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FUEL CELL FUNDAMENTALS

Third Edition

RYAN O’HAYRE Department of Metallurgical and Materials Engineering Colorado School of Mines [PhD, Materials Science and Engineering, Stanford University]

SUK-WON CHA School of Mechanical and Aerospace Engineering Seoul National University [PhD, Mechanical Engineering, Stanford University]

WHITNEY G. COLELLA The G.W.C. Whiting School of Engineering, and The Energy, Environment, Sustainability and Health Institute The Johns Hopkins University Gaia Energy Research Institute [Doctorate, Engineering Science, The University of Oxford]

FRITZ B. PRINZ R.H. Adams Professor of Engineering Departments of Mechanical Engineering and Material Science and Engineering Stanford University

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10 9 8 7 6 5 4 3 2 1

To the parents who nurtured us.

To the teachers who inspired us.

CONTENTS

PREFACE xi

ACKNOWLEDGMENTS xiii

NOMENCLATURE xvii

I FUEL CELL PRINCIPLES

1 Introduction 3

1.1 What Is a Fuel Cell? / 3 1.2 A Simple Fuel Cell / 6 1.3 Fuel Cell Advantages / 8 1.4 Fuel Cell Disadvantages / 11 1.5 Fuel Cell Types / 12 1.6 Basic Fuel Cell Operation / 14 1.7 Fuel Cell Performance / 18 1.8 Characterization and Modeling / 20 1.9 Fuel Cell Technology / 21 1.10 Fuel Cells and the Environment / 21 1.11 Chapter Summary / 22

Chapter Exercises / 23

vi CONTENTS

2 Fuel Cell Thermodynamics 25

2.1 Thermodynamics Review / 25

2.2 Heat Potential of a Fuel: Enthalpy of Reaction / 34

2.3 Work Potential of a Fuel: Gibbs Free Energy / 37

2.4 Predicting Reversible Voltage of a Fuel Cell under Non-Standard-State Conditions / 47

2.5 Fuel Cell Efficiency / 60

2.6 Thermal and Mass Balances in Fuel Cells / 65

2.7 Thermodynamics of Reversible Fuel Cells / 67

2.8 Chapter Summary / 71

Chapter Exercises / 72

3 Fuel Cell Reaction Kinetics 77

3.1 Introduction to Electrode Kinetics / 77

3.2 Why Charge Transfer Reactions Have an Activation Energy / 82

3.3 Activation Energy Determines Reaction Rate / 84

3.4 Calculating Net Rate of a Reaction / 85

3.5 Rate of Reaction at Equilibrium: Exchange Current Density / 86

3.6 Potential of a Reaction at Equilibrium: Galvani Potential / 87

3.7 Potential and Rate: Butler–Volmer Equation / 89

3.8 Exchange Currents and Electrocatalysis: How to Improve Kinetic Performance / 94

3.9 Simplified Activation Kinetics: Tafel Equation / 97

3.10 Different Fuel Cell Reactions Produce Different Kinetics / 100

3.11 Catalyst–Electrode Design / 103

3.12 Quantum Mechanics: Framework for Understanding Catalysis in Fuel Cells / 104

3.13 The Sabatier Principle for Catalyst Selection / 107

3.14 Connecting the Butler–Volmer and Nernst Equations (Optional) / 108

3.15 Chapter Summary / 112

Chapter Exercises / 113

4 Fuel Cell Charge Transport 117

4.1 Charges Move in Response to Forces / 117

4.2 Charge Transport Results in a Voltage Loss / 121

4.3 Characteristics of Fuel Cell Charge Transport Resistance / 124

4.4 Physical Meaning of Conductivity / 128

4.5 Review of Fuel Cell Electrolyte Classes / 132

CONTENTS vii

4.6 More on Diffusivity and Conductivity (Optional) / 153

4.7 Why Electrical Driving Forces Dominate Charge Transport (Optional) / 160

4.8 Quantum Mechanics–Based Simulation of Ion Conduction in Oxide Electrolytes (Optional) / 161

4.9 Chapter Summary / 163

Chapter Exercises / 164

5 Fuel Cell Mass Transport 167

5.1 Transport in Electrode versus Flow Structure / 168

5.2 Transport in Electrode: Diffusive Transport / 170

5.3 Transport in Flow Structures: Convective Transport / 183

5.4 Chapter Summary / 199

Chapter Exercises / 200

6 Fuel Cell Modeling 203

6.1 Putting It All Together: A Basic Fuel Cell Model / 203

6.2 A 1D Fuel Cell Model / 206

6.3 Fuel Cell Models Based on Computational Fluid Dynamics (Optional) / 227

6.4 Chapter Summary / 230

Chapter Exercises / 231

7 Fuel Cell Characterization 237

7.1 What Do We Want to Characterize? / 238

7.2 Overview of Characterization Techniques / 239

7.3 In Situ Electrochemical Characterization Techniques / 240

7.4 Ex Situ Characterization Techniques / 265

7.5 Chapter Summary / 268

Chapter Exercises / 269

II FUEL CELL TECHNOLOGY

8 Overview of Fuel Cell Types 273

8.1 Introduction / 273

8.2 Phosphoric Acid Fuel Cell / 274

8.3 Polymer Electrolyte Membrane Fuel Cell / 275

8.4 Alkaline Fuel Cell / 278

8.5 Molten Carbonate Fuel Cell / 280

viii CONTENTS

8.6 Solid-Oxide Fuel Cell / 282

8.7 Other Fuel Cells / 284

8.8 Summary Comparison / 298

8.9 Chapter Summary / 299

Chapter Exercises / 301

9 PEMFC and SOFC Materials 303

9.1 PEMFC Electrolyte Materials / 304

9.2 PEMFC Electrode/Catalyst Materials / 308

9.3 SOFC Electrolyte Materials / 317

9.4 SOFC Electrode/Catalyst Materials / 326

9.5 Material Stability, Durability, and Lifetime / 336

9.6 Chapter Summary / 340

Chapter Exercises / 342

10 Overview of Fuel Cell Systems 347

10.1 Fuel Cell Subsystem / 348

10.2 Thermal Management Subsystem / 353

10.3 Fuel Delivery/Processing Subsystem / 357

10.4 Power Electronics Subsystem / 364

10.5 Case Study of Fuel Cell System Design: Stationary Combined Heat and Power Systems / 369

10.6 Case Study of Fuel Cell System Design: Sizing a Portable Fuel Cell / 383

10.7 Chapter Summary / 387

Chapter Exercises / 389

11 Fuel Processing Subsystem Design 393

11.1 Fuel Reforming Overview / 394

11.2 Water Gas Shift Reactors / 409

11.3 Carbon Monoxide Clean-Up / 411

11.4 Reformer and Processor Efficiency Losses / 414

11.5 Reactor Design for Fuel Reformers and Processors / 416

11.6 Chapter Summary / 417

Chapter Exercises / 419

CONTENTS ix

12 Thermal Management Subsystem Design 423

12.1 Overview of Pinch Point Analysis Steps / 424

12.2 Chapter Summary / 440

Chapter Exercises / 441

13 Fuel Cell System Design 447

13.1 Fuel Cell Design Via Computational Fluid Dynamics / 447

13.2 Fuel Cell System Design: A Case Study / 462

13.3 Chapter Summary / 476

Chapter Exercises / 477

14 Environmental Impact of Fuel Cells 481

14.1 Life Cycle Assessment / 481

14.2 Important Emissions for LCA / 490

14.3 Emissions Related to Global Warming / 490

14.4 Emissions Related to Air Pollution / 502

14.5 Analyzing Entire Scenarios with LCA / 507

14.6 Chapter Summary / 510

Chapter Exercises / 511

A Constants and Conversions 517

B Thermodynamic Data 519

C Standard Electrode Potentials at 25∘C 529

D Quantum Mechanics 531

D.1 Atomic Orbitals / 533

D.2 Postulates of Quantum Mechanics / 534

D.3 One-Dimensional Electron Gas / 536

D.4 Analogy to Column Buckling / 537

D.5 Hydrogen Atom / 538

D.6 Multielectron Systems / 540

D.7 Density Functional Theory / 540

x CONTENTS

E Periodic Table of the Elements 543

F Suggested Further Reading 545

G Important Equations 547

H Answers to Selected Chapter Exercises 551

BIBLIOGRAPHY 555

INDEX 565

PREFACE

Imagine driving home in a fuel cell car with nothing but pure water dripping from the tailpipe. Imagine a laptop computer that runs for 30 hours on a single charge. Imagine a world where air pollution emissions are a fraction of that from present-day automobiles and power plants. These dreams motivate today’s fuel cell research. While some dreams (like cities chock-full of ultra-low-emission fuel cell cars) may be distant, others (like a 30-hour fuel cell laptop) may be closer than you think.

By taking fuel cells from the dream world to the real world, this book teaches you the science behind the technology. This book focuses on the questions “how” and “why.” Inside you will find straightforward descriptions of how fuel cells work, why they offer the potential for high efficiency, and how their unique advantages can best be used. Emphasis is placed on the fundamental scientific principles that govern fuel cell operation. These principles remain constant and universally applicable, regardless of fuel cell type or technology.

Following this philosophy, the first part, “Fuel Cell Principles,” is devoted to basic fuel cell physics. Illustrated diagrams, examples, text boxes, and homework questions are all designed to impart a unified, intuitive understanding of fuel cells. Of course, no treatment of fuel cells is complete without at least a brief discussion of the practical aspects of fuel cell technology. This is the aim of the second part of the book, “Fuel Cell Technology.” Informative diagrams, tables, and examples provide an engaging review of the major fuel cell technologies. In this half of the book, you will learn how to select the right fuel cell for a given application and how to design a complete system. Finally, you will learn how to assess the potential environmental impact of fuel cell technology.

xii PREFACE

Comments or questions? Suggestions for improving the book? Found a typo, think our explanations could be improved, want to make a suggestion about other important con- cepts to discuss, or have we got it all wrong? Please send us your feedback by emailing us at [email protected]. We will take your suggestions into consideration for the next edition. Our website http://groups.yahoo.com/group/fcf3 posts these discussions, fliers for the book, and additional educational materials. Thank you.

ACKNOWLEDGMENTS

The authors would like to thank their friends and colleagues at Stanford University and the former Rapid Prototyping Laboratory (RPL), now the Nano-Prototyping Laboratory (NPL), for their support, critiques, comments, and enthusiasm.Without you, this text would not have been written! The beautiful figures and illustrations featured in this textbook were crafted primarily by Marily Mallison, with additional illustrations by Dr. Michael Sanders—their artistic touch is greatly appreciated!

The authors would like to thank the Deans of the Stanford School of Engineering, Jim Plummer and Channing Robertson, and John Bravman, Vice Provost Undergraduate Educa- tion, for the support that made this book possible.Wewould also like to acknowledgeHonda R&D, its representatives J. Araki, T. Kawanabe, Y. Fujisawa, Y. Kawaguchi, Y. Higuchi, T. Kubota, N. Kuriyama, Y. Saito, J. Sasahara, and H. Tsuru, and Stanford’s Global Climate and Energy Project (GCEP) community for creating an atmosphere conducive to studying and researching new forms of power generation. All members of RPL/NPL are recog- nized for stimulating discussions. Special thanks to Dr. Tim Holme for his innumerable contributions, including his careful review of the text, integration work, nomenclature and equation summaries, and the appendixes. Thanks also to Professor Rojana Pornprasertsuk, who developed the wonderful quantum simulation images for Chapter 3 and Appendix D. The authors are grateful to Professor Yong-il Park for his help in the literature survey of Chapter 9 and Rami Elkhatib for his significant contributions in writing this section. Profes- sor Juliet Risner deserves gratitude for her beautiful editing job, and Professor Hong Huang deserves thanks for content contribution. Dr. Jeremy Cheng, Dr. Kevin Crabb, Professor Turgut Gur, Shannon Miller, Masafumi Nakamura, and A. J. Simon also provided signifi- cant editorial advice. Thanks to Dr. Young-Seok Jee, Dr. Daeheung Lee, Dr. Yeageun Lee,

xiii

xiv ACKNOWLEDGMENTS

Dr. Wonjong Yu, and Dr. Yusung Kim for their contributions to Chapters 6 and 13. Spe- cial thanks to Rusty Powell and Derick Reimanis for their careful editing contributions to the second edition. Finally, thanks to colleagues at the Colorado School of Mines (CSM), including Bob Kee and Neal Sullivan for their helpful discussions and for a decade’s worth of students at CSM for catching typos and identifying areas in need for clarification for this third edition.

We would like to extend our gratitude to Professor Stephen H. Schneider, Professor Terry Root, Dr. Michael Mastrandrea, Mrs. Patricia Mastrandrea, Dr. Gerard Ketafani, and Dr. Jonathan Koomey. We would also like to thank the technical research staff within the U.S. Department of Energy (DOE) complex, including researchers at DOE national lab- oratories [Sandia National Laboratories (SNL), Lawrence Berkeley National Laboratory (LBNL), Argonne National Laboratory (ANL), the National Renewable Energy Laboratory (NREL), and Lawrence Livermore National Laboratory (LLNL), among others]. We would also like to thank research participants within the International Energy Agency (IEA) Sta- tionary Fuel Cell Annex, the American Institute of Chemical Engineers (AICHE) Transport and Energy Processes Division (TEP), and the National Academy of Engineering (NAE) Frontiers of Engineering (FOE) program.

For intellectually stimulating discussions on energy system design, we also would like to thank Dr. Salvador Aceves (LLNL), Dr. Katherine Ayers (ProtonOnsite Inc.), Professor Nigel Brandon (Imperial College London), Mr. Tom Brown (California State University Northridge), Dr. Viviana Cigolotti [Energy and Sustainable Economic Development (ENEA)], Professor Peter Dobson [University of Oxford (Oxon)], Dr. Elango Elangovan (Ceramatec Inc.), Professor Ferhal Erhun, Dr. Angelo Esposito (European Institute for Energy Research), Dr. Hossein Ghezel-Ayagh [FuelCell Energy Inc. (FCE)], Dr. Lorenz Gubler [Paul Scherrer Institut (PSI)], Dr. Monjid Hamdan (Giner Inc.), Dr. Joseph J. Hartvigsen (Ceramatec Inc.), Professor Michael Hickner (The Pennsylvania State University), Professor Ben Hobbs (Johns Hopkins University), Professor Daniel M. Kammen [University of California at Berkeley (UCB)], Professor Jon Koomey, Dr. Scott Larsen (New York State Energy Research and Development Authority), Mr. Bruce Lin (EnerVault Inc.), Dr. Ludwig Lipp (FCE), Dr. Bernard Liu (National Cheng Kung University), Professor V. K. Mathur (University of New Hampshire), Dr. Marianne Mintz (ANL), Professor Catherine Mitchell (University of Exeter), Dr. Cortney Mittelsteadt (Giner Inc.), Dr. Yasunobu Mizutani (ToHo Gas Co. Ltd.), John Molburg (Argonne National Laboratory), Dr. Angelo Moreno [Italian National Agency for New Technologies, Energy and Sustainable Economic Development (ENEA)], Professor Vincenzo Mulone (University of Rome Tor Vergata), Dr. Jim O’Brien (Idaho National Laboratory), Professor Joan Ogden (University of California at Davis), Dr. Pinakin Patel (FCE), Dr. Randy Petri (Versa Power Inc.), Professor Bruno Pollet (University of Ulster), Dr. Peter Rieke [Pacific Northwest National Laboratory (PNNL)], Dr. Subhash C. Singhal (PNNL), Professor Colin Snowdon (Oxon), Professor Robert Socolow (Princeton University), Mr. Keith Spitznagel (KAS Energy Services LLC), Professor Robert Steinberger-Wilckens (University of Birmingham), Dr. Jeffry Stevenson (PNNL), Professor Richard Stone (Oxon), Professor Etim Ubong (Kettering University), Professor Eric D. Wachsman (University of Maryland), Professor Xia Wang (Oakland University), and Professor Yingru Zhao (Xiamen University).

ACKNOWLEDGMENTS xv

Fritz B. Prinz wants to thank his wife, Gertrud, and his children, Marie-Helene and Benedikt, for their love, support, and patience.

Whitney G. Colella would like to thank her friends and family, especially the Bakers, Birchards, Chens, Colellas, Culvers, Efthimiades, Hoffmans, Jaquintas, Judges, Louies, Mavrovitis, Omlands, Pandolfis, Panwalkers, Qualtieris, Scales, Smiths, Spielers, Tepers, Thananarts, Tragers, Wasleys, and Wegmans.

Suk-Won Cha wishes to thank Unjung, William, and Sophia for their constant support, love, and understanding.

Ryan O’Hayre sends his thanks and gratitude to Lisa for her friendship, encouragement, confidence, support, and love. Thanks also to Kendra, Arthur, Morgan, little Anna, and little Robert. Ryan has always wanted to write a book … probably something about dragons and adventure. Well, things have a funny way of working out, and although he ended up writing about fuel cells, he had to put the dragons in somewhere.…

NOMENCLATURE

Symbol Meaning Common Units

A Area cm2

Ac Catalyst area coefficient Dimensionless a Activity Dimensionless ASR Area specific resistance Ω ⋅ cm2 C Capacitance F Cdl Double-layer capacitance F c∗ Concentration at reaction surface mol∕cm2 c Concentration mol∕m3 c Constant describing how mass transport affects

concentration losses V

cp Heat capacity J∕mol ⋅ K D Diffusivity cm2∕s E Electric field V∕cm E Thermodynamic ideal voltage V Ethermo Thermodynamic ideal voltage V ET Temperature-dependent thermodynamic voltage at

reference concentration V

F Helmholtz free energy J, J∕mol F Faraday constant 96, 485 C∕mol Fk Generalized force N f Reaction rate constant Hz, s−1

f Friction factor Dimensionless

xvii

xviii NOMENCLATURE

Symbol Meaning Common Units

G, g Gibbs free energy J, J∕mol g Acceleration due to gravity m∕s2 ΔG‡ Activation energy barrier J∕mol, J ΔGact Activation energy barrier J∕mol, J H Heat J H, h Enthalpy J, J∕mol HC Gas channel thickness cm HE Diffusion layer thickness cm h Planck’s constant 6.63 × 10−34 J ⋅ s ℏ Reduced Planck constant, h∕2𝜋 1.05 × 10−34 J ⋅ s hm Mass transfer convection coefficient m∕s i Current A J Molar flux, molar reaction rate mol∕cm2 ⋅ s Ĵ Mass flux g∕cm2 ⋅ s, kg∕m2 ⋅ s JC Convective mass flux kg∕m2 ⋅ s j Current density A∕cm2 j0 Exchange current density A∕cm2 j00 Exchange current density at reference

concentration A∕cm2

jL Limiting current density A∕cm2 jleak Fuel leakage current A∕cm2 k Boltzmann’s constant 1.38 × 10−23 J∕K L Length m M Molar mass g∕mol, kg∕mol M Mass flow rate kg∕s Mik Generalized coupling coefficient between

force and flux Varies

m Mass kg mcp Heat capacity flow rate kW∕kg ⋅ ∘C N Number of moles Dimensionless NA Avogadro’s number 6.02 × 1023 mol−1 n Number of electrons transferred in the reaction Dimensionless ng Number of moles of gas Dimensionless P Power or power density W or W∕cm2 P Pressure bar, atm, Pa Q Heat J, J∕mol Q Charge C Qh Adsorption charge C∕cm2 Qm Adsorption charge for smooth catalyst surface C∕cm2 q Fundamental charge 1.60 × 10−19 C R Ideal gas constant 8.314 J∕mol ⋅ K R Resistance Ω Rf Faradaic resistance Ω

NOMENCLATURE xix

Symbol Meaning Common Units

Re Reynolds number Dimensionless S, s Entropy J∕K, J∕mol ⋅ K S∕C Steam-to-carbon ratio Dimensionless Sh Sherwood number Dimensionless T Temperature K, ∘C t Thickness cm U Internal energy J, J∕mol u Mobility cm2∕V ⋅ s ū Mean flow velocity cm∕s, m∕s V Voltage V V Volume L, cm3

V Reaction rate per unit area mol∕cm2 ⋅ s 𝑣 Velocity cm∕s 𝑣 Hopping rate s−1, Hz 𝑣 Molar flow rate mol∕s, mol∕min W Work J, J∕mol X Parasitic power load W x Mole fraction Dimensionless x𝑣 Vacancy fraction mol vacancies∕mol sites yx Yield of element X Dimensionless Z Impedance Ω z Height cm

Greek Symbols

Symbol Meaning Common Units

𝛼 Charge transfer coefficient Dimensionless 𝛼 Coefficient for CO2 equivalent Dimensionless 𝛼∗ Channel aspect ratio Dimensionless 𝛽 Coefficient for CO2 equivalent Dimensionless 𝛾 Activity coefficient Dimensionless Δ Denotes change in quantity Dimensionless 𝛿 Diffusion layer thickness m, cm 𝜀 Efficiency Dimensionless 𝜀FP Efficiency of fuel processor Dimensionless 𝜀FR Efficiency of fuel reformer Dimensionless 𝜀H Efficiency of heat recovery Dimensionless 𝜀O Efficiency overall Dimensionless 𝜀R Efficiency, electrical Dimensionless 𝜀 Porosity Dimensionless �̇� Strain rate s−1

xx NOMENCLATURE

Symbol Meaning Common Units

𝜂 Overvoltage V 𝜂act Activation overvoltage V 𝜂conc Concentration overvoltage V 𝜂ohmic Ohmic overvoltage V λ Stoichiometric coefficient Dimensionless λ Water content Dimensionless 𝜇 Viscosity kg ⋅ m/s 𝜇 Chemical potential J, J/mol �̃� Electrochemical potential J, J/mol 𝜌 Resistivity Ω cm 𝜌 Density kg∕cm3, kg∕m3 𝜎 Conductivity S∕cm, (Ω ⋅ cm)−1 𝜎 Warburg coefficient Ω∕s0.5 𝜏 Mean free time s 𝜏 Shear stress Pa 𝜑 Electrical potential V 𝜑 Phase factor Dimensionless 𝜔 Angular frequency (𝜔 = 2𝜋f ) rad/s

Superscripts

Symbol Meaning

0 Denotes standard or reference state eff Effective property

Subscripts

Symbol Meaning

diff Diffusion E, e, elec Electrical (e.g., Pe,Welec) f Quantity of formation (e.g., ΔHf ) (HHV) Higher heating value (LHV) Lower heating value i Species i P Product P Parasitic R Reactant rxn Change in a reaction (e.g., ΔHrxn) SK Stack SYS System

Nafion is a registered trademark of E.I. du Pont de Nemours and Company. PureCell is a registered trademark of UTC Fuel Cells, Inc. Honda FCX is a registered trademark of Honda Motor Co., Ltd. Home Energy System is a registered trademark of Honda Motor Co., Ltd. Gaussian is a registered trademark of Gaussian, Inc.

PART I

FUEL CELL PRINCIPLES

CHAPTER 1

INTRODUCTION

You are about to embark on a journey into the world of fuel cells and electrochemistry. This chapter will act as a roadmap for your travels, setting the stage for the rest of the book. In broad terms, this chapter will acquaint you with fuel cells: what they are, how they work, and what significant advantages and disadvantages they present. From this starting point, the subsequent chapters will lead you onward in your journey as you acquire a fundamental understanding of fuel cell principles.

1.1 WHAT IS A FUEL CELL?

You can think of a fuel cell as a “factory” that takes fuel as input and produces electricity as output. (See Figure 1.1.) Like a factory, a fuel cell will continue to churn out product (electricity) as long as raw material (fuel) is supplied. This is the key difference between a fuel cell and a battery. While both rely on electrochemistry to work their magic, a fuel cell is not consumed when it produces electricity. It is really a factory, a shell, which transforms the chemical energy stored in a fuel into electrical energy.

Viewed this way, combustion engines are also “chemical factories.” Combustion engines also take the chemical energy stored in a fuel and transform it into useful mechanical or electrical energy. So what is the difference between a combustion engine and a fuel cell?

In a conventional combustion engine, fuel is burned, releasing heat. Consider the sim- plest example, the combustion of hydrogen:

H2 + 1 2 O2 ⇌ H2O (1.1)

4 INTRODUCTION

Electricity

Fuel cell H2O(1/g)O2(g)

H2(g)

Figure 1.1. General concept of a (H2–O2) fuel cell.

On the molecular scale, collisions between hydrogen molecules and oxygen molecules result in a reaction. The hydrogen molecules are oxidized, producing water and releasing heat. Specifically, at the atomic scale, in amatter of picoseconds, hydrogen–hydrogen bonds and oxygen–oxygen bonds are broken, while hydrogen–oxygen bonds are formed. These bonds are broken and formed by the transfer of electrons between themolecules. The energy of the product water bonding configuration is lower than the bonding configurations of the initial hydrogen and oxygen gases. This energy difference is released as heat. Although the energy difference between the initial and final states occurs by a reconfiguration of electrons as they move from one bonding state to another, this energy is recoverable only as heat because the bonding reconfiguration occurs in picoseconds at an intimate, subatomic scale. (See Figure 1.2.) To produce electricity, this heat energy must be converted into mechanical energy, and then the mechanical energy must be converted into electrical energy. Going through all these steps is potentially complex and inefficient.

Consider an alternative solution: to produce electricity directly from the chemical reac- tion by somehow harnessing the electrons as they move from high-energy reactant bonds

Reaction progress

P ot

en tia

l e ne

rg y

Products (H2O)

Reactants (H2/O2)

32 41

H2 H2

H2O

H2O O2

Figure 1.2. Schematic of H2–O2 combustion reaction. (Arrows indicate the relative motion of the molecules participating in the reaction.) Starting with the reactant H2–O2 gases (1), hydrogen–hydrogen and oxygen–oxygen bondsmust first be broken, requiring energy input (2) before hydrogen–oxygen bonds are formed, leading to energy output (3, 4).

WHAT IS A FUEL CELL? 5

to low-energy product bonds. In fact, this is exactly what a fuel cell does. But the question is, how do we harness electrons that reconfigure in picoseconds at subatomic length scales? The answer is to spatially separate the hydrogen and oxygen reactants so that the electron transfer necessary to complete the bonding reconfiguration occurs over a greatly extended length scale. Then, as the electrons move from the fuel species to the oxidant species, they can be harnessed as an electrical current.

BONDS AND ENERGY

Atoms are social creatures. They almost always prefer to be together instead of alone. When atoms come together, they form bonds, lowering their total energy. Figure 1.3 shows a typical energy–distance curve for a hydrogen–hydrogen bond. When the hydro- gen atoms are far apart from one another (1), no bond exists and the system has high energy. As the hydrogen atoms approach one another, the system energy is lowered until the most stable bonding configuration (2) is reached. Further overlap between the atoms is energetically unfavorable because the repulsive forces between the nuclei begin to dominate (3). Remember:

• Energy is released when a bond is formed. • Energy is absorbed when a bond is broken.

For a reaction to result in a net release of energy, the energy released by the formation of the product bonds must be more than the energy absorbed to break the reactant bonds.

Internuclear distance (pm)

P ot

en tia

l e ne

rg y

(K J/

m ol

)

20010074

–100

–200

–300

–400 –436

–500

Figure 1.3. Bonding energy versus internuclear separation for hydrogen–hydrogen bond: (1) no bond exists; (2) most stable bonding configuration; (3) further overlap unfavorable due to inter- nuclear repulsion.

6 INTRODUCTION

1.2 A SIMPLE FUEL CELL

In a fuel cell, the hydrogen combustion reaction is split into two electrochemical half reac- tions:

H2 ⇌ 2H + + 2e− (1.2)

1 2 O2 + 2H+ + 2e− ⇌ H2O (1.3)

By spatially separating these reactions, the electrons transferred from the fuel are forced to flow through an external circuit (thus constituting an electric current) and do useful work before they can complete the reaction.

Spatial separation is accomplished by employing an electrolyte. An electrolyte is a mate- rial that allows ions (charged atoms) to flow but not electrons. At a minimum, a fuel cell must possess two electrodes, where the two electrochemical half reactions occur, separated by an electrolyte.

Figure 1.4 shows an example of an extremely simple H2–O2 fuel cell. This fuel cell consists of two platinum electrodes dipped into sulfuric acid (an aqueous acid electrolyte). Hydrogen gas, bubbled across the left electrode, is split into protons (H+) and electrons following Equation 1.2. The protons can flow through the electrolyte (the sulfuric acid is like a “sea” of H+), but the electrons cannot. Instead, the electrons flow from left to right through a piece of wire that connects the two platinum electrodes. Note that the resulting current, as it is traditionally defined, is in the opposite direction. When the electrons reach the right electrode, they recombinewith protons and bubbling oxygen gas to producewater following Equation 1.3. If a load (e.g., a light bulb) is introduced along the path of the electrons, the flowing electrons will provide power to the load, causing the light bulb to glow. Our fuel cell

H2 O2

e–

H+

Figure 1.4. A simple fuel cell.

A SIMPLE FUEL CELL 7

is producing electricity! The first fuel cell, invented by William Grove in 1839, probably looked a lot like the one discussed here.

ENERGY, POWER, ENERGY DENSITY, AND POWER DENSITY

To understand how a fuel cell compares to a combustion engine or a battery, several quantitative metrics, or figures of merit, are required. The most common figures of merit used to compare energy conversion systems are power density and energy density.

