results have shown that excellent operation can be obtained with only a gel electrolyte to immobilize the substances. However, it is necessary to pay attention to the mechanism of electrode starvation when employing gels because the S= ion can be depleted in the δ layer, resulting in high resistance and little charge transfer. The risk with gel electrolytes is that a large amount of the total reagents in the needed oxidation state can be trapped too far from the reaction sites for prolonged periods of time, becoming essentially unavailable for electron exchange.

The Concentration Cell


6.6 Some Interesting Aspects of Concentration Cells 1. All active materials employed have some solubility, enabling all solids that are formed during the cycling processes to be returned to new and uniform positions within the cell. 2. The cell is symmetric in the materials sense, i.e., the active materials are the same throughout the cell. However, they are not symmetric with regard to oxidation state population densities. 3. There are only two oxidation states available with materials selected, hence the symmetry at discharged condition. 4. Ionic, energy-storing components are stored within electronically conductive, high surface area pores of electrodes, thus enabling high coulombic capacity. 5. To further increase coulombic capacity, active materials are stored in quantities that exceed their solubility in the electrolytes. These precipitated components are deposited and stored also within the pore structure of electrodes so that they may be readily available for re-solubilization and subsequent participation in the electrolytic, energy producing reactions. 6. To obtain and maintain high electric potentials, the concentrations of reducing and oxidizing agents are replenished by the respective precipitation of components. An example of this is during the discharge mode of a cell at the positive electrode. As the sodium ions arrive, the sulfur stored in that electrode solubilizes and generates sulfur ions in association with the newly arrived sodium ions, thus maintaining a high sulfide ion concentration within the close (Helmholtz) region to contribute to cell voltage. A similar, but opposite, process takes place at the negative electrode where solid sodium sulfide salt is precipitated out of solution. 7. All active components are electronically non-conductive to prevent internal short circuit situations that are encountered with metal plating, dendrite growth, etc.


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For the sake of simplicity, let us replace activities with concentrations R and L for the right and left sides, respectively. Now the expression for net cell potential, E, can be represented as


RT R Fe L Fe ln . nF R Fe L Fe


Examining this cell more closely will reveal some serious limitations regarding maximum attainable voltages and energy densities. Even if we had a perfect barrier (separator with zero resistance and zero unwanted diffusion or transport), the first limitation would be due to the maximum solubility of the iron salts. They are limited to about 3 molar at room temperature. That means that the limitation would be 1.5 molar solution on each side at the start (assuming equal volumes on either side). Looking now at the voltages that might be expected, we see that, if upon charging we went from 1.4 M to 2.9 M of ferrous on the left side, ferrous concentration would be down on the right side to 0.1 M, and, correspondingly, the ferric ion concentrations would be up on the left to 2.9 M and down to 0.1 M on the right. A total charge transfer could be calculated on the basis of cell volumes. Substituting these values into the Nernst equation above, we get


0.06 log

2.9 2.9 0.06 log 840 0.15 volts. 0.10 0.10


Even if we were able to achieve concentrations of 0.01 M and 3 M on the respective sides, the maximum voltage realizable would be about 0.25 volts. Let’s now look at the total electric charge available. Taking the maximum molarity change of 1.5 and a volume of 1 cm2 per cell side, the maximum charge transfer would be amp-hours at an average voltage of 0.15 volts. Obviously, this needs to be improved. The greatest limitations to attaining higher voltages and energy densities are due to two principal factors: 1. Limitations of solubility of reagents 2. Subsequent limitations to concentration differentials Hence, we cannot be limited by the solubility of reagents if we want high ED. We must, therefore, synthesize another means of keeping the reagents around and available. Reagents are, in this example of an iron cell, ferrous and ferric compounds. If we try to increase bulk electrolyte concentration,

The Concentration Cell


the reagents will merely precipitate out and fall to the bottom. However, an actual concentration cell would not have a bottom as such because very porous, conductive materials would occupy most of the intervening space in a cell. In order to overcome these factors, we may resort to storing reagents in an adsorbed or “pseudo-solid” form within electrode structures themselves. This single feature enables us to go well beyond solubility limitations as well as developing and maintaining concentration ratios for higher voltages. It is especially possible to achieve these ends because reagents in solid form are not metallic and do not conduct electronically. Figure 6.12 shows, in principle, this approach method. The entire intervening space between electrodes and the separator is occupied by extremely high surface area and electronically conductive carbon. These micro or nano-porous particles are selected for their conductivity range, pore size, and pore distribution. The carbon employed has an available surface area measured as between 1,500 to 2,500 m2 per gram. This corresponds to a structure whose walls are in the range of 2 to 6 atomic diameters thick. Much of the reagents are stored in an interstitial state on the carbon structure. Estimates based on these kinds of data and experimental findings indicate it might be possible to attain performances equivalent to 10 to 15 molar solutions. Neg


Conductive carbon substrates

Microporous conductive carbon particles


Figure 6.12 Simple non-mobile electrolyte concentration cell.


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Also, by the self-constricted, controlled ionization constants within these structures, it appears that we are able to attain effective concentration ratios of 1010 to 1014 from each oxidation species. In looking at the basic equation for the voltage, it can be seen that, in order to achieve a cell potential of even 1.2 volts, the logarithm term in the expression for voltage must be as high as 20. An additional benefit is the reduction of diffusion losses (increased charge retention) because much of the reagents are stored in an interstitial solid state. Such cells and their electrodes must be fabricated and prepared before assembly to utilize such compactness. There is a balance between undissociated and ionized molecular ferrous and ferric compounds as part of the storage and transport release processes. Our understanding of this and various other processes and energy level change mechanisms is not yet clear, but progress is being made. The attractiveness of these cells is their simplicity and their reversibility for an indefinite number of times.

6.7 Concentration Cell Storage Mechanisms that Employ Sulfur In the CIR system it is necessary to store reagents in very concentrated form at the electrode surfaces themselves. The reagents can be stored as ions in the form of soluble compounds or as their solids ready to go into solution as needed during charging/discharging modes. Bulk storage of reactants, oxidized and reduced state ions, in the electrolyte as dissolved compounds provides very low voltages and specific energy storage density. Most compounds of useable materials, such as those of iron, sulfur, bromine, copper, etc., are limited to about 4 molar at standard temperatures and pressures. Such limitations in solubility give rise to small energy densities. The cells with which we are concerned here are classified as cells with liquid junctions and in which transference takes place. The sulfur system will be our first analytic model for determining the balance of materials at the beginning and end of a charge cycle. If we were to regard only the soluble forms of the sulfur salts, e.g., sodium, potassium, or lithium compounds, the maximum ED attainable is calculated as follows. Let us assume as a first approximation that no polysulfides are formed and that the process is strictly between sodium monosulfide and elemental

The Concentration Cell


sulfur. Also, we need to start any charge cycle with the same materials with their concentrations equal on both sides of a cell. The minimum concentrations for balance would be 2 Na2 // 2S (fully charged), where // indicates a divider or ionically conducting separator between cell compartments. Also, we will assume for simplicity that the two cell compartments are equal in volume and that only sodium ions are transported across the separator. Let us take this step to discharge. The discharged situation would be Na2S + 2S // Na2S + S, which is a symmetrical balance with no difference in concentration on either side of Na2S or S. The net transfer of charge per liter per side would be 50 AH. The next workable concentration for balance is an even number of moles, or 4Na2S // 4S fully charged, 2S + 2Na2S // 2Na2S + 2S discharged state. The net transfer per liter per side here would be 100 AH. The next, of course, would be 6Na2S // 6S charged, 3Na2S + 3S // 3Na2S + 3S discharged. Net charge transfer per 2 liters total of volume is 150 AH. In generalized form, if n is any integer, then the process proceeds as above as 2nNa2S + 2nS // 2nNa2S + 2nS charged, and the net charge transfer per total liter volume is simply n × 50 AH. Let us examine briefly the structure of the final reactants. If the highest possible polymers were formed upon discharge, they would be the dimers, or Na2S2. If the cell electrodes were unable to effectively (reversibly) store elemental sulfur, then we would need to resort to the maximum polysulfide at the charged state. That seems to be the pentasulfide, Na2S5. Then the cell at total charge might be xNa2S // yNa2S5, and thus limited in reactant weight by the necessity for higher ratio of sodium ions per sulfur atoms present. Let us start with a one molar solution of the pentasulfide on one side of the cell and the appropriate molar monosulfide on the opposite side so that we conclude at the end of discharge with the same materials at the same concentrations on each side. The steps toward total discharge are as follows: 1. 2. 3. 4. 5.

XNa2S // Na2S5 (X−1)Na2S // Na2S4 + Na2S (X−2)Na2S // Na2S3 + 2Na2S (X−3)Na2S // Na2S2 + 3Na2S (X−4)Na2S // 5Na2S


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Hence, the value of X = 9 makes the balance exact at the discharge end. The total charge from this 9 molar mono-sulfide solution would be 200 AH, and the total weight exclusive of water would be 862 grams. The ED for the dry salts at 1 volt is then 105 WH/lb. Now we shall examine the possible mechanisms for storing the materials in their two states of oxidation. The goal is to structure a system such that a maximum of concentration difference can be achieved between S−2 and S on the two opposite electrodes. It is important to sustain that high voltage over the longest portion of the discharge, or charge delivery.

6.8 Species Balance There are three explanations possible that, at this point, could describe the situation of affairs within the cell at any state of charge. First, let us assume that all species remain solution at all times, that the only charge carrier within the cell is Na+ ions, and that there is no net migration of sulfur from one cell side into the other. Vertical bars, ||, enclose concentrations. The two sides of a cell (opposite the separator) are indicated by subscripts a and b:

|Sa | |S b | constant,


|Sa 2 | |S b2 | constant.



The above also assumes that the sodium sulfide exits within the pores of the carbon electrodes as ionized molecules. This assumption seems not to satisfy the very high cell potentials experimentally measured in the laboratory with solute concentrations in the range of 2 to 3 molar. We will later examine the diffusion rates and possible high concentration differentials in such a configuration. Secondly, it seems more likely, however, that the species sulfur, polysulfide or monosulfide, would partially be in either a solid state or in an adsorbed state on the pore surfaces. If so, the balance of materials may be expressed as

Sa 2


S b2








The Concentration Cell


where the subscripts relate to the cell sides a and b, and i and s indicate ionic or solid state, respectively. The brackets contain the absolute quantities of the respective ions, or atoms, in either the a or b side of the cell. There is also the possibility that these materials may exist not as solids but in the form of an “interfacial state.” See “Physical Chemistry” by E. A. Moelwyn-Hughes, Pergamon Press, 1957, pages 894–950.

6.9 Electrode Surface Potentials It is important to consider whether the cell configuration is one with or without transport in order to account for any liquid junction potentials supplementing the electrode voltages. Nernst’s relationship does not account for the very high electric potentials attained in laboratory test cells solely on the basis of measured bulk electrolyte concentration differences on either side of cell separators. Since there are no other significant reactions taking place the electrodes must be experiencing ionic concentrations ratios orders of magnitude greater than the bulk electrolyte ratios. The search then begins for a plausible explanation of how a cell with no other energy related processes taking place, other than the difference in concentration of the two oxidation states of the same chemical element, can produce such high potentials and sustain high electrical charge densities. If there are no other processes involved, then we must look toward some mechanism whereby the chemical species in question are able to be accumulated at the electrode surfaces and give rise to such high voltages. At the present time, it is speculated that the specific ionic species, in this case the ferric and ferrous ions, are injected by the process of electrosorption directly into the pore regions of the microporous carbon at very high effective concentrations. In the case of a concentration cell based on iron chemistry even though the maximum concentrations of the compounds of ferric and ferrous chlorides are limited to not much over 3 molar at room temperature, the population densities of the adsorbed and stored ferrous and ferric ions become equivalent to extremely high activities seen by the electrodes. In order to accomplish this, more energy than what is normally needed would be required to separate the oxidation states during the “charging” process. This condition, for the maintenance of conservation principles, has been observed during cell cycling in terms of the voltamp inputs and outputs.


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There are undoubtedly many processes and mechanisms that take place at the electrode surfaces that call for greater understanding and quantifying. Let us first list and then examine the physical and chemical activity possibilities. Omitting the chances that there are any net or permanent chemical changes occurring in the electrical cycling of the cell, the following are possible, reversible processes: • Exceeding salt solubility at electrodes during charging, resulting in solid compounds at the surfaces with attendant free energy changes (energy of ionization and dissolution) • The creation of immense concentration ratios of S= in the sulfur based cell and of Fe++ and Fe+++ ions in solution at electrode surfaces in iron cells, which are far greater than could exist in the bulk electrolyte perhaps in an interstitial state • Adsorption of iron ions, and perhaps the salt compoundsthemselves, within the carbon porous structure, which may be Langmuir or van der Waals processes, depending on whether they are attached as electrically charged components or as neutral molecules with dipole moments.

6.10 Further Examination of Concentration Ratios In examining the cell from a practical viewpoint, it is obvious that most inorganic salt compounds are soluble in water only to the extent of 2 to 4 molar concentrations. If we evaluate the performance that can be expected from simple, two compartment concentration cells, depending on bulk concentration differences, the performance as practical storage cells is quite low. For a concentration ratio of 10:1 of a divalent active species, such as sulfide, the cell potential is ~0.03 volts. A concentration ratio of 100:1 results in a potential of 0.06 volts. These are exceedingly low values for a device to be practical, and the major portion of the stored electrical charge would be delivered at substantially lower voltages than the above. The maximum charge transfer per liter of 3 molar solution to 0.03 molarity is only slightly more than the transfer from 3 molar to 0.30 molar. In the experiments conducted that relate to this invention, potentials of 1.0 volt and higher are regularly attained and sustained. This development provides a mechanism for increasing the voltages and charge

The Concentration Cell


densities of a concentration cell by over an order of magnitude above that which are realized simply from differences in bulk electrolyte concentration differences of chemical species on opposite sides of a cell membrane separator. The voltages obtained are associated with concentration ratios of well over 1000:1. An extremely important aspect of this development concerns the method of collecting and storing the differentials in concentrations at the electrodes. In the method employed here, the substances that produce the electric potentials are electrolytically produced ions injected into the micro-pores of both cathode and anode. Activated carbon, especially micro-pore coconut charcoal, is the most effective means for storing most molecular species in an available and reversible fashion. The cell reagents are stored by means of adsorption of the van der Waals type. The carbon structure appears unaltered even after 10,000 cycles of charge/discharge events. Activated carbon has a surface area of approximately 1,000 to 2,000 square meters per gram of accessible surface, depending on electrolyte conditions and physical properties of the solute to be adsorbed. Assuming about 103 m2, or 107 cm2 area per gm, and an average molecular area for sulfur or sodium polysulfide of 3 × 10−8 cm2, the number of molecules that could occupy the surface area of one gram of carbon as a mono-molecular layer is in the order of 1022 molecules, or about 2 × 10−2 moles. That number corresponds to about 0.1 ampere-hours of electric charge for a divalent species, such as sulfur. The bulk density of the active carbon used is in the range of 0.60 gm/ cm3, and its void space is about 60%. Hence, about 60% of the space occupied by 1 cm3 of charcoal would be filled by electrolyte in a concentration cell that employs such a material as the major electrode structure. From the preceding estimates, about 0.006 ampere-hours of charge can be stored per cm3 of electrode volume. The potential at the electrode surface is dictated by the concentration of the specific ion that is in the immediate vicinity (Helmholtz layer) rather than the bulk concentration that is some significant distance away. Returning to the above estimate of the number of moles adsorbed per gram of carbon, we find that the quantity of material in one cm3 of carbon is ~2 × 10−2 moles. That quantity of molecules, if dissolved in the 0.6 cm3 volume, would correspond to a 40 molar solution. In actuality, the electrodes encounter a much higher effective concentration because of the manner in which the species exists at the interface. The necessity for introducing the specific reagents into the active carbon pores by direct ion migration by means of an electric field gradient in the


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charging process can be explained as follows. The rate of molecular diffusion, J, due to a concentration gradient is simply



dc grams cm 2 sec 1 , dx


where D is the diffusion coefficient, and c is the concentration of a particular molecular species in solution. Generally, D has a value of about 2 × 10−5 cm2 sec−1 for most salts in water solution. The thermal diffusion of solute into the pores of activated carbon would have to take place only as a result of the concentration gradient generated by the adsorption rate out of solution of the specific chemical species. That rate is quite slow. In effect, it would give rise to only a very small value of dc/dx in the above expression. The rate of adsorption of molecules from solution is directly proportional to the number of adsorption sites that remain available and the concentration of the chemical species in the immediate vicinity of the carbon surface. This is expressed by the familiar equation

Rate of adsorption, Ra = (1 – θ)ka,


where θ is the fraction of the total number of available sites that are occupied by the adsorbents on the carbon surface, and ka is an experimental constant of the tests. If the storage of a substance by adsorption depends solely on thermal diffusion as the driv ing force, the process is very slow and is severely counteracted by diffusion in the opposite direction as sites are increasingly occupied. However, if an electric current is applied to the ionized salt solution with a current density of 0.01 amps/cm2, then the migration rate of the species will be in the range of 1016 ions per second, or about 3 × 10−7 gm sec−1 cm2. That is a much greater rate than can be expected from the thermal diffusion rate estimated above.