To understand energy density and power density, you first need to understand the difference between energy and power:

Energy is defined as the ability to do work. Energy is usually measured in joules (J) or calories (cal).

Power is defined as the rate at which energy is expended or produced. In other words, power represents the intensity of energy use or production. Power is a rate. The typical unit of power, the watt (W), represents the amount of energy used or produced per second (1 W = 1 J∕s).

From the above discussion, it is obvious that energy is the product of power and time:

Energy = power × time (1.4)

Although the International System of Units (SI) uses the joule as the unit of energy, you will often see energy expressed in terms of watt-hours (Wh) or kilowatt-hours (kWh). These units arise when the units of power (e.g., watts) are multiplied by a length of time (e.g., hours) as in Equation 1.4. Obviously, watt-hours can be converted to joules or vice versa using simple arithmetic:

1Wh × 3600s∕h × 1 (J∕s)∕W = 3600J (1.5)

Refer to Appendix A for a list of some of the more common unit conversions for energy and power. For portable fuel cells and other mobile energy conversion devices, power density and energy density are more important than power and energy because they provide information about how big a system needs to be to deliver a certain amount of energy or power. Power density refers to the amount of power that can be produced by a device per unit mass or volume. Energy density refers to the total energy capacity available to the system per unit mass or volume.

Volumetric power density is the amount of power that can be supplied by a device per unit volume. Typical units are W∕cm3 or kW∕m3.

Gravimetric power density (or specific power) is the amount of power that can be sup- plied by a device per unit mass. Typical units are W/g or kW/kg.

8 INTRODUCTION

Volumetric energy density is the amount of energy that is available to a device per unit volume. Typical units are Wh∕cm3 or kWh∕m3.

Gravimetric energy density (or specific energy) is the amount of energy that is available to a device per unit mass. Typical units are Wh∕g or kWh∕kg.

1.3 FUEL CELL ADVANTAGES

Because fuel cells are “factories” that produce electricity as long as they are supplied with fuel, they share some characteristics in common with combustion engines. Because fuel

(a)

2 3

Fuel cell, battery

Combustion engine

Chemical energy

Mechanical energy

Heat energy

Electrical energy

Battery

Fuel tank

Fuel cell or combustion

engine

Work out

(b)

Figure 1.5. Schematic comparison of fuel cells, batteries, and combustion engines. (a) Fuel cells and batteries produce electricity directly from chemical energy. In contrast, combustion engines first convert chemical energy into heat, then mechanical energy, and finally electricity (alternatively, the mechanical energy can sometimes be used directly). (b) In batteries, power and capacity are typically intertwined—the battery is both the energy storage and the energy conversion device. In contrast, fuel cells and combustion engines allow independent scaling between power (determined by the fuel cell or engine size) and capacity (determined by the fuel tank size).

FUEL CELL ADVANTAGES 9

cells are electrochemical energy conversion devices that rely on electrochemistry to work their magic, they share some characteristics in common with primary batteries. In fact, fuel cells combine many of the advantages of both engines and batteries.

Since fuel cells produce electricity directly from chemical energy, they are often far more efficient than combustion engines. Fuel cells can be all solid state and mechanically ideal, meaning no moving parts. This yields the potential for highly reliable and long-lasting systems. A lack of moving parts also means that fuel cells are silent. Also, undesirable products such as NOx, SOx, and particulate emissions are virtually zero.

Unlike batteries, fuel cells allow easy independent scaling between power (determined by the fuel cell size) and capacity (determined by the fuel reservoir size). In batteries, power and capacity are often convoluted. Batteries scale poorly at large sizes, whereas fuel cells scale well from the 1-W range (cell phone) to the megawatt range (power plant). Fuel cells offer potentially higher energy densities than batteries and can be quickly recharged by refu- eling, whereas batteries must be thrown away or plugged in for a time-consuming recharge. Figure 1.5 schematically illustrates the similarities and differences between fuel cells, bat- teries, and combustion engines.

FUEL CELLS VERSUS SOLAR CELLS VERSUS BATTERIES

Fuel cells, solar cells, and batteries all produce electrical power by converting either chemical energy (fuel cells, batteries) or solar energy (solar cells) to a direct-current (DC) flow of electricity. The key features of these three devices are compared in Figure 1.6 using the analogy of buckets filled with water. In all three devices, the electrical output power is determined by the operating voltage (the height of water in the bucket) and current density (the amount of water flowing out the spigot at the bottom of the bucket).

Fuel cells and solar cells can be viewed as “open” thermodynamic systems that oper- ate at a thermodynamic steady state. In other words, the operating voltage of a fuel cell (or a solar cell) remains constant in time so long as it is continually supplied with fuel (or photons) from an external source. In Figure 1.6, this is shown by the fact that the water in the fuel cell and solar cell buckets is continually replenished from the top at the same rate that it flows out the spigot in the bottom, resulting in a constant water level (constant operating voltage).

In contrast, most batteries are closed thermodynamic systems that contain a finite and exhaustible internal supply of chemical energy (reactants). As these reactants deplete, the voltage of the battery generally decreases over time. In Figure 1.6, this is shown by the fact that the water in the battery bucket is not replenished, resulting in a decreasing water level (decreasing operating voltage) with time as the battery is discharged. It is important to point out that battery voltage does not decrease linearly during discharge. During discharge, batteries pass through voltage plateaus where the voltage remains more or less constant for a significant part of the discharge cycle. This phenomenon is captured by the strange shape of the battery “bucket.”

10 INTRODUCTION

Figure 1.6. Fuel cells versus solar cells versus batteries. This schematic diagram provides another way to look at the similarities and differences between three common energy conversion technologies that provide electricity as an output.

FUEL CELL DISADVANTAGES 11

In addition to the thermodynamic operating differences between fuel cells, solar cells, and batteries, Figure 1.6 also shows that fuel cells typically operate at much higher cur- rent densities than solar cells or batteries. This characteristic places great importance on using low-resistance materials in fuel cells to minimize ohmic (“IR”) losses. We will learn more about minimizing ohmic losses in Chapter 4 of this textbook!

1.4 FUEL CELL DISADVANTAGES

While fuel cells present intriguing advantages, they also possess some serious disadvan- tages. Cost represents a major barrier to fuel cell implementation. Because of prohibitive costs, fuel cell technology is currently only economically competitive in a few highly spe- cialized applications (e.g., onboard the Space Shuttle orbiter). Power density is another significant limitation. Power density expresses how much power a fuel cell can produce per unit volume (volumetric power density) or per unit mass (gravimetric power density). Although fuel cell power densities have improved dramatically over the past decades, fur- ther improvements are required if fuel cells are to compete in portable and automotive applications. Combustion engines and batteries generally outperform fuel cells on a volu- metric power density basis; on a gravimetric power density basis, the race is much closer. (See Figure 1.7.)

Fuel availability and storage pose further problems. Fuel cells work best on hydrogen gas, a fuel that is not widely available, has a low volumetric energy density, and is difficult

100

1000

10000

0.01 0.1 1 10

G ra

vi m

et ri

c p

o w

er d

en si

ty (

W /k

g )

Fuel cell (portable) IC engines

(automotive)

Lead-acid battery

IC engine (portable)

Li-ion battery

Fuel cell (automotive)

Volumetric Power Density (kW/L) IC = Internal Combustion

Figure 1.7. Power density comparison of selected technologies (approximate ranges).

12 INTRODUCTION

0 5 10 15 20 25 30 35

Volumetric Energy Density (MJ/L)

G ra

vi m

et ri

c en

er g

y d

en si

ty (

M J/

kg )

Hydrogen, 7500PSI (including system) Hydrogen, liquid

(including system)

Hydrogen, metal hydride (low)

Hydrogen, metal hydride (high)

Methanol

Ethanol

Gasoline

Hydrogen, 3500PSI (including system)

Figure 1.8. Energy density comparison of selected fuels (lower heating value).

to store. (See Figure 1.8.) Alternative fuels (e.g., gasoline, methanol, formic acid) are dif- ficult to use directly and usually require reforming. These problems can reduce fuel cell performance and increase the requirements for ancillary equipment. Thus, although gaso- line looks like an attractive fuel from an energy density standpoint, it is not well suited to fuel cell use.

Additional fuel cell limitations include operational temperature compatibility concerns, susceptibility to environmental poisons, and durability under start–stop cycling. These sig- nificant disadvantages will not be easy to overcome. Fuel cell adoption will be severely limited unless technological solutions can be developed to hurdle these barriers.

1.5 FUEL CELL TYPES

There are five major types of fuel cells, differentiated from one another by their electrolyte:

1. Phosphoric acid fuel cell (PAFC)

2. Polymer electrolyte membrane fuel cell (PEMFC)

3. Alkaline fuel cell (AFC)

4. Molten carbonate fuel cell (MCFC)

5. Solid-oxide fuel cell (SOFC)

FUEL CELL TYPES 13

TABLE 1.1. Description of Major Fuel Cell Types

PEMFC PAFC AFC MCFC SOFC

Polymer Liquid H3PO4 Liquid KOH Molten Electrolyte membrane (immobilized) (immobilized) carbonate Ceramic

Charge carrier H+ H+ OH− CO3 2− O2−

Operating temperature

80∘C 200∘C 60–220∘C 650∘C 600–1000∘C

Catalyst Platinum Platinum Platinum Nickel Perovskites (ceramic)

Cell components Carbon based Carbon based Carbon based Stainless based

Ceramic based

Fuel compatibility H2, methanol H2 H2 H2, CH4 H2, CH4, CO

While all five fuel cell types are based on the same underlying electrochemical princi- ples, they all operate at different temperature regimens, incorporate different materials, and often differ in their fuel tolerance and performance characteristics, as shown in Table 1.1. Most of the examples in this book focus on PEMFCs or SOFCs. We will briefly contrast these two fuel cell types.

• PEMFCs employ a thin polymermembrane as an electrolyte (the membrane looks and feels a lot like plastic wrap). The most common PEMFC electrolyte is a membrane material called NafionTM. Protons are the ionic charge carrier in a PEMFCmembrane. As we have already seen, the electrochemical half reactions in an H2–O2 PEMFC are

H2 → 2H + + 2e−

1 2 O2 + 2H+ + 2e− → H2O

(1.6)

PEMFCs are attractive for many applications because they operate at low temperature and have high power density.

• SOFCs employ a thin ceramic membrane as an electrolyte. Oxygen ions (O2–) are the ionic charge carrier in an SOFC membrane. The most common SOFC electrolyte is an oxide material called yttria-stabilized zirconia (YSZ). In an H2–O2 SOFC, the electrochemical half reactions are

H2 + O2− → H2O + 2e−

1 2 O2 + 2e− → O2−

(1.7)

To function properly, SOFCs must operate at high temperatures (>600∘C). They are attractive for stationary applications because they are highly efficient and fuel flexible.

14 INTRODUCTION

Note how changing the mobile charge carrier dramatically changes the fuel cell reaction chemistry. In a PEMFC, the half reactions are mediated by the movement of protons (H+), and water is produced at the cathode. In a SOFC, the half reactions are mediated by the motion of oxygen ions (O2–), and water is produced at the anode. Note in Table 1.1 how other fuel cell types use OH– or CO3

2– as ionic charge carriers. These fuel cell types will also exhibit different reaction chemistries, leading to unique advantages and disadvantages.

Part I of this book introduces the basic underlying principles that govern all fuel cell devices. What you learn here will be equally applicable to a PEMFC, a SOFC, or any other fuel cell for that matter. Part II discusses thematerials and technology-specific aspects of the five major fuel cell types, while also delving into fuel cell system issues such as stacking, fuel processing, control, and environmental impact.

1.6 BASIC FUEL CELL OPERATION

The current (electricity) produced by a fuel cell scales with the size of the reaction area where the reactants, the electrode, and the electrolyte meet. In other words, doubling a fuel cell’s area approximately doubles the amount of current produced.

Although this trend seems intuitive, the explanation comes from a deeper understanding of the fundamental principles involved in the electrochemical generation of electricity. As we have discussed, fuel cells produce electricity by converting a primary energy source (a fuel) into a flow of electrons. This conversion necessarily involves an energy transfer step, where the energy from the fuel source is passed along to the electrons constituting

Anode

Electrolyte

Cathode

Hydrogen

Oxygen

Figure 1.9. Simplified planar anode–electrolyte–cathode structure of a fuel cell.

BASIC FUEL CELL OPERATION 15

the electric current. This transfer has a finite rate and must occur at an interface or reaction surface. Thus, the amount of electricity produced scales with the amount of reaction surface area or interfacial area available for the energy transfer. Larger surface areas translate into larger currents.

To provide large reaction surfaces that maximize surface-to-volume ratios, fuel cells are usually made into thin, planar structures, as shown in Figure 1.9. The electrodes are highly porous to further increase the reaction surface area and ensure good gas access. One side of the planar structure is provisioned with fuel (the anode electrode), while the other side is provisioned with oxidant (the cathode electrode). A thin electrolyte layer spatially separates the fuel and oxidant electrodes and ensures that the two individual half reactions occur in isolation from one another. Compare this planar fuel cell structure with the simple fuel cell discussed earlier in Figure 1.4. While the two devices look quite different, noticeable similarities exist between them.

ANODE = OXIDATION; CATHODE = REDUCTION

To understand any discussion of electrochemistry, it is essential to have a clear concept of the terms oxidation, reduction, anode, and cathode.

Oxidation and Reduction

• Oxidation refers to a process in which electrons are removed from a species. Elec- trons are liberated by the reaction.

• Reduction refers to a process in which electrons are added to a species. Electrons are consumed by the reaction.

For example, consider the electrochemical half reactions that occur in an H2–O2 fuel cell:

H2 → 2H + + 2e− (1.8)

1 2 O2 + 2H+ + 2e− → H2O (1.9)

The hydrogen reaction is an oxidation reaction because electrons are being liber- ated by the reaction. The oxygen reaction is a reduction reaction because electrons are being consumed by the reaction. The preceding electrochemical half reactions are therefore known as the hydrogen oxidation reaction (HOR) and the oxygen reduction reaction (ORR).

Anode and Cathode

• Anode refers to an electrode where oxidation is taking place. More generally, the anode of any two-port device, such as a diode or resistor, is the electrode where electrons flow out.

• Cathode refers to an electrode where reduction is taking place. More generally, the cathode is the electrode where electrons flow in.

16 INTRODUCTION

For a hydrogen–oxygen fuel cell:

• The anode is the electrode where the HOR takes place. • The cathode is the electrode where the ORR takes place.

Note that the above definitions have nothing to do with which electrode is the positive electrode or which electrode is the negative electrode. Be careful! Anodes and cathodes can be either positive or negative. For a galvanic cell (a cell that produces electricity, like a fuel cell), the anode is the negative electrode and the cathode is the positive elec- trode. For an electrolytic cell (a cell that consumes electricity), the anode is the positive electrode and the cathode is the negative electrode.

Just remember anode= oxidation, cathode = reduction, and you will always be right!

Figure 1.10 shows a detailed, cross-sectional view of a planar fuel cell. Using this figure as a map, we will now embark on a brief journey through the major steps involved in pro- ducing electricity in a fuel cell. Sequentially, as numbered on the drawing, these steps are as follows:

1. Reactant delivery (transport) into the fuel cell

2. Electrochemical reaction

3. Ionic conduction through the electrolyte and electronic conduction through the exter- nal circuit

4. Product removal from the fuel cell

2 23

Figure 1.10. Cross section of fuel cell illustrating major steps in electrochemical generation of electricity: (1) reactant transport, (2) electrochemical reaction, (3) ionic and electronic conduction, (4) product removal.

BASIC FUEL CELL OPERATION 17

By the end of this book, you will understand the physics behind each of these steps in detail. For now, however, we’ll just take a quick tour.

Step 1: Reactant Transport. For a fuel cell to produce electricity, it must be continually supplied with fuel and oxidant. This seemingly simple task can be quite complicated. When a fuel cell is operated at high current, its demand for reactants is voracious. If the reactants are not supplied to the fuel cell quickly enough, the device will “starve.” Efficient delivery of reactants is most effectively accomplished by using flow field plates in combinationwith porous electrode structures. Flowfield plates containmany fine channels or grooves to carry the gas flow and distribute it over the surface of the fuel cell. The shape, size, and pattern of flow channels can significantly affect the performance of the fuel cell. Understanding how flow structures and porous elec- trode geometries influence fuel cell performance is an exercise in mass transport, diffusion, and fluid mechanics. The materials aspects of flow structures and elec- trodes are equally important. Components are held to stringent materials property constraints that include very specific electrical, thermal, mechanical, and corrosion requirements. The details of reactant transport and flow field design are covered in Chapter 5.

Step 2: Electrochemical Reaction. Once the reactants are delivered to the electrodes, they must undergo electrochemical reaction. The current generated by the fuel cell is directly related to how fast the electrochemical reactions proceed. Fast electrochem- ical reactions result in a high current output from the fuel cell. Sluggish reactions result in low current output. Obviously, high current output is desirable. Therefore, catalysts are generally used to increase the speed and efficiency of the electrochemi- cal reactions. Fuel cell performance critically depends on choosing the right catalyst and carefully designing the reaction zones. Often, the kinetics of the electrochemical reactions represent the single greatest limitation to fuel cell performance. The details of electrochemical reaction kinetics are covered in Chapter 3.

Step 3: Ionic (and Electronic) Conduction. The electrochemical reactions occurring in step 2 either produce or consume ions and electrons. Ions produced at one electrode must be consumed at the other electrode. The same holds for electrons. To main- tain charge balance, these ions and electrons must therefore be transported from the locations where they are generated to the locations where they are consumed. For electrons this transport process is rather easy. As long as an electrically conduc- tive path exists, the electrons will be able to flow from one electrode to the other. In the simple fuel cell in Figure 1.4, for example, a wire provides a path for electrons between the two electrodes. For ions, however, transport tends to be more difficult. Fundamentally, this is because ions are much larger and more massive than elec- trons. An electrolyte must be used to provide a pathway for the ions to flow. In many electrolytes, ions move via “hopping” mechanisms. Compared to electron transport, this process is far less efficient. Therefore, ionic transport can represent a significant resistance loss, reducing fuel cell performance. To combat this effect, the electrolytes in technological fuel cells are made as thin as possible to minimize the distance

18 INTRODUCTION

over which ionic conduction must occur. The details of ionic conduction are covered in Chapter 4.

Step 4: Product Removal. In addition to electricity, all fuel cell reactions will generate at least one product species. The H2–O2 fuel cell generates water. Hydrocarbon fuel cells will typically generate water and carbon dioxide (CO2). If these products are not removed from the fuel cell, they will build up over time and eventually “strangle” the fuel cell, preventing new fuel and oxidant from being able to react. Fortunately, the act of delivering reactants into the fuel cell often assists the removal of product species out of the fuel cell. The same mass transport, diffusion, and fluid mechanics issues that are important in optimizing reactant delivery (step 1) can be applied to product removal. Often, product removal is not a significant problem and is frequently over- looked. However, for certain fuel cells (e.g., PEMFC) “flooding” byproduct water can be a major issue. Because product removal depends on the same physical principles and processes that govern reactant transport, it is also treated in Chapter 5.

1.7 FUEL CELL PERFORMANCE

The performance of a fuel cell device can be summarized with a graph of its current–voltage characteristics. This graph, called a current–voltage (i–V) curve, shows the voltage output of the fuel cell for a given current output. An example of a typical i–V curve for a PEMFC is shown in Figure 1.11. Note that the current has been normalized by the area of the fuel cell, giving a current density (in amperes per square centimeter). Because a larger fuel cell

Ideal (thermodynamic) fuel cell voltage (Chapter 2)

Activation region

Ohmic region

Mass transport

region (Chapter 3) (Chapter 4) (Chapter 5)

F ue

l c el

l v ol

ta ge

( V

)

Current density (A/cm2)

Figure 1.11. Schematic of fuel cell i–V curve. In contrast to the ideal, thermodynamically predicted voltage of a fuel cell (dashed line), the real voltage of a fuel cell is lower (solid line) due to unavoid- able losses. Three major losses influence the shape of this i–V curve; they will be described in Chapters 3–5.

FUEL CELL PERFORMANCE 19

can produce more electricity than a smaller fuel cell, i–V curves are normalized by fuel cell area to make results comparable.

An ideal fuel cell would supply any amount of current (as long as it is supplied with sufficient fuel), while maintaining a constant voltage determined by thermodynamics. In practice, however, the actual voltage output of a real fuel cell is less than the ideal thermo- dynamically predicted voltage. Furthermore, the more current that is drawn from a real fuel cell, the lower the voltage output of the cell, limiting the total power that can be delivered. The power (P) delivered by a fuel cell is given by the product of current and voltage:

P = iV (1.10)

A fuel cell power density curve, which gives the power density delivered by a fuel cell as a function of the current density, can be constructed from the information in a fuel cell i–V curve. The power density curve is produced by multiplying the voltage at each point on the i–V curve by the corresponding current density. An example of combined fuel cell i–V and power density curves is provided in Figure 1.12. Fuel cell voltage is given on the left-hand y-axis, while power density is given on the right-hand y-axis.

F ue

l c el

l v ol

ta ge

( V

)

Current density (A/cm2)

F ue

l c el

l p ow

er d

en si

ty (

W /c

m 2 )

0 0 0.2 1.00.4 0.6 0.8 1.2 1.4

0.2

0.4

0.6

0.8

1.0

1.2

0.6

0.4

0.2

0.1

0.3

0.5

0.7

i-V curve

Power density curve

Figure 1.12. Combined fuel cell i–V and power density curves. The power density curve is constructed from the i–V curve by multiplying the voltage at each point on the i–V curve by the cor- responding current density. Fuel cell power density increases with increasing current density, reaches a maximum, and then falls at still higher current densities. Fuel cells are designed to operate at or below the power density maximum. At current densities below the power density maximum, voltage efficiency improves but power density falls. At current densities above the power density maximum, both voltage efficiency and power density fall.

20 INTRODUCTION

The current supplied by a fuel cell is directly proportional to the amount of fuel con- sumed (each mole of fuel provides n moles of electrons). Therefore, as fuel cell voltage decreases, the electric power produced per unit of fuel also decreases. In this way, fuel cell voltage can be seen as a measure of fuel cell efficiency. In other words, you can think of the fuel cell voltage axis as an “efficiency axis.” Maintaining high fuel cell voltage, even under high current loads, is therefore critical to the successful implementation of the technology.

Unfortunately, it is hard to maintain a high fuel cell voltage under the current load. The voltage output of a real fuel cell is less than the thermodynamically predicted voltage output due to irreversible losses. The more current that is drawn from the cell, the greater these losses. There are three major types of fuel cell losses, which give a fuel cell i–V curve its characteristic shape. Each of these losses is associated with one of the basic fuel cell steps discussed in the previous section:

1. Activation losses (losses due to electrochemical reaction)

2. Ohmic losses (losses due to ionic and electronic conduction)

3. Concentration losses (losses due to mass transport)

The real voltage output for a fuel cell can thus be written by starting with the thermody- namically predicted voltage output of the fuel cell and then subtracting the voltage drops due to the various losses:

V = Ethermo − 𝜂act − 𝜂ohmic − 𝜂conc (1.11) where

V = real output voltage of fuel cell Ethermo = thermodynamically predicted fuel cell voltage output; this will be the subject

of Chapter 2 𝜂act = activation losses due to reaction kinetics; this will be the subject of Chapter 3

𝜂ohmic = ohmic losses from ionic and electronic conduction; this will be the subject of Chapter 4

𝜂conc = concentration losses due to mass transport; this will be the subject of Chapter 5

The three major losses each contribute to the characteristic shape of the fuel cell i–V curve. As shown in Figure 1.11, the activation losses mostly affect the initial part of the curve, the ohmic losses are most apparent in the middle section of the curve, and the con- centration losses are most significant in the tail of the i–V curve.

Equation 1.11 sets the stage for the next six chapters of this book. As you progress through these chapters, you will be armed with the tools needed to understand the major losses in fuel cell devices. Using Equation 1.11 as a starting point, you will eventually be able to characterize and model the performance of real fuel cell devices.

1.8 CHARACTERIZATION AND MODELING

Characterization and modeling are pivotal to the development and advancement of fuel cell technology. By assimilating theory and experiment, careful characterization and modeling

FUEL CELLS AND THE ENVIRONMENT 21

studies allow us to better understand how fuel cells work, often paving the way toward further improvements.

Because these subjects provide great insight, each has been given a chapter in this book. Fuel cell modeling is covered in Chapter 6. Fuel cell characterization techniques are covered in Chapter 7. These chapters will yield a practical understanding of how fuel cells are tested, how to diagnose their performance, and how to develop simple mathematical models to predict fuel cell behavior.

1.9 FUEL CELL TECHNOLOGY

The first half of this book is devoted to understanding the fundamental principles underly- ing fuel cells. However, no treatment of fuel cells is complete without a discussion of the practical aspects of fuel cell technology. This is the aim of the second part of this book. A series of chapters will introduce the major considerations for fuel cell stacking and system design, as well as specific technological aspects related to each of the five major fuel cell types. You will gain insight into the state of the art in fuel cell materials and fuel cell design as well as a historical perspective on the development of practical fuel cell technology.

1.10 FUEL CELLS AND THE ENVIRONMENT

If employed correctly, fuel cells are environmentally friendly. In fact, this may be their single greatest advantage over other energy conversion technologies. However, the envi- ronmental impact of fuel cells depends strongly on the context of their use. If they are not deployed wisely, fuel cells may be no better than our current fossil energy conversion sys- tem! In the final chapter of this book, youwill learn to evaluate possible fuel cell deployment scenarios. Using a technique known as process chain analysis, you will be able to identify promising fuel cell futures.

One such future, referred to as the “hydrogen economy,” is illustrated in Figure 1.13. In this figure, H2 fuel cells are coupled with electrolyzers and renewable energy conver- sion technologies (such as wind and solar power) to provide a completely closed-loop, pollution-free energy economy. In such a system, fuel cells would play a prominent role, with a primary benefit being their dispatchability. When the sun is shining or the wind is blowing, the electricity produced from solar and wind energy can be used to power cities directly, while producing extra hydrogen on the side via electrolysis. Anytime the wind stops or night falls, however, the fuel cells can be dispatched to provide on-demand power by converting the stored hydrogen into electricity. In such a system, fossil fuels are com- pletely eliminated.

Currently, it is unclear when, if ever, the hydrogen economy will become a reality. Var- ious studies have examined the technical and economic hurdles that stand in the way of the hydrogen economy. While many of these studies differ on the details, it is clear that the transition to a hydrogen economy would be difficult, costly, and lengthy. Do not count on it happening anytime soon. In the meantime, we have a fossil fuel world. Even in a fossil fuel world, however, it is important to realize that fuel cells can provide increased

22 INTRODUCTION

Sun

Wind power

Solar power H2

storage

Water

Electrolyzer Fuel cell

O2 O2

Figure 1.13. Schematic of hydrogen economy dream.

efficiency, greater scaling flexibility, reduced emissions, and other advantages compared to conventional power technologies. Fuel cells have found, and will continue to find, niche applications. These applications should continue to drive forward progress for decades to come, with or without the hydrogen economy dream.

1.11 CHAPTER SUMMARY

The purpose of this chapter was to set the stage for learning about fuel cells and to give a broad overview of fuel cell technology.

• A fuel cell is a direct electrochemical energy conversion device. It directly converts energy from one form (chemical energy) into another form (electrical energy) through electrochemistry.

• Unlike a battery, a fuel cell cannot be depleted. It is a “factory” that will continue to generate electricity as long as fuel is supplied.

• At a minimum, a fuel cell must contain two electrodes (an anode and a cathode) sep- arated by an electrolyte.

• Fuel cell power is determined by fuel cell size. Fuel cell capacity (energy capacity) is determined by the fuel reservoir size.

• There are five major fuel cell types, differentiated by their electrolyte. • Electrochemical systems must contain two coupled half reactions: an oxidation reac-

tion and a reduction reaction. An oxidation reaction liberates electrons. A reduction reaction consumes electrons.

• Oxidation occurs at the anode electrode. Reduction occurs at the cathode electrode. • The four major steps in the generation of electricity in a fuel cell are (1) reactant

transport, (2) electrochemical reaction, (3) ionic (and electronic) conduction, and (4) product removal.

CHAPTER EXERCISES 23

• Fuel cell performance can be assessed by current–voltage curves. Current–voltage curves show the voltage output of a fuel cell for a given current load.

• Ideal fuel cell performance is dictated by thermodynamics. • Real fuel cell performance is always less than ideal fuel cell performance due to losses.

The major types of loss are (1) activation loss, (2) ohmic loss, and (3) concentration loss.

CHAPTER EXERCISES

Review Questions

1.1 List three major advantages and three major disadvantages of fuel cells compared to other power conversion devices. Discuss at least two potential applications where the unique attributes of fuel cells make them attractive.

1.2 In general, do you think a portable fuel cell would be better for an application requiring low power but high capacity (long run time) or high power but small capacity (short run time)? Explain.

1.3 Label the following reactions as oxidation or reduction reactions: (a) Cu → Cu2+ + 2e−

(b) 2H+ + 2e− → H2 (c) O2− → 1

2 O2 + 2e−

(d) CH4 + 4O2− → CO2 + 2H2O + 8e−

(e) O2− + CO → CO2 + 2e−

(f) 1 2 O2 + H2O + 2e− → 2(OH)−

(g) H2 + 2(OH)− → 2H2O + 2e−

1.4 From the reactions listed in problem 1.3 (or their reverse), write three complete and balanced pairs of electrochemical half reactions. For each pair of reactions, identify which reaction is the cathode reaction and which reaction is the anode reaction.

1.5 Consider the relative volumetric and gravimetric energy densities of 7500 psi com- pressed H2 versus liquid H2. Which would probably be the better candidate for a fuel cell bus? Hint: Bus efficiency strongly depends on gross vehicle weight.

1.6 Describe the four major steps in the generation of electricity within a fuel cell. Describe the potential reasons for loss in fuel cell performance for each step.

Calculations

1.7 Energy is released when hydrogen and oxygen react to produce water. This energy comes from the fact that the final hydrogen–oxygen bonds represent a lower total energy state than the original hydrogen–hydrogen and oxygen–oxygen bonds. Calcu- late how much energy (in kilojoules per mole of product) is released by the reaction

H2 + 1 2 O2 ⇌ H2O (1.12)

24 INTRODUCTION

at constant pressure and given the following standard bond enthalpies. Standard bond enthalpies denote the enthalpy absorbed when bonds are broken at standard temper- ature and pressure (298 K and 1 atm).

Standard Bond Enthalpies

H–H = 432 kJ∕mol O = O = 494 kJ∕mol H–O = 460 kJ∕mol

1.8 Consider a fuel cell vehicle. The vehicle draws 30 kW of power at 60 mph and is 40% efficient at rated power. (It converts 40% of the energy stored in the hydrogen fuel to electric power.) You are asked to size the fuel cell system so that a driver can go at least 300 miles at 60 mph before refueling. Specify the minimum volume and mass requirements for the fuel cell system (fuel cell + fuel tank) given the following information:

• Fuel cell power density: 1 kW∕L, 500 W∕kg • Fuel tank energy density (compressed hydrogen): 4 MJ∕L, 8 MJ∕kg

1.9 For the fuel cell i–V curve shown in Figure 1.11, sketch the approximate correspond- ing current density–power density curve.

1.10 A cylindrical metal hydride container measures 9 cm in diameter, is 42.5 cm in length, and has a mass of 7 kg. Themetal hydride container has a capacity of 900 normal liters of hydrogen. Using the lower heating value of hydrogen (244 kJ∕mol), determine the energy density.

(a) 3.6 kWh/L

(b) 3.6 MWh/ L

(d) 1.0 kWh/ L

CHAPTER 2

FUEL CELL THERMODYNAMICS

Thermodynamics is the study of energetics; the study of the transformation of energy from one form to another. Since fuel cells are energy conversion devices, fuel cell thermodynam- ics is key to understanding the conversion of chemical energy into electrical energy. For fuel cells, thermodynamics can predict whether a candidate fuel cell reaction is energetically spontaneous. Furthermore, thermodynamics places upper bound limits on the maximum electrical potential that can be generated in a reaction. Thus, thermodynamics yields the theoretical boundaries of what is possible with a fuel cell; it gives the “ideal case.”

Any real fuel cell will perform at or below its thermodynamic limit. Understanding real fuel cell performance requires a knowledge of kinetics in addition to thermodynamics. This chapter covers the thermodynamics of fuel cells. Subsequent chapters will cover the major kinetic limitations on fuel cell performance, defining practical performance.

2.1 THERMODYNAMICS REVIEW

This section presents a brief review of the main tenets of thermodynamics. These basic the- ories are typically taught in an introductory thermodynamics course. Next, these concepts are extended to include parameters that are needed to understand fuel cell behavior. Readers are advised to consult a thermodynamics book if additional review is required.

2.1.1 What Is Thermodynamics?

It is no secret that no one really understands the meaning of popular thermodynamic quan- tities. For example, Nobel Prize–winning physicist Richard Feynman wrote in his Lectures

26 FUEL CELL THERMODYNAMICS

on Physics: “It is important to realize that in modern physics today, we have no knowl- edge of what energy is” [1]. We have even less intuition about terms such as enthalpy and free energy. The fundamental assumptions of thermodynamics are based on human experi- ence. Assumptions are the best we can do. We assume that energy can never be created or destroyed (first law of thermodynamics) only because it fits with everything experienced in human existence. Nevertheless, no one knows why it should be so.

If we accept a few of these fundamental assumptions, however, we can develop a self-consistent mathematical description that tells us how important quantities such as energy, temperature, pressure, and volume are related. This is really all that thermody- namics is; it is an elaborate bookkeeping scheme that allows us to track the properties of systems in a self-consistent manner, starting from a few basic assumptions or “laws.”

2.1.2 Internal Energy

A fuel cell converts energy stored within a fuel into other, more useful forms of energy. The total intrinsic energy of a fuel (or of any substance) is quantified by a property known as internal energy (U). Internal energy is the energy associated with microscopic movement and interaction between particles on the atomic and molecular scales. It is separated in scale from the macroscopic ordered energy associated with moving objects. For example, a tank of H2 gas sitting on a table has no apparent energy. However, the H2 gas actually has significant internal energy (see Figure 2.1); on the microscopic scale it is a whirlwind of molecules traveling hundreds of meters per second. Internal energy is also associated with the chemical bonds between the hydrogen atoms. A fuel cell can convert only a portion of the internal energy associated with a tank of H2 gas into electrical energy. The limits on

Macroscopic view

Microscopic viewH2 tank

Figure 2.1. Although this tank of H2 gas has no apparent macroscopic energy, it has significant inter- nal energy. Internal energy is associated with microscopic movement (kinetic energy) and interactions between particles (chemical/potential energy) on the atomic scale.

THERMODYNAMICS REVIEW 27

how much of the internal energy of the H2 gas can be transformed into electrical energy are established by the first and second laws of thermodynamics.

2.1.3 First Law

The first law of thermodynamics is also known as the law of conservation of energy—energy can never be created or destroyed—as expressed by the equation

d(Energy)univ = d(Energy)system + d(Energy)surroundings = 0 (2.1)

Viewed another way, this equation states that any change in the energy of a system must be fully accounted for by energy transfer to the surroundings:

d(Energy)system = −d(Energy)surroundings (2.2)

There are two ways that energy can be transferred between a closed system and its sur- roundings: via heat (Q) orwork (W). This allows us to write the first law in its more familiar form:

dU = dQ − dW (2.3)

This expression states that the change in the internal energy of a closed system (dU)must be equal to the heat transferred to the system (dQ)minus the work done by the system (dW). To develop this expression from Equation 2.2, we have substituted dU for d(Energy)system; if we choose the proper reference frame, then all energy changes in a system are manifested as internal energy changes. Note that we define positive work as work done by the system on the surroundings.

For now, we will assume that only mechanical work is done by a system. Mechanical work is accomplished by the expansion of a system against a pressure. It is given by

(dW)mech = pdV (2.4)

where p is the pressure and dV is the volume change. Later, when we talk about fuel cell thermodynamics, we will consider the electrical work done by a system. For now, how- ever, we ignore electrical work. Considering only mechanical work, we can rewrite the expression for the internal energy change of a system as

dU = dQ − pdV (2.5)

2.1.4 Second Law

The second law of thermodynamics introduces the concept of entropy. Entropy is deter- mined by the number of possible microstates accessible to a system, or, in other words, the number of possible ways of configuring a system. For this reason, entropy can be thought

28 FUEL CELL THERMODYNAMICS

of as a measure of “disorder,” since an increasing entropy indicates an increasing number of ways of configuring a system. For an isolated system (the simplest case)

S = k logΩ (2.6)

where S is the total entropy of the system, k is Boltzmann’s constant, and Ω denotes the number of possible microstates accessible to the system.

WORK AND HEAT

In contrast to internal energy, work and heat are not properties of matter or of any partic- ular system (e.g., substance or body). They represent energy in transit, in other words, energy that is transferred between substances or bodies.

In the case of work, this transfer of energy is accomplished by the application of a force over a distance. Heat, on the other hand, is transferred between substances whenever they have different thermal energies, as manifested by differences in their temperature.

Due to repercussions of the second law (which we will discuss momentarily), work is often called themost “noble” form of energy; it is the universal donor. Energy, in the form of work, can be converted into any other form of energy at 100% theoretical efficiency. In contrast, heat is the most “ignoble” form of energy; it is the universal acceptor. Any form of energy can eventually be 100% dissipated to the environment as heat, but heat can never be 100% converted back to more noble forms of energy such as work.

The nobility of work versus heat illustrates one of the central differences between fuel cells and combustion engines. A combustion engine burns fuel to produce heat and then converts some of this heat into work. Because it first converts energy into heat, the combustion engine destroys some of the work potential of the fuel. This unfortu- nate destruction of work potential is called the “thermal bottleneck.” Because a fuel cell bypasses the heat step, it avoids the thermal bottleneck.

Microstates can best be understood with an example. Consider the “perfect” system of 100 identical atoms shown in Figure 2.2a. There is only one possible microstate, or configuration, for this system. This is because the 100 atoms are exactly identical and indistinguishable from one another. If we were to “switch” the first and the second atoms, the system would look exactly the same. The entropy of this perfect 100-atom crystal is therefore zero (S = k log 1 = 0). Now consider Figure 2.2b, where three atoms have been removed from their original locations and placed on the surface of the crystal. Any three atoms could have been removed from the crystal, and depending on which atoms were removed, the final configuration of the system would be different. In this case, there are many microstates available to the system. (Figure 2.2b represents just one of them.) We can calculate the number of microstates available to the system by evaluating the number of possible ways there are to take N atoms from a total of Z atoms:

Ω ≡ Z(Z − 1)(Z − 2) · · · (Z − N + 1) N!

= Z! (Z − N)!(N!)

(2.7)

THERMODYNAMICS REVIEW 29

(a) (b)

Figure 2.2. (a) The entropy of this 100-atom perfect crystal is zero because there is only one possible way to arrange the atoms to produce this configuration. (b) When three atoms are removed from the crystal and placed on the surface, the entropy increases. This is because there are many possible ways to configure a system of 100 atoms where 3 have been removed.

In Figure 2.2b, there are 100 atoms. The number of ways to take 3 atoms from 100 is

Ω = 100! 97!3!

== 1.62 × 105 (2.8)

This yields S = 7.19 × 10−23 J∕K. Except for extremely simple systems like the one in this example, it is impossible to

calculate entropy exactly. Instead, a system’s entropy is usually inferred based on how heat transfer causes the entropy of the system to change. For a reversible transfer of heat at constant pressure, the entropy of a system will change as

dS = dQrev T

(2.9)

where dS is the entropy change in the system associated with a reversible transfer of heat (dQrev) at a constant temperature (T). In other words, “dumping” energy, including heat, into a system causes its entropy to increase. Essentially, by providing additional energy to the system, we enable it to access additional microstates, causing its entropy to increase. For an irreversible transfer of heat, the entropy increase will be even larger than that dictated by Equation 2.9. This is a key statement of the second law of thermodynamics.

The most widely known form of the second law acknowledges that the entropy of a system and its surroundings must increase or at least remain zero for any process:

dSuniv ≥ 0 (2.10) This inequality, when combined with the first law of thermodynamics, allows us to sep-

arate thermodynamically “spontaneous” processes from “nonspontaneous” processes.

2.1.5 Thermodynamic Potentials

Based on the first and second laws of thermodynamics, we can write down “rules” to specify how energy can be transferred from one form to another. These rules are called thermody- namic potentials. You are already familiar with one thermodynamic potential: the internal

30 FUEL CELL THERMODYNAMICS

energy of a system. We can combine results from the first and the second laws of thermo- dynamics (Equations 2.3 and 2.9) to arrive at an equation for internal energy that is based on the variation of two independent variables, entropy S and volume V:

dU = T dS − pdV (2.11)

Remember, T dS represents the reversible heat transfer and p dV is the mechanical work. As mentioned above, from this equation we can conclude that U, the internal energy of a system, is a function of entropy and volume:

U = U(S,V) (2.12)

We can also derive the following useful relations, which show how the dependent vari- ables T and p are related to variations in the independent variables (S and V):(dU

) V = T (2.13)

(dU dV

) S = −p (2.14)

Unfortunately, S and V are not easily measurable in most experiments. (There is no such thing as an “entropy meter.”) Therefore, a new thermodynamic potential is needed equivalent to U but depending on quantities that are more readily measured than S and V. Temperature T and pressure p fall into this category. Happily, there is a simple math- ematical way to accomplish this conversion using a Legendre transform. A step-by-step transformation ofU begins by defining a new thermodynamic potentialG(T , p) as follows:

G = U − (dU dS

) V S −

(dU dV

) S V (2.15)

Since we know that (dU∕dS)V = T and (dU∕dV)S = −p, we obtain

G = U − TS + pV (2.16)

This function is called the Gibbs free energy. Let us show that G is indeed a function of the temperature and the pressure. The variation of G (mathematically dG) results in

dG = dU − T dS − SdT + pdV + V dp (2.17)

Since we know that dU = T dS – p dV , we can see that

dG = −SdT + V dp (2.18)

So, the Gibbs free energy is nothing more than a thermodynamic description of a system that depends on T and p instead of S and V .

THERMODYNAMICS REVIEW 31

What if we want a potential that depends on S and p? No problem! Remember that U is a function of S and V . To get a thermodynamic potential that is a function of S and p, we need only to transform U with respect to V this time. Analogously to Equation 2.15, we define this new thermodynamic potential H as

H = U − (dU dV

) S V (2.19)

Again, since (dU∕dV)S = – p, we obtain

H = U + pV (2.20)

where H is called enthalpy. Through differentiation, we can show that H is a function of S and p:

dH = dU + pdV + V dp (2.21)

Again, dU = T dS – p dV; so dH = T dS + V dp (2.22)

Thus far, we have defined three thermodynamic potentials: U(S, V), H(S, p), and G(T , p). Defining a fourth and final thermodynamic potential that depends on temperature and volume, F(T , V), completes the symmetry:

F = U − TS (2.23)

where F is the Helmholtz free energy. We leave it to the reader to show that

dF = −SdT − pdV (2.24)

A summary of these four thermodynamic potentials is provided in Figure 2.3. This mnemonic diagram, originally suggested by Schroeder [2], can help you keep track of the relationships between the thermodynamic potentials. Loosely, the four potentials are defined as follows:

• Internal Energy (U). The energy needed to create a system in the absence of changes in temperature or volume.

• Enthalpy (H). The energy needed to create a system plus the work needed to make room for it (from zero volume).

• Helmholtz Free Energy (F). The energy needed to create a system minus the energy that you can get from the system’s environment due to spontaneous heat transfer (at constant temperature).

• Gibbs Free Energy (G). The energy needed to create a system and make room for it minus the energy that you can get from the environment due to heat transfer. In other words, G represents the net energy cost for a system created at a constant envi- ronmental temperature T from a negligible initial volume after subtracting what the environment automatically supplied.

32 FUEL CELL THERMODYNAMICS

U Internalenergy

H Enthalpy

F Helmholtzfree energy

G Gibbs free energy

U = energy needed to create a system

H = energy needed to create a system plus the work needed to make room for it

H = U + pV G = U + pV –TS

F = U –TS

G = total energy to create a system and make room for it minus the energy provided by the environment

F = energy needed to create a system minus the energy provided by the environment

–TS

+pV

Figure 2.3. Pictorial summary of the four thermodynamic potentials. They relate to one another by offsets of the “energy from the environment” term TS and the “expansion work” term pV. Use this dia- gram to help remember the relationships. Copyright © 2000 by AddisonWesley Longman. Reprinted by permission of Pearson Education, Inc. (Figure 5.2, p. 151, fromAn Introduction to Thermal Physics by Daniele V. Schroeder [2]).

2.1.6 Molar Quantities

Typical notation distinguishes between intrinsic and extrinsic variables. Intrinsic quantities such as temperature and pressure do not scale with the system size; extrinsic quantities such as internal energy and entropy do scale with system size. For example, if the size of a box of gas molecules is doubled and the number of molecules in the box doubles, then the internal energy and entropy double, while the temperature and pressure remain constant. It is conventional to denote intrinsic quantities with a lowercase letter (p) and extrinsic quantities with an uppercase letter (U).

Molar quantities such as û, the internal energy per mole of gas (units of kilojoules per mole), are intrinsic. It is often useful to calculate energy changes due to a reaction on a per-mole basis:

Δĝrxn,Δŝrxn,Δ�̂�rxn

The Δ symbol denotes a change during a thermodynamic process (such as a reaction), calculated as final state–initial state. Therefore, a negative energy change means energy is released during a process: A negative volume change means the volume decreases during

THERMODYNAMICS REVIEW 33

a process. For example, the overall reaction in a H2–O2 fuel cell,

H2 + 1 2 O2 → H2O (2.25)

has Δĝrxn = −237 kJ∕mol H2 at room temperature and pressure. For every mole of H2 gas consumed (or every 1/2 mol of O2 gas consumed or mole of H2O produced), the Gibbs free-energy change is –237 kJ. If 5 mol of O2 gas is reacted, the extrinsic Gibbs free-energy change (ΔGrxn) would be

5 mol O2 × (

1 mol H2 (1∕2) mol O2

) × ( −237 kJ mol H2

) = −2370 kJ (2.26)

Of course the intrinsic (per-mole) Gibbs free energy of this reaction is still Δĝrxn = −237 kJ∕mol H2.

2.1.7 Standard State

Because most thermodynamic quantities depend on temperature and pressure, it is conve- nient to reference everything to a standard set of conditions. This set of conditions is called the standard state. There are two common types of standard conditions:

The thermodynamic standard state describes the standard set of conditions under which reference values of thermodynamic quantities are typically given. Standard-state conditions specify that all reactant and product species are present in their pure, most stable forms at unit activity. (Activity is discussed in Section 2.4.3.) Standard-state conditions are designated by a superscript zero. For example, Δĥ0 represents an enthalpy change under standard-state thermodynamic conditions. Importantly, there is no “standard temperature” in the definition of thermodynamic standard-state conditions. However, since most tables list standard-state thermodynamic quantities at 25∘C (298.15 K), this temperature is usually implied. At temperatures other than 25∘C, it is sometimes necessary to apply temperature corrections to Δĥ0 and Δŝ0 values obtained at 25∘C, although it is frequently approximated that these values change only slightly with temperature, and hence this issue can be ignored. For temperatures far from 25∘C, however, this approximation should not be made. You will have the opportunity to explore this issue in Example 2.1 and problem 2.9.

It should be noted that Δĝ0 changes much more strongly with temperature (as shown in Equation 2.39) and thereforeΔĝ0 values should always be adjusted by tem- perature using at least the linear dependence predicted by Equation 2.39. The use of this linear temperature dependence is shown in Example 2.2.

Standard temperature and pressure, or STP, is the standard condition most typically associated with gas law calculations. STP conditions are taken as room temperature (298.15 K) and atmospheric pressure. (Standard-state pressure is actually defined as 1 bar= 100 kPa. Atmospheric pressure is taken as 1 atm = 101.325 kPa. These slight differences are usually ignored.)

34 FUEL CELL THERMODYNAMICS

2.1.8 Reversibility

We frequently use the term “reversible” when talking about the thermodynamics of fuel cells. Reversible implies equilibrium. A reversible fuel cell voltage is the voltage produced by a fuel cell at thermodynamic equilibrium. A process is thermodynamically reversible when an infinitesimal reversal in the driving force causes it to reverse direction; such a system is always at equilibrium.

Equations relating to reversible fuel cell voltages only apply to equilibrium conditions. As soon as current is drawn from a fuel cell, equilibrium is lost and reversible fuel cell volt- age equations no longer apply. To distinguish between reversible and nonreversible fuel cell voltages in this book, wewill use the symbolsE andV, whereE represents a reversible (ther- modynamically predicted) fuel cell voltage and V represents an operational (nonreversible) fuel cell voltage.

2.2 HEAT POTENTIAL OF A FUEL: ENTHALPY OF REACTION

Now that we have reviewed general thermodynamics, the exciting work begins. We will now apply what we know about thermodynamics to fuel cells. Remember, the goal of a fuel cell is to extract the internal energy from a fuel and convert it into more useful forms of energy. What is the maximum amount of energy that we can extract from a fuel? The maximum depends on whether we extract energy from the fuel in the form of heat or work. As is shown in this section, the maximum heat energy that can be extracted from a fuel is given by the fuel’s enthalpy of reaction (for a constant-pressure process).

Recall the differential expression for enthalpy (Equation 2.22):

dH = T dS + V dp (2.27)

For a constant-pressure process (dp = 0), Equation 2.27 reduces to

dH = T dS (2.28)

Here, dH is the same as the heat transferred (dQ) in a reversible process. For this reason, we can think of enthalpy as a measure of the heat potential of a system under constant-pressure conditions. In other words, for a constant-pressure reaction, the enthalpy change expresses the amount of heat that could be evolved by the reaction. From where does this heat origi- nate? Expressing dH in terms of dU at constant pressure provides the answer:

dH = T dS = dU + dW (2.29)

From this expression, we see that the heat evolved by a reaction is due to changes in the internal energy of the system, after accounting for any energy that goes toward work. The

HEAT POTENTIAL OF A FUEL: ENTHALPY OF REACTION 35

internal energy change in the system is largely due to the reconfiguration of chemical bonds. For example, as discussed in the previous chapter, burning hydrogen releases heat due to molecular bonding reconfigurations. The product water rests at a lower internal energy state than the initial hydrogen and oxygen reactants. After accounting for the energy that goes toward work, the rest of the internal energy difference is transformed into heat during the reaction. The situation is analogous to a ball rolling down a hill; the potential energy of the ball is converted into kinetic energy as it rolls from the high-potential-energy initial state to the low-potential-energy final state.

The enthalpy change associated with a combustion reaction is called the heat of com- bustion. The name heat of combustion indicates the close tie between enthalpy and heat potential for constant-pressure chemical reactions. More generally, the enthalpy change associated with any chemical reaction is called the enthalpy of reaction or heat of reaction. We use the more general term enthalpy of reaction (ΔHrxn or Δĥrxn) in this text.

2.2.1 Calculating Reaction Enthalpies

Since reaction enthalpies are associated mainly with the reconfiguration of chemical bonds during a reaction, they can be calculated by considering the bond enthalpy differences between the reactants and products. For example, in problem 1.7, we approximated how much heat is released in the H2 combustion reaction by comparing the enthalpies of the reactant O–O and H–H bonds to the product H–O bonds.

Bond enthalpy calculations are somewhat awkward and give only rudimentary approxi- mations. Therefore, enthalpy-of-reaction values are normally calculated by computing the formation enthalpy differences between reactants and products. A standard-state formation enthalpy Δĥ0f (i) tells how much enthalpy is required to form 1 mol of chemical species i at STP from the reference species. For a general reaction

aA + bB → mM + nN (2.30)

where A and B are reactants; M and N are products; and a, b, m, n represent the number of moles of A, B, M, and N, respectively; Δĥ0rxn may be calculated as

Δĥ0rxn = [ mΔĥ0f (M) + nΔĥ

0 f (N)

] − [ aΔĥ0f (A) + bΔĥ

0 f (B)

] (2.31)

Thus, the enthalpy of reaction is computed from the difference between the molar weighted reactant and product formation enthalpies. Note that enthalpy changes (like all energy changes) are computed in the form of final state–initial state, or in other words, products–reactants.

An expression analogous to Equation 2.31 may be written for the standard-state entropy of a reaction, Δŝ0rxn, using standard entropy values ŝ0 for the species taking part in the reaction. See Example 2.1 for details.

36 FUEL CELL THERMODYNAMICS

Example 2.1 A direct methanol fuel cell uses methanol as fuel instead of hydrogen. Calculate the Δĥ0rxn and Δŝ0rxn for the methanol combustion reaction:

CH3OH(liq) + 3 2 O2 → CO2 + 2H2O(liq) (2.32)

Solution: From Appendix B, the Δĥ0f and ŝ 0 values for CH3OH, O2, CO2, and H2O

are given in the following table.

Chemical Species Δĥ0f (kJ/mol) ŝ 0 [J/(mol⋅K)]

CH3OH(liq) –238.5 127.19 O2 0 205.00 CO2 –393.51 213.79 H2O(liq) –285.83 69.95

Following Equation 2.31, the Δĥ0rxn for methanol combustion is calculated as

Δĥ0rxn = [ 2Δĥ0f (H2O(liq)) + Δĥ

0 f (CO2)

] − [ 3 2 Δĥ0f (O2) + Δĥ

0 f (CH3OH(liq))

] = [2(−285.83) + (−393.51)] −

[ 3 2 (0) + (−238.5)

] = −726.67 kJ∕mol (2.33)

Similarly, Δŝ0rxn is calculated as

Δŝ0rxn = [ 2ŝ0(H2O(liq)) + ŝ0(CO2)

] − [ 3 2 ŝ0(O2) + ŝ0(CH3OH(liq))

] = [2(69.95) + (213.79)] −

[ 3 2 (205.00) + (127.19)

] = −81.00 J∕(mol ⋅ K) (2.34)

2.2.2 Temperature Dependence of Enthalpy

The amount of heat energy that a substance can absorb changes with temperature. It fol- lows that a substance’s formation enthalpy also changes with temperature. The variation of enthalpy with temperature is described by a substance’s heat capacity:

Δĥf = Δĥ0f + ∫ T

cp(T)dT (2.35)

where Δĥf is the formation enthalpy of the substance at an arbitrary temperature T, Δĥ0f is the reference formation enthalpy of the substance at T0 = 298.15 K, and cp(T) is the

WORK POTENTIAL OF A FUEL: GIBBS FREE ENERGY 37

constant-pressure heat capacity of the substance (which itself may be a function of tempera- ture). If phase changes occur along the path between T0 and T, extra cautionmust be taken to make sure that the enthalpy changes associated with these phase changes are also included.

In a similar manner, the entropy of a substance also varies with temperature. Again, this variation is described by the substance’s heat capacity:

ŝ = ŝ0 + ∫ T

cp(T) T

dT (2.36)

From Equations 2.31, 2.35, and 2.36, Δĥrxn and Δŝrxn for any reaction at any tempera- ture can be calculated as long as the basic thermodynamic data (Δĥ0f , ŝ

0, cp) are provided. Appendix B provides a collection of basic thermodynamic data for a variety of chemical species relevant to fuel cells.

Since heat capacity effects are generally minor,Δĥ0f and ŝ 0 values are usually assumed to

be independent of temperature, simplifying thermodynamic calculations. See Example 2.2 for an illustration.

In a perfect world, we could harness all of the enthalpy released by a chemical reaction to do useful work. Unfortunately, thermodynamics tells us that this is not possible. Only a portion of the energy evolved by a chemical reaction can be converted into useful work. For electrochemical systems (i.e., fuel cells), the Gibbs free energy gives the maximum amount of energy that is available to do electrical work.

2.3 WORK POTENTIAL OF A FUEL: GIBBS FREE ENERGY

Recall from Section 2.1.5 that the Gibbs free energy can be considered to be the net energy required to create a system and make room for it minus the energy received from the envi- ronment due to spontaneous heat transfer. Thus, G represents the energy that you had to transfer to create the system. (The environment also transferred some energy via heat, but G subtracts this contribution out.) IfG represents the net energy you had to transfer to create the system, then G should also represent the maximum energy that you could ever get back out of the system. In other words, the Gibbs free energy represents the exploitable energy potential, or work potential, of the system.

2.3.1 Calculating Gibbs Free Energies

Since the Gibbs free energy is the key to the work potential of a reaction, it is necessary to calculate Δĝrxn values as we calculated Δĥrxn and Δŝrxn values. In fact, we can calcu- late Δĝrxn values directly from Δĥrxn and Δŝrxn values. Recalling how G is defined, it is apparent that G already contains H, since G = U + PV − TS and H = U + PV . We can therefore define the Gibbs free energy as

G = H − TS (2.37)

38 FUEL CELL THERMODYNAMICS

Differentiating this expression gives

dG = dH − T dS − SdT (2.38)

Holding temperature constant (isothermal process, dT = 0) and writing this relationship in terms of molar quantities give

Δĝ = Δĥ − TΔŝ (2.39)

Thus, for an isothermal reaction, we can computeΔĝ in terms ofΔĥ andΔŝ. The isothermal reaction assumption means that temperature is constant during the reaction. However, it is important to realize that we can still use Equation 2.39 to calculate Δĝ values at different reaction temperatures.

Example 2.2 Determine the approximate temperature at which the following reac- tion is no longer spontaneous:

CO + H2O(g) → CO2 + H2 (2.40)

Solution: To answer this question, we need to calculate the Gibbs free energy for this reaction as a function of temperature and then solve for the temperature at which the Gibbs free energy for this reaction goes to zero:

Δĝrxn(T) = Δĥrxn(T) − TΔŝrxn(T) = 0 (2.41)

To get an approximate answer, we can assume that Δĥrxn and Δŝrxn are indepen- dent of temperature (heat capacity effects are ignored). In this case, the temperature dependence of Δĝrxn is approximated as

Δĝrxn(T) = Δĥ0rxn − TΔŝ0rxn (2.42)

From Appendix B, the Δĥ0f and ŝ 0 values for CO, CO2, H2, and H2O are given in

the table below.