6.11 Empirical Results with Small Laboratory Cells The voltages experimentally obtained in single cells do not conform to the simple expression (RT/nF) × ln(c1/c2). The values are much greater than predicted. The main question is why. Since the cell is symmetrical, voltages are kept low, and since there are no sources of energy other than concentration differences on opposite sides of the separator, the operation would appear to be a simple concentration cell.

The Concentration Cell


Let us examine some of the cells’ performances. Doing so requires the following data: • Cell electrolyte capacity ~ 12 cc per side • Electrolyte composition and AH capacity to same composition on either side • Some test data, i.e., AH input and voltages obtained • Empirical potentials agree with the concentration relationship even when assuming no loss due to diffusion, etc. Solutions have been 2.5 molar in Na2S with added sulfur of 100 gm per 100 cc of 2.5 molar monosulfide solution. This would make the solution at full charge Na2S//Na2S4. However, this does not result in 0.05 AH capacity per 1 cc of total solution (both sides of cell). The empirical shapes of the charge/ discharge curves for the single cell-porous secondary cells do not conform to concentration cell mathematical predictions. A 10 cell module with no significant storage on porous electrodes did give the performance predicted by concentration cell math. A significant question is whether the van der Waals “forces” provide additional energy for discharge in the porous electrodes, or whether it simply provides for a higher apparent concentration differential. Also, there is the issue of the thermodynamics (energetics) associated with adsorption. Heat is usually liberated when molecules drop into the adsorbed state. If so, how does that affect the storage of energy? Usually, heat must be applied to free adsorbed state molecules. Hence, that cannot contribute to the cell potential. Perhaps it contributes indirectly by making the apparent concentration greater by holding onto species in greater surface densities. Is it possible that simply the huge increase in concentration as charging proceeds would account for this? Let’s assume that the electrolyte is the large “storage volume” of a cell for the reagents to reside in as a reservoir. Upon charging, these species are concentrated in very small volumes on the porous electrode surfaces and appear considerably more concentrated than the simple equations that employ the reservoir electrolyte volumes in the log relationship. Generally, the energy (evolved) of adsorption is low, about 10 kcal/mole adsorbed, which is about the same given off when gas condenses to liquid. Thus, the energy of adsorption is to be subtracted from the potentials realized via concentration differences. To explore single layer and multilayer adsorption, reviewing the Freundlich and Langmuir equations would be helpful. Langmuir relations apply best to saturated situations. Equilibrium is reached when desorption


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rates are equal to adsorption rates. One of the adsorbents can also be the solvent, taking up space in the adsorbent surfaces. Now we will discuss what the electrode experiences in terms of electrochemical potentials within these layers. Is it strictly a concentration voltage, or are there other processes involved? For treatments of energies of interfacial states, etc., see Moelwyn and Hughes, pages 922 et. seq. Some factors that influence the shapes of curves of cell behavior are listed below: 1. Electrical current, upon charging, “forces” the ions or molecules to become more concentrated than can be accomplished otherwise. The greater the current density, the greater the concentration of ion densities. 2. As proximity of these ions increase, the greater are the van der Waal “forces” that tend to further concentrate them. 3. As the population density of the species increases, the rate of adsorption increases at the electrode porous surfaces. 4. At least the “apparent” concentration differences between cell electrodes are enhanced. 5. We should address the reagent ion concentration seen by the electrode as the number of ions per unit volume in the immediate vicinity of the electrode rather than the calculations based on the “bulk” concentration of the cell on either side. 6. The voltages observed across a cell are well beyond what the simple macroscopic calculations predict. They might then be due simply to a temporarily high build-up caused by slow reagent diffusion away from the electrodes during charging. Then, the dwell time of these voltages would be very much shorter than observed in laboratory cells. If the above is true, then we must determine the specific volume for the electrode so that the concentration of reagents can be computed. So, let us see if we can approximate the performance of a cell as if it had small volume but molarity in the ranges of 10, 20, to 100. A good number to use for effective molecular diameters in these types of calculations, as obtained from diffusion and viscosity measurements, is about 5 × 10−8 cm. Then, examine the adsorption process on the basis of sorption rates due to thermal motion of molecules impinging on the outer porous electrode surfaces. Consider how movement under a voltage gradient impressed across the cell would change anything in terms of capture probability or desorption rates.

The Concentration Cell


For additional information about mean free paths and activation energies see pages 443, 448, and 348–350 of Daniels and Alberty. For some straightforward mathematical treatment of adsorption and capture probability see page 523 et. seq. Similarly, the Langmuir isotherms should lead the way for energy and electric potential relations in adsorption processes. Koreyta, in his book on electrochemistry, provides some inputs on pages 226, 227, (Helmholtz layer and Gibbs free energy of adsorption) and 246 on electrode processes and adsorption. He also discusses the solvent as being a structure less dielectric in which ions move about and interact. Now consider the diffusion rate of a molecule or ionic species at room temperature within a solvent such as water. We can estimate this strictly from the classical kinetic theory. Then, see how many will diffuse across a unit area per unit time and compare this with the flow rate of an ion if a voltage gradient were imposed across that same area. The diffusion coefficient D is defined in the relationship dq/dt = D (dc/ dx), where the linear concentration gradient is in the direction normal to some unit area. The problem here in determining the amount of net flow, or number of molecules diffusing into the porous charcoal sites, is that there is no thickness layer to assign. Hence, we might just look at the desorption rate for the solid surface (and make a guess as to its surface volume) and solve for a diffusion layer thickness that would match adsorption rate data and diffusion. For example, if we plot the rate dc/dt for a given solution and molecular species and then look for where the curves of adsorption rates and diffusion rates intersect, it should tell us what the initial conditions are. As adsorption progresses, the rate of adsorption will decrease because sites are being filled and desorption becomes a significant factor in the ultimate equilibrium. Moving boundary layer data with ionic solutions provides some useful insight into ion mobility as well as ionic velocities. The migration or diffusion of ions under the influence of a voltage gradient follows the same mathematical format as Fick’s law for molecular diffusion. One simple configuration of a concentration cell employs the two soluble oxidation states of iron. In the case of metallic materials, we are much less interested in using their elemental state because of the consequent possibilities of short-circuiting a cell via metallic dendrites and gas evolution, especially of free metals on electrode surfaces in an acidic electrolyte. The reaction of ferrous/ferric solutions discussed previously does operate very well as a concentration dependent cell, but care must be exercised during “charging” to maintain voltages below the hydrogen evolution level, usually in the vicinity of 1.2 volts in the acidic electrolyte. However, this problem can be avoided by resorting to non-aqueous solvents as


Energy Storage 2nd Edition

electrolytes. Such cells employing organic solvents generally have higher electrical resistance, and the salts will usually be less soluble. Limited solubility is not necessarily a serious problem if the cells are designed to operate beyond that point with solids in the electrolyte.

6.12 Iron/Iron Concentration Cell Properties Balancing the concentrations of components in a ferric/ferrous cell for the basic reactions that take place at either electrode is simply

Fe+3 + e–




The components on either side of a functioning concentration cell can be represented as shown below. The composition of the reagents must be the same on both sides of a cell separator at the end of discharge. At the beginning of discharge (charged state), the conditions are 2FeCl2 // 2 FeCl3. The cell could be represented at the end of discharge as FeCl3 + FeCl2 // FeCl2 + FeCl3 symmetrical. We can now calculate the maximum available energy density for this process, assuming a maximum of 1 volt potential difference in aqueous solutions to keep below the hydrogen evolution voltage in an acidic environment. The total molecular weight, ignoring that of water, is 2 × (56 + 2 × 35) + 2 × (56 + 3 × 35) = 574. Since there is a single charge carrier, there is 26.6 AH per 454/574 = 0.79 lb of reagents. This works out to about 1 volt × 26.6 × 0.79 = 21 Wh/lb divided by 2 as an average load voltage, or about 11 Wh/lb or dry reagents. The actual number is quite a bit lower when accounting for weight of water, or other solvent, and the dead weights of electrodes and enclosure. Despite the low ED, the system may be useful because of its low cost, high safety, and reliability of performance in stationary applications. The above estimates presume that the electrical charge carrier is the chloride ion, Cl−. A second version of this concentration cell could depend on positive ions as charge carriers. Perhaps the best would be the hydrogen ion, H+, because of its high mobility and consequent higher conductance in solution. A cation, or positive ion transfer membrane, could be employed in the cell for best separation and lowest diffusion losses. However, a micro porous membrane could serve well also. The cell composition at full charge might look like 2FeCl2 + HCl // 2FeCl3. At the end of discharge, the components on either side of the

The Concentration Cell


separator would be the same with the exception that HCl has been transposed, or FeCl2 + FeCl3 // FeCl2 + FeCl3 + HCl. The ED of this process is even less than that of the preceding because we must account for the additional weights of HCl in the reaction.

6.13 The Mechanisms of Energy Storage Cells There are a few chemical elements and their compounds that lend themselves well to such processes. Preference at this time is for sulfur, and it is the material about which we have the most empirical data. The reactions occurring at each electrode during both charge and discharge are S + 2e S–2. The element and its alkali–metal compounds are cheap, plentiful, safe, well behaved at room temperature, and have very useful properties such as the ability to readily form “polymers.” These polymers are merely sulfur atoms attached to the sulfide components such as the following:

Na2S + S


Na2S2 + S


Na2S3 + S

Na2S4 and so forth


All of the above are very soluble in water and other polar solvents. Please note, however, that the sodium ion has nothing to do with the potential producing mechanisms. It is merely the cation part of the compound that could just as easily be served by K, or NH4. In order to have conduction within a cell and not have an accumulation of electrical charge on either electrode, the transport mechanisms must be accompanied by suitable oxidation/reduction processes for the exchange of electrons in an external circuit. Figure 6.13 below shows a very simple

Figure 6.13 Cell diagram.






Energy Storage 2nd Edition

cell of two electrodes, an ion transfer separator, and two equal volumes, V, of electrolytes. The cell voltage is proportional to the logarithm of the concentration ratios in the immediate vicinity of the electrode surfaces. However, for the sake of mathematical simplicity, let us assume that the cell potential, E, can be expressed as

C E K 1 C2

Q1 K

V Q2



Q1 Q2





i dt,



i dt


The simplified descriptive approach above avoids mathematical complexities that contribute virtually nothing to the main arguments of cell operation. A plot of the voltage versus charge before substitution of the last expressions is a straight line, as depicted in Figure 6.14. As the cell is charged from its totally discharged state, in which all concentrations are equal to Qo, more sulfides accumulate on one side (−), and the sulfide ions are depleted on the other (+) side.


Max. Q

Figure 6.14 Simple linear dependence of voltage on concentration.

The Concentration Cell


However, if we substitute idt for the variable Q, the dependency of E upon Q is transformed from a linear dependency to a different shape. Making the substitutions, the subsequent approximations lead to the following equation for E:






Q o iΔt Q o iΔt


To simplify further, we may just let iΔt = x to see how the parameters are interdependent. So, we obtain

a x , a x



where a also represents Qo. Differentiating the above with respect to x,

dE dx

d (a x)(a x )



(a x ) . (a x) (a x)2


A plot of E versus x looks like the illustration in Figure 6.15, in which the voltage changes increasingly rapidly as x increases and approaches the value a. This is hardly a linear relationship. It is obvious that, as x increases, the voltage is large for smaller intervals of x as x approaches a. But, at these higher potentials, the ion carrier

E dE



Figure 6.15 Shape of the functional relationship in equation 6.39.


Energy Storage 2nd Edition

population density grows quite small, meaning that less charge is available at higher voltages until x = a, when no further conduction is possible. In practice, we would want to modify this behavior in a cell because only a small portion of the total charge would be delivered at the higher electric potentials. Herein lies the rather important, if not critical, part of this approach. First, we want to have as high a concentration of our reagents, in this case S−2 and S, as possible from the very start. This is partly accomplished by having some of the reagents Na2S and S initially in solid form within microporous carbon electrodes. Remembering that only the sulfide ion S−2 counts in the math for determining potential, we want high concentrations of such ions along with the necessary sulfur molecules to be available at both electrodes. Normally, this availability is limited at any time because of the solubility of the salt, Na2S. Therefore, we store a fair amount of it in “solid form” at the electrodes. Initially the cell has equal amounts of both sulfur and sulfides at each electrode. As charging progresses, sulfides are created at one electrode and sulfur at the other. A maximum value of sulfides at one electrode is reached, and continued charging results in solids coming out on that electrode stored for later discharge. Similarly, at the opposite electrode sulfur is deposited for later discharge. When discharge begins, the cell voltage is quite high because of the concentration differential that develops. As the cell is discharged, some of the sulfides become solubilized, replacing the loss of sulfides on the concentrated side and, thus, keeping the potential up. On the opposite side during discharge, the transferred sodium ions that are produced result in continued diminishment of sulfide ions and the release of sulfur. Thus, we have a reservoir of ions beyond solubility that is available to the electrolyte/electrode interface at some rate determined by diffusion constants and dynamics of electron transfer and solubilization rates. The discharge currents must be kept low enough so that the supply mechanism can keep up with the demands and maintain high potentials. This is the case because inordinately high voltages have been obtained with cells that have been provided with such structures at the electrode surfaces. Cells with no porous carbon provisions generate potentials in the range of at most 0.03 to 0.08 volts even after prolonged charging. Smooth electrodes give the expected performance shown by the Nernst equation. The thermodynamics are certainly correct, but practical performance is immensely altered with storage, probably by virtue of interfacial forces of the van der Waals type. The exact mechanisms of how the reagents are stored are not entirely understood, that is whether they are stored as solids adjacent to the electrode surfaces or actually adsorbed as ionic materials, etc. It makes a

The Concentration Cell


difference when one wishes to perform analyses or optimize parameters. Let us assume for the moment that, after solution saturation of the sulfides and polysulfides have been attained, further charging results in solids accumulation. Then we must consider the rates at which solution and dissolution takes place. Let us assume further that the rates with which the species are removed as solids and returned as solutes are simply proportional to the concentrations Ca and Cb of the respective species, sulfide and sulfur. Then the rates R at which the sulfides and sulfur solidify can be expressed as the following functions:

Ra = fa (C1, Ca, i) Rb = fb (C2, Cb, i)


Obviously, the rate of removal of sulfide will be directly proportional to its concentration in solution, inversely proportional to the amount already in solid form at the electrodes, and to some extent due to the electrical current density. The exact form of the above functions can be better estimated with more modeling of the rate mechanisms. The very reproducible and consistent experimental evidence that we have obtained to date with hundreds of cycles and many lab cells would indicate, according to the Nernst equation, that the electrodes are seeing concentration ratios as high as 1020 or more for us to be able to obtain 1.5 to 2.0 volts:

The Nernst Voltage, E = 0.059 Log(C1/C2)


A fascinating aspect of this system is that it does not depend on the voltages of any couple such as Zn/Br (1.8 volts), Zn/Cu (0.8 volts), etc. The materials are the same on both sides of the cell only at different concentrations. Diffusion of unwanted materials from one side to another does not result in irreversible run down of the cell. Aging mechanisms are scant, no gasses evolve, and pH stays constant. The carnival prize seems to be worth the effort to squeeze maximum ED out of it. The oxidation and reduction at the two electrodes necessary to get an electrical output is simply S + 2e = S=. Iron will do the same thing between ferrous and ferric. The ionic species, sulfide ion, is stored within the porous electrode structure at effective concentrations that are much greater than what can be achieved in solution. The sulfide ion storage is at electrode surfaces. Some of that storage is probably in the form of solid sodium sulfide immediately adjacent to electrode surfaces and available to those surfaces as they


Energy Storage 2nd Edition

return into solution. Some storage probably takes place in ionic form as adsorbed materials. Charge retention is a function primarily of the solubilization rates of reactants and diffusion across a membrane barrier. Dependence on the latter should be minimized because of the unavoidably encountered trade off between molecular diffusion and electrical conduction.