Chemical Species Δĥ0f ( kJ/mol) ŝ 0 [J/(mol⋅K)]

CO –110.53 197.66 CO2 –393.51 213.79 H2 0 130.68 H2O(g) –241.83 188.84

WORK POTENTIAL OF A FUEL: GIBBS FREE ENERGY 39

Following Equation 2.31, Δĥ0rxn is calculated as

Δĥ0rxn = [ Δĥ0f (CO2) + Δĥ

0 f (H2)

] − [ Δĥ0f (CO) + Δĥ

0 f (H2O)

] = [(−393.51) + (0)] − [(−110.53) + (−241.83)]

= −41.15 kJ∕mol (2.43)

Similarly, Δŝ0rxn is calculated as

Δŝ0rxn = [ ŝ0(CO2) + ŝ0(H2)

] − [ ŝ0(CO) + ŝ0(H2O)

] = [(213.79) + (130.68)] − [(197.66) + (188.84)]

= −42.03 J∕(mol ⋅ K) (2.44) This gives

Δĝrxn(T) = −41.15 kJ∕mol − T[−0.04203 kJ∕(mol ⋅ K)] (2.45)

Examining this expression, it is apparent that at low temperatures the enthalpy term will dominate over the entropy term, and the free energy will be negative. How- ever, as the temperature increases, entropy eventually wins and the reaction ceases to be spontaneous. Setting this equation equal to zero and solving for T give us the temperature where the reaction ceases to be spontaneous:

− 41.15 kJ∕mol + T[0.04203 kJ∕(mol ⋅ K)] = 0 T ≈ 979K ≈ 706∘C (2.46)

This reaction is known as the water gas shift reaction. It is important for high-temperature internal reforming of direct hydrocarbon fuel cells. These fuel cells run on simple hydrocarbon fuels (such as methane) in addition to hydrogen gas. Since these fuels contain carbon, carbon monoxide is often produced. The water gas shift reaction allows additional H2 fuel to be created from the CO stream. However, if the fuel cell is run above 700∘C, the water gas shift reaction is thermodynamically unfa- vorable. Therefore, operating a high-temperature direct hydrocarbon fuel cell requires a delicate balance between the thermodynamics of the reactions (which are more favorable at lower temperatures) and the kinetics of the reactions (which improve at higher temperatures). This balance is discussed in greater detail in Chapter 11.

2.3.2 Relationship between Gibbs Free Energy and Electrical Work

Now that we know how to calculate Δg, we can determine the work potential of a fuel cell. For fuel cells, recall that we are specifically interested in electrical work. Let us find the maximum amount of electrical work that we can extract from a fuel cell reaction.

40 FUEL CELL THERMODYNAMICS

From Equation 2.17, remember that we define a change in Gibbs free energy as

dG = dU − T dS − SdT + pdV + V dp (2.47)

As we have done previously, we can insert the expression for dU based on the first law of thermodynamics (Equation 2.3) into this equation. However, this time we expand the work term in dU to include both mechanical work and electrical work:

dU = T dS − dW

= T dS − (pdV + dWelec) (2.48)

which yields dG = −SdT + V dp − dWelec (2.49)

For a constant-temperature, constant-pressure process (dT , dp = 0) this reduces to

dG = −dWelec (2.50)

Thus, the maximum electrical work that a system can perform in a constant-temperature, constant-pressure process is given by the negative of the Gibbs free-energy difference for the process. For a reaction using molar quantities, this equation can be written as

Welec = −Δgrxn (2.51)

Again, remember that the constant-temperature, constant-pressure assumption used here is not really as restrictive as it seems. The only limitation is that the temperature and pres- sure do not vary during the reaction process. Since fuel cells usually operate at constant temperature and pressure, this assumption is reasonable. It is important to realize that the expression derived above is valid for different values of temperature and pressure as long as these values are not changing during the reaction. We could apply this equation for T = 200 K and p = 1 atm or just as validly for T = 400 K and p = 5 atm. Later, we will examine how such steps in temperature and pressure (think of them as changes in the oper- ating conditions from one fixed state to a new fixed state) affect the maximum electrical work available from the fuel cell.

OPERATION OF A THERMODYNAMIC ENGINE AT CONSTANT TEMPERATURE AND PRESSURE (OPTIONAL)

The thermodynamics of fuel cell operation can be analyzed just like any other thermody- namic (or heat) engine. In the case of a fuel cell, steady-state operation typically occurs under constant-pressure (isobaric) and constant-temperature (isothermal) environments. Figure 2.4 describes the operation of this heat engine.

WORK POTENTIAL OF A FUEL: GIBBS FREE ENERGY 41

Reactants at T0 , p0

Products at T0 , p0

Isothermal and isobaric environment at T0 , p0

External work, W

Heat flux, Qrev

A thermodynamic device (engine) with internal

chemical reaction

Figure 2.4. Diagram of a reversible thermodynamic engine (or heat engine) operating under con- stant pressure and temperature. Reactants and products enter and exit from the engine at constant pressure and temperature, respectively. The engine generates external work using the chemical (heat) energy of reactants. Also, the engine releases unused chemical energy to the isothermal and isobaric environment.

Reactants at ambient temperature and pressure T0 and p0 enter the engine. At this time, the reactants carry a total chemical (heat) energy or enthalpy of HReactant(T0, p0). After the chemical reactions take place in the engine, products exit from the engine at ambient temperature and pressure T0 and p0 carrying HProduct(T0, p0). The engine generates external work, W, using the heat energy from the chemical reaction. At the same time, the engine releases unused heat, Q(= −Qrev), to the environment at ambient temperature T0.

Assuming no accumulation of energy in the device in steady state, we can write an equation for the heat and energy balance of the system using the first law of thermody- namics:

HReactants(T0, p0) = HProducts(T0, p0) − Qrev +W (2.52)

After rearranging the equation for W, we obtain

W = HReactants(T0, p0) − HProducts(T0, p0) + Qrev = −ΔH(T0, p0) + Qrev

(2.53)

Since the engine is thermodynamically reversible, we obtain the following equation from the second law of thermodynamics:

dS(T0, p0) = dQrev T0

(2.54)

42 FUEL CELL THERMODYNAMICS

Integrating both sides and solving for Qrev, we have

∫ dS(T0, p0) = SProducts(T0, p0) − SReactants(T0, p0) = ΔS(T0, p0)

= ∫ dQrev T0

= Qrev T0

Qrev = T0ΔS(T0, p0)

(2.55)

Plugging Equation 2.55 into 2.53 and solving for W, we have

W = −ΔH(T0, p0) + Qrev = −ΔH(T0, p0) + T0ΔS(T0, p0)

= −ΔG(T0, p0)

(2.56)

Thus, any thermodynamic engine at steady state can generate a maximum amount of work equivalent to the Gibbs free energy if it operates under isobaric (constant-pressure) and isothermal (constant-temperature) conditions. The fuel cell is one type of thermo- dynamic engine that can generate work,W, in electrical form under this condition. This result is not surprising, since we have already learned that maximum available thermo- dynamic work potential under this condition is equal to the Gibbs energy in the system.

2.3.3 Relationship between Gibbs Free Energy and Reaction Spontaneity

In addition to determining the maximum amount of electrical work that can be extracted from a reaction, the Gibbs free energy is also useful in determining the spontaneity of a reaction. Obviously, ifΔG is zero, then no electrical work can be extracted from a reaction. Worse yet, if ΔG is greater than zero, then work must be input for a reaction to occur. Therefore, the sign of ΔG indicates whether or not a reaction is spontaneous:

ΔG > 0 Nonspontaneous (energetically unfavorable) ΔG = 0 Equilibrium ΔG < 0 Spontaneous (energetically favorable)

A spontaneous reaction is energetically favorable; it is a “downhill” process. Although spontaneous reactions are energetically favorable, spontaneity is no guarantee that a reac- tion will occur, nor does it indicate how fast a reaction will occur. Many spontaneous reactions do not occur because they are impeded by kinetic barriers. For example, at STP, the conversion of diamond to graphite is energetically favorable (ΔG < 0). Fortunately for diamond lovers, kinetic barriers prevent this conversion from occurring. Fuel cells, too, are constrained by kinetics. The rate at which electricity can be produced from a fuel cell is lim- ited by several kinetic phenomena. These phenomena are covered in Chapters 3–5. Before

WORK POTENTIAL OF A FUEL: GIBBS FREE ENERGY 43

we get to kinetics, however, you need to understand how the electrical work capacity of a fuel cell is translated into a cell voltage.

2.3.4 Relationship between Gibbs Free Energy and Voltage

The potential of a system to perform electrical work is measured by voltage (also called electrical potential). The electrical work done by moving a charge Q, measured in coulombs, through an electrical potential difference E in volts is

Welec = EQ (2.57)

If the charge is assumed to be carried by electrons, then

Q = nF (2.58)

where n is number of moles of electrons transferred and F is Faraday’s constant. Combining Equations 2.51, 2.57, and 2.58 yields

Δĝ = −nFE (2.59)

Thus, the Gibbs free energy sets the magnitude of the reversible voltage for an electro- chemical reaction. For example, in a hydrogen–oxygen fuel cell, the reaction

H2 + 1 2 O2 ⇌ H2O (2.60)

has a Gibbs free-energy change of –237 kJ/mol under standard-state conditions for liq- uid water product. The reversible voltage generated by a hydrogen–oxygen fuel cell under standard-state conditions is thus

E0 = − Δĝ0rxn nF

= − −237, 000 J∕mol

(2 mol e−∕mol reactant)(96, 485 C∕mol) = +1.23 V

(2.61)

where E0 is the standard-state reversible voltage andΔĝ0rxn is the standard-state free-energy change for the reaction.

At STP, thermodynamics dictates that the highest voltage attainable from a H2–O2 fuel cell is 1.23 V. If we need 10 V, forget about it. In other words, the chemistry of the fuel cell sets the reversible cell voltage. By picking a different fuel cell chemistry, we could establish a different reversible cell voltage. However, most feasible fuel cell reactions have reversible cell voltages in the range of 0.8–1.5 V. To get 10 V from fuel cells, we usually have to stack several cells together in series.

44 FUEL CELL THERMODYNAMICS

TABLE 2.1. Selected List of Standard Electrode Potentials

Electrode Reaction E0 (V)

Fe2+ + 2e− ⇌ Fe −0.440 CO2 + 2H+ + 2e− ⇌ CHOOH(aq) −0.196 2H+ + 2e− ⇌ H2 +0.000 CO2 + 6H+ + 6e− ⇌ CH3OH + H2O +0.03 O2 + 4H+ + 4e− ⇌ 2H2O +1.229

2.3.5 Standard Electrode Potentials: Computing Reversible Voltages

Although we learned how to calculate cell voltage using Equation 2.59, the cell poten- tials of many reactions have already been calculated for us in standard electrode potential tables. It is often easier to determine reversible voltages using these electrode potential tables. Standard electrode potential tables compare the standard-state reversible voltages of various electrochemical half reactions relative to the hydrogen reduction reaction. In these tables, the standard-state potential of the hydrogen reduction reaction is defined as zero, thus making it easy to compare other reactions.

To illustrate the concept of electrode potentials, a brief list is presented in Table 2.1. A more complete set of electrode potentials is provided in Appendix C.

To find the standard-state voltage produced by a complete electrochemical system, we simply sum all the potentials in the circuit:

E0cell = ∑

E0half reactions (2.62)

THE QUANTITY nF

When studying fuel cells or other electrochemical systems, we will frequently encounter expressions containing the quantity nF. This quantity is our bridge from the world of thermodynamics (where we talk about moles of chemical species) to the world of elec- trochemistry (where we talk about current and voltage). In fact, the quantity nF expresses one of the most fundamental aspects of electrochemistry: the quantized transfer of elec- trons, in the form of an electrical current, between reacting chemical species. In any electrochemical reaction, there exists an integer correspondence between the moles of chemical species reacting and the moles of electrons transferred. For example, in the H2–O2 fuel cell reaction, 2 mol of electrons is transferred for every mole of H2 gas reacted. In this case, n = 2. To convert this molar quantity of electrons to a quantity of charge, we must multiply n by Avogadro’s number (NA = 6.022 × 1023 electrons∕mol) to get the number of electrons and then multiply by the charge per electron (q = 1.60 × 10–19 C∕electron) to get the total charge. Thus we have

Q = nNAq = nF (2.63)

WORK POTENTIAL OF A FUEL: GIBBS FREE ENERGY 45

What we call Faraday’s constant is really the quantity NA×q:

F = NA × q = (6.022 × 1023electrons∕mol) × (1.60 × 10–19C∕electron)

= 96, 485 C∕mol

Interestingly, the fact that Faraday’s constant is a large number has important techno- logical repercussions. Because F is large, a little chemistry produces a lot of electricity. This relationship is one of the factors that make fuel cells technologically feasible.

Students are often confused whether they should base the number of moles of elec- trons transferred (n) in a reaction on a per-mole reactant basis, per-mole product basis, or so on. The answer is that it does not matter as long as you are consistent. For example, consider the reaction

A + 2B → C + 2e− ΔGrxn (2.64)

In this reaction, n = 2 per mole of A reacted, or per mole of C produced, or per 2 mol of B reacted. If instead n is desired per mole of B reacted, then the reaction stoichiometry must be adjusted as

1 2 A + B → 1

2 C + e− 1

2 ΔGrxn (2.65)

Now, per mole of B reacted, n = 1. Also n = 1 per 1/2 mol of A reacted or per 1/2 mol of C produced. However, keep in mind that the Gibbs free energy for reaction 2.65 is now 1 2 ΔG of the original reaction. As long as n and ΔG are kept consistent with the reaction

stoichiometry, you should not suffer any confusion.

For example, the standard-state potential of the hydrogen–oxygen fuel cell is deter- mined by

H2 → 2H + + 2e− E0 = −0.000

+ 1 2 (O2 + 4H+ + 4e− → 2H2O) E0 = +1.229

= H2 + 1 2 O2 → H2O E

0 cell = +1.229

Note that we multiply the O2 reaction by 1/2 to get the correct stoichiometry. However, do not multiply the E0 values by 1/2. The E0 values are independent of reaction amounts. Note also that in this calculation we reverse the direction of the hydrogen reaction (in a hydrogen–oxygen fuel cell, hydrogen is oxidized, not reduced). When we reverse the direc- tion of a reaction, we reverse the sign of its potential. For the hydrogen reaction, this makes no difference, since+0.000 V = –0.000 V. However, the standard-state potential of the iron oxidation reaction, for example,

Fe ⇌ Fe2+ + 2e− (2.66)

would be +0.440 V. A complete electrochemical reaction generally consists of two half reactions, a reduction

reaction and an oxidation reaction. However, electrode potential tables list all reactions as

46 FUEL CELL THERMODYNAMICS

reduction reactions. For a set of coupled half reactions, how do we know which reaction will spontaneously proceed as the reduction reaction and which reaction will proceed as the oxidation reaction? The answer is found by comparing the size of the electrode potentials for the reactions. Because electrode potentials really represent free energies, increasing potential indicates increasing “reaction strength.” For a matched pair of electrochemical half reactions, the reaction with the larger electrode potential will occur as written, while the reaction with the smaller electrode potential will occur opposite as written. For example, consider the Fe2+–H+ reaction couple from the list above. Because the hydrogen reduction reaction has a larger electrode potential compared to the iron reduction reaction (0 V > –0.440 V), the hydrogen reduction reaction will occur as written. The iron reaction will proceed in the opposite direction as written:

2H+ + 2e− → H2 E0 = +0.000 Fe → Fe2+ + 2e− E0 = +0.440

Fe + 2H+ → Fe2+ + H2 E0 = +0.440

Thus, thermodynamics predicts that in this system iron will be spontaneously oxidized to Fe2+ and hydrogen gas will be evolved, with a net cell potential of +0.440 V. This is the thermodynamically spontaneous reaction direction under standard-state conditions. Any thermodynamically spontaneous electrochemical reactionwill have a positive cell potential. Of course, the reaction could be made to occur in the reverse direction if an external voltage greater than 0.440 V is applied to the cell. In this case, a power supply would be doing work to the cell in order to overcome the thermodynamics of the system.

Example 2.3 A direct methanol fuel cell uses methanol (CH3OH) as fuel instead of hydrogen:

CH3OH + 3 2 O2 → CO2 + 2H2O (2.67)

Calculate the standard-state reversible potential for a direct methanol fuel cell.

Solution: We break this overall reaction into two electrochemical half reactions:

CH3OH + H2O ⇌ CO2 + 6H+ + 6e− E0 = −0.03 3 2 (O2 + 4H+ + 4e− ⇌ 2H2O) E0 = +1.229

CH3OH + 3 2 O2 → CO2 + 2H2O E0 = +1.199

Thus, the net cell potential for a methanol fuel cell is +1.199 V—almost the same as for a H2–O2 fuel cell. Note that although we multiplied the oxygen reduction reac- tion by 3

2 to get a balanced reaction, we did not multiply the E0 value by 3

2 . The E0

values are independent of reaction amounts.

PREDICTING REVERSIBLE VOLTAGE OF A FUEL CELL UNDER NON-STANDARD-STATE CONDITIONS 47

2.4 PREDICTING REVERSIBLE VOLTAGE OF A FUEL CELL UNDER NON-STANDARD-STATE CONDITIONS

Standard-state reversible fuel cell voltages (E0 values) are only useful under standard-state conditions (room temperature, atmospheric pressure, unit activities of all species). Fuel cells are frequently operated under conditions that vary greatly from the standard state. For example, high-temperature fuel cells operate at 700–1000∘C, automotive fuel cells often operate under 3–5 atm of pressure, and almost all fuel cells cope with variations in the concentration (and therefore activity) of reactant species.

In the following sections, we systematically define how reversible fuel cell voltages are affected by departures from the standard state. First, the influence of temperature on the reversible fuel cell voltage will be explored, then the influence of pressure. Finally, con- tributions from species activity (concentration) will be delineated, which will result in the formulation of the Nernst equation. In the end, we will have thermodynamic tools to predict the reversible voltage of a fuel cell under any arbitrary set of conditions.

2.4.1 Reversible Voltage Variation with Temperature

To understand how the reversible voltage varies with temperature, we need to go back to our original differential expression for the Gibbs free energy:

dG = −SdT + V dp (2.68)

from which we can write (dG dT

) p = −S (2.69)

For molar reaction quantities, this becomes( d(Δĝ) dT

) p

= −Δŝ (2.70)

We have previously shown that the Gibbs free energy is related to the reversible cell voltage by

Δĝ = −nFE (2.71)

Combining Equations 2.70 and 2.71 allows us to express how the reversible cell voltage varies as a function of temperature: (dE

) p = Δŝ

nF (2.72)

48 FUEL CELL THERMODYNAMICS

We define ET as the reversible cell voltage at an arbitrary temperature T. At constant pressure, ET can be calculated by

ET = E0 + Δŝ nF

(T − T0) (2.73)

Generally, we assume Δŝ to be independent of temperature. If a more accurate value of ET is required, it may be calculated by integrating the heat-capacity-related temperature dependence of Δŝ.

As Equation 2.73 indicates, ifΔŝ for a chemical reaction is positive, thenET will increase with temperature. If Δŝ is negative, then ET will decrease with temperature. For most fuel cell reactions Δŝ is negative; therefore reversible fuel cell voltages tend to decrease with increasing temperature.

For example, consider our familiar H2–O2 fuel cell. As can be calculated from the data in Appendix B, Δŝrxn = −44.34 J/(mol⋅K) (for H2O(g) as product). The variation of cell voltage with temperature is approximated as

ET = E0 + −44.34J∕(mol ⋅ K)

(2)(96, 485) (T − T0)

= E0 − (2.298 × 10−4 V∕K)(T − T0) (2.74)

Thus, for every 100 degrees increase in cell temperature, there is an approximate 23-mV decrease in cell voltage. A H2–O2 SOFC operating at 1000 K would have a reversible volt- age of around 1.07 V. The temperature variation for the electrochemical oxidation of a number of different fuels is given in Figure 2.5.

Since most reversible fuel cell voltages decrease with increasing temperature, should we operate a fuel cell at the lowest temperature possible? The answer is NO! As you will learn in Chapters 3 and 4, kinetic losses tend to decrease with increasing temperature. Therefore, real fuel cell performance typically increases with increasing temperature even though the thermodynamically reversible voltage decreases.

2.4.2 Reversible Voltage Variation with Pressure

Like temperature effects, the pressure effects on cell voltage may also be calculated starting from the differential expression for the Gibbs free energy:

dG = −SdT + V dp (2.75)

This time, we note ( dG dp

) T

= V (2.76)

Written for molar reaction quantities, this becomes( d(Δĝ) dp

) T

= Δ�̂� (2.77)

PREDICTING REVERSIBLE VOLTAGE OF A FUEL CELL UNDER NON-STANDARD-STATE CONDITIONS 49

Temperature (°C)

S ta

nd ar

d po

te nt

ia l (

V )

0.9 100

0.95

1.00

1.05

1.10

1.15

1.20

1.25

1.30

Temperature (K)

200 300 400 500 600 700 800 900 1000

300 400 500 600 700 800 900 100011001200

CH3OH CO

H2 H2O(l)2 2 C CO

C CO

CH4

C2H4

H2 H2O(g)

Figure 2.5. Reversible voltage (ET ) versus temperature for electrochemical oxidation of a variety of fuels. (After Broers and Ketelaar [3].)

We have previously shown that the Gibbs free energy is related to the reversible cell voltage by

Δĝ = −nFE (2.78)

Substituting this equation into Equation 2.77 allows us to express how the reversible cell voltage varies as a function of pressure:(

dE dp

) T

= −Δ�̂� nF

(2.79)

In other words, the variation of the reversible cell voltage with pressure is related to the volume change of the reaction. If the volume change of the reaction is negative (if fewer moles of gas are generated by the reaction than consumed, for instance), then the cell volt- age will increase with increasing pressure. This is an example of Le Chatelier’s principle: Increasing the pressure of the system favors the reaction direction that relieves the stress on the system.

Usually, only gas species produce an appreciable volume change. Assuming that the ideal gas law applies, we can write Equation 2.79 as(

dE dp

) T

= − ΔngRT nFp

(2.80)

where Δng represents the change in the total number of moles of gas upon reaction. If np is the number of product moles of gas and nr is the number of reactant moles of gas, then Δng = np – nr.

50 FUEL CELL THERMODYNAMICS

Pressure, like temperature, turns out to have a minimal effect on reversible voltage. As you will see in a forthcoming example, pressurizing a H2–O2 fuel cell to 3 atm H2 and 5 atm O2 increases the reversible voltage by only 15 mV.

2.4.3 Reversible Voltage Variation with Concentration: Nernst Equation

To understand how the reversible voltage varies with concentration, we need to introduce the concept of chemical potential. Chemical potential measures how the Gibbs free energy of a system changes as the chemistry of the system changes. Each chemical species in a system is assigned a chemical potential. Formally

𝜇𝛼i = ( 𝜕G 𝜕ni

) T , p,nj≠i

(2.81)

where 𝜇𝛼i is the chemical potential of species i in phase α and (𝜕G∕𝜕ni)T , p,nj≠i expresses how much the Gibbs free energy of the system changes for an infinitesimal increase in the quantity of species i (while temperature, pressure, and the quantities of all other species in the system are held constant). When we change the amounts (concentrations) of chem- ical species in a fuel cell, we are changing the free energy of the system. This change in free energy in turn changes the reversible voltage of the fuel cell. Understanding chemical potential is key to understanding how changes in concentration affect the reversible voltage.

Chemical potential is related to concentration through activity a:

𝜇i = 𝜇0i + RT ln ai (2.82)

where 𝜇0i is the reference chemical potential of species i at standard-state conditions and ai is the activity of species i. The activity of a species depends on its chemical nature:

• For an ideal gas, ai = pi∕p0, where pi is the partial pressure of the gas and p0 is the standard-state pressure (1 atm). For example, the activity of oxygen in air at 1 atm is approximately 0.21. The activity of oxygen in air pressurized to 2 atm would be 0.42. Since we accept p0 = 1 atm, we are often lazy and write ai = pi, recognizing that pi is a unitless gas partial pressure.

• For a nonideal gas, ai = 𝛾(pi∕p0), where 𝛾 is an activity coefficient describing the departure from ideality (0 < 𝛾i < 1).

• For a dilute (ideal) solution, ai = ci∕c0, where ci is the molar concentration of the species and c0 is the standard-state concentration (1 M = 1 mol/L). For example, the activity of Na+ ions in 0.1 M NaCl is 0.10.

• For nonideal solutions, ai = 𝛾(ci∕c0). Again, we use 𝛾 to describe departures from ideality (0 < γ < 1).

• For pure components, ai = 1. For example, the activity of gold in a chunk of pure gold is 1. The activity of platinum in a platinum electrode is 1. The activity of liquid water is usually taken as 1.

• For electrons in metals, ai = 1.

PREDICTING REVERSIBLE VOLTAGE OF A FUEL CELL UNDER NON-STANDARD-STATE CONDITIONS 51

Combining Equations 2.81 and 2.82, it is possible to calculate changes in the Gibbs free energy for a system of i chemical species by

dG = ∑ i

𝜇i dni = ∑ i

(𝜇0i + RT ln ai)dni (2.83)

WHAT IS CHEMICAL POTENTIAL?

Recall from Section 2.1.5 that U, F, H, and G are extrinsic quantities and therefore scale with the size or number of atoms in the system. In our initial discussions of these thermodynamic potentials, however, this explicit composition dependence was not included. Initially, we defined each thermodynamic potential using two independent variables only. In order to accommodate the thermodynamic dependence on the number of atoms in a system, we must explicitly add ni (the number of atoms or molecules of species i) as a third variable. Thus, the four thermodynamic potentials actually depend on three independent variables as U = U(S,V , ni), G = G(T , p, ni), H = H(S, p, ni), and F = F(T ,V , ni).

The quantity that describes how U, F, H, and G depend on ni is called the chemical potential, 𝜇i. The chemical potential has a logarithmic dependence on the concentra- tion (number per volume) or the activity (normalized concentration) of species i in a system:

𝜇i = 𝜇0i + RT ln ai

This logarithmic dependence can be understood based on the relative impact of adding atoms when a system is small compared to when a system is large. When a thermodynamic system is very small, that is, the number of species in the system is low, adding or subtracting a few particles will have a big impact on the activ- ity and hence the chemical potential. Conversely, if the number of species in the system is very large, a small change in the number of species will not have a big impact on the activity or the chemical potential. In other words, the magnitude of change in chemical potential depends on how many atoms or molecules of species i are present. This “size sensitivity” is captured by the mathematical form of the chemical potential, which incorporates the composition dependence inside a natural logarithm.

As will be discussed soon in Section 2.4.4, the concept of the chemical potential needs to be further expanded when dealing with charged particles. Charged particles are sensitive not only to chemical composition but also to electric fields. In this situation, we can formally expand the concept of chemical potential into electrochemical potential by adding the electrostatic potential of the charged particles to the chemical potential. In its most basic definition, the electrochemical potential represents the work required to assemble 1 mol of ions from some standard state and bring it to a defined chemical concentration and electrical potential.

52 FUEL CELL THERMODYNAMICS

Consider an arbitrary chemical reaction placed on a molar basis for species A in the form

1A + bB ⇌ mM + nN (2.84)

where A and B are reactants, M and N are products, and l, b, m, and n represent the number of moles of A, B,M, and N, respectively. On amolar basis for species A,Δĝ for this reaction may be calculated from the chemical potentials of the various species participating in the reaction (assuming a single phase):

Δĝ = (m𝜇0M + n𝜇 0 N) − (𝜇

0 A + b𝜇

0 B) + RT ln

amMa n N

a1Aa b B

(2.85)

Recognizing that the lumped standard-state chemical potential terms represent the standard-state molar free-energy change for the reaction, Δĝ0, the equation can be simplified to a final form:

Δĝ = Δĝ∘ + RT ln amMa

n N

a1Aa b B

(2.86)

This equation, called the van’t Hoff isotherm, tells how the Gibbs free energy of a system changes as a function of the activities (read concentrations or gas pressures) of the reactant and product species.

From previous thermodynamic explorations (Section 2.3.4), we know that the Gibbs free energy and the reversible cell voltage are related:

Δĝ = −nFE (2.87)

Combining Equations 2.86 and 2.87 allows us to see how the reversible cell voltage varies as a function of chemical activity:

E = E0 − RT nF

ln amMa

n N

a1Aa b B

(2.88)

For a system with an arbitrary number of product and reactant species, this equation takes the general form

E = E0 − RT nF

∏ a𝑣iproducts∏ a𝑣ireactants

(2.89)

Always take care to raise the activity of each species by its corresponding stoichiomet- ric coefficient (𝑣i). For example, if a reaction involves 2Na+, the activity of Na+ must be raised to the power of 2 (e.g., a2

Na+ ). Importantly, only chemical species that are actually

participating as reactants or products in the electrochemical reaction appear in the Nernst equation (e.g., O2, H2, and H2O for a H2 fuel cell). The activities or partial pressures of unreactive, inert, or diluent species (such as N2 in air) should not be included.

PREDICTING REVERSIBLE VOLTAGE OF A FUEL CELL UNDER NON-STANDARD-STATE CONDITIONS 53

This important result is known as the Nernst equation. The Nernst equation outlines how reversible electrochemical cell voltages vary as a function of species concentration, gas pressure, and so on. This equation is the centerpiece of fuel cell thermodynamics. Remem- ber it forever.