6.14 Operational Models of Sulfide Based Cells It is now necessary to develop some simple working models of the processes within a cell during both charging and discharging modes at each electrode. It will also be necessary to make some simplifying assumptions in our first attempt to explain what is happening at interfaces and at boundaries between phases. There are only three forms in which the reagents can exist in the immediate vicinity of the electrodes: (1) as solids of composition Na2Sx and s, (2) in solution with water as Na+ and S= ions or dissolved as polysulfides (unionized sulfur attached to the Na2S molecule), and (3) in the adsorbed state on the porous carbon electrodes as ionic species or as polysulfides. This situation is illustrated in Figure 6.16 with some assigned designations for the concentrations of these various quantities. The diagram is strictly a schematic form and does not necessarily display the actual physical structure and relative locations of the components. Certain additional presumptions must be made at this point in order to move forward with an analysis. These presumptions can and will be modified as we learn about what takes place inside the cell. When possible, we will take the path of greatest simplicity, Occam’s Razor approach. As a first model for analytical purposes, let us assume only two nonporous electrodes and a membrane that separates the two compartments. This is shown in Figure 6.17 below. The storage of reactants and reagents takes place in this first model solely in the electrolyte as dissolved components. The initial concentration, Co, of the active reagent, Na2S in this case, is simply


Qo , V


where V is the volume for electrolytes on each side of the cell. If we designate I as the electric current (ionic) passing through the cell upon charging,

The Concentration Cell


Solids Electrode

Electrolyte Adsorbed layers (Interstitial)

Figure 6.16 Sulfur species at electrode vicinities. Na+ S=




Adsorbed in electrode carbon

Na2S [Na+, S=] Solution S

Figure 6.17 Storage and flow model – first balance analysis.

then the rate, dQ/dt, with which sulfide ions are produced from dissolved sulfur (polysulfides) in the (+) side of the cell is as follows:

dQ k iI dt I dC ki V dt


There are four principal mechanisms that we know are involved in the operation of a cell. At the negative electrode during charging, they are the following: • The generation of sulfide ions by migration of positive sodium ions toward the negative electrode • The diffusion of ions across the barrier from one side of the cell to the other by virtue of concentration differentials of those specific species


Energy Storage 2nd Edition • The adsorption and desorption of molecules onto and out of the surfaces of porous carbon electrodes • As solute concentrations rise within an electrolyte, some portion will come out of solution (precipitate) as solids.

There are some other secondary issues associated with cell operation, such as concentration gradients within cell compartments, the number of water molecules adsorbed onto electrode surfaces with the solutes, and whether the adsorbents are still “in solution” as ionic species or as undissociated molecules. However, for the present we will ignore these matters to keep the operational explanations and descriptions as simple as possible.

6.15 Storage Solely in Bulk Electrolyte To further shorten and make more specific the ensuing descriptions, let us treat the sulfur/sulfide system as the one in question, and let us be concerned, at the beginning, solely with the processes within the negative side of the cell. That then defines the species with which we are concerned – the S= ion and the S molecule. During charging, sulfur is reduced to sulfide with the acquisition of two electrons at the negative electrode. At the positive electrode, sulfides are oxidized to sulfur by giving up two electrons, a remarkably simple and symmetrical cell. Looking strictly at the flow balance of S= ions generated at the electrode and diffusing across the barrier per unit area of working electrode and barrier, Ra = generation rate of S= ions = KaI, and Rb = loss rate by diffusion (Fick’s first law) = Kb(C1 − C2), where C1 and C2 are respectively the S= concentrations in the negative and the positive sides. Net rate, R, of accumulation of S= in the negative side is

R = Ra – Rb.


Putting the above in terms of electric current and ion quantities, we get the following: at any time, t, the amount of S= ions present in the negative and positive sides are Q1 and Q2, respectively, and the sum of these amounts is always Qo. Thus,


dQ1 dt

K b (C1 C 2)

Kb Kb (Q1 Q 2) (2 Q1 Q o). (6.46) V V

The Concentration Cell


Initially in a discharged cell Q1 = Q2, and as the cell is charged by the electric current Q1 becomes larger until it reaches a value great enough such that the rate of charging equals the loss of S= ions via diffusion across the barrier. This value is attained when Ra = Rb, or

Kb (2 Q1 Q o ). V

K a I K b (C1 C 2 )


The concentration differential attainable in such a simple cell is severely limited by diffusion losses, and the cell potential is limited by the maximum concentrations that can be provided for the electrodes. The total electric charge available at higher cell potentials would be quite small, as is evidenced by the algebraic comparisons below of charge density at different voltages. Since the sum of the sulfide ions present within the cell is a constant, we shall designate it Qo. Then, it follows that Q1 + Q2 = Qo. As Q1 approaches Qo, Q2 approaches zero. If we continue, for the purposes of mathematical simplicity, in letting the cell potential, Ec, be directly proportional to the cell concentration ratios rather than its more representative form given by Nernst, (the relationships will be corrected later after the present arguments for dynamic cell balance are established) then we can state the following:



C2 C2


Q1 Q2


Q1 . Q o Q1


Upon charging or discharging the energy, Ψ, we can evaluate the integral between two different sets of limits and compare the numerical results for energy input or output over the respective voltage ranges. The energy over any interval of time or voltage may simply be represented as

Ψ = Ec × Q.


Expressed in a more useful manner, the incremental change in energy because Ec dQ = Q dEc is




Q1 dQ1 . Q o Q1



Energy Storage 2nd Edition

The integral of this equation is

∫ dΨ = –k[Q1 + Qo ln (Qo + Q1)].


Putting the above expression in terms of voltage, we obtain the more convenient form

EQo = Q1 (k + E) EQ o . (k E)



And the differential of energy assumes the form



E (k E)

dE .


Integrating this last expression in terms of E, we get

E dE = Q o

E (k E)

dE E k ln(E + k)


Now, to illustrate that there is less energy available over any voltage interval upon discharge at the higher voltages, we need to merely look at the derivative of Ψ to see that as E becomes very large the slope of the curve of Ψ decreases and approaches a constant value of Qo. At a later time, the more representative Nernst expression can be substituted for cell potential, E, i.e.,


kRT ln

C1 . C2


At present, the exact form of the voltage dependence on concentrations is not important to establishing the argument for dynamic stability of the cell.

The Concentration Cell


6.16 More on Storage of Reagents in Adsorbed State In our previous approach to describing a dynamic balance of ion flow we considered only the rate of species generation via electric current counterbalanced by the loss of those species across to the opposite side of a cell by diffusion through a membrane. Membrane impedance to diffusion loss is limited mainly by a compromise with electrical conductivity through the same separator. Let’s introduce an additional and primary method of materials storage – adsorption. This is accomplished by employing microporous carbon attached to the surface of both electrodes. In the negative electrode, S= ions are stored in the form of the Na2S compound, and in the positive electrode the S is stored as either Na2S unionized, Na2Sx, or as the element S. Actually, S is stored at both electrodes during the entire process as a reservoir of reagents during charging at the negative electrode and for discharging at the positive electrode. However, the presence of S at either electrode does not enter into the thermodynamic relationship for electric potential. With such a modification of cell design we are able to develop much higher effective ionic concentrations. Let me use a fictitious configuration below to illustrate the essence of such improvement. If it were possible to establish a very small compartment with volume, δ, per unit area immediately adjacent to each electrode by means of an idealized barrier, as shown in Figure 6.18, we would be able to build electrolytically gigantic concentrations of species supplied by the much larger reservoirs of volume V. From our earlier relationship regarding potentials and concentration ratios (we will stay for a short while longer with the linear model rather than the more representative logarithmic one), the cell potential as a function of charge now becomes associated only with the concentrations in the

δ Immediate region to electrode surfaces V



Figure 6.18 Hypothetical compartments at electrode surfaces.


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region δ. Before we can proceed further with modeling the cell it is necessary to more closely examine the processes at the electrodes within the δ region. (See Figure 6.19). In the previous simple situation of flat electrodes and a single membrane, we assumed perfect and instantaneous mixing of electrolyte solutes. If we make that assumption again in this new model, we encounter some difficulties. Let us look at what happens at the negative electrode (realizing that a similar situation takes place at the positive side). In order for the reduction of sulfur to sulfide ions to occur, it is necessary to have a continuous supply of sulfur immediately available at the negative electrode. This supply can exist as solid sulfur in the electrode or as polysulfide ions in solution. In order to be able to charge the (−) electrode, sulfur must have either been previously stored within the δ compartment, or it must be made available as a stream from the large V reservoir. A mechanism is needed to keep the S= ions in the immediate neighborhood of the (−) electrode in order to retard their drifting away into the bulk solution. The latter explanation is based on the small compartment having free access via molecular diffusion to the entire electrolyte. That would make the situation identical to “storage in bulk electrolyte,” as described in the first case. In addition, high concentrations cannot be accumulated because of mixing. Then, other than the storage of sulfur prior to charging, there is no method we can visualize in which the δ region can sustain the high concentrations necessary for high electrode potentials.




S 2Na+ + S + 2e– Na2S


2Na+ + S=

Charging process

Figure 6.19 Representation of processes at electrode surface regions.

The Concentration Cell


However, it is possible to accumulate extremely high sulfide concentrations by adsorption within the electrodes during the charging process. An example of high surface area carbon shows that as much as 2,000 square meters of area is available per gram of activated carbon. Looking at what that might mean in terms of molecules of storage capability, we perform the following quick approximations. Since the average, effective molecular diameters are in the order of 2 − 5 × 10−8 cm, the average area of small molecules is in the order of 10−15 cm2. The available area of activated carbon surfaces is 2 × 103 m2 = 2 × 107 cm2. Dividing the two numbers gives us about 2 × 1022 molecules or atoms stored per gram of active (microporous) carbon. An interesting note here concerns the wall thickness of the porous carbon. One gram of carbon is 1/12 of a molecular weight. Since there are about 6 × 1023 molecules per gram equivalent weight of substance (Avogadro’s number), about 5 × 1022 molecules of carbon are present to do the adsorbing, or about 3 molecules of carbon per molecule of adsorbent. If we further assume an agreement with the Langmuir model of molecular surface bonding via a Van der Waals type of force, then the adsorbent is a monomolecular layer, and the carbon wall structure is on average not much more than 2 to 3 carbon atoms thick. The average specific gravity of such activated carbon is in the range of 1 gram per cm2. Then we see that about 2 × 1022 molecules, or ionic species, are stored per cm3 volume of porous electrode. To find the storage capabilities of this electrode in terms of electrical charge (amp-hours), we look at the relationship provided by Faraday. The Faraday equivalent is 96,500 coulombs per equivalent, or about 105 coulombs per single valence molecule. Since sulfur has two charges, that number becomes 2 × 105 coulombs. Since there are about 2 × 1022/6 × 1023 = 0.03 Avogadro numbers of S atoms stored, it requires in the order of 0.03 × 2 × 105 = 6 × 103 coulombs to electrolytically transport that number of dual charged ionic species across the cell. That number is 6 × 103 ampere seconds, or about 1.7 ampere-hours of charge (per cm3 of electrode). That is a fairly high charge density for an adsorbent electrode. There is significant evidence that adsorption of molecular species in solution will occur in more than one molecular layer (Freundlich and Langmuir’s adsorption isotherms). Now we can return to the consideration of rate processes. Superposed on the previous situation of bulk storage, the rates of adsorption/desorption must be taken into account. Again, as a first approximation, let us use the Langmuir expression regarding adsorption isotherms. As discussed earlier,


Energy Storage 2nd Edition

this approximation does not account for changes in adsorptivity as the surface sites become more occupied, and the ratio of the coefficients αa and αd, the adsoption and desorption, in the relationships of previous equations is not constant.

Rate of adsorption = αa (1 – θ)C Rate of desorption = αdθ


If we re-examine equations (6.25) and (6.26), perhaps we can put all the variable quantities of S= ions in terms of the amount adsorbed, Qa. We know that at time zero the following conditions exist: • • • • •

Qa = 0 P=0 C1 = C o Cδ = C1 = Co Q1 = Qo

Also, at t = 0, the rate of change, or increase of Qa is

dQ a dt




We can next proceed to conclude that the electric potential of the cell can be approximated as a function of the ratio of adsorbed species on each of the two electrodes, or

E = 0.059ln

[Q a ] [Q a ]



6.17 Energy Density An interesting estimate is the energy density available from a cubic centimeter of elemental sulfur in the charge transfer of one volt. The specific gravity of the sulfur atoms is about 2 grams per cm3, and its atomic weight is 32. With two electrons per atom transfer upon oxidation/ reduction, about 100,000 coulombs × 1/32 × 2 would be the charge exchange

The Concentration Cell


per gram of sulfur to S=. However, since 2 grams are present per cm3, the math works out to about 1.2 × 104 coulombs per cm3. Since 1 coulomb is an ampere-second, the charge transfer is about 3  amp-hours, which corresponds with 3 watt-hours of energy. Since the cells are symmetrical and there is an equal amount of material on either side, these numbers need to be divided by 2. Accounting for both sides of the cells, this works out to a maximum energy density, assuming an average potential of only 1 volt of about 24  Wh per in3, or over 41 kWh per cubic foot of cells. That is a very respectable number, even neglecting weights of water, electrodes, case, etc. Also, these cells might be able to operate at levels of many volts, thus increasing the energy proportionately.

6.18 Observations Regarding Electrical Behavior To provide a general idea of the shape of some typical charge/discharge curves, the following graphs (Figures 6.20–6.25) qualitatively show how these cells behave. Actual data with current and voltage values are supplied later in the book. For the present, these do show the nature of the cells and clearly show that they do not behave in any linear fashion. The discharge curve for constant current, constant load, or any other control is not “flat” with time. Efficiency of charge and discharge can be very high if the difference between open circuit and charging/ discharging potentials are maintained small and, perhaps, constant as diagrammed in the last drawing. The power supply and load should be designed to follow these curves for best efficiency.

Constant charge voltage, Ec E

Time, t Voltage limited charging

Figure 6.20 Voltage limited charging.


Energy Storage 2nd Edition


Time, t

Figure 6.21 Constant current charging.


Time, t

Figure 6.22 Fixed load discharge from “overcharge”.


Time, t

Figure 6.23 Constant current discharge.

The Concentration Cell


ΔE Open circuit E Charging voltage

Time, t



Load Current, Amps

Cell voltage, Volts

Figure 6.24 Maintaining a constant charge to open circuit differential.

Time, Hours

Figure 6.25 Discharge at constant power output.

6.19 Concluding Comments The following is only a general outline of the basic rationale behind the development of concentration cells. These types of cells do offer a new and different class of phenomenon that can be engineered into practical devices for the storage of energy. Even though the above discussion is centered on the element sulfur, there are many other compounds that could serve the same purpose. Sulfur has ebb, chosen here primarily because it has been experimentally studied most extensively and because its physical and chemical properties lend themselves to experiment easily. There are no problems with high oxidation rates, toxicity, solubility, etc., with which


Energy Storage 2nd Edition

to cope. Iron will serve the same purpose as sulfur. The ferric, Fe3, and the ferrous, Fe+2, states are analogous to the S= and S states of the elements. Iron poses one problem of inconvenience in the laboratory, which is its tendency to oxidize in the presence of air. However, that is certainly not a problem in the development of practical hardware. In order to achieve high working potentials of over 2 volts, a non-aqueous electrolyte must be employed to avoid the decomposition of water during cell charging. Again, that is not a problem in a reasonably well-equipped laboratory. There are numerous stable and compatible non-aqueous solvents that can be employed with good conductivities. One concern is the necessity for the occlusion of air and, particularly, water vapor because most of such solvents tend to be quite hygroscopic. The diagram shows both cations, M+, as well as anions, X−, migrating from the respective electrodes with opposite electrical polarity as charge carriers during the charge mode. Upon arrival at the (−) electrode, the Ma ions are reduced to Mb form and acquire electron(s) from the power supply in the external circuit. In the positive electrode side of the cell X−, ions associate with the Ma ions, whose charge is greater formed at that surface. During the discharge mode, exactly the opposite of these processes occurs, and stored electrical energy as complex concentration differences in X− and M+ are restored. Discharge at constant power output Lab cell of 0.50 cubic inch active volume



0.6 0.5


Cell volts

Volts 0.4

1.5 0.3 1

0.2 0.1 0


Amps 0




0.8 Hours

Figure 6.26 Volt-amp data.





Load current, Amps


The Concentration Cell



Typical Performance Characteristics

The plot in Figure 6.26 shows typical test bench results of small engineering cells with an electrode area of 10 square inches. The curve is for the discharge mode of operation at constant power delivery to a load. Total charge capacity of the cell in this instance is about 0.40 amp-hour. Voltage and current are continuously changing to maintain constant power at 0.10 watt. External power management circuits are employed to achieve this type of performance.

6.21 Sulfide/Sulfur Half Cell Balance The information contained in the following text, graphs, and mathematical development concerns the properties of a symmetrical electrochemical cell employing the basic and reversible reaction at both electrodes. The electrolyte is an aqueous or other suitable solvent such as alcohol or a solution of an alkali sulfide salt such as (NH4)2S, Na2S, or K2S. Since such sulfide salts solubilize sulfur, there are no solids present during normal operation of the cell. A microporous membrane with ionic selectivity is employed as the separator between (−) and (+) electrolytes. In this instance, both Na+ as well as S−2 ions migrate in opposite directions as dictated by the electrical polarity of the electrodes. The rate of such transport for these ions is determined by their respective mobility through the solutions. The polysulfide Na2Sx is the state common to both sides of the cell at the totally discharged stage. There are m-moles of each compound in solution on each side. When charging begins, higher polysulfides are generated at the (+) electrode, and lower sulfides are produced at the (−) electrode. If posilytes and negalytes have equal volumes, then at full charge the (−) side will have a saturated solution of the maximum solubilized sulfur, or Na2S5, as electrolyte, and the (+) side will be a maximum concentration of Na2S electrolyte. If the concentration and/or volume of the (+) side is greater than that of the (−) side, then some free, solid sulfur may be deposited onto the (−) electrode surfaces at full charge. Such a configuration results in an increase in energy storage capacity of the cell since there would be less sodium and, perhaps, water needed for cell operation.


Energy Storage 2nd Edition

6.22 General Cell Attributes The primary reason for pursuing a cell of this type is its potentially long life and maintenance-free operation. Since both sides of the cell contain the same chemical species, there is no possibility of degradation of performance or structure with time or cycling. The electrical potential in the cell is derived from the difference in concentration of one chemical species. In this instance, it is the sulfur/sulfide couple. Initially (cell in the “discharged state”), the concentrations of sulfur and sulfide ions are the same on either side of the cell. Each side is separated from the other by a microporous or ion selective membrane. Attributes of the cell include the following: • • • • • •

Benign chemical environment No maintenance Unlimited shelf life Unlimited cycle life No gas production Versatile system for either sealed unit or circulating electrolyte designs • Very inexpensive and abundantly available chemical reagents • Simple and inexpensive cell construction

It would seem that the advantages of indefinitely long life, abuse withstanding, and low cost would more than compensate for the above limitations. The high dependence of cell voltage on the state of charge is not different from that of employing compressed gas, rotating mass (flywheel), or capacitive storage since their working potential is similarly dependent on the amount of stored energy at any moment in time.

6.23 Electrolyte Information The three salts cited earlier have high solubility in water and in alcohol. Some data is provided in Table 6.3. Voltage produced within the cell is due to the concentration ratio of chemical species at the electrodes and does not depend on the absolute concentrations of solutes. It is important to operate cells at high salt concentrations in order to minimize water, electrode, and cell structure weights, and to maximize energy density. Consider a cell that employs a near optimized electrolyte composition such that all materials balance in the ion transfer processes. If we wish to

The Concentration Cell


Table 6.3 Salt data. Salt

Molecular wt.





200 to 500 g/L

4 to 8 ohm-cm



>800 g/L

3 to 6



>1000 g/L

6 to 10

From: Handbook of chemistry & physics, 1981, CRC Co.

keep all components in solution, then the electrolytes are as follows at the beginning of discharge: (−) side Na2S5 // 5 Na2S (+) side. As discharge proceeds (assuming only Na+ ions are transported), compositions of each cell side come, as indicated in the following steps, all the way up to reverse total charge. Table 6.4 shows how the cell potential decreases as discharge progresses and how the electrical polarity reverses if carried further by an external power supply. The flow of electrons during “discharge,” in such a symmetrical cell, is from the negative electrode to the external load. When zero potential is reached at the same concentrations of S and S=, the flow of electrons goes to the negative electrode from the power supply during the “reverse charge” shown below. There are 26 AH of charge per liter per gram equivalent weight. Thus, there are 156 AH transferred for the 6 moles of sodium ions. This corresponds to 0.156 AH/cc of total electrolyte, or about 9.36 AM per cc. For a cell with 1 in2 electrode area and a total spacing of 0.020 in, its electrolyte volume would be 0.020 in3 = 0.328 cc. This cell would then have a charge capacity of 9.36 AM/cc × 0.328 cc = 3.07 AM. Table 6.4 Cell potential decreases. Cell potential

(−) Side

(+) Side

0.02 volts




Na2S4 + Na2S

3Na2S + Na2S2

Na2S3 + 2Na2S

Na2S3 + 2Na2S


Na2S2 + 3Na2S

Na2S4 + Na2S




Discharged 0.00 Charging


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Higher capacities can be achieved if the polymerization of sulfur can proceed further or if free sulfur is allowed to accumulate. Also, there are additional small voltage contributions from the formation energy of the polysulfides. For example, the following are some measured values from Oxidation Potentials by W. M. Latimer:





S 2 2 2e E = 0.48 v




S3 2 2e E = 0.49 v,


which gives differences of 0.01 volts per “polymerization” stage. If we were to postulate that sulfur would polymerize to the S7 state, we would have a series of concentration and corresponding oxidation exchanges such as the following:

Na2 S7 // 7 Na2S ~ +0.25 volts Na2 S6 + Na2 S // 5 Na2S + Na2S2 ~ +0.14 Na2 S5 + 2Na2 S // 3 Na2S + 2 Na2 S2 ~ +0.065 Discharge Na2S4 + 3 Na2S // Na2S + 3 Na2S2 0.00 Charge Na2S3 + 4 Na2S // 2 Na2S2 + Na2S3 ~ –0.065 Na2S2 + 5 Na2S // Na2S3 + Na2S4 ~ –0.14 7 Na2S // Na2S7 0.00


However, it is not necessary to rely on polymerization greater than 4 or 5 for sulfur since any sulfur on the (−) side would eventually be solubilized as the cell discharges. The first chemical equation of the above group could be written as

2S+ Na2S5 // 7 Na2S,


The Concentration Cell


with the same performance results. As a matter of convenience, and to reduce the number of characters in these chemical balance equations, the sulfur is all expressed as polymer attachments. The manner of implementing this increase in AH capacity is via plating out free sulfur if necessary and increasing concentration or quantity (volume) of the polysulfide side at the beginning of discharge. As one can see, if there is an excess (stoichiometrically) of polysulfides on the (−) side, then free sulfur can be generated on the (+) side during charge, thus giving rise to a very small concentration of Na+ and S–2 ions in that electrolyte. The resultant concentration ratio of ions across the separator would be very large. However, such a large ratio would be dissipated quickly upon discharge, as will be seen in the ensuing equations.

6.24 Concentration Cell Mechanism and Associated Mathematics For any chemical reaction of the common form

aA + bB = gG + hH,


the lower case letters are the numerical ratios of the upper case reactants. The free energy change for the reaction is expressed as the following:

ΔF = RT ln

a G g a Hh a Aa a B b

RT ln

a G g a Hh a Aa a B b



Or, it can be represented as

ΔF = –RT ln K + RT ln Q,


in which K represents the equilibrium constant for activities in the equilibrium state, and Q represents the activity quotient or ratio of activities of the products to those of the reactants. Since, and if reactants and products are at unit activity, ln(Q) = 0 and E = E0. Thus, E = E0 − RT/nFln(Q).


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For most situations of interest here, the expression becomes

E = E0

RT a ln A , nF aB


where aA and aB are the activity coefficients for ions A and B. For those conditions encountered in these electrochemical cells, the ratio of activity coefficients is equal to their concentration ratio, or in this instance of a sulfide/polysulfide concentration cell,

E = E0

C 2 RT ln 1(S ) . nF C 2(S 2 )


In addition, we have the potential due to the sodium ion concentration differences, or

E = E0

C RT ln 1(Na nF C 2(Na





where C1 and C2 are the concentrations of the same ionic species on each side, respectively, of a cell. Now, let us examine a specific cell configuration for analysis and study. We will choose a unit cell with an active frontal area of 1 in2 and an electrolyte thickness of 0.010 inch on each side of the separator. This cell has a total electrolyte volume of 0.020 in3 × 16.4 cc/in3 = 0.328 cc. An initial solution of concentration of 9 molar has Na2S in the (−) side and the equivalent of Na2S9 polysulfide in the (+) side. These may be in the form of undissolved monosulfide and undissolved sulfur on the (+) electrode.

6.25 Calculated Performance Data Returning to the expression for concentration potential, we must relate the concentrations to the specifics of the cell volume and the dynamics of electric current flow. In general, the concentrations C1 and C2 may be written

The Concentration Cell


as C1 = Q1/V and C2 = Q2/V, where V is the volume on each side of the membrane. During discharge at any time, t, after full charge, the quantities of reagents in each side are as follows:

Q1o = Q1o – ∫ idt, Q2 = Q2o + ∫ idt.


The reason for normalizing the performance model is to relate the discharge rate to capacity, cell resistance, voltage polarization rates, and eventually diffusion limiting considerations. Thus, our model is a one square inch area configuration with electrolyte spacing of 0.010 inches. Its volume is 0.328 cc. Concentration of initially charged solution in the (−) side is 9 molar Na2S. The transfer of 18 normals of Na+ ions would give a total of 28.1 ampminutes per cc of total solution (Table 6.5). The (+) side has a concentration of 1 molar 2Na+ ions corresponding to 28.1/9 = 3.12 AM/cc. If this is indeed the initial condition of the cell at the start of discharge, then we can set up a mathematical relationship to evaluate voltages as discharge progresses. Now, it is necessary to set the initial concentrations and then calculate potentials versus discharge time. Then the ampere-minute quantities in the 0.328 cc volume cell become

Q1 = 9.2 – ∫ idt, Q2 = 1.02 + ∫ idt


The complete expression for the cell potential at various stages of discharge is determined by

Table 6.5 Charge equivalents. Charged state



AM/0.328 cc

(A) Na2S5//5Na2S




(B) NaS//7NaS




(C) Na2S9//9Na2S




(D) Na2S11//11Na2S





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9.2 RT ln nF 1.02

idt idt

8.314 298.1 9.2 0.02Δt ln , (6.71) 96, 500 1.02 0.02Δt

where we have employed 0.02 amp as constant discharge current, n = 1 for sodium ions, and operation is at standard temperatures. This expression has been programmed into a Lotus spreadsheet and a sequential method was employed to estimate voltages, energies, and power output during discharge. Since there is some uncertainty as to the sources of voltages between concentration ratios of Na+, S−2, and the formation of polysulfides during concentration changes, we will look at the type of performance obtainable for different assumptions. Empirical data is available for single cell in shallow cycling operation. The rather high potentials realized at early stages of charging suggest that there are numerous mechanisms present as well as some significantly steep concentration gradients established at the surfaces of electrodes, even at low current densities. In attempting to estimate performance via mathematical modeling, we have choices to make regarding the range of molarity change of ionic species, the initial concentration ratio (depth of dilution of sodium ions), and whether a multiplier is appropriate to the voltage equation in order to account for the observed shape and magnitude of empirical voltage discharge curves. Three different cases (A, B, and C) will be selected for graphing. Case (A) is a situation of least energy capacity, where the multiplier = 1, 5 molarity of Na2S, and initial ratio of concentration is 5:1. The equation describing the voltage versus discharge time is

E 0.0257 ln

9.2 0.02Δt 1.02 0.02Δt



depending on the extent to which the cell is charged beyond its above stipulated stoichiometric ratios. Case (B) involves a mid-range energy density cell, where multiplier = 5, and m = 7 molarity of Na2S, and initial concentration ratio = 70:1. Based on experimental evidence, it seems that a multiplier of value 5 for the entire concentration potential equation is reasonable and consistent with experimental results. In addition to the energy of formation of

The Concentration Cell


the polysulfides, concentration differentials are generated within the cell for sulfur, sulfide, and sodium ions. The figures to follow are graphical representations of a cross section of numerical cell value combinations of the factors discussed above. The more general equation for voltage may be given as

E m 0.0257 ln

(1.02 M d) 0.02Δt (1.02 d ) 0.02Δt



We arrive at the initial concentrations of the (−) and (+) reagents as follows. If the ratio desired is R, and the increment, d, is added to the (+) side, at the decrement of the (−) side is


1.02M d . 1.02 d


Hence, if we wish a ratio of 70:1, about 10 times as that obtained, and if the concentrations were 5 molar and 1 molar, respectively, then solving the above for d gives 70(1.02 + d) = 1.02(7) – d, and d = 0.905, and the new equation for voltage is

E 5 0.0257 ln

8.05 0.02Δt 0.115 0.02Δt



Case (C) involves a high capacity cell configuration, with multiplier = 5, an 11 molarity (+) solution, and an initial concentration ratio = 1100:1. The values calculate as follows: 1100(1.02 + d) = 1.02(11) – d, and d = 0.99, and the voltage, E, becomes

E 5 0.0257 ln

12.21 0.02Δt 0.03 0.02Δt



6.26 Another S/S−2 Cell Balance Analysis Method Perhaps another more direct and simple method of showing the materials balance and estimating the energy density of a concentration cell


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is that shown below for the sulfur/sulfide cell. We can assume that the process will no longer be limited to the maximum amount of sulfur that the polysulfide can solubilize. As an idealized example, the initial condition for a fully charged cell is Na2S // S, or more generally aNa2S // bS, where a and b are whole numbers of moles. In order for the process to balance at zero charge (complete discharged) state, a = b, and a > 1. We can now compute the maximum charge stored per unit weight of reactants in this concentration cell. The simplest example would be 2Na2S // 2S at full charge, and Na2S + S // Na2S + S at total discharge, with a transfer of 50 AH per total molecular weight of reactants. This amounts to 2(78) + 2(32) = 220 gm with a charge transfer of 50 AH giving as energy density 50 AH/220 gm × 454 gm/lb = 103 AH/lb of dry materials. It is possible to further generalize the analysis for the cell processes wherein the sulfur is always attached to the sodium polysulfide molecules. Since the details of interim stages of complexing can be readily known, we will assume the following steps in the charge transfer and discharge of a cell that begin with the polysulfide on one side and the monosulfide on the opposite side. Let us take the pentasulfide as the largest size complex available. The cell configuration and reactions become those shown below. Starting with the fully charged state as before, but with the bisulfide on one side and the monosulfide on the other, aNa2S // bNa2S2. The smallest value for a is 3 since it is necessary to remove two 2Na atoms from the monosulfide in order to meet the conditions of no free sulfur on either side of the cell. Without going through the approximation sequences, the numerical ratio that results functions to make both sides of the cell identical after discharge is a = 3, b = 2, 3Na2S // 2Na2S2 Charged, and Na2S + Na2S2 // 2Na2S + Na2S2 Discharged. The total gram molecular weight of both sides is 234 + 220 = 454, and the charge transferred by 2Na+ ions is 50 AH. The charge density of dry salt is simply 50 AH × 454 gm/lb /454 gm = 50 AH/lb. If we start out with the trisulfide, the reaction balance, etc. are aNa2S // bNa2S3 Charged, a = 3, b = 1, 3Na2S // Na2S3, and Na2S + Na2S2 // Na2S + Na2S2. Total weight = 3 × 78 + 142 = 376, and the charge density = 50 AH × 454/376 = 60 AH/lb. The cell reaction making use of the next higher initial polymer of sulfur and sulfide is as follows: aNa2S // bNa2S4, a = 5, b = 1, 5Na2S // Na2S4, 3Na2S + Na2S2 // Na2S + Na2S3, and Na2S + 2Na2S2 // 2Na2S + Na2S2.