As an example of the utility of this equation, we will apply it to the familiar hydrogen– oxygen fuel cell reaction:

H2 + 1 2 O2 ⇌ H2O (2.90)

We write the Nernst equation for this reaction as

E = E0 − RT 2F

ln aH2O

aH2a 1∕2 O2

(2.91)

Following our activity guidelines, we replace the activities of hydrogen and oxygen gases by their unitless partial pressures (aH2 = pH2 , aO2 = pO2). If the fuel cell is operated below 100∘C, so that liquid water is produced, we set the activity of water to unity (aH2O = 1). This yields

E = E0 − RT 2F

ln 1

pH2p 1∕2 O2

(2.92)

From this equation, it is apparent that pressurizing the fuel cell in order to increase the reactant gas partial pressures will increase the reversible voltage. However, because the pressure terms appear within a natural logarithm, the voltage improvements are slight. For example, if we operate a room temperature H2–O2 fuel cell on 3 atm pure H2 and 5 atm air, thermodynamics predicts a reversible cell voltage of 1.254 V:

E = 1.229 − (8.314)(298.15) (2)(96, 485)

ln 1

(3)(5 × 0.21)1∕2

= 1.254 V (2.93)

PRESSURE, TEMPERATURE, AND NERNST EQUATION

The Nernst equation accounts for the same pressure effects that were previously dis- cussed in Section 2.4.2. Either Equation 2.89 or Equation 2.79 can be used to determine how the reversible voltage varies with pressure. If you use one, do not also use the other. The Nernst equation allows you to calculate voltage effects directly in terms of reactant and product pressures, while Equation 2.79 requires the volume change for the reaction (which you will have to express in terms of reactant gas pressures using the ideal gas law). The Nernst equation is generally more convenient.

Although temperature enters into the Nernst equation as a variable, the Nernst equation does not fully account for how the reversible voltage varies with temperature.

54 FUEL CELL THERMODYNAMICS

At an arbitrary temperature T ≠ T0, the Nernst equation must be modified as

E = ET − RT nF

∏ a𝑣iproducts∏ a𝑣ireactants

(2.94)

where ET is given from Equation 2.73 as

ET = E0 + Δŝ nF

(T − T0) (2.95)

Thus, the full expression describing how the reversible cell voltage varies with tem- perature, pressure, and activity can be written as

E = E0 + Δŝ nF

(T − T0) − RT nF

∏ a𝑣iproducts∏ a𝑣ireactants

(2.96)

In summary, to properly account for both temperature and pressure changes, make sure to use Equation 2.96 or Equations 2.73 and 2.79.

This is not much of an increase for all the extra work of pressurizing the fuel cell stack! From a thermodynamic perspective it is not worth the trouble; however, as you will learn in Chapters 3 and 5, there may be kinetic reasons to pressurize a fuel cell.

In contrast, what does the Nernst equation indicate about low-pressure operation? Per- haps we are worried that almost all fuel cells operate on air instead of pure oxygen. Air is only about 21% oxygen, so at 1 atm, the partial pressure of oxygen in air is only 0.21. How much does this affect the reversible voltage of a room temperature H2–O2 fuel cell?

E = 1.229 − (8.314)(298.15) (2)(96, 485)

ln 1

(1)(0.21)1∕2

= 1.219V (2.97)

Operation in air drops the reversible voltage by only 10 mV. Again, kinetic factors can introduce more deleterious penalties for air operation. However, as far as thermodynamics is concerned, air operation is not a problem.

2.4.4 Concentration Cells

The curious phenomenon of the concentration cell highlights some of the most fascinating implications of the Nernst equation. In a concentration cell, the same chemical species is present at both electrodes but at different concentrations. Amazingly, such a cell will develop a voltage because the concentration (activity) of the chemical species is different

PREDICTING REVERSIBLE VOLTAGE OF A FUEL CELL UNDER NON-STANDARD-STATE CONDITIONS 55

H2 (100 atm)

H2 (10–8atm)

Porous Pt electrode

Electrolyte membrane

_ +

H+ 2H+ + 2e– H2H2 2H

+ + 2e–

Figure 2.6. Hydrogen concentration cell. A high-pressure hydrogen compartment and a low-pressure hydrogen compartment are separated by a platinum–electrolyte–platinum membrane structure. This device will develop a voltage due to the difference in the chemical potential of hydrogen between the two compartments.

at one electrode versus the other electrode. For example, a salt water battery consisting of salt water at one electrode and freshwater at the other will produce a voltage because the concentration of salt differs at the two electrodes.

As a second example, consider the hydrogen concentration cell shown in Figure 2.6, which consists of a pressurized hydrogen fuel compartment and an evacuated ultra-low- pressure vacuum compartment separated by a composite platinum–electrolyte–platinum membrane structure. This “hydrogen fuel cell” contains no oxygen to react with the hydro- gen, yet it will still produce a significant voltage. Thus, you could even use this fuel cell in outer space, where oxygen is unavailable. The thermodynamic voltage produced by the cell is related to the concentration of hydrogen in the fuel compartment relative to the vacuum compartment. For example, if the hydrogen fuel compartment is pressurized to 100 atm H2 and the vacuum compartment is evacuated to 10–8 atm (presumably what remains will be mostly H2), then this device will exhibit a voltage as determined by the Nernst equation:

E = 0 − (8.314)(298.15) (2)(96, 485)

ln 10−8

100

= 0.296V (2.98)

At room temperature, we can extract almost 0.3 V just by exploiting a difference in hydrogen concentration. How is this possible? A voltage develops because the chemi- cal potential of the hydrogen on one side of the membrane is dramatically different from the chemical potential of the hydrogen on the other side of the membrane. Driven by the chemical potential gradient, some of the hydrogen in the fuel compartment decomposes on the platinum catalyst electrode to protons and electrons. The protons flow through the

56 FUEL CELL THERMODYNAMICS

electrolyte to the vacuum compartment, where they react with electrons in the second plat- inum catalyst electrode to reproduce hydrogen gas. If the two platinum electrodes are not connected, then very quickly excess electrons will accumulate on the fuel side, while elec- trons will be depleted on the vacuum side, setting up an electrical potential gradient. This electrical potential gradient retards further movement of hydrogen from the fuel compart- ment to the vacuum compartment. Equilibrium is established when this electrical potential gradient builds up sufficiently to exactly balance the chemical potential gradient. (This is very similar to the “built-in voltage” that occurs at semiconductor p–n junctions.) The chem- ical potential difference created by the vastly different hydrogen concentrations at the two electrodes is offset by the development of an electrical potential, which is equal but opposite in magnitude. The concept of chemical and electrical potentials offsetting one another to maintain thermodynamic equilibrium is summarized by a quantity called the electrochem- ical potential:

�̃� = 𝜇i + ziF𝜙i = 𝜇0i +RT ln ai + ziF𝜙i (2.99)

where �̃�i is the electrochemical potential of species i,𝜇i is the chemical potential of species i, zi is the charge number on the species (e.g., ze− = −1, zCu2+ = +2), F is Faraday’s constant, and 𝜙i is the electrical potential experienced by species i. At equilibrium, the net change in the electrochemical potential for the species taking part in the system must be zero; in other words, the chemical and electrical potentials offset one another. For a reaction(∑

𝑣i�̃�i

) products

−

(∑ i

𝑣i�̃�i

) reactants

= 0

(at equilibrium)(∑ i

𝑣i𝜇i

) products

−

(∑ i

𝑣i𝜇i

) reactants

= −ziFΔ𝜙i

(2.100)

Compare this to Equation 2.59. Do you see how these two equations are really expressing the same thing? Following procedures analogous to Equations 2.82, 2.83, 2.84–2.86, we can rederive the Nernst equation from the basis of the electrochemical potential:

�̃�i = 𝜇0i + RT ln ai + ziF𝜙i = 0 (2.101)

The trick to rederiving the Nernst equation is to write out the change in electrochemical potential for the reactants being converted into products while also including the change in electrochemical potential for the electrons as they move from the anode to the cathode. Solving for the difference in the electrical potential for the electrons at the cathode versus the anode (Δ𝜙e− ) gives the cell potential E. If nmoles of electrons move from the anode to the cathode per mole of chemical reaction, then

Δ𝜙e− = E = − Δĝ0

nF − RT

nF ln

∏ a𝑣iproducts∏ a𝑣ireactants

(2.102)

PREDICTING REVERSIBLE VOLTAGE OF A FUEL CELL UNDER NON-STANDARD-STATE CONDITIONS 57

which gives

E = E0 − RT nF

∏ a𝑣iproducts∏ a𝑣ireactants

(2.103)

The details of this derivation are left as a homework problem at the end of this chapter. Based on this discussion of concentration cells, you should see that it is possible to

think of an H2–O2 fuel cell as simply a hydrogen concentration cell. Oxygen is used at the cathode merely as a convenient way to chemically “tie up” hydrogen. The O2 gas keeps the cathode concentration of hydrogen to extremely low effective levels, allowing a significant thermodynamic voltage to be produced.

ELECTROCHEMICAL EQUILIBRIUM

This dialogue box provides additional details on the calculation of electrochemical equi- libria. As an example, we will derive the Nernst equation for the Cu2+ concentration cell illustrated in Figure 2.7.

1 2

Cu2+ SO42–

e– e–

11 2 1 2 CueCu +

–+ –+ + 22222 eCuCu

[Cu12+] = 10M [Cu22+] = 10–5M

Figure 2.7. Copper concentration cell.

In this concentration cell, we have two electrolyte baths containing different con- centrations of Cu2+ ions (with counterbalancing SO4

2– ions for ionic charge balance),

58 FUEL CELL THERMODYNAMICS

connected by an SO4 2– conducting salt bridge. Copper electrodes are placed in both

baths, and a voltage potential difference is established between the two electrodes, which exactly counterbalances the chemical potential difference caused by the Cu2+ concen- tration difference between the two baths. Because of the high concentration of Cu2+ ions in bath 1, we have the reaction

Cu2+1 + 2e − 1 → Cu1 (2.104)

Copper ions precipitate from solution, consuming electrons in the process and leaving the electrode positively charged. In bath 2, the opposite reaction occurs due to the low concentration of Cu2+ ions:

Cu2 → Cu 2+ 2 + 2e

− 2 (2.105)

Copper dissolves from electrode 2, which therefore builds up a negative charge. The buildup of charge between electrodes 1 and 2 proceeds until the voltage is sufficiently large to exactly offset the chemical potential difference due to the Cu2+ ion imbalance between baths 1 and 2. At this point, electrochemical equilibrium has been established.

In order to mathematically describe this electrochemical equilibrium, we must employ Equation 2.100. The overall reaction occurring in this concentration cell is

Cu2 + Cu2+1 + 2e − 1 → Cu

2+ 2 + 2e

− 2 + Cu1 (2.106)

This is simply the sum of the two half-cell reactions above. The next step is to write the electrochemical potentials for each of the species in this overall reaction (following Equation 2.99):

�̃�1 Cu2+

= 𝜇0 Cu2+

+ RT ln a1 Cu2+

+ 2F𝜑1 Cu2+

�̃�2 Cu2+

= 𝜇0 Cu2+

+ RT ln a2 Cu2+

+ 2F𝜑2 Cu2+

�̃�1e− = 𝜇0e− + RT ln a1e− − 1F𝜑1e− = 𝜇0e− − 1F𝜑1e−

�̃�2e− = 𝜇0e− + RT ln a2e− − 1F𝜑2e− = 𝜇0e− − 1F𝜑2e−

�̃�1Cu = 𝜇 0 Cu + RT ln a

1 Cu = 𝜇

0 Cu

�̃�2Cu = 𝜇 0 Cu + RT ln a

2 Cu = 𝜇

0 Cu (2.107)

In writing these equations, we’ve used z = +2 for Cu2+ ions, and z = –1 for e–. The activity of electrons in metals is defined as 1, as is the activity of a pure component (Cu), so these terms vanish from the equations. We now apply Equations 2.113 to the overall reaction 2.106, yielding

(𝜇0Cu + 𝜇 0 Cu2+

+ RT ln a2 Cu2+

+ 2F𝜑2 Cu2+

+ 2𝜇0e− − 2F𝜑2e−)

− (𝜇0Cu + 𝜇 0 Cu2+

+ RT ln a1 Cu2+

+ 2F𝜑1 Cu2+

+ 2𝜇0e− − 2F𝜑1e−) = 0 (2.108)

PREDICTING REVERSIBLE VOLTAGE OF A FUEL CELL UNDER NON-STANDARD-STATE CONDITIONS 59

Note that we have multiplied the electrochemical potentials of the electron terms by 2 since in each case the stoichiometric coefficient for electrons is 2. Canceling terms and rearranging the equation yields

RT ln a1 Cu2+

+ 2F𝜑1 Cu2+

− RT ln a2 Cu2+

− 2F𝜑2 Cu2+

= 2F(𝜑1e− − 𝜑2e−) (2.109)

Now comes an important point: The salt bridge connecting the two baths maintains ionic charge equilibrium. In other words, when Cu2+ ions are consumed in bath 1 and created in bath 2, the ion bridge allows counterbalancing SO4

2– ions to move from bath 1 to bath 2, thereby maintaining zero net ionic charge in both baths. Mathematically, this means 𝜑1

Cu2+ = 𝜑2

Cu2+ . Applying this final simplification yields

RT ln a1 Cu2+

a2 Cu2+

= 2F(𝜑1e− − 𝜑2e−) = 2FE (2.110)

where (𝜑1e− − 𝜑2e−) represents the equilibrium electrical potential (voltage) difference established between the two electrodes due to the Cu2+ ion concentration difference between the two baths. This final equation result is the Nernst equation for this concen- tration cell.

2.4.5 Summary

Let us briefly summarize the effects of non-standard-state conditions on reversible elec- trochemical cell voltages. In the past few pages, we have used classical thermodynam- ics to predict how changes in temperature, pressure, and chemical composition affect the reversible voltages of fuel cells. (Incidentally, these relations are equally applicable to all electrochemical systems, not just fuel cells.)

• The variation of the reversible cell voltage with temperature is(dE dT

) p = Δŝ

nF (2.111)

• The variation of the reversible cell voltage with pressure is( dE dp

) T

= − ΔngRT nFp

= −Δ�̂� nF

(2.112)

• The variation of the reversible cell voltage with chemical activity (chemical compo- sition, concentration, etc.) is given by the Nernst equation:

E = E0 − RT nF

∏ a𝑣iproducts∏ a𝑣ireactants

(2.113)

60 FUEL CELL THERMODYNAMICS

The Nernst equation accounts for the pressure effects on reversible cell voltage (it supersedes Equation 2.112) but does not fully account for the temperature effects. When T ≠ T0, E0 in the Nernst equation should be replaced by ET. Importantly, only electrochemically active species appear in the Nernst equation (e.g., O2, H2, and H2O for a H2 fuel cell). The activities or partial pressures of unreactive, inert, or diluent species (such as N2 in air) should not be included.

These equations give us the ability to predict the reversible voltage of a fuel cell under an arbitrary set of conditions.

2.5 FUEL CELL EFFICIENCY

For any energy conversion device, efficiency is of great importance. Central to a discussion of efficiency are the concepts of “ideal” (or reversible) efficiency and “real” (or practical) efficiency. Although you might be tempted to think that the ideal efficiency of a fuel cell should be 100%, this is not true. Just as thermodynamics tells us that the electrical work available from a fuel cell is limited by ΔG, the ideal efficiency of a fuel cell is also limited by ΔG. The story for real fuel cell efficiency is even worse. A real fuel cell must always be less efficient than an ideal fuel cell because real fuel cells incur nonideal irreversible losses during operation. A discussion of real fuel cell efficiency motivates forthcoming chapters, where these non-thermodynamic losses are discussed.

2.5.1 Ideal Reversible Fuel Cell Efficiency

We define the efficiency, 𝜀, of a conversion process as the amount of useful energy that can be extracted from the process relative to the total energy evolved by that process:

𝜀 = useful energy

total energy (2.114)

If we wish to extract work from a chemical reaction, the efficiency is

𝜀 = work Δĥ

(2.115)

For a fuel cell, recall that the maximum amount of energy available to do work is given by the Gibbs free energy. Thus, the reversible efficiency of a fuel cell can be written as

𝜀thermo, fc = Δĝ Δĥ

(2.116)

At room temperature and pressure, the H2–O2 fuel cell has Δĝ0 = −237.17 kJ/mol and Δĥ0HHV = −285.83 kJ/mol. This yields a 83% reversible HHV efficiency for the H2–O2 fuel cell at STP:

𝜀thermo, fc = −237.17 −285.83

= 0.83 (2.117)

FUEL CELL EFFICIENCY 61

In contrast to a fuel cell, the maximum theoretical efficiency of a conventional heat/expansion engine is described by the Carnot cycle. This efficiency may be derived from classical thermodynamics. We do not repeat the derivation here, but we provide the result:

𝜀Carnot = TH − TL

TH (2.118)

HIGHER HEATING VALUE EFFICIENCY

To convert water from the liquid to the vapor state requires heat input. The quantity of heat required is called the latent heat of vaporization. Due to this latent heat of vapor- ization, the Δĥrxn for a hydrogen–oxygen fuel cell is significantly different, depending on whether vapor or liquid water product is assumed. When liquid water is produced, Δĥ0rxn = −286 kJ/mol; when water vapor is produced, Δĥ0rxn = −241 kJ/mol. Basically, the difference between these two numbers tells us that more total heat is recoverable if the product water can be condensed to the liquid form. The extra heat recovered by condensing steam to liquid water is precisely the latent heat of vaporization. Because condensation to liquid water results in more heat recovery, the Δĥ0rxn involving liquid water is called the higher heating value (HHV), while the Δĥ0rxn involving water vapor is called the lower heating value (LHV).

Which of these values should be used in computing a fuel cell’s efficiency? The most equitable calculations of fuel cell efficiency use the HHV. Using the HHV instead of the LHV is appropriate because it acknowledges the true total heat that could theoretically be recovered from the hydrogen combustion reaction. Use of the LHV will result in higher, but perhaps misleading, efficiency numbers.

All calculations and examples in this bookwill make use of the HHV. Thus, we should rewrite Equation 2.116 to explicitly reflect this fact:

𝜀thermo, fc = Δĝ

ΔĥHHV (2.119)

In these efficiency calculations, it is important to note that Δĝ should still be calcu- lated by properly accounting for phase transitions. Thus, for a hydrogen–oxygen fuel cell operating above 100∘C, the calculation of Δĝ should use formation enthalpies and entropies for water vapor. Below 100∘C, the calculation of ΔĥHHV should use the for- mation enthalpies and entropies for liquid water. You should recognize that calculating Δĝ based on water vapor above 100∘C, while simultaneously using ΔĥHHV (based on liquid water) for efficiency calculations, does not represent a contradiction. What this calculation says is that, in a fuel cell operating above 100∘C, we are losing the ability to convert the latent heat of vaporization of the product water into useful work.

In this expression, TH is the maximum temperature of the heat engine and TL is the rejection temperature of the heat engine. For a heat engine that operates at 400∘C (673 K) and rejects heat at 50∘C (323 K), the reversible efficiency is 52%.

62 FUEL CELL THERMODYNAMICS

Carnot cycle

H2/O2 fuel cell

R ev

er si

bl e

ef fic

ie nc

y (H

H V

)

Temperature (°C)

Temperature (K)

300 400

400

500 600 700 800 900 1000

1100 1200 1.00

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20 100 200 300

Figure 2.8. Reversible HHV efficiency of H2–O2 fuel cell compared to reversible efficiency of heat engine (Carnot cycle, rejection temperature 273.15 K). Fuel cells hold a significant thermodynamic efficiency advantage at low temperature but lose this advantage at higher temperatures. The kink in the fuel cell efficiency curve at 100∘C arises from the entropy difference between liquid water and water vapor (consider the H2O(l) vs. H2O(g) curves from Figure 2.5).

From the Carnot equation, it is apparent that the reversible efficiency of a heat engine improves as the operating temperature increases. In contrast, the reversible efficiency of a fuel cell tends to decrease as the operating temperature increases.

As an example, the reversible HHV efficiency of an H2–O2 fuel cell is compared to the reversible efficiency of a heat engine as a function of temperature in Figure 2.8. Fuel cells hold a significant thermodynamic efficiency advantage at low temperature but lose this advantage at higher temperatures. Note the kink in the fuel cell efficiency curve at 100∘C. This change in slope arises from the entropy difference between liquid water and water vapor.

2.5.2 Real (Practical) Fuel Cell Efficiency

As mentioned previously, the real efficiency of a fuel cell must always be less than the reversible thermodynamic efficiency. The two major reasons are:

1. Voltage losses

2. Fuel utilization losses

FUEL CELL EFFICIENCY 63

The real efficiency of a fuel cell, 𝜀real, may be calculated as

𝜀real = (𝜀thermo) × (𝜀voltage) × (𝜀fuel) (2.120)

where 𝜀thermo is the reversible thermodynamic efficiency of the fuel cell, 𝜀voltage is the volt- age efficiency of the fuel cell, and 𝜀fuel is the fuel utilization efficiency of the fuel cell. Each of these terms is briefly discussed:

• The reversible thermodynamic efficiency, 𝜀thermo, was described in the previous section. It reflects how, even under ideal conditions, not all the enthalpy contained in the fuel can be exploited to perform useful work.

• The voltage efficiency of the fuel cell, 𝜀voltage, incorporates the losses due to irre- versible kinetic effects in the fuel cell. Recall from Section 1.7 that these losses are captured in the operational i–V curve of the fuel cell. The voltage efficiency of a fuel cell is the ratio of the real operating voltage of the fuel cell (V) to the thermodynami- cally reversible voltage of the fuel cell (E):

𝜀voltage = V E

(2.121)

Note that the operating voltage of a fuel cell depends on the current (i) drawn from the fuel cell, as given by the i–V curve. Therefore, 𝜀voltage will change depending on the current drawn from the cell. The higher the current load, the lower the voltage efficiency. Therefore, fuel cells aremost efficient at low load. This is in direct contrast to combustion engines, which are generally most efficient at maximum load.

• The fuel utilization efficiency, 𝜀fuel, accounts for the fact that not all of the fuel pro- vided to a fuel cell will participate in the electrochemical reaction. Some of the fuel may undergo side reactions that do not produce electric power. Some of the fuel will simply flow through the fuel cell without ever reacting. The fuel utilization efficiency, then, is the ratio of the fuel used by the cell to generate electric current versus the total fuel provided to the fuel cell. If i is the current generated by the fuel cell (A) and 𝑣fuel is the rate at which fuel is supplied to the fuel cell (mol/s), then

𝜀fuel = i∕nF 𝑣fuel

(2.122)

If an overabundance of fuel is supplied to a fuel cell, it will be wasted, as reflected in 𝜀fuel. Fuel cells are typically operated in either a constant-flow-rate condition, or a constant-stoichiometry condition. In the constant-flow-rate condition, a constant amount of fuel is supplied to the cell regardless of how much it actually needs at a particular current density. Typically, sufficient fuel is provided to ensure that the cell is not starved at maximum current density. However, this means that significant amounts of fuel will be wasted when the fuel cell is operating at lower current densities.

More often, the supply of fuel to a fuel cell is adjusted according to the current so that the fuel cell is always supplied with just a bit more fuel than it needs at any load. Fuel cells operated in this manner are at constant stoichiometry. For example, a fuel

64 FUEL CELL THERMODYNAMICS

cell supplied with 1.5 timesmore fuel than would be required for 100% fuel utilization is operating at 1.5 times stoichiometric. (The stoichiometric factor λ for this fuel cell is 1.5.) For fuel cells operating under a stoichiometric condition, fuel utilization is independent of current, and we can write the fuel utilization efficiency as

𝜀fuel = 1 𝜆

where 𝜆 = ( 𝜈fuel

i∕nF

) (2.123)

Combining effects of thermodynamics, irreversible kinetic losses, and fuel utilization losses, we can write the practical efficiency of a real fuel cell as

𝜀real =

( Δĝ

ΔĥHHV

)(V E

)( i∕nF 𝑣fuel

) (2.124)

For a fuel cell operating under a constant-stoichiometry condition, this equation simpli- fies to

𝜀real =

( Δĝ

ΔĥHHV

)(V E

)(1 𝜆

) (2.125)

As illustrated in Figure 2.9, operation under a constant-stoichiometry condition versus a constant-flow-rate condition has significant repercussions on fuel cell efficiency. Under a

Current density (A/cm2)

C el

l v ol

ta ge

( V

)

E ffi

ci en

j–V curve

ε, constant stoichiometry (λ=1.1)

ε, constant flow rate (110% max fuel consumption)

0.8

0.6

0.4

0.2

0 0

0.8

0.6

0.4

0.2

0.1

0.3

0.5

0.5 1.5

0.7

0.9

Figure 2.9. Fuel cell efficiency under constant-stoichiometry versus constant-flow-rate conditions. Under a constant-stoichiometry condition (λ = 1.1), the fuel cell efficiency curve follows the fuel cell j–V curve, and efficiency is highest at low current density. Under a constant-flow-rate condition (in this case, 110% of the rate required at maximum current), fuel cell efficiency is poor at low current densities (because most of the fuel is wasted) and reaches a maximum at high current densities when most of the fuel is used.

THERMAL AND MASS BALANCES IN FUEL CELLS 65

constant-stoichiometry condition, the fuel cell efficiency curve follows the shape of the fuel cell j–V curve (because the fuel flow rate is constantly adjusted to match the fuel cell current), and therefore efficiency is highest at low current density. In contrast, under a constant-flow condition, efficiency is lowest at low current density because most of the fuel is wasted. In general, then, constant-stoichiometry operation is preferred under most circumstances, but this requires a system control scheme so that the fuel flow rate can be continuously adjusted to match the fuel cell current.

2.6 THERMAL AND MASS BALANCES IN FUEL CELLS

A fuel cell is an energy conversion device, not an energy creation device (energy cre- ation would violate the first law of thermodynamics). A fuel cell converts chemical energy into electrical energy (with some inevitable waste heat, dictated, as we have learned, by entropy and the second law of thermodynamics). A hydrogen fuel cell, for example, con- sumes hydrogen and oxygen to generate water, heat, and electricity. Although hydrogen and oxygen are consumed during operation, water, heat, and electricity are produced in corre- spondingly proportionate quantities such that the laws of energy and mass conservation are maintained. It is important to be able to account for the exact quantities of fuel, oxidant, water, heat, and electricity entering and/or leaving a fuel cell. Fortunately, this thermal and mass balance accounting can be straightforwardly conducted by applying the laws of mass and energy conservation.

FromEquation 2.63, the rate of consumption of reactant, 𝜈 (mol/s), in a fuel cell is related to the current, i, via

i = Q∕s = nF𝑣 (2.126)

If we know the enthalpy of the reactant fuel, Δĥ (J/mol), the rate of energy input, Pin (J/s), into the fuel cell is

Pin = |Δĥ|𝑣 = Ph + Pe = Ph + V × i (2.127) Here Ph (J∕s), Pe (J∕s), V (V), and i (A) stand for the heat production rate, output

electrical power, operating voltage, and operating current of the fuel cell, respectively. Equation 2.127 is a simple but important energy balance equation that describes how the input fuel energy into a fuel cell is converted into a mixture of electrical energy and heat. Combining Equations 2.126 and 2.127, we have

Ph = Pin − Pe = |Δĥ|𝑣 − V × i = ⎛⎜⎜⎝ 𝜆 |||Δĥ||| nF

− V ⎞⎟⎟⎠ × i = (𝜆EH − V) × i

(2.128)

where λ is the stoichiometry factor. Recall from the previous section of this chapter that λ describes how much fuel is delivered to the fuel cell compared to the stoichiometric amount required for operation at current i (𝜆 = nF𝑣∕i). From this equation, we can determine how much heat a fuel cell generates when it produces electricity at a specified

66 FUEL CELL THERMODYNAMICS

voltage, V, and current, i. The term EH = |Δĥ| nF

in Equation 2 is known as the “thermoneu- tral voltage.” EH represents an “ideal” voltage calculated from the enthalpy of reaction, similarly to how the ideal reversible voltage of a fuel cell (E0 = |Δĝ|

nF ) is calculated from the

Gibbs free energy of reaction. Even though EH does not have any direct physical meaning in a fuel cell, it is extremely useful for calculating the magnitude of heat release from a fuel cell. The difference between reaction enthalpy input into the fuel cell and electrical power output from the fuel cell must be dissipated as heat. By converting the reaction enthalpy term into a “hypothetical” voltage, this heat loss can then be schematically represented on the fuel cell j–V curve as shown in Figure 2.10.

As an example, for a hydrogen fuel cell at STP, we can calculate

EH = |Δĥ| nF

= 286, 000 J∕mol

2 × 96, 485 C∕mol = 1.48 V

If this fuel cell is operating at 0.7 V and 10 A under STP conditions with 100% fuel utiliza- tion (λ = 1), it generates 7W of electrical power (Pe = 0.7 V × 10 A = 7 W) and 7.8W of heat [Ph = (1.48 V – 0.7 V) × 10 A = 7.8 W using Equation 2.128]. As is the case with many practical fuel cells, this fuel cell actually generates more heat than power!