The Concentration Cell


Total weight is 390 + 174 = 564. Since there are four Na+ ions transferred from fully charged to symmetrical distribution of ions at discharge, the charge density is 100 × 454/564 = 80 AH/lb. Taking the pentasulfide as the last or highest complex, the cell parameters become aNa2S // bNa2S5, a = 7, b = 1, 7Na2S // Na2S5, 5Na2S + Na2S2 // Na2S + Na2S4, 3Na2S + 2Na2S2 // 2Na2S + Na2S3, Na2S + 3Na2S2 // 3Na2S + Na2S2. Total weight is now 546 + 206 = 752. There are three transfers of 2Na+ ions, hence the charge density is now 150 AH × 454/752 = 90 AH/lb. If it were possible in a practical cell to utilize higher complexes, the charge density would merely approach the maximum value of 103 AH/ lb. In order to compute the energy density of such cells, it is necessary to multiply the charge density by an appropriate voltage. Since the cell potential is so dependent on the state of charge, a reasonable value of working cell voltage over the entire range of charge storage would be half of the full open circuit voltage of 1.0 to 1.2 volts, or about 0.5 to 0.6 volts. Hence, the maximum energy density of the cell, assuming no water (solvent) weight or other contributions to inefficiencies, would be about 52 to 62 WH/lb of reactants. The operating open circuit potential is purposely limited to between 1.0 and 1.2 volts to prevent the evolution of hydrogen gas at electrode surfaces. H2 evolution would necessitate the periodic readjustment of electrolyte composition and the venting of cells and would eventually result in mechanical erosion of electrodes. Another approach to preventing gas generation at electrodes is the employment of non-aqueous solvents such as absolute alcohol, pyridine, DMSO, and nitriles.

6.27 A Different Example of a Concentration Cell, Fe+2/Fe+3 The principles employed in concentration cell design are not restricted to the use of sulfur and numerous polysulfides such as those of potassium, ammonium, lithium, etc. In fact, the above concentration cell approach to energy storage can make use of numerous other materials that have properties suitable to practical methods of implementation and different characteristics that may make them more applicable to certain uses. These materials include the use of the elements iron, bromine, iodine, and chromium. Their behavior as electrochemical species is well known and readily available.


Energy Storage 2nd Edition

The balance relations for the iron concentration cell are as follows. We will make use of the two oxidation states of iron, Fe2 and Fe+3 ions. Their solubility is such that high concentrations (two to four molar, respectively) of these are easily attained in water. Potentials during charge must be kept below that for the formation of free iron, Fe0. That potential in water solutions is about 1.2 volts. The reaction of interest to us here is of the form aFeCl2 + bHCl // cFeCl3 + dHCl fully charged state. The charge carrier is the hydrogen, H+, in this cell. A cation exchange membrane, or a microporous separator, is employed in this cell. In order for the reaction to proceed and have a symmetrical situation on both sides of the separator, i.e., no further oxidation/reduction energetics remain, the minimum values for the coefficients a, b, c, and d are 2, 1, 2, and 1. Thus, the initial and final states are 2FeCl2 + HCl // 2FeCl3, and FeCl2 + FeCl3 // FeCl2 + FeCl3 + HCl. There is only one charge carrier (exchange) per such step. Hence, the total weight of reagents is 252 + 36 + 322 = 910 gm. The charge density is then 25 AH × 454/910 = 13 AH/lb. Even though the energy density is not as attractive as that of the sulfide system, there are some outstanding features such as extremely low cost of materials, low hazard, and no chance of solids deposition if potentials are kept below that for Fe+2 + 2e− → Fe0.

6.28 Performance Calculations Based on Nernst Potentials A series of graphs and calculated data plots has been generated based on a simplified model of the volt-amp behavior of a basic cell. This was done in order to acquire a preliminary appreciation of the type of behavior that can be expected from actual cells, which were simultaneously fabricated and laboratory tested. Such calculations, along with graphs of the voltage versus current-time or state of charge of a cell, enable us to determine whether our actual test cells are operating in an expected fashion, or whether there are some discrepancies between actual behavior and our simplified understanding of the mechanisms involved. Even if the data might be quantitatively different from that calculated, the main importance in these types of investigations is whether the qualitative aspects are as predicted, or in other words, are the shapes and general behavior of the two groups of data, i.e., the calculated versus the empirical curves.

The Concentration Cell


Despite the absence of many factors in the modeling such as diffusion and coulombic inefficiencies, it was found that the agreement between the two is quite close. To best illustrate these initial attempts to analytically represent cell performance, a few simplified models follow.


Constant Current Discharge

We will now look at the behavior of a cell when confined to discharging at a constant current by providing a suitably varying electrical load at its terminals. The mathematical formula for the value of cell voltage as it varies with the state of charge, on which the plot is based, is given as



an 1 , bn 1


where the terms a and b are the concentrations of the active reagent in the opposite sides of the cell. In this case, that reagent is the sulfur ion, S=. The subscripts n and n−1 are employed to designate the sequence in time of the terms. For example, the voltage at any time, t, indicated by subscript n is found from the value of an, which occurs at that same time, and the value of an−1, which occurred at the previous time interval. This arithmetical approximation enables us to make the evaluations in a spreadsheet program such as Lotus or Excel. The evaluations can thus be made piecewise, and plots can be generated from the calculated data. It is also assumed here that the volumes on either side of the cell are the same and normalized to give a one-to-one correlation between AH and change in concentrations. To complete the mathematical description, the relationships for the a and b terms are as follows:

an = a – 0.05 (tn – tn–1) bn = bn–1 + 0.05 (tn – tn–1)


A constant current of 0.05 amps is almost arbitrarily assumed here per square inch of electrode surface area, and the time interval in obtaining readings of voltage is expressed as tn – tn−1 in the expressions (6.78).


Energy Storage 2nd Edition Calculated — voltage versus remaining charge


Cell volts




0.5 20



50 60 70 80 Percentage of remaining charge




Figure 6.27 Calculated – voltage versus remaining charge.

The amount of electrical charge, Qn, removed from a cell over a time interval tn – tn−1 is given as

Qn = Qn–1 – 0.05 (tn – tn–1).


Figure 6.27 is such a graph where the cell potential, E, is plotted against the percentage of remaining electrical charge within the cell. A distinct disadvantage of concentration cells is the high dependence of voltage on SOC. However, this characteristic can be largely overcome, as discussed earlier in this chapter and later in Chapter 8, by having most of the reagents (sulfide ions) in the solid state at the electrodes with controlled dissolution as charging or discharging progresses, depending on which electrode polarity is involved.


Constant Power Discharge

Another informative graph that shows the behavior of cells, when operated to deliver constant power during discharge, is obtained from the simple assumptions and relationships shown below. In these analyses, the predominant shape of the curves is logarithmic because of the fundamental equation for voltage as set forth in Nernst’s equation. As before, the voltage is proportional to the log of the ratio of concentrations of specific ions on each cell side, or

E ln

a b

. n


The Concentration Cell


Similar to equations (6.78) and (6.79), the expressions for a, b, and Q now are as follows:

an = an–1 – P (tn – tn–1)/En–1 bn = bn–1 + P (tn – tn–1)/En–1 Qn = Qn–1 – P (tn – tn–1)/En–1


The total electrical charge that has passed through the load at constant power of course is N

Q total

Qi ,



where N is the total number of samples taken over the time interval of interest. A chart of such a discharge is shown in Figure 6.28. The major portion of the electrical energy, or charge, is delivered at the lower levels of voltage because of the very rapid decline of the ratios a/b from fully charged. The only method we know of to reduce this rapid decline is the introduction of a stabilizing feature. This would maintain high a/b values by storing the major portion of charge in the form of unionized (solid) reagents at the electrode sites and then gradually replenish the ionic

Cell volts

2 1.5 1 0.5 0



2 3 4 Discharge time - hours Cell volts

% of charge remaining

Figure 6.28 Voltage versus discharge time @ constant power.



110 100 90 80 70 60 50 40 30 20 10 0

Percentage of remaining charge

Calculated - voltage versus discharge time 2.5

Energy Storage 2nd Edition


reagents by controlling their precipitation and dissolution at the respective electrodes.

6.29 Empirical Data Countless laboratory cells and arrays based on concentration differentials have been constructed and tested. Most of these test cells were either sulfide or ferric/ferrous based chemistry systems. Numerous other systems such as bromine/bromide, iodine/iodide, and cupric/cuprous cells have been explored as well. The electrical results, as one would expect, obtained are very similar to those with the sulfide devices. However, other problems were encountered with these and other alternative cells associated with chemical stability (attack of surrounding materials of construction), volatility, costs, and incompatibility with surroundings. A graph of the empirical results, or electrical behavior of such cells, is shown in Figure 6.29. It is typical of simple cells with no provisions made in their design for diffusion limitations, solidification of reagents, or any other performance controls. These early cells illustrate the very basic characteristics one can expect from this class of concentration cells. These curves were generated by discharging cells across a constant resistive load – not the optimum arrangement for either high efficiency or good

Single cell cycling data Sodium sulfide cell with 5 sq. in. area




0.3 1 34.35




0.5 16.44



0 0



600 Elapsed minutes Volts



UU porous carbon & sybron separator

Figure 6.29 UU carbon, sybron separator, spec. Grav. soln, +1.2.


0.1 0 1200




The Concentration Cell


performance. However, they do demonstrate the general characteristics of voltage rise and decay during charging and discharging, respectively. One of the main purposes of such extended cell testing is to learn a bit more about life and cycling limitations. Over 5,000 cycles have been accumulated on single cell laboratory devices with no discernable deterioration in performance or chemical composition of the electrolyte or erosion of electrodes, thus making predictions of cycle life rather difficult with no degradation data from which to extrapolate.

7 Thermodynamics of Concentration Cells Concentration cells are all “redox” in nature because, as is necessarily the case, the reduction and oxidation of chemical species occur within the device for the transference of energy. However, in this instance the redox label does not imply that these devices or cells are full flow electrolyte design with provisions for removing or replacing electrolytes. They are best operated as stationary (static) electrolyte systems. The particulars of design and physical configuration depend upon the intended applications. For example, it may be desirable to use close spacing between electrodes for higher power density application requirement devices, thus proportionately reducing their energy density.

7.1 Thermodynamic Background The fundamental principles upon which the CIR system is based are probably best identified from the following thermodynamic considerations. Here, we quickly review some of the important and relevant thermodynamic functions upon which the analysis of cell behavior is based. In order to arrive at the intended expression for the electrical potential of a CIR cell, we will begin with the familiar equation of state for an ideal gas:

PV = nRT,


where P and V are pressure and volume, respectively, of a quantity of gas consisting of n moles. T is temperature in degrees, absolute Kelvin, and R is the universal gas constant with a value, in the MKS system of units, of 8.31 joules/mole-degree. To prepare for the relationship for cell potentials, we return to some fundamental concepts. The Gibbs function, G, for any chemical system is given as

G = H − TS,


Ralph Zito and Haleh Ardebili. Energy Storage 2nd Edition: A New Approach, (163–174) © 2019 Scrivener Publishing LLC



Energy Storage 2nd Edition

where H is the enthalpy, and S is the entropy of the system. Taking the total derivative of G, we obtain

dG = dH − SdT − TdS.


Also, the enthalpy, H, is defined as H = U + PV, where U is the internal energy of a system. To place the above in more useable terms, we now proceed through the next manipulations aimed at obtaining a final, more useable expression for the free energy (Gibbs function) and the chemical potential. The total derivative of H is

dH = dU + PdV + VdP.


For neighboring reversible processes,

dU = TdS − PdV


Substituting for dU in the equation for dH, we get

dH = TdS − PdV + PdV = VdP, or


dU = TdS + PdV


Substituting the following into the equation for dG,

dG = VdP − SdT,


since T is constant for isothermal processes, the differential free energy becomes

dG = VdP.


Integrating, we obtain the more familiar expression P2

G nRT P1

dP P


P2 . P1


The idea of chemical potential, u, is crucial to the study and analysis of electrochemical cells. Here we simply introduce the chemical potential, u,

Thermodynamics of Concentration Cells 165 in the following fashion. For a specific constituent, k, of a phase state, uk, is the potential as described by


G , nk


for a multi-component system. If we have but one component, in solution, then

U = dG/dn,


and the chemical potential is the molal Gibbs function and is a function of T and P, or u = G/n, where n = moles. Now, addressing the electrochemical cell subject, let us convert pressure, P, for an ideal gas into the equivalent parameters for an ionic solution. Additional intermediate terms must be introduced in order to bridge any gap in the continuity of reasoning of thermodynamic equilibrium. The concepts of fugacity, f, and activity, a, have proven to have a significant benefit. If Fm is the partial molal energy and a measure of the escaping tendency, for example, of a vapor from its liquid phase, then the fugacity has been defined by the relationship below:

Fm = RTln(f/fo) + B(T),


where B is a constant serving as reference point. If fo refers to some standard state, then P


VdP=RTln Po

f . fo


A more commonly employed parameter in electrolytic solutions is the “activity” of the solution in the immediate vicinity of an electrode. This is usually represented as a = f/fo, and it is usually directly proportional to the molality of dilute solutions because the solute behaves closely to that of a free gas. Hence, we may finally obtain as the free energy, or Gibbs function, of the solution

G = RTln(a).



Energy Storage 2nd Edition

In determining the voltage, E, associated with chemical potential in electrochemical cells, the free energy may be simply divided by the number of electrical charges that are being transferred in a particular process. In the case of the transference of n moles of singly charged ions at an electrode, the relationship assumes the following form:


RT ln(activities), zF


where F is Faraday’s number of 96,500 coulombs per equivalent. In general, the equation may be put into the format

E Eo

RT oxidized form ln , zF reduced form


where z is the difference in electric charge of the ions in question.

7.2 The CIR Cell Configurations of the CIR cells may employ two or three of the available oxidation states of an element or compound. A cell is diagrammatically represented in equation 7.18 that makes use of two oxidation states of a metallic element, M. It will be assumed here for illustrative purposes that the oxidation states are M+i and M+j. Then, we may represent a “materially symmetrical” cell employing these agents as

Pt; M+i, M+j || M+j, M+i; Pt.


Platinum electrodes are shown in the diagramed cell. They merely represent electrode properties with non-reactivity properties (no chemical participation in the cell processes) and high electronic conduction. In actuality, carbon structures are employed in practical cell designs because of their much lower cost and because the carbon physical properties are better suited to the optimum operation of these cells. This cell has two chemically inert electrodes, each in its own electrolyte with electrolytes separated by a porous plate. In this case, the separator might be either a microporous membrane that simply retards the diffusion and consequent mixing of the electrolytes, or it might be perm-selective

Thermodynamics of Concentration Cells 167 (ion selective membrane). In this instance, the separator could be a cation membrane that permits (+) charged ions such as H+ to pass. The electrolyte is a solution consisting mainly of salts of the metal, M, and usually a simple acid. Charge transfer within such a cation type cell is principally via hydrogen ions. Very high cell conductivity can be achieved. The activities associated with each of the ionic species as shown in equations 7.19 and 7.20 are as follows: On the left side the activity

for M+i = a1, and for M+j = a2


On the right side the activity

for M+j = a3, and for M+i = a4


When substituting the activities in the Nernst equation, care must be taken with their order of appearance and algebraic sign of the terms. Perhaps the simplest way to calculate the voltages for a cell is to follow the above format of oxidized form/reduced form. First, we will convert to the log at base 10 for convenience of arithmetic. Then, the expression for voltage will become

E Eo

0.0592 z


log i

a oxidized a reduced




Consider the cell described above, in which both sides have identical inert electrodes and iron ions present. In this symmetrical instance Eo is zero. The voltage for this symmetrical cell is evaluated from the activities of the pertinent ions on either side of the separator for the two electrolytes. The voltage due to the differences in activities for the identified ionic components is found by calculating

E 0.00257 ln

a2 a1


a3 a4



Charge, z, is 1 for the ion transport number. As a result of our cell development the above expression is modified to take into account the preferential storage processes as part of cell design. Electrode processes have been modified to provide for a very selective storage and large activity coefficient enhancement. The design and operational techniques have produced charge storage at cell potentials well over an order


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of magnitude greater than ascribable to the unmodified Nernst equation. These may be effectively represented in the Nernst type expression as

E 0.0257 ln

a2 a1


a3 a4



where α and β are multipliers of the activities due to a dipolemultilayer storage specific to the molecule associated with that ion. During the development of the iron/redox system we have learned how to control the activities of each specific ion in the processes, and there is no metallic iron in its elemental state present in the concentration cell. Hence, it is possible to attain very high electric potential differences in these cells, which makes them practical as energy storage devices. There are no electronically conductive solids at any time present within the cells, other than the inert carbon electrodes. Any solids that might result from cell operating conditions are reversibly soluble and do not result in possible short-circuiting or other deleterious effects. This approach, with appropriate modifications in design geometry and inactive materials of construction, can be applied to a number of active materials that will dissociate in polar solvents and with two or more soluble oxidation states. A full flow electrolyte system has some distinct advantages despite its greater complexity and costs. These advantages include (1) the separation of energy density factors from those of power density, (2) an infinitely long charge retention, and (3) total reversibility, in principle. However, our tests in a limited development program have shown little adaptability of the CIR system to full flow electrolyte configurations. It appears that, in order to benefit from high concentration differences, the reactants are best electro-deposited into porous electrodes that occupy the entire space between the separator and the conductive electrodes. If a cell is designed to permit full flow past the electrode surfaces, then sulfide ions and polysulfides must molecularly diffuse into the depths of the carbon pores. That diffusion process is slow and gives rise to undesirably high concentrations in the bulk electrolyte with consequent diffusion losses through the cell separator. It may be possible with some clever composite designs of electrodes to overcome this deficiency and benefit both from concentration cell performance and the advantages of a full flow system. The mechanical complexities, costs, and size may well be worth it in some applications that have peculiar power and energy density demands or in which long charge retention times are required.

Thermodynamics of Concentration Cells 169 Nature offers few selections from its list of elements and compounds that have all the properties desirable in such redox systems. Among these desirable qualities for practicality are an ambient temperature operation, a low hazard, a plentiful supply of materials, benign behavior, and minimum side reactions that can result in operational failure. TRL has investigated and developed a number of redox-type of systems with varying success in the past. These include the zinc/bromine, iron/redox, and the polysulfide/ bromine secondary batteries. Other than the last system, they were all “half redox” cells because only one of the two components of the electrochemical couples are in solution at all times. Both the zinc/bromine and the iron/ redox cells suffer from all of the problems associated with the deposition and dissolution of solid metal onto the surface of an electrode. These virtually unsolvable problems include dendrite shorting, non-uniform plating, metal particle fall-off, passivation, and gas generation in acid solutions. These are very severe problems that have been scrutinized for many years. Unfortunately, these characteristic problems will always be present regardless of how successful some of the cures and improvements in performance become. The polysulfide/bromine system does offer the situation where all reagents are soluble at all times. However, the electrolyte is acidic on one side of the separator and basic on the other. Without some additional subsidiary processes, the pH differences are difficult to maintain. Also, there is the problem of gradual and irreversible diffusion of bromine and sulfide diffusion to opposite sides of a cell as cycling proceeds. TRL has contended with these problems with some degree of success. Unfortunately, these deleterious effects are present and lead to inexorable and ultimate cell failure. Then, it becomes necessary to resort to some complex external method to restore operation. The most attractive of the above cells, in terms of low cost, safety, and simplicity, is the iron/redox cell, wherein we make use of the three oxidation states, Fe0, Fe++, and Fe+++. The energy storing reaction for the couple Fe0/Fe+++ is

Fe0 + 2Fe+++ = 3Fe++ 1.2 volts.


After considerable effort, this couple has essentially been set aside as a practical means of energy storage, not because of its rather limited energy density (maximum of ~75 WH/lb for the dry reagents), but because of its unmanageability with regard to iron plating qualities and hydrogen evolution (hydrolysis and iron attack) in the necessarily low pH electrolyte. It is possible that a non-aqueous solvent can be employed, in which the iron salts are soluble and have a sufficiently low resistivity so that practical levels of power can be realized from such cells. However, as of this date no


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significant effort has been expended to search for likely solutions to the problems associated with iron in acidic, aqueous environments. We have all been searching for a means of electrochemically storing energy that avoids all of the above problems completely and with energy density sufficiently high to make it attractive for large-scale, stationary applications, such as load leveling and solar and wind power. On the basis of our collective past experiences with redox types of systems, we turned our attention to the possibilities of concentration cells. This type of cell has the very attractive property of symmetry. The materials of either side of the cell are the same but at different oxidation states. An example of a “materially symmetrical” cell is the vanadium redox battery. The vanadium redox cell is not a concentration cell as such. Rather, it is a cell that makes use of an electrochemical couple between three or four of the soluble oxidation states that are available for vanadium. There are problems of reversibility and high costs associated with the materials that tend to limit its application potentials. The CIR cells rely on only two soluble oxidation states of the same material, or compound, as in iron+2 and iron+3. The potential is easily calculated from the above relationships if we know the activity of the specific ions at the electrode surfaces. Returning to equation 7.23, the Nernst equation, we can calculate voltages from concentrations in dilute solutions where the activities are nearly the same as the concentrations. In order to measure the potential due to concentration differences, one would normally employ a cell with a reference electrode, such as H+/H2, that would provide a knownpotential at one end of the cell. However, for the sake of illustration, we can simply postulate a cell with the following conditions. In an operating concentration cell, we are primarily dependent upon the concentrations of the two primary reactants (oxidation states of iron) at their respective electrodes – in this case, the concentration of Fe++ and Fe+++ at the negative and positive electrodes, respectively. Due to the fact that the sum of all Fe++ and Fe+++ ions on both sides is constant at all times because, for example, as Fe++ ions are generated at one electrode the same number is removed at the opposite electrode, the expression for the voltage in this instance becomes

E 0.0257


a3 a1



c3 c1


where ϕ reflects the interdependence of Fe++ and Fe+++ concentrations in this cell.

Thermodynamics of Concentration Cells 171 It is significant to examine the accuracy of using concentrations in place of activities. Certainly, in very dilute solutions the numerical equivalence of activity with concentration is very close. In fact, for most dissolved materials the values of activity do not change very drastically with low concentrations of the particular chemical species. The terms c2 and c4 have cancelled, and the concentrations were substituted for activities. Let us look at a magnitude of the voltage one might expect from such a concentration cell. If the concentrations c3 and c1 are 1.0 and 0.01 molar, then the potential, E, would have an impractical, low maximum value of

0.0592(log 100) = 0.118 volts.


Proceeding further, it is obvious that a concentration ratio of reactants of 106:1 or more at their respective electrodes would be required to achieve an open circuit potential of 1.0 volts. This is a rather discouraging prediction of the practicality of concentration cells. Our preliminary lab experiments with two or three chemical systems did verify these data. However, when a charging cycle was performed with a cell with thick porous structure, potentials of over 1.2 volts were obtained. Furthermore, the cells were capable of delivering on a sustained basis a significant electric charge to the external circuit at these levels of voltages. Quite obviously, something else was taking place besides the bulk concentration differences in the electrolytes of a two-compartment cell. The micro- or nano-porous carbon particles are in intimate, physical, and electrical contact with the electrodes. This type of simple cell with inexpensive components and reagent materials prompted us to explore the mechanisms involved in the storage of charge at such unexpected high values. The same sort of behavior was attained with very similar structures employing compounds of sulfur. The electrical performances of the cells were consistent and repeatable. The sulfur-based cells employ alkali salts of sulfur such as sodium, potassium, and ammonium mono and polysulfides. The search begins for a plausible explanation of how a cell, with no other energy related processes taking place other than the difference in concentration of the two oxidation states of the same chemical element, can produce such high potentials and sustain high electrical charge densities. If there are no other processes involved, then we must look toward some mechanism whereby the chemical species in question are able to accumulate at the electrode surfaces and give rise to such high voltages. At present we speculate that the specific ionic species, in this case the ferric and ferrous ions, are injected directly into the pores regions of the


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microporous carbon at high effective concentrations by the process of electro-sorption. Even though the maximum concentration of the compounds of ferric and ferrous chlorides is limited to not much over 3 molar at room temperature, the population densities of the adsorbed and stored ferrous and ferric substances become equivalent to extremely high activities seen by the electrodes. In order to accomplish this, more energy is required than would normally be to separate the oxidation states during the “charging” process. This condition, for the maintenance of conservation principles, has been observed during cell cycling in terms of the volt/amp inputs and outputs. There are undoubtedly many processes and mechanisms taking place at the electrode surfaces that call for greater understanding and quantifying. Let us first list and then examine the physical and chemical activity possibilities. Omitting the chances that there are any net or permanent chemical changes occurring in the electrical cycling of the cell, the following are possible, reversible processes: • Exceeding salt solubility at electrodes during charging, resulting in solid compounds at the surfaces with attendant free energy changes (energy of ionization and dissolution) • The creation of immense concentration ratios of Fe++ and Fe+++ ions in solution at electrode surfaces • The adsorption of iron ions and, perhaps, of the salt compounds themselves within the carbon porous structure, which may be Langmuir type processes or Van der Waal’s, depending on whether they are attached as electrically charged components or as neutral molecules with dipole moments. Meanwhile, the following is an encapsulated glance at the energies of solution, or the dissolution for the ferrous and ferric chloride salts present in the cell being discussed. A perhaps overly simplified, but nevertheless indicative, view is suggested here. Consider the free energies (integral heat of dilution) to infinite dilution as given by Latimer’s “Oxidation Potentials”: • • • • •

Fe++ in aqueous solution ~ −20 kcal/mole Fe+++ ~ −2.53 Cl− ~ −31.35 FeCl2 in solid, crystal form ~ −72.2 FeCl3 ~ −80.4

If, in the reduction of ferric to ferrous ions at the negative electrode during the charging process, the generation of ferrous salts occurs too

Thermodynamics of Concentration Cells 173 rapidly for it to remain in solution, then the process of Fe+++ + e = Fe++ is accompanied by the precipitation of ferrous chloride into the pores of the electrode, i.e.,

FeCl2 = Fe++ + 2Cl− .


Then, there is another exchange of energy involving the heat of solution. Its net value is ΔF = −72 + 20.3 + 2(31.35) = + 7.8 kcal/mole at the positive electrode when charging. Similarly, one can estimate the free energy at the negative electrode. The net reaction would appear to be

FeCl3 = Fe+++ + 3Cl− .


Here, ΔF = −80.4 + 2.53 + 3(31.35) = −16.8 kcal/mole. This result can be related to an electric potential by the relationship below:

ΔF(net) = −nE × 23,060,


where E is in volts, and n is the number of charge changes on the ion. Upon adding the free energy changes, we see that the voltages associated with this dissolution are in the vicinity of 0.3+ volts, a not insignificant amount. Or, we may represent the processes at the two opposite electrodes during charging as

Fe+++ + 3Cl− + e = FeCl2 + Cl−.


Free energy for this reduction and dissolution is

ΔF = −2.53 − 3(31.35) + 72.2 + 31.35 ~ + 7 kcal,


Fe++ + 3Cl− = FeCl3 + e,


ΔF = −20 − 3(31.35) + 80.4 + 31.35 ~ −2.3 kcal.




Also, it is important to examine the dynamics and kinetics of the transport and molecular diffusion.

8 Polysulfide – Diffusion Analysis The ensuing analysis was prompted by problems encountered with the sulfur (−) electrode while developing a bromine/sulfide secondary cell for load leveling application possibilities. The lives of our composite carbon/ plastic, negative electrodes were severely limited upon extensive cycling. After much research and experimentation, the puzzle of limited electrode cycle life was solved. The problem was caused by the erosion of carbon particles from the composite structure by the formation of hydrogen gas just under the surface of the somewhat porous plate. The polymer (plastic) component adheres very well to the total electrode structure and has a physical continuity that does not characterize the very frangible carbon components. After extended operation of this sort with gasses ejecting carbon, the electrode acquires a very high interface resistance and ceases operating as a functioning electrode, producing mono-sulfides due to starvation in the depths of the holes left by the separated carbon. It is most important that we try to take into consideration any and all possible processes and mechanisms in our analysis of the experimental observations. It must also be understood that much of what is offered as explanations for cell behavior is in the realm of speculation – to be verified as we continue with these empirical studies. The dynamics of molecular diffusion, ionic conduction, and the oxidationreduction processes that occur at the electrode/electrolyte interfaces are considered here with a view toward simplifying and qualifying. In concentration cells, these are considerations of prime importance in the study of the balance of materials transport. Certain operational models must be assumed for such an analysis to be initiated and progress. As we learn more and attempt to match the theoretical modeling to empirical data, some of these assumptions may be abandoned or changed in order to provide a better picture of what actually occurs within a cell. In concentration cells, in which the situation usually involves the movement of uncharged molecules as well as ionic species toward and away from electrodes necessary to its operation, a great amount of attention must be Ralph Zito and Haleh Ardebili. Energy Storage 2nd Edition: A New Approach, (175–226) © 2019 Scrivener Publishing LLC



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Table 8.1 The transport of various species. Transport mechanism

Toward electrode

Away from electrode

Electric field assist



Neutral diffusion



Electric field retard


devoted to these transport mechanisms (Table 8.1). In the pages to follow, a straightforward examination of the ionic and molecular transport processes that occur within an electrochemical cell are examined. The subject cell makes use of sulfur/polysulfide reactions at electrode surfaces. It is the intention that this simple exploration of the diffusion balances will enable us to better assess cell performance and charge capacity for not only the sulfide cell but for other concentration cell systems based on different chemistry. The sections to follow discuss the various mechanisms that take place within electrochemical cells of this type. Various mechanisms take place within electrochemical cells of this type and present a preliminary analysis of current density distribution and reagent diffusion at the surface of porous electrodes. This initial review is intended primarily to provide a basis for some further attempts at characterizing the kinetics that occur at the surface of the negative electrode.