Because heat generation in fuel cells is significant, heat removal must almost always be designed into fuel cell systems. Heat can be removed from a fuel cell by (1) coolant flowing through the fuel cell, (2) unused but heated fuel and oxidant exiting the fuel cell, and/or (3) heat conduction or radiation from the fuel cell to the environment. Heat management in fuel cells is discussed in more detail in Chapter 12.

iout

Current (A)

V ol

ta ge

( V

)

EH-V

iin

Figure 2.10. Thermal balance in a fuel cell. The difference between the operation voltage V and an “imaginary” thermoneutral voltage calculated from the enthalpy of reaction (EH = |Δĥ|

nF ) represents

the total energy loss in a fuel cell. This energy is converted to heat. The input, consumption, and output fluxes of reactants can be converted to equivalent currents to satisfy mass balance.

THERMODYNAMICS OF REVERSIBLE FUEL CELLS 67

Most fuel cells are supplied with more fuel and oxidant than they consume. Excess fuel and oxidant are provided to the cell because depletion effects inside a fuel cell can degrade performance or even permanently damage fuel cell structures. Unused reactants simply exit the fuel cell, carrying some of the fuel cell’s heat with them. For a given species, overall mass balance requires that the amount coming out of the fuel cell must be equal to the amount going into the fuel cell plus or minus any amount which is produced/consumed within the fuel cell:

𝑣out = 𝑣in ± i nF

(2.129)

Here, vin (mol/s) and vout (mol/s) represent the molar input flow rate and output flow rate of a species, respectively, and the i/nF term accounts for production/consumption of that species within the fuel cell; the negative sign applies if the species is consumed in the fuel cell, while the positive sign applies if the species is produced within the fuel cell.

For example, consider a H2/air fuel cell that generates 1000 kA and is supplied with air at 20 mol/s. Using Equation 2.129, we can find the oxygen output flux from the fuel cell:

𝑣O2,out = 𝑣O2,in − i nF

= 𝑣Air,in ×𝑤O2 − i nF

= 20mol∕s × 0.21 − 1, 000, 000A 4 × 96, 485C∕mol

= 1.6mol∕s (2.130)

Here, 𝑤O2 represents the molar fraction of oxygen in air (=0.21). Please note that n = 4 in this calculation since one O2 molecule accepts four electrons. In comparison, the water generation rate (or hydrogen consumption rate) for this fuel cell would be(

𝑣H2O = 𝑣H2 = i nF

= 1000 kA 2 × 96485 C∕mol

) = 5.18 mol∕s

The input and output flow rate of reactants can be converted to equivalent current using Equation 2.126 and plotted in the polarization curve (see Figure 2.10). For example, air supply at 20 mol/s would be sufficient to generate up to 1621 kA (nF𝑣O2 = nF𝑣air𝑤O2 = 4 × 96, 485 C∕s × 20 mol∕s × 0.21 = 1621 kA) for a hydrogen fuel cell. Since the fuel cell generates 1000 kA with this supply of oxygen but could generate as much as 1621 kA, the air stoichiometric factor must be 1.62 (1621 kA/1000 kA = 1.62).

2.7 THERMODYNAMICS OF REVERSIBLE FUEL CELLS

Certain fuel cells can be designed to operate in either the forward or reverse direction. In other words, they can operate under the “fuel cell” mode, converting hydrogen and oxygen to water and electricity, or under the “electrolyzer mode,” converting water and electricity to hydrogen and oxygen. The two modes are contrasted in Equation 2.131 below:

Fuel cell mode: H2 + 1 2 O2 → H2O + Electricity

Electrolyzer mode: H2O + Electricity → H2 + 1 2 O2 (2.131)

68 FUEL CELL THERMODYNAMICS

A fuel cell that can run in both directions is known as a reversible fuel cell. Under the electrolysis mode, efficiency is calculated as the chemical energy (enthalpy) of the fuel produced by the system divided by the electrical energy supplied to the system. Thus the maximum ideal (thermodynamic) limit for electrolyzer efficiency is given by

𝜂thermo,electrolyzer = Δĥ Δĝ

(2.132)

For water electrolysis at room temperature and pressure, we have Δĝ0 = 237.17 kJ/mol and Δĥ0HHV = 286 kJ/mol, respectively (these are simply the reverse of the values for the fuel cell mode of operation). This implies a 120% reversible HHV efficiency for water electrolysis at STP!

𝜀thermo,electrolyzer = 286 237

= 1.2 (2.133)

How is it possible that the ideal thermodynamic efficiency for water electrolysis is greater than 100%? The answer comes from the direction of the irreversible entropic heat flow under the electrolyzer mode as compared to the fuel cell mode (i.e., the TΔs term). Under H2–O2 fuel cell operation, the amount of electricity produced (as given by Δg) is less than the amount of chemical energy supplied (as given by Δh) due to irreversible entropic heat losses to the environment (quantified by TΔs). However, in the electrolyzer mode, the situation is reversed. The amount of electricity required for electrolysis (as given by Δg) is less than the amount of chemical energy produced (as given by Δh) due to irreversible entropic heat contributions from the environment (quantified by TΔs). Thus, electrolysis has the potential to achieve greater than 100% efficiency (based on our definition of efficiency) because heat from the environment is used in the process of splitting water into hydrogen. This can be quantified if we substitute the relationship Δg = Δh – TΔs into Equation 2.132:

𝜂thermo,electrolyzer = Δĥ Δĝ

= Δĥ Δĥ − TΔs

(2.134)

It should be noted that the >100% thermodynamic efficiency for water electrolysis is not in violation of thermodynamic principles. In a reversible fuel cell, the entropic losses incurred under the fuel cell mode of operation exactly offset the entropic gains associ- ated with the electrolyzer mode of operation, such that the overall ideal thermodynamic round-trip efficiency involved in splitting water with electricity and then making electricity with the produced hydrogen is exactly 100%. In other words,

𝜀thermo,electrolyzer × 𝜀thermo, fc = 1.2 × 0.83 = 1.0 (2.135)

In reality, the actual efficiency of even very good electrolyzers is generally less than 100% for many of the same reasons that the practical efficiency of fuel cells is less than the ther- modynamic limit. These idealities cause the operating voltage of a practical electrolyzer to be higher than the ideal STP thermodynamic voltage of 1.23 V (typically 1.4 V or higher is applied for electrolysis), indicating that more electricity is required to split water than the ideal thermodynamic prediction. Meanwhile, the voltage that is produced when this

THERMODYNAMICS OF REVERSIBLE FUEL CELLS 69

hydrogen is consumed in the fuel cell mode is inevitably less than the ideal STP thermody- namic voltage of 1.23 V (typically less than 1 V). Thus, the practical round-trip efficiency of combined electrolysis + fuel cell operation is inevitably far less than 100%.

2.7.1 Heat Balance in Reversible Fuel Cells

In Section 2.6, we discussed fuel cell heat and mass balance. However, for a reversible fuel cell operating under the electrolysis mode operation, there are subtle heat balance differ- ences. Figure 2.11 illustrates these differences.

As discussed in Section 2.6, the heat balance of a fuel cell can be directly visualized on the j–V curve by comparing the operating voltage, V, versus the thermoneutral voltage, EH = |Δĥ|

nF . In the fuel cell mode, there is a net production of heat given by the difference

between EH and V. However, upon switching from the fuel cell mode to the electrolyzer mode, the situation reverses. At low electrolyzer current densities, there is a net heat con- sumption by the electrolyzer. The heat consumption of the electrolyzer can be visualized by the difference between electrical power supplied to the electrolyzer (as given by the oper- ating voltage V and current i) versus the chemical “power” produced by the electrolyzer (as given by EH and i):

Ph,electrolysis = Pe, in − Pchem, out = V × i − |Δĥ| inF = ( V − |Δĥ|

) × i = (V − EH) × i

(2.136)

In this analysis, the Faradaic efficiency of the electrolyzer is assumed to be 100%. This means that 100% of the current supplied to the electrolyzer is assumed to produce hydro- gen fuel.

As can be seen in this equation, and also in Figure 2.11, there is a net consumption of heat at low current densities when the operating voltage of the electrolyzer, V, is below the thermoneutral voltage, EH. However, above the thermoneutral voltage, net heat is pro- duced in the electrolysis mode because entropic heat consumption is more than offset by irreversible heat production due to activation, ohmic, and mass transport losses in the elec- trolyzer. Maintaining system temperature during electrolysis under endothermic (net heat consumption) conditions can be difficult. Thus, most electrolyzers are designed to operate at or above the thermoneutral voltage.

Figure 2.12 illustrates a final key difference between fuel cell and electrolysis modes of operation. As was illustrated in Figure 2.8, the ideal thermodynamic efficiency of a H2–O2 fuel cell decreases with increasing temperature due to increasing irreversible entropic losses (TΔs losses). As shown in Figure 2.12, the situation is reversed for an electrolyzer. Thus, the ideal thermodynamic efficiency of an electrolyzer increaseswith increasing temperature. At the same time, kinetic and mass transport losses tend to decrease at high temperatures (just as in fuel cell operation). Thus, for situations where high-quality waste heat is available, high-temperature electrolysis is an interesting option as it can provide the opportunity for high-efficiency operation.

70 FUEL CELL THERMODYNAMICS

Figure 2.11. Thermal balance in a reversible fuel cell illustrating both the fuel cell and electrolyzer domains of operation. Under fuel cell operation, the difference between the operation voltage V and

the thermoneutral voltage EH (EH = |Δĥ| nF

) represents the heat loss in the fuel cell. Under the elec- trolyzer mode of operation, there is a net consumption of heat at low current densities when the operating voltage of the electrolyzer, V, is below the thermoneutral voltage, EH. However, above the thermoneutral voltage, net heat is produced in the electrolysis mode because entropic heat consump- tion is fully offset by irreversible heat production due to activation, ohmic, and mass transport losses in the electrolyzer. Maintaining system temperature during electrolysis under endothermic (net heat consumption) conditions can be difficult. Thus, most electrolyzers are designed to operate at or above the thermoneutral voltage.

CHAPTER SUMMARY 71

Figure 2.12. Reversible HHV efficiency of H2O electrolysis compared to an H2–O2 fuel cell. The thermodynamic efficiency of electrolysis increases with increasing temperature, while thermody- namic fuel cell efficiency decreases with increasing temperature.

2.8 CHAPTER SUMMARY

The purpose of this chapter is to understand the theoretical limits to fuel cell performance by applying the principles of thermodynamics. The main points introduced in this chapter include the following:

• Thermodynamics provides the theoretical limits or ideal case for fuel cell perfor- mance.

• The heat potential of a fuel is given by the fuel’s heat of combustion or, more generally, the enthalpy of reaction.

• Not all of the heat potential of a fuel can be utilized to perform useful work. The work potential of the fuel is given by the Gibbs free energy, ΔG.

72 FUEL CELL THERMODYNAMICS

• Electrical energy can only be extracted from a spontaneous (“downhill”) chemical reaction. The magnitude of ΔG gives the amount of energy that is available (“free”) to do electrical work. Thus, the sign of ΔG indicates whether or not electrical work can be done, and the size of ΔG indicates how much electrical work can be done.

• The reversible voltage of a fuel cell, E, is related to the molar Gibbs free energy by Δĝ = −nFE.

• ΔG scales with reaction amount whereasΔĝ and E do not scale with reaction amount. • E varies with temperature as dE∕dT = Δŝ∕nF. For fuel cells,Δŝ is generally negative;

therefore, reversible fuel cell voltages tend to decrease with increasing temperature. E varies with pressure as dE∕dp = −ΔngRT∕(nFp) = −Δ�̂�∕nF

• The Nernst equation describes how E varies with reactant/product activities:

E = E0 − RT nF

∏ a𝑣iproducts∏ a𝑣ireactants

• The Nernst equation intrinsically includes the pressure effects on reversible cell volt- age but does not fully account for the temperature effects.

• Ideal HHV fuel cell efficiency 𝜀thermo = Δĝ∕ΔĥHHV. • Thermodynamic fuel cell efficiency generally decreases as temperature increases.

Contrast this to heat engines, for which thermodynamic efficiency generally increases as temperature increases.

• Real fuel cell efficiency is always less than the ideal thermodynamic efficiency. Major reasons are irreversible kinetic losses and fuel utilization losses. Total overall effi- ciency is given by the product of individual efficiencies.

• A fuel cell satisfies the laws of energy and mass conservation. Accordingly, the ther- mal and mass balance of a fuel cell can be obtained from input, output, and conversion fluxes of energy and mass in the fuel cell.

CHAPTER EXERCISES

Review Questions

2.1 If an isothermal reaction involving gases exhibits a large negative volume change, will the entropy change for the same reaction likely be negative or positive? Why?

2.2 (a) If Δĥ for a reaction is negative and Δŝ is positive, can you say anything about the spontaneity of the reaction? (b) What if Δĥ is negative and Δŝ is negative? (c) What if Δĥ is positive and Δŝ is negative? (d) What if Δĥ is positive and Δŝ is positive?

2.3 Reaction A has Δĝrxn = −100 kJ/mol. Reaction B has Δĝrxn = −200 kJ/mol. Can you say anything about the relative speeds (reaction rates) for these two reactions?

2.4 Why doesΔG for a reaction scale with reaction quantity but E does not? For example, ΔG0rxn for the combustion of 1 mol of hydrogen is 1 × –237 kJ∕mol = –237 kJ,

CHAPTER EXERCISES 73

while ΔG0rxn for the combustion of 2 mol of hydrogen is 2 × –237 kJ∕mol = –474 kJ. In both cases, however, the reversible cell voltage produced by the reaction, E0, is 1.23 V.

2.5 In general, will increasing the concentration (activity) of reactants increase or decrease the reversible cell voltage of an electrochemical system?

2.6 Derive the Nernst equation starting from Equation 2.101 for a general chemical reac- tion of the form

zeA + 1A + bB ⇌ mM + nN + zeC (2.137)

2.7 Can the thermodynamic efficiency of a fuel cell, as defined by 𝜀 = Δĝ∕Δĥ, ever be greater than unity? Explain why or why not. Consider all fuel cell chemistries, not just H2–O2 fuel cells.

2.8 Assume x moles per second of methanol and y moles per second of air are supplied to a direct methanol fuel cell (DMFC) generating a current of i amperes at a volt- age V (volts). (a) Write expressions for the output mass flux (mol/s) of methanol (𝑣MeOH, out), air (𝑣air, out), water (𝑣H2O, out), and carbon dioxide (𝑣CO2, out) using the given variables. (b) Write expressions for the stoichiometric factors for methanol (𝜆MeOH) and air (𝜆air) using the given variables. (Clearly indicate numeric values for n in all cases.)

Calculations

2.9 In Example 2.2, we assumed that Δĥrxn and Δŝrxn were independent of tempera- ture. We are now interested in determining how much of an error this assumption introduced into our solution. Rework Example 2.2 assuming constant-heat-capacity values for all species involved in the reaction. Heat capacity values are provided in the following table.

Chemical Species cp (J/mol⋅K)

CO 29.2 CO2 37.2 H2 28.8 H2O(g) 33.6

Note that a more accurate calculation is made by using temperature-dependent heat capacity equations. These equations generally use polynomial series to reflect how the heat capacity changes with temperature. Such calculations are tedious and are now mostly done via computer programs.

2.10 (a) If a fuel cell has a reversible voltage of E1 at p = p1 and T = T1, write an expres- sion for the temperature T2 that would be required to maintain the fuel cell voltage at E1 if the cell pressure is adjusted to p2. (b) For a H2–O2 fuel cell operating at room

74 FUEL CELL THERMODYNAMICS

temperature and atmospheric pressure (on pure oxygen), what temperature would be required to maintain the original reversible voltage if the operating pressure is reduced by one order of magnitude?

2.11 In Section 2.4.4, it was mentioned that you could think of a hydrogen–oxygen fuel cell as simply a hydrogen concentration cell, where oxygen is used to chemically “tie up” hydrogen at the cathode. Oxygen’s ability to chemically tie up hydrogen is mea- sured by the Gibbs free energy of the hydrogen–oxygen reaction. At STP (assuming air at the cathode), what is the effective hydrogen pressure that oxygen is able to chemically maintain at the cathode of a hydrogen–oxygen (air) fuel cell?

2.12 A typical H2–O2 PEMFC might operate at a voltage of 0.75 V and λ = 1.10. At STP, what is the efficiency of such a fuel cell (use HHV and assume pure oxygen at the cathode)?

2.13 A direct methanol fuel cell generates 1000 A at 0.3 V at STP. Methanol and air are supplied to the fuel cell at 0.003 and 0.03 mol/s, respectively. Calculate (a) the output mass flux (mol/s) of methanol (𝑣MeOH, out), air (𝑣air, out), water (𝑣H2O, out), and carbon dioxide (𝑣CO2, out); (b) the stoichiometric factors for methanol (𝜆MeOH) and air (𝜆air); and (c) the heat generation rate (J/s) for this fuel cell assuming Δĥrxn = –719.19 kJ/mol for methanol combustion at STP.

2.14 You are provided with a fuel cell that is designed to operate at j = 3 A∕cm2 and P = 1.5 W∕cm2. How much fuel cell active area (in cm2) is required to deliver 2 kW of electrical power? (This is approximately enough to provide power to the average American home.)

(a) 296.3 cm2

(b) 1333.3 cm2

(d) 666.6 cm2

2.15 For the fuel cell described above in problem 2.14, assuming operation on pure hydro- gen fuel, how much water would be produced during 24 hours of operation at P = 2 kW? (Recall: molar mass of water = 18 g/mol, density of water = 1 g/cm3.) (a) 0.49 L

(b) 10.7 L

(d) 66.3 L

2.16 Given a fuel cell with the following overall reaction: 3A(g) + 2B(g) → 2C(g), how will uniformly increasing the cell pressure affect the thermodynamic voltage?

(a) E decreases.

(b) E increases.

(d) This cannot be determined.

CHAPTER EXERCISES 75

2.17 Given the following half-cell reactions: 1. O2− + CO(g) → CO2(g) + 2e−

2. 2O2− → 4e− + O2(g) 3. 8e− + 2H2O(g) + CO2(g) → 4O2− + CH4(g) 4. 1

2 O2(g) + H2O(g) + 2e− → 2(OH)−

(a) Using two of these half reactions, write a balanced full-cell reaction for a fuel cell (consumes fuel and oxygen). Identify which reaction is occurring at the anode and which at the cathode.

(b) Using two of these half reactions, write a balanced full-cell reaction for an elec- trolysis cell (makes fuel and oxygen). Identify which reaction is occurring at the anode and which at the cathode.

2.18 A residential solid-oxide fuel cell is operated on methane (CH4) and is designed to provide the household with both heat and electricity.

(a) Assuming that the fuel cell is operated at j = 1 A∕cm2 and V = 0.6 V, howmuch fuel cell active area (in cm2) would be required to deliver 3 kW of electrical power? (This is approximately enough to provide power to the average American home.)

(b) At the fuel cell’s standard operating condition (750∘C, 1 atm), Δh and Δg for methane combustion are –802 and –801 kJ/mol, respectively. (Note: This is not a typo; Δh and Δg are almost equal for this reaction.) Assuming 100% fuel uti- lization, what is the rate of heat generation by the fuel cell (Pheat, in kW) when operated at j = 1 A∕cm2 and V = 0.6 V?

(c) Assuming 100% fuel utilization, how much water (in liters) would be produced during 24 hours of operation at Pelec = 3 kW? (Recall: molar mass of water = 18 g/mol, density of water = 1 g/cm3.)

(d) Given that the average American household water consumption is ∼200 gal/day (∼ 750 L∕day), would this fuel cell be able to supply the average American household’s entire daily water requirements in addition to its electrical power requirements? (Provide support for your answer.)

CHAPTER 3

FUEL CELL REACTION KINETICS

Having learned what is “ideally” possible with fuel cells in the previous chapter, our journey now enters the realm of the practical, beginning in this chapter with a discussion of fuel cell reaction kinetics. Fuel cell reaction kinetics discusses the nuts and bolts of how fuel cell reactions occur.

At the most fundamental level, a fuel cell reaction (or any electrochemical reaction) involves the transfer of electrons between an electrode surface and a chemical species adjacent to the electrode surface. In fuel cells, we harness thermodynamically favorable electron transfer processes to extract electrical energy (in the form of an electron current) from chemical energy. Previously, in Chapter 2, you learned how to distinguish thermo- dynamically favorable electrochemical reactions. Here, in Chapter 3, we study the kinetics of electrochemical reactions. In other words, we study the mechanisms by which electron transfer processes occur. Because each electrochemical reaction event results in the trans- fer of one or more electrons, the current produced by a fuel cell (number of electrons per time) depends on the rate of the electrochemical reaction (number of reactions per time). Increasing the rate of the electrochemical reaction is therefore crucial to improving fuel cell performance. Catalysis, electrode design, and other methods to increase the rate of the electrochemical reaction will be introduced.

3.1 INTRODUCTION TO ELECTRODE KINETICS

This section discusses a few basic concepts about electrochemical systems that tend to cause confusion. Crystallize these basic concepts in your mind and you will be on your way to understanding electrochemistry.

78 FUEL CELL REACTION KINETICS

3.1.1 Electrochemical Reactions Are Different from Chemical Reactions

All electrochemical reactions involve the transfer of charge (electrons) between an electrode and a chemical species. This distinguishes electrochemical reactions from chemical reac- tions. In chemical reactions, charge transfer occurs directly between two chemical species without the liberation of free electrons.

3.1.2 Electrochemical Processes Are Heterogeneous

Because electrochemistry deals with the transfer of charge between an electrode and a chemical species, electrochemical processes are necessarily heterogeneous. Electrochemi- cal reactions, like the HOR,

H2 ⇌ 2H + + 2e− (3.1)

can only take place at the interface between an electrode and an electrolyte. In Figure 3.1, it is obvious that hydrogen gas and protons cannot exist inside the metal electrode, while free electrons cannot exist within the electrolyte. Therefore, the reaction between hydrogen, protons, and electrons must occur where the electrode and electrolyte intersect.

3.1.3 Current Is a Rate

Because electrons are either generated or consumed by electrochemical reactions, the cur- rent i evolved by an electrochemical reaction is a direct measure of the rate of the electro- chemical reaction. The unit of current is the ampere; an ampere is a coulomb per second (C∕s). From Faraday’s law,

i = dQ dt

(3.2)

where Q is the charge (C) and t is time. Thus, current expresses the rate of charge transfer. If each electrochemical reaction event results in the transfer of n electrons, then

i = nFdN dt

= nF𝑣 (3.3)

where (dN∕dt = 𝑣) is the rate of the electrochemical reaction (mol∕s) and F is Faraday’s constant. (Faraday’s constant is necessary to convert a mole of electrons to a charge in coulombs.)

2e–

2H+

+ + + +

– – – –

–

– –

–

Electrode Electrolyte

Figure 3.1. Electrochemical reactions are heterogeneous. As this schematic shows, the HOR is a surface-limited reaction. It can take place only at the interface between an electrode and an electrolyte.

INTRODUCTION TO ELECTRODE KINETICS 79

Example 3.1 Assuming 100% fuel utilization, how much current can a fuel cell pro- duce if provisioned with 5 sccmH2 gas at STP? (1 sccm= 1 standard cubic centimeter per minute.) Assume sufficient oxidant is also supplied.

Solution: In this problem, we are provided with a volumetric flow rate of H2 gas. To get current, we need to convert volumetric flow rate into molar flow rate and then convert molar flow rate into current. Treating H2 as an ideal gas, the molar flow rate is related to the volumetric flow rate via the ideal gas law:

𝑣 = dN dt

= p(dV∕dt)

RT (3.4)

where 𝑣 is the molar flow rate and dV∕dt is the volumetric flow rate. At STP

𝑣 = dN dt

= (1 atm)(0.005 L∕min)

[0.082 L ⋅ atm∕(mol ⋅ K)](298.15 K) = 2.05 × 10−4 mol H2∕min (3.5)

Since 2 mol of electrons is transferred for every mole of H2 gas reacted, n = 2. Inserting n and dN∕dt into Equation 3.3 and converting from minutes to seconds give

i = nFdN dt

= (2)(96,485C∕mol)(2.05 × 10−4molH2∕min)(1min∕60s) = 0.659A (3.6)

Thus, a flow rate of 5 sccm H2 is sufficient to sustain 0.659 A of current, assuming 100% fuel utilization.

3.1.4 Charge Is an Amount

If we integrate a rate, we obtain an amount. Integrating Faraday’s law (Equation 3.2) gives

∫ t

0 i dt = Q = nFN (3.7)

The total amount of electricity produced, as measured by the accumulated charge Q in coulombs, is proportional to the number of moles of material processed in the electrochem- ical reaction.

Example 3.2 A fuel cell operates for 1 hour at 2 A current load and then operates for 2 more hours at 5 A current load. Calculate the total number of moles of H2 consumed by the fuel cell over the course of this operation. To what mass of H2 does this correspond? Assume 100% fuel utilization.

Solution: From the time–current profile that we are given, we can calculate the total amount of electricity produced by this fuel cell (as measured by the accumulated charge). Then, using Equation 3.7, we can calculate the total number of moles of H2 processed by the reaction.

80 FUEL CELL REACTION KINETICS

The total amount of electricity produced is calculated by integrating the current load profile over the operation time. For this particular example, the calculation is easy:

Qtot = i1t1 + i2t2 = (2A)(3600s) + (5A)(7200s) = 43,200C (3.8)

Since 2 mol of electrons is transferred for every mole of H2 reacted, n = 2. Thus, the total number of moles of H2 processed by this fuel cell is

NH2 = Qtot nF

= 43200C (2)(96,485C∕mol)

= 0.224molH2 (3.9)

Since the molar mass of H2 is approximately 2 g∕mol, this corresponds to about 0.448g of H2.

3.1.5 Current Density Is More Fundamental Than Current

Because electrochemical reactions only occur at interfaces, the current produced is usually directly proportional to the area of the interface. Doubling the interfacial area available for reaction should double the rate. Therefore, current density (current per unit area) is more fundamental than current; it allows the reactivity of different surfaces to be compared on a per-unit area basis. Current density j is usually expressed in units of amperes per square centimeter (A∕cm2):

j = i A

(3.10)

where A is the area. In a similar fashion to current density, electrochemical reaction rates can also be expressed on a per-unit-area basis. We give per-unit-area reaction rates the symbol J. Area-normalized reaction rates are usually expressed in units of moles per square centimeter per time (mol∕cm2 ⋅ s):

J = 1 A dN dt

= i nFA

= j

nF (3.11)

3.1.6 Potential Controls Electron Energy

Potential (voltage) is a measure of electron energy. According to band theory, the electron energy in a metal is measured by the Fermi level. By controlling the electrode potential, we control the electron energy in an electrochemical system (Fermi level), thereby influenc- ing the direction of a reaction. For example, consider a general electrochemical reaction occurring at an electrode between the oxidized (Ox) and reduced (Re) forms of a chemi- cal species:

Ox + e− ⇌ Re (3.12)

INTRODUCTION TO ELECTRODE KINETICS 81

Electrode Electrolyte ElectrolyteElectrode Electrolyte Electrode

e–

Equilibrium electrode potential

Positive (relative) electrode potential

Negative (relative) electrode potential

Fermi level

Increasing electron energy

Increasing electrode potential (Voltage)

Figure 3.2. Electrode potential can be manipulated to trigger reduction (left) or oxidation (right). The thermodynamic equilibrium electrode potential (middle) corresponds to the situation where the oxidation and reduction processes are balanced.

If the potential of the electrode is made relatively more negative than the equilibrium potential, the reaction will be biased toward the formation of Re. (Consider that a more negative electrode makes the electrode less “hospitable” to electrons, forcing electrons out of the electrode and onto the electroactive species.) On the other hand, if the electrode potential is made relatively more positive than the equilibrium potential, the reaction will be biased toward the formation of Ox. (A more positive electrode “attracts” electrons to the electrode, “pulling” them off of the electroactive species.) Figure 3.2 illustrates this concept schematically.

Using potential to control reactions is key to electrochemistry. Later in this chapter, we develop this principle more fully to understand how rate (and therefore the current produced by an electrochemical reaction) is related to cell voltage.

3.1.7 Reaction Rates Are Finite

It should be obvious that the rate of an electrochemical reaction, or any reaction for that matter, is finite. This means that the current produced by an electrochemical reaction is limited. Reaction rates are finite even if they are energetically “downhill” because an energy barrier (called an activation energy) impedes the conversion of reactants into products. As illustrated in Figure 3.3, in order for reactants to be converted into products, they must first make it over this activation “hill.” The probability that reactant species can make it over

82 FUEL CELL REACTION KINETICS

Reactants (H2 + O2)

F re

e en

er gy

Reaction progress

∆Grxn

∆G‡

Products (H2O)

Figure 3.3. An activation barrier (ΔG‡) impedes the conversion of reactants to products. Because of this barrier, the rate at which reactants are converted into products (the reaction rate) is limited.

this barrier determines the rate at which the reaction occurs. In the next section, we discuss why electrochemical reactions have activation barriers.