8.1 Polarization Voltages and Thermodynamics Voltage losses within a cell, other than those encountered by the electrolyte ionic resistance and electrode electronic resistance, can be classified into three categories: 1. Interface resistance, or ohmic over-potential is due to the resistance of “contact” between electrolyte and electrode. 2. Activation energy, or the rate with which a reaction proceeds, is associated with an energy barrier or activation over-voltage. 3. Concentration over-potential is a polarization voltage that is a function of the concentrations of reactants and reaction products. These factors may be derived and represented as follows. The Nernst equation for an electrode reaction of the form

aA + bB = cC + dD,


Polysulfide – Diffusion Analysis


is given in general terms as

E Eo

RT ln W, nF


where R is the universal gas constant, F is the Faraday number, and n is the charge on the ion. W is defined as


(a C )c (a D )d , (a A )a (a B )b


where the a-terms are the activities for the reactants and the reaction products. For the reaction of interest at the (−) electrode during the charge mode, (see equation (8.5)), equation (8.3) becomes


(a S )x 1 (a Na+ )2 x (a Na 2 S )2 x





Since the activities depend on concentration, or population density of the reactants, high over-voltages can be generated under severe starvation conditions at electrode surfaces. This possibility is especially real if cells are charged by constant current sources. Some cell operating conditions can lead to the evolution of hydrogen, thus tending to further complicate conditions and contribute to additional starvation of electrode sites. Our primary interest at this time is in the mechanisms associated with the negative (sulfur/sulfide) electrode and, more specifically, the processes involved in the charging mode. Since precisely the opposite occurs at the positive electrode during “charging” in a symmetrical concentration cell, the reverse processes can be analyzed to provide complete cell analysis.

8.2 Diffusion and Transport Processes at the (−) Electrode Surface The processes and reactions that occur at the (−) electrode during charge are more complex and “limiting” than those during the discharge mode. The solubilized sulfur attached to S= ions are supplied only by mechanical means


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or thermal diffusion against the (−) electrode repulsion. Sodium ions, on the other hand, migrate toward the (−) electrode via electric field attraction. Figure 8.1 shows the transport processes of interest as defined by us for further examination. For the primary energy storing reaction, shown below, to take place at the (−) electrode,

Na2Sx + 2(x − 1)Na+ + 2(x − 1)e−

2(x − 1)Na2S.


There must be adequate supply of not only Na+ ions to the surface, as demanded by the charging current density, but also a corresponding quantity of polysulfide. If these conditions are not met, then starvation at the electrode surface will occur, and if the voltages are high enough, water will be broken down and hydrogen will evolve in accordance with

1 NaOH+ H2 . 2

Na + H 2O+e


As shown in Figure 8.1, if some hydrogen is produced, then it further complicates the situation since it too must be removed from the reaction site to enable the reduction of sulfur to take place. To summarize, the transport of various species may be put into the following categories. There is the obvious necessity to provide high accessibility to the (−) electrode, especially during the charging mode since only the sodium and mono-sulfide ions are assisted by the electric field gradients and the polysulfide must be provided at the sites by forced flow for higher current densities. H2 SxS= Electrode

Diffusion processes

Na+ H2O S


Irregularly surfaced electronic conductor

Figure 8.1 Negative electrode in charging mode.

Electrolyte region

Polysulfide – Diffusion Analysis


8.3 Electrode Surface Properties, Holes, and Pores In order to enhance the electric current density of the frontal area of the electrode, it is quite reasonable to think in terms of increasing the total available working area by making the electrode surface porous or in terms of providing a high degree of irregularities on the surface, instead of a smooth, dense, non-porous electrode. As this direction is taken, the electrode area is increased, and the effective, or working, electric current density for any given frontal area current density is decreased. This would seem to be better since the demand rate for reagents to and from the electrode (micro-surfaces) is reduced, thus lessening the chances for starvation and high polarization potentials to appear. However, one must consider the fact that this increased surface area is not as readily available to the reagents as the smooth frontal area was, and the electric field strength within valleys and pores will be diminished. The following is a simple exercise as a very preliminary attempt to explore the interrelations between increased electrode “porosity” and its effect on performance. As a first step in the investigation, the shapes and intensities of electric fields and field gradients were examined as functions of geometry. Figure 8.2 shows some of the more direct and early approaches toward increasing electrode area per unit frontal (normal projection) of a flat plate electrode.





Figure 8.2 Shaped electrode fields.


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As can be seen in both cases above, the working surface area has been increased by either putting a portion of the electrode at an angle or by making an indentation in the electrode surface. Looking at the electric field, one sees that the field gradient is also changed, and the electric current will be modified accordingly as well. The question here is whether there is a net gain in performance and how much. Much has been done in the past regarding electric field configurations for various geometrical shapes of conductors. One excellent source is W. R. Smythe’s Static and Dynamic Electricity published in 1950 by McGraw-Hill Co. Unfortunately, even the simplest of geometry leads quickly to very lengthy and complex computations for field shapes. Consequently, we have taken a much more simplified approach merely in order to see trends in the results of providing pores and holes on the surface of electrodes. Smythe (see Bibliography reference 26, particularly Chapters I and V) and others have explored the solution of Laplace’s equation for plane conductors with infinite extension, or slotted planes. For example, the application of the Schwarz transformation to the computation of charge distribution, σ, in the x-direction on a plane conductor with a slot of width 2a gives the following expression:

E 1 2

x (x


1 2 2



a )

The conductive plane extends infinitely in the x and y directions, and the slot of width 2a extends indefinitely in the y-direction. In this case, an electric field exists in the z-direction, imposed by parallel plate conductors extending in the x and y-directions. An approach we have taken here to approximate the field strength within a hole or slot, as compared to the electric field intensity at the plate surface, is as follows. Figures 8.3a and 8.3b show configurations of parallel plate conductors as a means of establishing extreme conditions of geometry. The first drawing is for a conductor that has a very large area (slot width). In this case, the electric field is quite uniform within the slot, and the voltage linearly varies with the distance in the y-direction. In other words, the field strength, or voltage gradient, ϕ, is y

E y

E (L



Polysulfide – Diffusion Analysis


(a) L E λ


(b) L E λ


ΔE 2≤ ΔE1

Figure 8.3 (a) Electrodes with large area depressions. (b) Electrodes with narrow slots.

within the slot and E/L at the plate surface, whereas the field strength is close to zero in the very small slot shown in the second drawing. The electric field does not bend very much into the narrow hole. The field for this case is approximated by




both within the hole as well as at the surface of the plate. In essence, for deep and very narrow slots or holes there is no electric field within the depression. Now consider the situation where the size of the slot is in between the two above extremes, as shown in Figure 8.4. In this case, which more closely represents actuality, the electric potential drop, ΔE, from the top of the hole to the bottom can be expressed in numerous approximate ways. One very simple view is to let the portion, ΔE, which appears in the pore of the total field, E, between conductors, be further modified by a parameter, β, as follows:





where β is a multiplier or an adjustment factor dependent on the ratio of depth to width of the slot or hole. Perhaps, a workable representation of the


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Figure 8.4 Dimensioned hole or slot.

manner in which the field strength varies within a pore in accordance with the ratio of w to λ is w

1 e





An equally useable expression could be

w w


where α is a constant of proportionality. In both cases, equations (8.11) or (8.12), β goes to zero as w/λ, and β goes to unity as w/λ becomes large. For small pores, equation (8.12) can be approximated by β = (w/λ) α, and then the complete expression, equation (8.10), for small pores becomes


w L



Assuming a linear field, the voltage gradient within the hole is



w (L )



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The approximation is not valid in those cases where the pores, or electrode surface depressions, are shallow and have large areas. Since these are not the conditions of interest to us, and the mathematics are unnecessarily more complex to cover the entire range, this approach has been taken for the present. If the above is a reasonable first approximation for the ion attractive force producing fields within electrode pores or depressions, then the next step is to examine the magnitude of electric current densities that result from such surface irregularities.

8.4 Electric (Ionic) Current Density Estimates Ionic mobility, μ, in the y-direction is given as μ = (y/t) (1/ϕ), where t is time an ϕ is the electric field gradient, or ϕ = ∂E/∂y, and the ionic speed is

x t

x t


More useful to this analysis is the specific resistivity, ρ, of the electrolyte, expressed as








dq dt


and A= area normal to the current flow. By applying equation (8.13) and solving for the current, i, in the pore entrance where the electric field gradient, ϕλ, is as described in equation (8.12), we obtain




w A . (L )


Since the expressions are all normalized for a unit thickness of electrode into the plane of the page (the z-direction), area, A, is equal to w, and equation (8.14) becomes


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w2 (L )




This can be compared to the current of Ew/Lρ at the top surface of the plane conductor. It is seen that, as the width of the slot is made smaller and the depth, λ, is made larger, the current into the pore decreases, and the current densities, i/w, also change in accordance with

i w


w (L



into the pore, and

i w



at the top surface. As the electrode becomes more porous, i.e., more internal surface area per unit frontal area of the plane face, the current density will reduce proportionately. However, if we further assume, as another first approximation, that the current density within the hole is uniformly distributed along the entire available surface area of walls and bottom, then the expression for the current density, σp, within the pore becomes

i p



Ew 2 (2 w )(L



It should be also remembered that the specific resistivity, ρ, of the electrolyte within the pores will not be constant as the depletion of charge carriers continues with charging. When one integrates the total current across the face on the porous plate electrode, it also becomes apparent that the largest portion of the current flow is to the outermost surface of the electrode and not within the pores.

8.5 Diffusion and Supply of Reagents The supply of free sulfur to the (−) electrode takes place only by molecular or thermal diffusion or forced convection. Since attachments to negatively

Polysulfide – Diffusion Analysis


charged sulfide ions solubilize the free sulfur, the difficulty of supply is compounded by the electric repulsion of the (−) electrode. Perhaps, the greatest problem associated with over potentials arises from the difficulty of sulfur supply during charge. If we assume a uniform field gradient into a pore, then, in order to maintain a steady rate of supply, (dQ/dt)S, of SxS= during charge into the pore, the rate must be equal to the charging current into the pore with entrance area w, or

dQ dt




Figure 8.5 represents a pore with concentrations Ca and Cb outside and inside, respectively, of the pore. In accordance with Fick’s First Law, the rate of diffusion is directly proportional to the concentration gradient or difference across a boundary:

dQ dt



dC . dx


Applying the relationship above,

dQ dt

K a (C a C b )w

K a (C a C a R)w,



where R = Cb/Ca. In reality, the diffusion situation for the supply of SxS= is worse than described above. Since there is in fact concentration gradient all along the pore length, λ, the availability of sulfur diminishes at greater pore depth.


Neg charged


Electrolyte region

Figure 8.5 Carbon conductor electrode pore.



Carbon electrode


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The question that arises now is whether the value of R must be so small that, to maintain the balance of sulfur flow into the pore to match the conversion rate of sulfur into sulfide ions, severe polarization will be encountered and hydrogen gas will be produced. Under the best of conditions, and from equations (8.16) and (8.21), the rate match is

i w


w (L


K a (C a C a R) K a C a (1 R),


and that does not account for the electric repulsion forces of the (−) electrode. Solving for R in equation (8.26), we obtain


Ew K a C a (L )



Since no materials are lost (plated on) or generated (plated off) within the pores, only conversion takes place, and spent products must be removed and active reagents must be provided. Mono-sulfide is removed from the pore by diffusion processes but assisted by electric fields, whereas the same fields retard the supply of polysulfide. Composite expressions can be devised in order to describe the total migration rates of the various species by introducing some additional factors associated with coulombic forces.

8.6 Cell Dynamics 8.6.1 Electrode Processes Analyses The general expression describing the half-cell potential at the (−) electrode for the charging reaction, S + 2e− S=, as given by the Debeye-Huckel theory and the Nernst equation, is shown in equations (8.1) through (8.4). We will proceed to outline an initial approach to solving the problem or answering the questions that surround the shapes of the charging voltage curves for an LS-2 cell.

8.6.2 Polymeric Number Change It should be noted that the sulfur/sulfide concentration cell has an alkaline electrolyte due, in part, to the hydrolysis of the Na2S salt in accordance with the following reaction:

Polysulfide – Diffusion Analysis

Na2S + H2O

2NaOH + H2S

187 (8.28)

If a salt, AB, is dissolved in water, it will undergo some degree of hydrolysis. This can be expressed simply as

AB + H2O



The extent to which this occurs is usually measured as the degree of hydrolysis. In the case of a salt of a weak acid and strong base, as in this instance, the equilibrium constant, Kh, for hydrolysis may be expressed as


[OH ][HA] . [A ][H 2O]


The ionization equilibrium constant, K, is generally expressed as


[A ][B ] . [AB]


The ionization constant for NaOH is orders of magnitude greater than that of H2S, hence the resultant solution is quite basic. In the sulfide/polysulfide alkaline concentration cell, the important issues of the behavior of sulfide ion and sulfur atoms in the immediate vicinity of the electrodes become quite important. These concerns exist during the charging and discharging processes, as well. The situation is different than what is normally encountered in electrochemical cells because of the fact that both the oxidizing agent, S, and the reducing agent, S−2, are the same chemical specie. Furthermore, the output potential is not derived from a difference in chemical potential between two different species, but instead is a consequence of differences in concentrations of the same chemical element. To review, again, the elemental sulfur, S, is the oxidizer when being reduced by the addition of two electrons from an external voltage supply circuit at the negative electrode during charging. The exact opposite takes place at the electrode during the discharging mode when electrons are given up to the negative electrode and then subsequently conducted to an outside electric circuit load. We will now take a deeper look into the current density distribution and reagent diffusion at the surface of porous electrodes. This initial review is intended primarily to provide a basis for some further attempts at characterizing the kinetic actions that take place at the surface of the negative


Energy Storage 2nd Edition

electrode in a sulfide electrolyte. This analysis is independent of the processes that occur at the opposite, positive electrode because the electrolytes are assumed to be independent. The only transfer between electrolytes across a cation membrane in this analysis is presumed to be Na+, sodium ions. The initial form of the electrolyte when the cell is in the discharged state is presumed to be Na2S5, or the penta-sulfide form, which can be represented as Na2S · S4. There are then four sulfur atoms available as essentially free sulfur attached to the sodium mono-sulfide in solution. If we now consider the various sequences in which the sulfur becomes detached from the salt during the charging processes, the following seem to be the available choices. 1. Random generation of various polysulfides, Na2S · Sx, where x can have values from 0 to 4 – In this case, we would assume some statistical distribution function to describe the molecules in solution at any point in time. 2. To assume that the SxS= decreases in size by one sulfur atom at a time during charge and then promptly leaves the immediate electrode surface region to make way for the next SxS= ion until all SxS= species have been converted to Sx−1S= – The next stage in the reduction of size would take place until, via such systematic procedures, the electrolyte is converted to Na2S at full charge. 3. To assume that each SxS= goes all the way to S= in a series of consecutive steps and then leaves the electrode surface area – This assumption would state that there is always only a ratio of S= to S4S= with which to contend. In any event, if we let the availability of sulfur for reduction to sulfide be the same independent of the multiple attachment factor to sulfide, there should be little practical or analytical concern here as to the exact distribution of sulfur at any time during the charging process. Figure 8.6 shows the diffusion rates in which we are interested for this present analysis. The drawing diagrams the simple but essential aspects of a half-cell. The volume, V, indicated in the figure includes the entire negative electrolyte volume for a unit area of (−) electrode, including that which is stored in the reservoirs external to the cell. Hence, the distance between the membrane and the electrode is not representative of the true geometry. We will use method #3 above to represent a species in the analysis regarding the availability of S (or concentration of S) as independent of x, or the total concentration of S in the form of SxS= taken as |S|. Now we have only two kinds of anionic species in solution in the negative electrolyte of

Polysulfide – Diffusion Analysis


Volume, V SxS= Negative electrode S4S= electrolyte


Figure 8.6 Diffusion representation.

volume V, (other than OH− and any Br− ions that have migrated through from the (+) side), and they are S= and S4S=. Then, the diffusion processes to and from the electrode surface can be represented as shown in Figure 8.7. It should be noted that there is no storage (build up or depletion of reagents at the electrode surface) of reagents within some region of immediate proximity to the electrode surface. There are, in fact, six analytical approaches that we will try to take toward describing the kinetics at the electrode surface. These will be taken in succeeding steps and are listed below. • Case 1 – Flat surface electrode, no surface storage region, instantaneous and perfect mixing of reagents throughout the volume, V, no hydrogen generation Volume, V


Negative electrode SxS=


Figure 8.7 Diffusion to and from electrodes.