3.2 WHY CHARGE TRANSFER REACTIONS HAVE AN ACTIVATION ENERGY

Even reactions as elementary as the HOR actually consist of a series of even simpler basic steps. For example, the overall reaction H2 ⇌ 2H

+ + 2e− might occur by the following series of basic steps:

1. Mass transport of H2 gas to the electrode:( H2(bulk) → H2(near electrode)

) 2. Absorption of H2 onto the electrode surface:(

H2(near electrode) +M → M · · ·H2 )

3. Separation of the H2 molecule into two individually bound (chemisorbed) hydrogen atoms on the electrode surface:

(M · · ·H2) +M → 2(M · · ·H)

4. Transfer of electrons from the chemisorbed hydrogen atoms to the electrode, releas- ing H+ ions into the electrolyte:

2 × [ M · · ·H → (M + e−) + H+(near electrode)

] 5. Mass transport of the H+ ions away from the electrode:

2 × [ H+(near electrode) → H

+ (bulk electrolyte)

]

WHY CHARGE TRANSFER REACTIONS HAVE AN ACTIVATION ENERGY 83

Electrolyte

H+

…H

e–

Electrode (M)

Figure 3.4. Schematic of chemisorbed hydrogen charge transfer reaction. The reactant state, a chemisorbed hydrogen atom (M · · ·H), is shown at 1. Completion of the charge transfer reac- tion, as shown at 2, liberates a free electron into the metal and a free proton into the electrolyte ((M + e−) + H+).

Just as an army can only march as fast as its slowest member, the overall reaction rate will be limited by the slowest step in the series. Suppose that the overall reaction above is limited by the electron transfer step between chemisorbed hydrogen and the metal electrode surface (step 4 above). This step can be represented as

M · · ·H = (M + e−) + H+ (3.13)

In this equation, M · · ·H represents a hydrogen atom chemisorbed on the metal surface and (M + e−) represents a liberated metal surface site and a free electron in the metal. This reaction step is depicted physically in Figure 3.4, while Figure 3.5 illustrates the energetics. First consider curve 1 of Figure 3.5. This curve depicts the free energy of the chemisorbed atomic hydrogen, H, which increases with distance from the metal electrode surface. We know that atomic hydrogen is not very stable; stability improves with chemisorption of the atomic hydrogen to the metal electrode surface. Chemisorption to the metal surface allows the hydrogen to partially satisfy its bonding requirements, lowering its free energy. Separating the atomic hydrogen from the metal surface destroys this bond, thus increasing the free energy.

Now consider curve 2, which depicts the free energy of a H+ ion in the electrolyte. This curve shows that energy is required to bring the H+ ion toward the surface, working against the electrostatic repulsive forces between the charged ion and the anode surface. This energy increases dramatically as the H+ ion is brought closer and closer to the surface because it is energetically unfavorable (due to electrostatic repulsion) for the H+ ion to exist within the metal phase. The free energy of the H+ ion is lowest when it is deep within the electrolyte, far from the metal surface.

The “easiest” (minimum) energy path for the conversion of chemisorbed hydrogen to H+ and (M + e−) is given by the dark solid line in Figure 3.5. Note that this energy path necessarily involves overcoming a free-energy maximum. This maximum occurs because any deviation from the energetically stable reactant and product states involves an increase in free energy (as detailed by curves 1 and 2). The point marked a on the diagram is called the activated state. Species in the activated state have overcome the free-energy barrier; they can be converted into either products or reactants without further impediment.

84 FUEL CELL REACTION KINETICS

(M + e–) + H+

F re

e en

er gy

Distance from interface

∆Grxn

1 2

∆G1 ‡

∆G2 ‡

(M…H)

Figure 3.5. Schematic of energetics of chemisorbed hydrogen charge transfer reaction. Curve 1 shows the free energy of the reactant state ([M · · ·H]) as a function of the distance of separation between the H atom and the metal surface. Curve 2 shows the free energy of the product state ([M + e−) + H+]) as a function of the distance of separation between the H+ ion and the metal surface. The dark line denotes the “easiest” (minimum) energy path for the conversion of [M · · ·H] to [(M + e−) + H+]. The activated state is represented by a.

3.3 ACTIVATION ENERGY DETERMINES REACTION RATE

Only species in the activated state can undergo the transition from reactant to product. Therefore, the rate of conversion of reactants to products depends on the probability that a reactant species will find itself in the activated state. While it is beyond the scope of this book to treat theoretically, statistical mechanics arguments hold that the probability of finding a species in the activated state is exponentially dependent on the size of the activation barrier:

Pact = e −ΔG‡

1 ∕(RT) (3.14)

where Pact is the probability of finding a reactant species in the activated state, ΔG ‡ 1 is the

size of the energy barrier between the reactant and activated states, R is the gas constant, and T is the temperature (K). Starting from this probability, we can describe a reaction rate as a statistical process involving the number of reactant species available to participate in the reaction (per-unit reaction area), the probability of finding those reactant species in the activated state, and the frequency at which those activated species decay to form products:

J1 = c∗R × f1 × Pact

= c∗R f1e −ΔG‡

1 ∕(RT)

(3.15)

where J1 is the reaction rate in the forward direction (reactants→ products), c ∗ R is the reac-

tant surface concentration (mol∕cm2), and f1 is the decay rate to products. The decay rate

CALCULATING NET RATE OF A REACTION 85

to products is given by the lifetime of the activated species and the likelihood that it will convert to a product instead of back to a reactant. (A species in the activated state can “fall” either way.) More details on the decay rate are presented in a discussion box.

MORE ON TEMPERATURE EFFECTS (OPTIONAL)

At high overvoltage levels, increasing the temperature can actually decrease the current density. How is this possible?While increasing temperature increases j0, it has the oppo- site effect on the activation overvoltage. At high enough overvoltage levels, this “bad” temperature effect actually outweighs the “good” temperature effect. Since this rever- sal only occurs at high overvoltage levels, we can use the Tafel approximation of the Butler–Volmer equation to further discuss the situation:

j = j0e𝛼nF𝜂act∕(RT) (3.45)

If we then incorporate the temperature effect of j0 and lump all the non-temperature- dependent terms into a constant, A, we get

j = Ae−ΔG ‡ 1 ∕(RT)e𝛼nF𝜂act∕(RT) (3.46)

From this equation, it is apparent that the current density j will increase with increasing temperature when 𝛼nF𝜂act < ΔG

‡ 1, but the current density will decrease with

increasing temperature when 𝛼nF𝜂act > ΔG ‡ 1. In other words, for activation overvoltage

levels greater than ΔG‡1∕𝛼nF, increasing the temperature is no longer helpful; instead, it causes the current density to decrease.

100 FUEL CELL REACTION KINETICS

This subtle temperature effect is seldom seen experimentally. Other positive effects of increasing the temperature (such as improvements in ion conductivity and mass trans- port) usually outweigh this reaction kinetics effect. Nonetheless, the phenomenon pro- vides an interesting side note that highlights the complexity of electrochemical reaction kinetics.

3.10 DIFFERENT FUEL CELL REACTIONS PRODUCE DIFFERENT KINETICS

As was previously mentioned, the Butler–Volmer equation applies in general to all simple electrochemical reactions. Variations between reactions can be treated in terms of varia- tions in the kinetic parameters 𝛼, j0, and n. Sluggish reaction kinetics (low 𝛼 and j0 values) result in severe performance penalties, while fast reaction kinetics (high 𝛼 and j0 values) result in minor performance penalties. As an example, consider the basic H2–O2 fuel cell. In an H2–O2 fuel cell, the HOR kinetics are extremely fast, while the ORR kinetics are extremely slow. Therefore, the bulk of the activation overvoltage loss occurs at the cath- ode, where the ORR takes place. The difference between the anode and cathode activation losses in a typical low-temperature H2–O2 fuel cell is illustrated in Figure 3.13.

The ORR is sluggish because it is complicated. Completion of the ORR requires many individual steps and significant molecular reorganization. In comparison, the HOR is rela- tively straightforward. The contrast between H2 and O2 kinetics is highlighted in Tables 3.1 and 3.2, which present lists of j′0 values for the HOR and ORR at a variety of smooth metal surfaces. Although Pt surfaces are most active for both reactions, the j′0 values for the ORR are still at least six orders of magnitude lower than for the HOR. Furthermore, most fuel cells run on air instead of pure oxygen. Although you saw in the previous chapter that air operation does not cause a significant thermodynamic penalty, it does cause a significant kinetic penalty. Because the oxygen concentration shows up in either the Butler–Volmer equation or j0 (depending on which version of the Butler–Volmer equation you choose),

1000

1.2

0.5

Anode activation loss

Cathode activation loss

C el

l v ol

ta ge

( V

)

Current density (mA/cm2)

Theoretical EMF or ideal voltage

Figure 3.13. Relative contributions to activation loss fromH2–O2 fuel cell anode versus cathode. The bulk of the activation overvoltage loss occurs at the cathode due to the sluggishness of the oxygen reduction kinetics.

DIFFERENT FUEL CELL REACTIONS PRODUCE DIFFERENT KINETICS 101

TABLE 3.1. Standard-State (T ≈ 300 K, 1 atm) Exchange Current Densities for Hydrogen Oxidation Reaction on Various Metal Surfaces

Surface Electrolyte j′0 (A/cm 2)

Pt Acid 10−3

Pt Alkaline 10−4

Pd Acid 10−4

Rh Alkaline 10−4

Ir Acid 10−4

Ni Alkaline 10−4

Ni Acid 10−5

Ag Acid 10−5

W Acid 10−5

Au Acid 10−6

Fe Acid 10−6

Mo Acid 10−7

Ta Acid 10−7

Sn Acid 10−8

Al Acid 10−10

Cd Acid 10−12

Hg Acid 10−12

Note:Rounded to nearest decade. Values are normalized per real unit surface area of metal [4, 5].

operation in air (which is only approximately one-fifth oxygen) causes an additional 5× kinetic penalty compared to operation on pure oxygen.

Because the HOR is straightforward and kinetically fast, there is a significant kinetic advantage to using hydrogen fuel. When more complex hydrocarbon fuels are used, the anode kinetics become just as complicated and sluggish as the cathode kinetics, if not more so. Furthermore, fuels that involve carbon tend to generate undesirable intermedi- ates that “poison” the fuel cell. The most serious of these for low-temperature fuel cells is CO. Carbon monoxide permanently absorbs onto platinum, clogging up reaction sites. The CO-passivated Pt surface is thus poisoned, and the desired electrochemical reactions no longer occur.

Many of these kinetic problems are resolved in high-temperature fuel cells. For SOFCs, CO can act as a fuel rather than a poison. Furthermore, high temperature improves the oxy- gen kinetics, dramatically reducing the oxygen activation losses. The reactivity of hydrocar- bon fuels also improves. Even in high-temperature fuel cells, however, poisoning can occur, most notably sulfur poisoning and carbon “coking,” which occurs when carbon deposits that are left behind by hydrocarbon fuels build up on the electrode and catalyst surfaces.

102 FUEL CELL REACTION KINETICS

TABLE 3.2. Standard-State (T ≈ 300 K, 1 atm) Exchange Current Densities for Oxygen Reduction Reaction on Various Surfaces

Surface Electrolyte j′0 (A/cm 2)

Metal Surfaces in Acid Electrolyte

Pt Acid 10−9

Pd Acid 10−10

Ir Acid 10−11

Rh Acid 10−11

Au Acid 10−11

Pt Alloys in PEMFC

Pt–C Nafion 3 × 10−9

PtMn–C Nafion 6 × 10−9

PtCr–C Nafion 9 × 10−9

PtFe–C Nafion 7 × 10−9

PtCo–C Nafion 6 × 10−9

PtNi–C Nafion 5 × 10−9

Note: Values are normalized per real unit surface area of metal. The exchange current density for the ORR is orders of magnitude smaller than for the HOR, although the same group of metals shows the highest activity for both reactions. Pt alloys may show a slight performance enhancement over pure Pt in a PEMFC environment [6].

Not only do fuel cell reaction kinetics change depending on the type of fuel and tempera- ture used, but they also change depending on the type of electrolyte used. For example, the hydrogen oxidation reaction in a polymer electrolyte membrane (acidic) fuel cell, where H+ is the charge carrier, occurs as

H2 → 2H + + 2e− (3.47)

Compare this to the hydrogen oxidation reaction in an alkaline fuel cell (AFC), where OH– is the charge carrier:

H2 + 2OH− → 2H2O + 2e− (3.48)

Compare this, yet again, to the hydrogen oxidation reaction in a SOFC, where O2 − is

the charge carrier: H2 + O2− → H2O + 2e− (3.49)

The differences in reaction chemistry and temperature for these fuel cell types mean that different catalysts are used. For low-temperature acidic fuel cells (PEMFCs and PAFCs) a Pt-based catalyst is used. For AFCs, nickel-based catalysts can be used. For SOFCs, nickel-based or ceramic-based catalysts are used. For the interested reader, Sections 8.2–8.6

CATALYST–ELECTRODE DESIGN 103

cover some of the specifics about catalyst materials for various fuel cell types, and further details on catalyst materials are provided in Chapter 9.

3.11 CATALYST–ELECTRODE DESIGN

As we have seen, activation losses are minimized by maximizing the exchange current density. Since the exchange current density is a strong function of the catalyst material and the total reaction surface area, catalyst–electrode design focuses on these two parameters to achieve optimal performance.

To maximize reaction surface area, highly porous, nanostructured electrodes are fab- ricated to achieve intimate contact between gas-phase pores, the electrically conductive electrode, and the ion-conductive electrolyte. This nanostructuring is a deliberate attempt to maximize the total number of reaction sites in the fuel cell. In the fuel cell literature, these reaction sites are often called triple-phase zones or triple-phase boundaries (TPBs). This name refers to the fact that the fuel cell reactions can only occur where the three impor- tant phases—electrolyte, gas, and electrically connected catalyst regions—are in contact. The TPB is where all the action occurs! A simplified schematic of the TPBs is shown in Figure 3.14.

The second parameter, optimal catalyst material, is a function of the fuel cell chemistry and operating temperature, as previously discussed. Themajor requirements for an effective catalyst include:

• High mechanical strength • High electrical conductivity • Low corrosion • High porosity • Ease of manufacturability • High catalytic activity (high j0 )

For a PEMFC, platinum or Pt-based alloys are currently the best known catalysts. For highertemperature fuel cells, nickel- or ceramic-based catalysts are often used. As mentioned earlier, technology-specific catalyst selections are discussed in detail in

Gas pores Catalytic electrode particles

Electrolyte TPB’s

Figure 3.14. Simplified schematic of electrode–electolyte interface in a fuel cell, illustrating TPB reaction zones where catalytically active electrode particles, electrolyte phase, and gas pores intersect.

104 FUEL CELL REACTION KINETICS

Sections 8.2–8.6. Designing new catalysts is an area of intense research. In the next section, quantum mechanical approaches to catalyst simulation and design are briefly discussed.

Regardless of the type of catalyst, catalyst layer thickness is another variable that requires careful attention. In practice, the thickness of most fuel cell catalyst layers is between ∼10 and 50 μm.While a thin layer is preferred for better gas diffusion and catalyst utilization, a thick layer incorporates higher catalyst loading and presents more TPBs. Thus, catalyst layer optimization requires a delicate balance between mass transport and catalytic activity concerns.

Usually, the catalyst layer is reinforced by a thicker porous electrode support layer. In a PEMFC, this electrode support layer is called the gas diffusion layer (GDL). The GDL protects the often delicate catalyst structure, provides mechanical strength, allows easy gas access to the catalyst, and enhances electrical conductivity. Electrode supports typically range in thickness from 100 to 400 μm.Aswith the catalyst layer, a thinner electrode support generally provides better gas access but may also present increased electrical resistance or decreased mechanical strength.

The specifics of catalyst–electrode design vary by fuel cell type. Chapter 8 provides details for each of the main fuel cell types, while Chapter 9 provides more details about catalyst–electrode materials as well as design and fabrication approaches for polymer elec- trolyte membrane and solid-oxide fuel cells.

3.12 QUANTUM MECHANICS: FRAMEWORK FOR UNDERSTANDING CATALYSIS IN FUEL CELLS

Understanding the role of the catalyst in a fuel cell is crucial for designing next-generation fuel cell systems. As discussed in the previous section, virtually all PEMFCs today rely on the availability of platinum or platinum alloys as catalytic materials. Unfortunately, plat- inum is scarce and expensive. This is fueling the drive toward novel catalyst design.

Most catalysts to date have been discovered with a trial-and-error approach. Considering the vast space of materials combinations, however, it is quite likely that better catalysts are waiting to be discovered. Unfortunately, finding optimal catalysts by trial and error is too time consuming and expensive. Fortunately, a cost-effective systematic approach involving simulation followed by experimental verification has recently become possible. For fuel cells, this simulation approach may soon help identify novel material systems with equiva- lent or possibly better catalytic performance when compared to platinum. Modern quantum mechanical simulation tools will play a key role in this search. A rudimentary understanding of their capability will be important for the next generation of fuel cell scientists and engi- neers. In this section, we provide a glimpse into how quantum mechanics might contribute to the quest for new catalysts.

How exactly does a fuel cell catalyst work? Up to now, we have discussed catalysis from a continuum viewpoint. However, quantum-mechanics-based simulations can give us further insight. For example, consider the fuel cell anode from a quantum perspective. Hydrogen gas enters the fuel cell anode as a molecular species. As shown in Figure 3.15a, the hydrogen molecule consists of two hydrogen atoms strongly held together by an elec- tron bond. The three-dimensional (3D) surface drawn around the hydrogen molecule in

QUANTUM MECHANICS: FRAMEWORK FOR UNDERSTANDING CATALYSIS IN FUEL CELLS 105

(a) (c)

(d)(b)

Figure 3.15. Evolution of electron orbitals as a hydrogen molecule approaches a cluster of platinum atoms. (a) Platinum and hydrogen molecules are not yet interacting. (b, c) Atomic orbitals begin over- lapping and forming bonds. (d) Complete separation of hydrogen atoms occurs almost simultaneously with reaching the lowest energy configuration.

Figure 3.15a is a physical representation of the electron density in the molecule. In effect, the electron density distribution defines the spatial “extent” and “shape” of the molecule. Figure 3.15 was calculated using a quantum mechanical simulation technique known as density functional theory (DFT). Specifically, a commercially available tool called Gaus- sian1 was used, which is capable of determining the electron density and the minimum energy of a quantum system. It is only in the last decade that commercially available quan- tum tools like Gaussian have become widely available. They rely on the mathematical framework of quantum mechanics, the details of which are presented for the interested student in Appendix D.

In Figure 3.15b, we watch as the hydrogen molecule begins to interact with a platinum catalyst cluster. As the hydrogen molecule gets closer and closer (Figures 3.15b through d), bonds between the hydrogen molecule and the platinum atoms are formed. The new emerg- ing bonds between platinum and hydrogen lead to weakening of the hydrogen–hydrogen bond and ultimately to complete separation. Thus, the platinum catalyst facilitates the sepa- ration of the hydrogenmolecule into hydrogen atoms. In the absence of the platinum cluster, this reaction would not occur spontaneously; instead, significant energy input would be required to induce separation.

Each separated hydrogen atom in Figure 3.15d is sharing its electron with the platinum cluster. In the next reaction step, the hydrogen atoms must be removed from the platinum surface (as hydrogen ions), while leaving their electrons behind. The electrons can then be collected from the electrode and generate useful current. Inmost PEMFC environments, it is believed that the hydrogen ions are removed from the platinum surface by binding to water molecules, forming hydronium ions (H3O

+). Figure 3.16 illustrates this reaction sequence.

1Gaussian is a computational tool predicting energies, molecular structures, and vibrational frequencies of molecular systems by Gaussian Inc.

106 FUEL CELL REACTION KINETICS

(a)

(c)

(b)

Figure 3.16. Formation of hydronium. Water attaches to a positively charged proton on the platinum surface, forming a hydronium ion. The hydronium ion then desorbs from the surface. For simplicity only atomic nuclei (no electron orbitals) are shown.

Once a hydronium ion is formed, it may depart from the platinum surface. The forma- tion of hydronium and its subsequent detachment from the catalyst surface may require overcoming a small energy barrier. This energy can be provided by the random motion of surrounding water molecules or by the thermal vibration of the platinum surface. For a given temperature, the available thermal energy can be estimated as E ∼ kT , where k is Boltzmann’s constant (in eV∕K). Once the hydronium ion has departed, the platinum sur- face is available to participate in another reaction. A fresh hydrogen molecule can bind to the platinum surface and will be subject to the same set of reactions.

Figure 3.17 illustrates the situation at the fuel cell cathode. Figure 3.17a shows the p electron of an oxygen molecule approaching a platinum surface. Figure 3.17b indicates the bond formation of oxygen on the surface of the platinum cluster. As this figure indicates, splitting O2 on the surface of a platinum substrate does not occur as readily as for H2. The oxygen–oxygen bond is weakened but not destroyed after binding to platinum. The remaining bond strength is still 2.3 eV. In contrast, the bond strength of O2 without a plat- inum catalyst surface is 8.8 eV. Thus, significant energy is still required to complete the fuel cell reaction between this absorbed oxygen species and protons (hydronium ions) to

(a) (b)

Figure 3.17. (a) Oxygen molecule approaching a platinum catalyst surface. (b) Even after having reached lowest energy configuration via hybrid orbital formation, the oxygen molecule is not com- pletely separated into individual oxygen atoms.

THE SABATIER PRINCIPLE FOR CATALYST SELECTION 107

form water. This quantum mechanical picture provides an explanation for why the oxygen reaction occurs more slowly, and with greater losses, than the hydrogen reaction.

It is important to realize that the picture painted in these figures is necessarily simpli- fied. Various details, including the influence of voltage, platinum surface structure, and the involvement of additional water molecules, are ignored. For example, more sophisti- cated simulations of the cathode show that interactions of OH groups with partially broken oxygen molecules and protons further reduce the energy required for complete oxygen breakup.2 This mechanism is believed to occur in many low-temperature PEMFCs.

The Sabatier principle, discussed below, provides further qualitative insight into the fac- tors that affect catalytic activity and illustrates how next-generation quantum tools might be used to discover new catalyst materials.

3.13 THE SABATIER PRINCIPLE FOR CATALYST SELECTION

Choosing the right catalyst for a given chemical reaction such as the ORR at the cathode of a fuel cell or the HOR at the anode is critically important for making fuel cells competitive. As will be discussed in Chapter 9, many different metallic, alloy, and compound catalysts are under active investigation for both low-T and high-T fuel cells. Because of the nearly limitless range of potential ways to combine elements into new compounds and alloys, there are likely many more potentially promising catalysts just waiting to be discovered. In fact, the combination of materials and compositions is so large that scientists are beginning to rely more and more on computational methods to guide discovery. This transition has been triggered, in part, by the fact that computational power continues to grow exponen- tially, with commensurate reductions in costs, while the experimental discovery of feasible material alternatives is only becoming more time consuming and costly.

One computationally accessible qualitative principle that provides helpful insights into the trade-offs among different catalytic materials is the Sabatier principle. The Sabatier principle states that there is an optimum catalytic performance (catalytic activity) depending on the strength of adhesion between a catalyst and the reacting chemical species that it hosts. A catalytic surface that binds the reacting species too strongly will slow down the turnover frequency of reactants and reaction products. It “blocks” the surface. Alternatively, if the reacting species are hardly bound to the catalyst surface at all (i.e., the species is bound too weakly), the catalyst cannot do its job and few, if any, chemical reactions will occur. Catalytic activity can be quantified as the rate at which chemical reactions occur on the surface of a catalyst. It can be measured in moles of product produced per second per unit surface area (or per unit mass) of catalyst. Catalytic activity may also be quantified in terms of a more fundamental parameter known as turnover frequency, which is a measure of the rate of reaction (reactions per second) per individual catalytically active site.

The Sabatier principle can be uncovered by plotting turnover frequency (activity) versus adhesion strength as shown in Figure 3.18. When a number of different possible cata- lyst materials are plotted together in this fashion, a characteristic “volcano” type curve

2Also, the spin states of the electrons in platinum influence the energy required to break the oxygen bonds. See Appendix D for further explanations.

108 FUEL CELL REACTION KINETICS

Figure 3.18. This “volcano plot” shows that materials with intermediate reaction species absorption strength yield the highest catalytic activity for the oxygen reduction reaction. Platinum and palladium are high on the curve. Adapted from Ref. [6b].

is produced, with the maximum in catalytic activity occurring at an intermediate value of the reactant species adhesion strength. Because we can calculate the adhesion strength using quantum mechanics, volcano curves are now routinely reproduced and predicted by DFT calculations (Appendix D). This technique therefore holds significant promise for the discovery of improved catalytic materials in a cost-effective fashion. For detailed informa- tion we refer to the literature [6a].

3.14 CONNECTING THE BUTLER–VOLMER AND NERNST EQUATIONS (OPTIONAL)

As you have learned, in order to generate a net current in a fuel cell, a portion of the equi- librium electric potential that is built up at the anode and the cathode must be sacrificed, as shown in Figure 3.8. You have learned that this lost electrical potential can be represented as an activation overvoltage, 𝜂act. The Butler–Volmer equation nicely captures the behav- ior of a fuel cell both during operation (where the application of an activation overvoltage breaks the equilibrium to increase the forward current density, as shown in Figure 3.8) and at equilibrium (where 𝜂act, and hence j, is zero). In fact, the Butler–Volmer equation can describe the continuous transition of reaction kinetics from equilibrium to nonequilibrium and vice versa. From this observation, we can delve into an interesting discussion on the role of the Butler–Volmer equation in equilibrium—in other words, at zero current density.

Reviewing Section 2.4, you may recall that the Nernst equation describes the voltage of a fuel cell in equilibrium. As we have just discussed above, however, the Butler–Volmer equation also applies to a fuel cell in equilibrium, when j = 0. Thus, you should prob- ably guess that the Butler–Volmer equation must collapse to the Nernst equation under

CONNECTING THE BUTLER–VOLMER AND NERNST EQUATIONS (OPTIONAL) 109

equilibrium conditions. Your guess would be correct, and in this section, the relationship between these two equations is demonstrated.

To understand the relationship between the Nernst and Butler–Volmer equations, we have to include a description of the full reaction kinetics occurring at both the cathode and anode at the same time. To begin, let’s rewrite the Butler–Volmer equation from Section 3.7:

j = j0o

( C∗R C0∗R

exp𝛼nF𝜂∕(RT) − C∗P C0∗P

exp−(1−𝛼)nF𝜂∕(RT) )

(3.50)

This equation is the basic fundamental form of the Butler–Volmer equation. However, this equation assumes that only one reactant or product species is involved in the reaction. In this section, we will use (without derivation) a more general form of the Butler–Volmer equation that allows for more than one reactant or product species to be accommodated simultaneously:

j = j0o

(∏(C∗R,i C0∗R,i

)𝑣i exp𝛼nF𝜂∕(RT) −

∏(C∗P,i C0∗P,i

)𝑣i exp−(1−𝛼)nF𝜂∕(RT)

) (3.51)

In this expanded equation, the concentration of each species i may include an exponent term, 𝑣i, which reflects the number of molecules of that species involved in the reaction. We will use this equation to describe the reaction at the anode and cathode of a hydrogen fuel cell. Let’s write the half-cell reaction at the anode and the cathode, respectively.

Anode: H2 ↔ 2H

+ + 2e− (3.52)

Cathode: 2H+ + 2e− + 1

2 O2 ↔ H2O (3.53)

Using Equation 3.51, we can then write the reaction kinetics associated with each elec- trode’s reaction as follows:

Anode:

jA = jA0 ⎛⎜⎜⎝ C∗,AH2

C0∗,AH2

exp2𝛼 AF𝜂A∕(RT) −

( C∗,A H+

C0∗,A H+

)2( C∗,Ae−

C0∗,Ae−

)2 exp−2(1−𝛼

A)F𝜂A∕(RT) ⎞⎟⎟⎠ (3.54)

Cathode:

jC = jC0

⎛⎜⎜⎜⎝ (

C∗,C H+

C0∗,C H+

)2( C∗,Ce−

C0∗,Ce−

)2⎛⎜⎜⎝ C∗,CO2

C0∗,CO2

⎞⎟⎟⎠ 1 2

exp2𝛼 CF𝜂C∕(RT) −

C∗,CH2O

C0∗,CH2O exp−2(1−𝛼

C)F𝜂C∕(RT)

⎞⎟⎟⎟⎠ (3.55)

Here the superscripts A and C in the equations stand for the anode and the cathode, respectively.