Energy Storage 2nd Edition • Case 2 – Flat surface electrode, some reagent storage at the surface, instantaneous mixing of reagents within each of the two volumes, volume v is designated as the surface region storage, volume V is designated as the bulk (−) electrolyte, no hydrogen generation • Case 3 – Porous Surface electrode, some storage region, instantaneous mixing in both volumes, no hydrogen generation, smaller diffusion coefficient • Case 4 – Flat surface, some storage, some hydrogen generation, instantaneous mixing • Case 5 – Porous surface, some storage, some hydrogen generation, instantaneous mixing • Case 6 – Porous surface, some storage, some hydrogen generation, reagent concentration gradient at electrode surface volume, instantaneous mixing within volume V.

For perfect, instantaneous mixing there are no diffusion equations. Only the concentration changes as the various species are generated electrochemically. Referring to Figure 8.7, we see that the expression describing the half cell potential, E, is

E Eo

RT ln nZ


i s




and it is the final expression for this particular, somewhat unrealistic case of full access to the electrode surface, with no stagnant electrolyte region at the electrode surface. Activity coefficients for sulfide and sulfur are being researched in the chemical literature. Meanwhile, since we are primarily concerned here with functional relationships, we can proceed with a general analysis of the electrode processes. At present, our main interest is in the functional dependency of, and trends in, the voltages during cell operation. Figure 8.8 gives a few values of generally available activity coefficients for various compounds in aqueous solution (Latimer 1952) where γi = activity coefficient for the sulfide ions = aS−/|S=|, and γs = activity coefficient for the “free sulfur” = aSx/|S|. The concentrations of the species are further defined in this particular case as,


Qi V


Polysulfide – Diffusion Analysis






0 0






Concentration – molarity HBr






Figure 8.8 Activity coefficients for strong electrolytes from Latimer’s oxidation potentials.



Qs V


There is also a useable relationship between Qi and Qs. The total of the numbers of sulfide ions and sulfur atoms remains constant since one sulfide ion is produced for each sulfur atom lost during charging. Qi + Qs = Qo equals the sum of both species present initially, and Qs = 4Qi. At any time, t, later the expression assumes the form

q = Qo = qi + qs .


If we normalize the analysis for a unit of electrode area, then the volume in the main negative solution per unit area is V. Now it is necessary to address the rate, Rg, of sulfide ion generation during charge. This can be expressed as

Rg = αση,



Energy Storage 2nd Edition

where α = constant of proportionality, σ = current density, amps/ unit area, and η = coulombic efficiency. Thus, we may represent the rate of generation, dqi/dt, of S= ions as

dq i dt




which is also the exact rate of reduction of available sulfur. Since Qo + 4Qi = Qi by substitution between equations (8.37) and (8.32), we obtain


Qo . 5


Solving for qi by integrating equation (8.33),

qi = ασηt + k.


From the limits of the problem when t = 0,



Qo , 5


Qo 5


and the final expression becomes



It is now possible to place the values for qi and qs into equation (8.26) to solve for E as a function of time. Using the relationships in equations (8.27), (8.28) and (8.29), and substituting into equation (8.26),

E Eo

RT ln nZ


qi V

Qo q i s V



Polysulfide – Diffusion Analysis


Substituting the value of qi from equation (8.33) into equation (8.34),

E Eo

RT ln nZ




4Q o






is the final expression for this case of full access to the electrode surface and no stagnant region at the electrode surface. We have not yet found the activity coefficients for sulfide and sulfur in the literature. However, since we are primarily concerned here with functional relationships, the details of specific values for the activities can be left for a later time when we become interested in more dependable, quantitative evaluations of these expressions to compare with experimental results obtained in the laboratory. At present, our main interest is in the functional dependency of, and trends in, the voltages during cell operation. It appears that the chemical potentials of neutral and S= ions and neutral S are independent of whether they are “polymerized” or not, and that the effective reaction is S + 2e−  S=. While this is an approximation, it is quite reasonable given that the halfcell potentials for the reactions given below are so close together.




S= S 2




S= S3





0.48 volts

Since pH is usually very high, reactions of the form HS− H+ + S= and the presence of HS− ions are ignored for the present. Figure 8.8 is a plot of a few values of generally available activity coefficients for various compounds in aqueous solution (Latimer 1952). An interesting graph is the plot of half-cell voltage versus time as expressed in equation (8.37). In order to generate such a plot, the values of a few constants are needed. The values are listed below. Some assumptions or simplifications have been made in a few coefficients outside of the values of the physical constants. • R = universal gas constant = 1.99 cal deg−1 mole−1 = 8.3 joules deg−1 mole−1


Energy Storage 2nd Edition • α = conversion constant = mole/52 amp-hours = 0.0019 mole amp−1 hour−1 = 5.34 × 10−6 moles amp−1 sec−1 • η = coulombic efficiency = 1.0 • γi = γs = as an interim assumption • V = storage volume in liters per unit area of electrode • n = number of electrons transferred per event • Eo = 0.50 volts half cell voltage at standard conditions • Z = Faradays number = 96,500 coulombs/equiv = 26 AH/equiv.

Now we need to assess the values of σ, Qi, and Qs on the basis of a particular current density. Since most of the experiments performed at TRL with single cells of a 25 in2 area were performed between 4 and 6 amps, we will take 0.03 amps per cm2 as the current density. A total charging time of 5 hours is also commensurate with most of the data accumulated to date. Thus, a total charge of 0.15AH/cm2 corresponds to the capacity of the electrolyte of volume, V. Now, there are Qs = 0.15AH/(52AH/mole) = 2.9 × 10−3 moles of available S, and Qi = Qs/4 = 7.2 × 10−4 moles of S=. Substituting the above values into equation (8.37), the following is obtained:

E Eo

7.2 10 0.013 ln 2.9 10


1.6 10 7 t volts. 1.6 10 7 t



Figure 8.9 is a plot of equation (8.38) of half-cell voltage versus time in hours. Equation (8.38) then has the form

E Eo

7.2 10 0.013 ln 2.9 10

4 3

5.76 10 4 t . 5.76 10 4 t


Ohmic resistance of the electrolyte and any interface resistance have not been included in the plot. As can be seen, the voltage is below 0.5 volts at the beginning of charge when there are relatively few sulfides present, as compared to the available sulfur. As charging progresses, the potential rises fairly linearly until the concentration of sulfides becomes great and that of available sulfur has diminished significantly. In theory, the cell potential would rise to extremely high values if the charging were continued in the constant current mode. In reality, as the charging potential rises high enough the decomposition of water would take place, and hydrogen would evolve.

Polysulfide – Diffusion Analysis




0.58 0.57 0.56

Half cell voltage

0.55 0.54 0.53 0.52 0.51 0.5 0.49 0.48 0.47

0 Anal–2.doc



4 Time – hours




Figure 8.9 Half-cell potential versus time.

The factors influencing the shape of the plot include the following: • relative values of γi and γs • the manner in which γi and γs change with or depend on the concentration of the electrolyte • electric current density If, for example, the ratio γi/γs were to be greater or less than unity, as was assumed in the exercise above, the general shape of the voltage versus time curve does not change. Rather, it is displaced on the vertical axis accordingly, as shown in Figure 8.10. The data calculated and plotted above assumes that the activity coefficient ratios are constant throughout the range of solution concentrations. In reality, they probably do vary between the two extremes of very dilute to very concentrated. Figure 8.11a graphs two sample functions of the ratio of r = γi/γs. For the low-mid values, the ratio r1 = γi/γs starts at 1.5 and goes down to 0.5 midway, and then it returns to 1.5 at the end of charge. In the second instance, r2 = γi/γs starts at 0.5 and goes up through values of 1.5 and returns to 0.5 at the end of charge. The purpose is to observe the effect of widely varying ratios, r, upon the charging curve for the half-cell.


Energy Storage 2nd Edition Two different values or activity coefficient ratios 0.6 0.58

Half cell voltage

0.56 0.54 0.52 0.5 0.48 0.46 0.44






4 Time – hours

Ratio = 2





Ratio = 0.5

Figure 8.10 Half-cell potential versus time.

Figure 8.11b gives plots of the cell voltage for the two r-functions above. The shapes change noticeably with such a spread of values, one with a peak in the mid-concentration range and the other with a saddle in mid-range. The preceding analysis was performed for the very simplified situation, Case 1, where there is not a boundary layer on the electrode and the idealized perfect mixing of electrolyte components. The analysis continues with the circumstances described in Case 2. This next stage in the development of descriptive modeling more closely approximates actual conditions within a cell. It is expected that these exercises will significantly improve our investigations for electrode structures with lower polarization losses and will explain the seemingly peculiar voltage versus time curves and their changing behavior with cycling. A graph would plot the half-cell voltage versus time, as expressed in equation (8.35). In order to do so, the values of a few constants are needed. They are set as the following: • R = universal gas constant = 1.99 cal deg.−1 mole−1 • 8.3 joules deg.−1 mole−1 • α = conversion constant = mole/52 amp-hours = 0.0019 mole amp−1 hr−1= 5.34 × 10-6 moles amp−1 sec−1

Polysulfide – Diffusion Analysis


Plots for non-constant activity coefficient ratios

(a) 1.6


Ratio value, r









r1 low mid values



r2 high mid values

Plots for non-constant activity coefficient ratios


Half cell voltage

40 60 80 State charge in percentage

0.59 0.58 0.57 0.56 0.55 0.54 0.53 0.52 0.51 0.5 0.49 0.48 0.47 0.46





3 Time – hours

r1 high mid values




r2 low mid values

Figure 8.11 (a) Activity coefficient ratios versus charge state. (b) Half potential versus time.

• • • • •

η = coulombic efficiency = 1.0 γi = γs = as an interim assumption V = storage volume in liters per unit area of electrode n = electrical charge per ion = 2 equiv./mole Z = Faradays number = 96,500 coulombs/equiv. = 26 AH/equiv.


Energy Storage 2nd Edition

By substituting the above values into equation (8.35), the voltage values of half-cell potentials can be obtained.

8.7 Further Analysis of Electrode Behavior 8.7.1 Flat Electrode with Some Storage Properties The earlier analysis assumed simplified conditions, in which no delayed boundary region existed at the electrode surface for reagent storage and instantaneous solvent mixing. The purpose of this analysis and report is to present a sequence of studies aimed at understanding the fundamental processes associated with cell behavior at the negative electrode, in particular. The bromine electrode is comparatively well behaved and presents no inexplicable or unwanted electrical characteristics. The conditions under which this next stage in the analysis has been done are described fully in the text. The results are realistic and provide valuable assistance in understanding the rate process balances. A constant current mode of charging was assumed, and the coulombic efficiency is set at 100% all the way through. In reality, the current efficiency very much depends on current density and reagent concentrations. In the next sequence, we will explore cell behavior when current efficiency is treated as a function of both current density and polysulfide concentration in the immediate vicinity of the (−) electrode. The conditions we will examine are as follows: • Case 2 – Flat surface electrode, some reagent storage at the surface, instantaneous mixing of reagents within each of the two volumes, volume, v, is designated as the surface region storage, volume, V, is the bulk (−) electrolyte, no hydrogen generation. Figure 8.12 illustrates the regions in question and the distribution of reagents at any time within these two regions. The concentrations of sulfur and sulfide are considered uniform at all times throughout the respective regions. In other words, the concentration of SxS= for each value of x is assumed to be uniform in volume, v, and volume, V, separately. The chart in Figure 8.12 illustrates this idea that the concentration of each species is constant at all times in the cell regions. Figure 8.13 depicts the rate processes for the gain and loss of sulfide ions and available sulfur for each of the two cell regions.

Polysulfide – Diffusion Analysis (–) Electrode


Stagnant Electrolyte reservoir V Region V Region Boundary

Concentration of S= ions

During charge Initial concentration


Figure 8.12 Flat electrode with two associated storage regions.

Sulfide ions and available sulfur are entering and leaving the regions across the boundary (artificial separation of the two regions) by diffusion. Also, sulfide ions are generated at the rate Rg(−) into the boundary layer at the electrode surface by electrolysis (reduction). Available sulfur is lost at the rate Rg(S) to the region by the very same electrolysis process. Hence,

Rg (−) = −Rg (S) = R


is the generation rate. Figure 8.14 identifies the various species within the solutions. In order to minimize the complexity of the symbols, the ionic species in each of the two regions have been given the following designations for the amount of reagents present at any instant of time during the charging process: • qib = quantity of S= ions in the boundary layer at the electrode surface • qsb = quantity of free sulfur within the electrode surface boundary layer


Energy Storage 2nd Edition Boundary region vol. = v

Reservoir region vol. = v





km qir

kp qsr





Figure 8.13 The rates of gain and loss of sulfide ions.









Figure 8.14 Identification of ionic specie regions.

• qir = quantity of sulfide ions present in the reservoir region • qsr = quantity of free sulfur in the reservoir region. The rate, Rg(−), with which sulfide ions are being generated by the charging process is

Rg( )

dq ib dt



Polysulfide – Diffusion Analysis


and the rate with which free sulfur is reduced is

dq ib dt

R g (S)



The net rate, dqir/dt, of increase in sulfide ions in the reservoir region (volume, V) due to sulfide ions leaving and entering the boundary region via diffusion of both monosulfides and polysulfides into and from the reservoir is represented as

dq ir dt

q ib k m v b

q sb k p v b

q ir k m V r

q sr k p V r


The rate, dqsr/dt, of increase in free sulfur in the reservoir region due to diffusion into and from the boundary is

dq sr dt

q sb k p v b

q sr k p , V r


where λ is a diffusion path length. In volume, v, or the boundary region, the rate balance equations are

dq ib dt


q ir q k m +k p sr V r V r


q ib v b


q sb v b



dq sb dt


q sr kp V r

q sb kp. v b


The diffusion constants represented by k for the sodium mono-sulfide and the sodium penta-sulfide molecules should be self-explanatory. There are four diffusion terms in the expression for dqib/dt above because available sulfide ions are transported via both the polysulfides and the mono-sulfides. Since immediate mixing in each of the regions is assumed to take place, no provision has been made here to account for any repulsive electrical forces of the (−) electrode acting upon the different ionized components. Hence, in this treatment any deficiency or surplus of ionic species at the


Energy Storage 2nd Edition

immediate surface of the electrode is accounted for in the layer referred to as the boundary region. Since we are concerned at this time only with the neutral molecular diffusion of the reagents, and not ionized charge layers, the diffusion constant, km, and kp, for Na2S, and Na2S5 respectively, are considered the same whether they are in the boundary region diffusing away from or toward the (−) electrode or in the reservoir region drifting in either direction. Now it is possible to solve for a set of simultaneous differential equations that describe the concentration balances at any time during charging. There are four variables and four equations:

qib + qir + qsb + qsr = Qo


Since volume, v, is so much smaller than V,

qib + qsb