110 FUEL CELL REACTION KINETICS

In analogy to Figure 3.8, which illustrated the activation process at a single electrode, Figure 3.19 illustrates the situation when both the anode and the cathode are combined together. Please note that the activation overvoltage at each electrode can be adjusted inde- pendently and that they are typically not equal to one another, 𝜂A ≠ 𝜂C. In steady state, although the anode and cathode activation voltages are not necessarily equal, the current through the anode and the cathode should be equal (jA = jC = j). If you carefully examine Equations 3.54 and 3.55, you can see that this condition can be achieved by the adjust- ment of a few important parameters such as concentrations of protons (C∗,A

H+ , C∗,C

H+ ), elec-

trons (C∗,Ae− , C ∗,C e− ), hydrogen (C

∗,A H2

), oxygen (C∗,CO2 ), and water (C∗,CH2O

), or the overvoltages

(𝜂A, 𝜂C). Some of these parameters, such as C∗,AH2 , C ∗,C O2

, and C∗,CH2O , may be specified by the

composition of the gas streams delivered to the fuel cell. Let us now consider what happens at equilibrium, when jA = jC = j = 0. Under this con-

dition, Equation 3.54 becomes

0 = jA0 ⎛⎜⎜⎝ C∗,AH2

C0∗,AH2

exp2𝛼F𝜂 A∕(RT) −

( C∗,A H+

C0∗,A H+

)2( C∗,Ae−

C0∗,Ae−

)2 exp−2(1−𝛼

A)F𝜂A∕(RT) ⎞⎟⎟⎠ (3.56)

After rearranging this, we obtain

C∗,AH2

C0∗,AH2

exp2𝛼F𝜂 A∕(RT) =

( C∗,A H+

C0∗,A H+

)2( C∗,Ae−

C0∗,Ae−

)2 exp−2(1−𝛼

A)F𝜂A∕(RT) (3.57)

Applying the natural logarithm function to both sides of the equation yields

ln ⎛⎜⎜⎝ C∗,AH2

C0∗,AH2

⎞⎟⎟⎠ + 2𝛼AF𝜂A

RT = ln

( C∗,A H+

C0∗,A H+

)2 + ln

( C∗,Ae−

C0∗,Ae−

)2 − 2(1 − 𝛼

A)F𝜂A

RT (3.58)

After rearranging, we obtain

2F𝜂A

RT = − ln

⎛⎜⎜⎝ C∗,AH2

C0∗,AH2

⎞⎟⎟⎠ + ln (

C∗,A H+

C0∗,A H+

)2 + ln

( C∗,Ae−

C0∗,Ae−

)2 (3.59)

Or, upon using the definition of activity,

𝜂A = RT 2F

(− ln(a∗,AH2 ) + ln (a ∗,A H+

)2 + ln (a∗,Ae− ) 2) (3.60)

In a similar fashion, we obtain the following starting from Equation 3.55 for the cathode:

𝜂C = RT 2F

( − ln

( a∗,C H+

)2 − ln (a∗,Ce− )

2 − ln (a∗,CO2 ) 1 2 + ln(a∗,CH2O)

) (3.61)

CONNECTING THE BUTLER–VOLMER AND NERNST EQUATIONS (OPTIONAL) 111

2H++2e– 2H++2e–+

H2O

∆Grxn,anode

∆Grxn,cathodeC he

m ic

al fr

ee e

ne rg

y F

re e

en er

gy F

re e

en er

Anode CathodeElectrolyte

∆G‡1,anode

Anode CathodeElectrolyte

–nF∆φcathode

–nF∆φanode

–αnFηanode

ΔG‡anode

∆G‡2,anode

ΔG‡cathode

∆G‡1,cathode ∆G‡2,cathode

Transport through conductors

–nF anodeη

–nF cathodeη

–nF anodeη

–nF anodeη –nF anodeη

–nF cathodeη

–αnF cathodeη

O2 1 2

Figure 3.19. The overvoltage at the anode and the cathode modify the activation energy of each electrode according to the current. At steady state, the current at the anode and the cathode should be equal. Overvoltage and species concentrations are determined by satisfying this condition.

112 FUEL CELL REACTION KINETICS

Now we will combine Equations 3.60 and 3.61 by adding them:

𝜂A + 𝜂C = RT 2F

⎛⎜⎜⎜⎝ln a∗,CH2O

a∗,AH2

( a∗,CO2

) 1 2

− ln

( a∗,C H+

a∗,A H+

)2 − ln

( a∗,Ce−

a∗,Ae−

)2⎞⎟⎟⎟⎠ (3.62) Please remember that this equation describes the activation overvoltage of a fuel cell at

its “equilibrium state” or zero current density. Accordingly, this overvoltage should be the difference between the actual voltage and the reference voltage of the fuel cell (𝜂A + 𝜂C = E0 − E). Now we have

E = E0 − RT 2F

⎛⎜⎜⎜⎝ln a∗,CH2O

a∗,AH2

( a∗,AO2

) 1 2

− ln

( a∗,C H+

a∗,A H+

)2 − ln

( a∗,Ce−

a∗,Ae−

)2⎞⎟⎟⎟⎠ (3.63) This equation is actually the Nernst equation, although it has two additional terms

accounting for the concentration gradient of protons and electrons across the electrolyte. In Section 2.4.4, we calculated the Nernst voltage from the hydrogen concentration gradient across the electrolyte. Similarly, a concentration gradient of protons and electrons can generate a Nernst voltage. Typically, the proton and electron activity terms can be neglected (at equilibrium, the activity of protons and electrons within the electrolyte will be approximately uniform), resulting in the simple Nernst equation for hydrogen and oxygen reactants.

TheNernst equation describes the relationship between the voltage and the concentration of species in a given electrochemical reaction at equilibrium. The Butler–Volmer equation does the same under nonequilibrium conditions. The analysis presented above shows that the Nernst equation is really just a special form of the Butler–Volmer equation when the cur- rent density is zero—or, in other words, when an electrochemical reaction is at equilibrium.

3.15 CHAPTER SUMMARY

The purpose of this chapter is to explain how fuel cell reaction processes lead to perfor- mance losses. The study of reaction processes is called reaction kinetics, and the voltage loss caused by kinetic limitations is known as an activation loss.

• Electrochemical reactions involve the transfer of electrons and occur at surfaces. • Because electrochemical reactions involve electron transfer, the current generated is

a measure of the reaction rate. • Because electrochemical reactions occur at surfaces, the rate (current) is proportional

to the reaction surface area. • Current density is more fundamental than current. We use current density (current per

unit area) to normalize the effects of system size. • An activation barrier impedes the conversion of reactants to products (and vice versa).

CHAPTER EXERCISES 113

• A portion of the fuel cell voltage is sacrificed to lower the activation barrier, thus increasing the rate at which reactants are converted into products and the current den- sity generated by the reaction.

• The sacrificed (lost) voltage is known as activation overvoltage 𝜂act. • The relationship between the current density output and the activation overvoltage

is exponential. It is described by the Butler–Volmer equation: j = j0(e𝛼nF𝜂act∕(RT) − e−(1−𝛼)nF𝜂act∕(RT)).

• The exchange current density j0 measures the equilibrium rate at which reactant and product species are exchanged in the absence of an activation overvoltage. A high j0 indicates a facile reaction, while a low j0 indicates a sluggish reaction.

• Activation overvoltage losses are minimized by maximizing j0. There are four major ways to increase j0: (1) increase reactant concentration, (2) increase reaction tempera- ture, (3) decrease the activation barrier (by employing a catalyst), and (4) increase the number of reaction sites (by fabricating high-surface-area electrodes and 3D struc- tured reaction interfaces).

• Fuel cells are usually operated at relatively high current densities (high activation overvoltages). At high activation overvoltage, fuel cell kinetics can be approximated by a simplified version of the Butler–Volmer equation: j = j0e𝛼nF𝜂act∕(RT). In a gener- alized logarithmic form, this is known as the Tafel equation 𝜂act = a + b log j, where b is the Tafel slope.

• For a H2–O2 fuel cell, the hydrogen (anode) kinetics are generally facile and produce only a small activation loss. In contrast, the oxygen kinetics are sluggish and lead to a significant activation loss (at low temperature).

• The details of fuel cell reaction kinetics are dependent on the fuel, electrolyte chem- istry, and operation temperature. For low-T fuel cells, Pt is commonly used as a catalyst. High-T fuel cells employ nickel- or ceramic-based catalysts.

• The main requirements for an effective fuel cell catalyst are (1) activity, (2) conduc- tivity, and (3) stability (specifically thermal, mechanical, and chemical stability in the fuel cell environment).

• To increase j0, fuel cell catalyst–electrodes are designed to maximize the number of reaction sites per unit area. Increasing the number of reaction sites means maximizing triple-phase boundary regions, where the electrolyte, reactant, and catalytically active electrode phases meet. The best catalyst–electrodes are carefully optimized, porous, high-surface-area structures.

CHAPTER EXERCISES

Review Questions

3.1 This problem is composed of three parts: (a) For the reaction

1 2 O2 + 2H+ + 2e− ⇌ H2O

114 FUEL CELL REACTION KINETICS

the standard electrode potential is+1.23 V. Under standard-state conditions, if the electrode potential is reduced to 1.0 V, will this bias the reaction in the forward or reverse direction?

(b) For the reaction H2 ⇌ 2H

+ + 2e−

the standard electrode potential is 0.0 V. Under standard-state conditions, if the electrode potential is increased to 0.10 V, will this bias the reaction in the forward or reverse direction?

(c) Considering your answers to parts (a) and (b), in an H2–O2fuel cell, if we increase the overall rate of the fuel cell reaction,

H2 + 1 2 O2 ⇌ H2O

which is made up of the half reactions

H2 ⇌ 2H + + 2e−

1 2 O2 + 2H+ + 2e− ⇌ H2O

what happens to the potential difference (voltage output) for the reaction?

3.2 Figure 3.7 presented one possible case for the voltage profile of a fuel cell. Draw two other possible voltage profiles that yield the same overall cell voltage but show vastly different individual Galvani potentials. Is it possible for one of the Galvani potentials to be negative yet still have the overall cell voltage be positive?

3.3 What is 𝛼? Assuming that the Galvani potential varies linearly across a reaction inter- face, sketch free-energy curves that result in situations where 𝛼 < 0.5, 𝛼 = 0.5, and 𝛼 > 0.5.

3.4 What does the exchange current density represent?

3.5 (a) In the Tafel equation, how is the Tafel slope b related to 𝛼? (Remember that the Tafel equation is defined using log instead of ln.)

(b) How is the intercept a related to the exchange current density?

3.6 For a SOFC (where the charge carrier in the electrolyte is O2–), CO is considered a fuel rather than a poison. Write an electrochemical half reaction showing how CO can be utilized as a fuel in the SOFC.

3.7 List the major requirements for an effective fuel cell catalyst material. List the major requirements for an effective fuel cell catalyst–electrode structure.

3.8 In Section 3.14, the half-cell reactions at both the anode and the cathode were assumed to involve the transfer of two electrons. Instead, we could describe these reactions as single-electron transfer reactions:

Anode: 1 2 H2 ↔ H

+ + e−

Cathode: H+ + e− + 1 4 O2 ↔

1 2 H2O

CHAPTER EXERCISES 115

Starting from these one-electron half-cell reactions, show that we can still obtain Equation 3.63 using Equation 3.51 for a fuel cell at equilibrium.

3.9 The half-cell reactions in a hydrogen fuel cell are sometimes described usingmultistep processes such as

Anode: H2 ↔ 2H + + 2e−

Cathode: 2H+ + 2e− + O2 ↔ H2O2,ad

H2O2,ad ↔ H2O + 1 2 O2

Starting with these multistep half-cell reactions, show that we can still obtain Equation 3.63 using Equation 3.51 for a fuel cell at equilibrium.

3.10 Consider the following generic, simple half-cell reaction at the anode of a fuel cell:

Anode: R ↔ P

Then the Butler–Volmer equation for this reaction is

j = j0

( C∗R C0∗R

exp𝛼nF𝜂∕(RT) − C∗P C0∗P

exp−(1−𝛼)nF𝜂∕(RT) )

(a) If the concentrations of the reactant (C∗∗R ) and product (C ∗∗ P ) species at zero current

density (or equilibrium) are not equal to the reference concentrations (C0∗R and C0∗P ), find the activation overvoltage of the anode at equilibrium.

(b) Let’s define a new overvoltage as 𝜂′ = 𝜂 − 𝜂A where 𝜂A is the overvoltage obtained from (a). (Note that 𝜂′ becomes zero at equilibrium.) Rewrite the Butler–Volmer equation using 𝜂′. Show that this equation also takes a form of the Butler–Volmer equation if we use the equilibrium concentrations (C∗∗R and C

∗∗ P ) as reference con-

centrations. What is the exchange current density in this equation?

Calculations

3.11 Consider two electrochemical reactions. Reaction A results in the transfer of 2 mol of electrons per mole of reactant and generates a current of 5 A on an electrode 2 cm2 in area. Reaction B results in the transfer of 3 mol of electrons per mole of reactant and generates a current of 15 A on an electrode 5 cm2 in area. What are the net reaction rates for reactions A and B (in moles of reactant per square centimeter per second)? Which reaction has the higher net reaction rate?

3.12 This problem has several parts: (a) If a portable electronic device draws 1 A current at a voltage of 2.5 V, what is the

power requirement for the device?

(b) You have designed a fuel cell that delivers 1 A at 0.5 V. How many of your fuel cells are required to supply the above portable electronic device with its necessary voltage and current requirements?

116 FUEL CELL REACTION KINETICS

(c) You would like the portable electronic device to have an operating lifetime of 100 h. Assuming 100% fuel utilization, what is the minimum amount of H2 fuel (in grams) required?

(d) If this H2 fuel is stored as a compressed gas at 500 atm, what volume would it occupy (assume ideal gas, room temperature)? If it is stored as a metal hydride at 5 wt % hydrogen, what volume would it occupy? (Assume the metal hydride has a density of 10 g/cm3.)

(e) If the fuel cell used methanol (CH3OH) fuel instead of H2, what would be the minimum amount (in grams) of methanol required for 100 h of life again assum- ing 100% fuel utilization? Methanol has a molecular mass of 32 g/mol. What would be the corresponding volume of liquid methanol fuel (the density of liquid methanol is 0.79 g/cm3)?

3.13 Everything else being equal, write a general expression showing how the exchange current density for a reaction changes as a function of temperature [e.g., write an expression for j0 (T) at an arbitrary temperature T as a function of j0 (T0) at a reference temperature T0]. If a reaction has j0 = 10−8A∕cm2 at 300 K and j0 = 10−4A∕cm2 at 600 K, what is ΔG‡ for the reaction? Assume that the preexponent portion of j0 is temperature independent.

3.14 (a) Everything else being equal, write a general expression showing how the exchange current density varies as a function of reactant concentration.

(b) Use this result and your answer from problem 3.13 to answer the following ques- tion: For a reaction with ΔG‡ = 20 kJ∕mol, what temperature change (starting from 300 K) has the same effect on j0 as increasing the reactant concentration by one order of magnitude? Assume that the preexponent portion of j0 is temperature independent.

3.15 All else being equal, at a given activation overvoltage, which effect produces a greater increase in the net current density for a reaction: doubling the temperature (in degrees Kelvin) or halving the activation barrier? Defend your answer with an equation. Assume that the preexponent portion of j0 is temperature independent.

3.16 Estimate the thermal energy required to separate molecular oxygen with and without a platinum catalyst. Convert this energy into temperature (degrees centigrade) and comment on the role of platinum as a catalyst in a PEMFC.

CHAPTER 4

FUEL CELL CHARGE TRANSPORT

The previous chapter on reaction kinetics detailed one of the most pivotal steps in the elec- trochemical generation of electricity: the production and consumption of charge via elec- trochemical half reactions. In this chapter, we address an equally important step in the electrochemical generation of electricity: charge transport. Charge transport “completes the circuit” in an electrochemical system, moving charges from the electrode where they are produced to the electrode where they are consumed.

There are two major types of charged species: electrons and ions. Since both electrons and ions are involved in electrochemical reactions, both types of charge must be transported. The transport of electrons versus ions is fundamentally different, primarily due to the large difference in mass between the two. In most fuel cells, ion charge transport is far more difficult than electron charge transport; therefore, we are mainly concerned with ionic conductivity.

As you will discover, resistance to charge transport results in a voltage loss for fuel cells. Because this voltage loss obeys Ohm’s law, it is called an ohmic, or IR, loss. Ohmic fuel cell losses are minimized by making electrolytes as thin as possible and employing high-conductivity materials. The search for high-ionic-conductivity materials will lead to a discussion of the fundamental mechanisms of ionic charge transport and a review of the most important electrolyte material classes.

4.1 CHARGES MOVE IN RESPONSE TO FORCES

The rate at which charges move through a material is quantified in terms of flux (denoted with the symbol J). Flux measures how much of a given quantity flows through a material per unit area per unit time. Figure 4.1 illustrates the concept of flux: Imagine water flow- ing down this tube at a volumetric flow rate of 10 L/s. If we divide the flow rate by the

117

118 FUEL CELL CHARGE TRANSPORT

Figure 4.1. Schematic of flux. Imagine water flowing down this tube at a volumetric flow rate of 10 L/s. Dividing this flow rate by the cross-sectional area of the tube (A) gives the flux JA of water moving down the tube. Generally, flux is measured in molar rather than volumetric quantities, so in this example the liters of water should be converted to moles.

cross-sectional area of the tube (A), we get the volumetric flux JA of water moving down the tube. In other words, JA gives the per-unit-area flow rate of water through the tube. Be careful! Remember that flux and flow rate are not the same thing. By computing a flux, we are normalizing the flow rate by a cross-sectional area.

The most common type of flux is a molar flux (typical units are mol/cm2 ⋅ s). Charge flux is a special type of flux that measures the amount of charge that flows through a material per unit area per unit time. Typical units for charge flux are C/cm2 ⋅ s = A∕cm2. From these units, you may recognize that charge flux is the same thing as current density. To denote that charge flux represents a current density and carries different units than molar flux, we give it the symbol j. The quantity ziF is required to convert from molar flux J to charge flux j, where zi is the charge number for the charge-carrying species (e.g., zi is +1 for Na+, –2 for O2–, etc.) and F is Faraday’s constant:

j = ziFJ (4.1)

ELIMINATE CONFUSION BETWEEN zi AND n

As we move from the discussion of electrochemical kinetics (Chapter 3) to a discussion of charge transport (Chapter 4), it is important to recognize the difference between the quantities zi and n. The quantity n, which we have used throughout the book, refers to the number of electrons transferred during an electrochemical reaction. For example, in the electrochemical half reaction

H2 → 2H + + 2e−

two electrons are transferred per mole of H2 gas reacted, and therefore n = 2. In contrast, the quantity zi, which we introduce here in Chapter 4, refers to the amount of charge carried by a charged species. For the charged species H+, as an example, zi = +1, while for the charged species e–, zi = −1.

CHARGES MOVE IN RESPONSE TO FORCES 119

In all materials, a force must be acting on the charge carriers (i.e., the mobile electrons or ions in the material) for charge transport to occur. If there is no force acting on the charge carriers, there is no reason for them to move! The governing equation for transport can be generalized (in one dimension) as

Ji = ∑ k

MikFk (4.2)

Where Ji represents a flux of species i, the Fk’s represent the k different forces acting on i, and the Mik’s are the coupling coefficients between force and flux. The coupling coefficients reflect the relative ability of a species to respond to a given force with movement as well as the effective strength of the driving force itself. The coupling coefficients are therefore a property both of the species that is moving and the material through which it is moving. This general equation is valid for any type of transport (charge, heat, mass, etc.). In fuel cells, there are three major driving forces that give rise to charge transport: electrical driving forces (as represented by an electrical potential gradient dV∕dx), chemical driving forces (as represented by a chemical potential gradient d𝜇∕dx), and mechanical driving forces (as represented by a pressure gradient dP∕dx).

As an example of how these forces give rise to charge transport in a fuel cell, consider our familiar hydrogen–oxygen PEMFC (see Figure 4.2). As hydrogen reacts in this fuel

CathodeElectrolyteAnode

e–

+–

2OH2

+–

e–

H+

Figure 4.2. In a H2–O2 fuel cell, accumulation of protons/electrons at the anode and depletion of protons/electrons at the cathode lead to voltage gradients which drive charge transport. The elec- trons move from the negatively charged anode electrode to the positively charged cathode electrode. The protons move from the (relatively) positively charged anode side of the electrolyte to the (rela- tively) negatively charged cathode side of the electrolyte. The relative charge in the electrolyte at the anode versus the cathode arises due to differences in the concentration of protons. This concentration difference can also contribute to proton transport between the anode and cathode.

120 FUEL CELL CHARGE TRANSPORT

cell, protons and electrons accumulate at the anode, while protons and electrons are con- sumed at the cathode. The accumulation/depletion of electrons at the two electrodes creates a voltage gradient, which drives the transport of electrons from the anode to the cathode. In the electrolyte, accumulation/depletion of protons creates both a voltage gradient and a concentration gradient. These coupled gradients then drive the transport of protons from the anode to the cathode.

In the metal electrodes, only a voltage gradient drives electron charge transport. How- ever, in the electrolyte, both a concentration (chemical potential) gradient and a voltage (electrical potential) gradient drive ion transport. How do we know which of these two driving forces is more important? In almost all situations, the electrical driving force domi- nates fuel cell ion transport. In other words, the electrical effect of the accumulated/depleted protons is far more important for charge transport than the chemical concentration effect of the accumulated/depleted protons. The underlying reasons why electrical driving forces dominate fuel cell charge transport are explained for the interested reader in an optional section near the end of this chapter (see Section 4.7).

For the case where charge transport is dominated by electrical driving forces, Equation 4.2 can be rewritten as

j = 𝜎 dV dx

(4.3)

where j represents the charge flux (not molar flux), dV∕dx is the electric field providing the driving force for charge transport, and 𝜎 is the conductivity, which measures the propensity of a material to permit charge flow in response to an electric field. This important application of Equation 4.2 simplifies the terms of fuel cell charge transport. In certain rare situations, both the concentration effects and electric potential effects may become important; in these cases, the charge transport equations become considerably more difficult.

Comparing Equation 4.3 to Equation 4.2, it is apparent that conductivity 𝜎 is nothing more than the name of the coupling coefficient that describes how flux and electrical driv- ing forces are related. The relevant coupling coefficient that describes transport due to a chemical potential (concentration) gradient is called diffusivity. For transport due to a pres- sure gradient, the relevant coupling coefficient is called viscosity. These transport processes are summarized in Table 4.1 using molar flux quantities.

TABLE 4.1. Summary of Transport Processes Relevant to Charge Transport

Transport Process Driving Force Coupling Coefficient Equation

Conduction Electrical potential gradient, dV∕dx

Conductivity 𝜎 J = 𝜎|zi|F dVdx Diffusion Concentration gradient, dc∕dx Diffusivity D J = −Ddc

Convection Pressure gradient, dp∕dx Viscosity 𝜇 J = Gc 𝜇

Note: The transport equation for convection in this table is based on Poiseuille’s law, where G is a geometric constant and c is the concentration of the transported species. Convection flux is often calculated simply as J = 𝑣ci, where v is the transport velocity.

CHARGE TRANSPORT RESULTS IN A VOLTAGE LOSS 121

4.2 CHARGE TRANSPORT RESULTS IN A VOLTAGE LOSS

Unfortunately, charge transport is not a lossless process. It occurs at a cost. For fuel cells, the penalty for charge transport is a loss in cell voltage. Why does charge transport result in a voltage loss? The answer is because fuel cell conductors are not perfect—they have an intrinsic resistance to charge flow.

Consider the uniform conductor pictured in Figure 4.3. This conductor has a constant cross-sectional area A and length L. Applying this example conductor geometry to our charge transport equation 4.3 produces

j = 𝜎V L

(4.4)

Solving for V yields

V = j (L 𝜎

) (4.5)

You might recognize that this equation is similar to Ohm’s law: V = iR. In fact, since charge flux (current density) and current are related by i = jA, we can rewrite Equation 4.5 as

V = i ( L A𝜎

) = iR (4.6)

where we identify the quantity L∕A𝜎 as the resistance R of our conductor. The voltage V in this equation represents the voltage which must be applied in order to transport charge at a rate given by i. Thus, this voltage represents a loss: It is the voltage that is expended, or sacrificed, in order to accomplish charge transport. This voltage loss arises due to our conductor’s intrinsic resistance to charge transport, as embodied by 1/𝜎.

Area = A Length = L

V = jL/σ = iRV

0 0 x L

R = L/Aσ

Figure 4.3. Illustration of charge transport along a uniform conductor of cross-sectional area A, length L, and conductivity 𝜎. A voltage gradient dV/dx drives the transport of charge down the con- ductor. From the charge transport equation j = 𝜎(dV∕dx) and the conductor geometry, we can derive Ohm’s law: V = iR. The resistance of the conductor is dependent on the conductor’s geometry and conductivity: R = L∕𝜎A.

122 FUEL CELL CHARGE TRANSPORT

Because this voltage loss obey’s Ohm’s law, we call it an “ohmic” loss. Like the activa- tion overvoltage loss (𝜂act) introduced in the previous chapter, we give this voltage loss the symbol η. Specifically, we label it 𝜂ohmic to distinguish it from 𝜂act. Rewriting Equation 4.6 to reflect our nomenclature and explicitly including both the electronic (Relec) and ionic

V ol

ta ge

( V

)

Distance (x)

Anode Electrolyte Cathode

(a)

(b)

ηact,C

ηact,A V

(c)

V ηohmic

Distance (x)

V ol

ta ge

( V

)

Anode Electrolyte Cathode

V ol

ta ge

( V

)

Anode Electrolyte Cathode

Distance (x)

Figure 4.4. (a) Hypothetical voltage profile of a fuel cell at thermodynamic equilibrium (recall Figure 3.7). The thermodynamic voltage of the fuel cell is given byE0. (b) Effect of anode and cathode activation losses on the fuel cell voltage profile (recall Figure 3.9). (c) Effect of ohmic losses on fuel cell voltage profile. Although the overall fuel cell voltage increases from the anode to the cathode, the cell voltage must decrease between the anode side of the electrolyte and the cathode side of the electrolyte to provide a driving force for charge transport.

CHARGE TRANSPORT RESULTS IN A VOLTAGE LOSS 123

(Rionic) contributions to fuel cell resistance gives

𝜂ohmic = iRohmic = i(Relec + Rionic) (4.7)

Because ionic charge transport tends to be more difficult than electronic charge transport, the ionic contribution to Rohmic tends to dominate.

The direction of the voltage gradient in an operating fuel cell electrolyte can often seem nonintuitive. As Figure 4.4c illustrates, although overall fuel cell voltage increases from the anode to the cathode, the cell voltage must decrease between the anode side of the electrolyte and the cathode side of the electrolyte to provide a driving force for charge transport.

Example 4.1 A 10-cm2 PEMFC employs an electrolyte membrane with a conduc- tivity of 0.10 Ω−1 ⋅ cm−1. For this fuel cell, Relec has been determined to be 0.005 Ω. Assuming the only other contribution to cell resistance comes from the electrolyte membrane, determine the ohmic voltage loss (𝜂ohmic) for the fuel cell at a current density of 1 A∕cm2 in the following cases: (a) the electrolyte membrane is 100 𝜇m thick; (b) the electrolyte membrane is 50 𝜇m thick.

Solution: We need to calculate Rionic based on the electrolyte dimensions and then use Equation 4.7 to calculate 𝜂ohmic. Since the fuel cell has an area of 10 cm

2, the current i of the fuel cell is 10 A:

i = jA = 1 A∕cm2 × 10 cm2 = 10 A (4.8)

From Equation 4.6 we can calculate Rionic for the two cases (a), (b) given in this problem:

Case (a): Rionic = L 𝜎A

= 0.01 cm (0.10 Ω−1 ⋅ cm−1)(10 cm2)

= 0.01 Ω

Case (b): Rionic = 0.005 cm

(0.10 Ω−1 ⋅ cm−1)(10 cm2) = 0.005 Ω

(4.9)

Inserting these values into Equation 4.7 and using i = 10 A gives the following values for 𝜂ohmic:

Case (a): 𝜂ohmic = i(Relec + Rionic) = 10 A(0.005 Ω + 0.01 Ω) = 0.15 V

Case (b): 𝜂ohmic = 10 A(0.005 Ω + 0.005 Ω) = 0.10 V (4.10)

With everything else equal, making the membrane thinner reduces the ohmic loss! However, note that the payoff does not scale directly with membrane thickness. Although the membrane thickness was cut in half in this example, the ohmic loss was only reduced by one-third. This occurs because not all of the fuel cell’s resistance contributions come from the electrolyte.

124 FUEL CELL CHARGE TRANSPORT

4.3 CHARACTERISTICS OF FUEL CELL CHARGE TRANSPORT RESISTANCE

As Equation 4.7 implies, charge transport linearly decreases fuel cell operating voltage as current increases. Figure 4.5 illustrates this effect. Obviously, if fuel cell resistance is decreased, fuel cell performance will improve.

Fuel cell resistance exhibits several important properties. First, resistance is geometry dependent, as Equation 4.6 clearly implies. Fuel cell resistance scales with area: To nor- malize out this effect, area-specific resistances are used to compare fuel cells of different sizes. Fuel cell resistance also scales with thickness; for this reason, fuel cell electrolytes are generally made as thin as possible. Additionally, fuel cell resistances are additive; resis- tance losses occurring at different locations within a fuel cell can be summed together in series. An investigation of the various contributions to fuel cell resistance reveals that the ionic (electrolyte) component to fuel cell resistance usually dominates. Thus, performance improvements may be won by the development of better ion conductors. Each of these important points will now be addressed.

4.3.1 Resistance Scales with Area

Since fuel cells are generally compared on a per-unit-area basis using current density instead of current, it is generally necessary to use area-normalized fuel cell resistances when dis- cussing ohmic losses. Area-normalized resistance, also known as area-specific resistance (ASR), carries units of Ω ⋅ cm2. By using ASR, ohmic losses can be calculated from current density via

𝜂ohmic = j(ASRohmic) (4.11)

Theoretical EMF or ideal voltage

1.0

C el

l v ol

ta ge

( V

)

1.2

Current (A)

0.5

Ohmic loss: ηohmic= iRohmic

Rohmic = 0.50 Ω

Rohmic = 0.75 Ω

Rohmic = 1.0 Ω

Figure 4.5. Effect of ohmic loss on fuel cell performance. Charge transport resistance contributes a linear decrease in fuel cell operating voltage as determined by Ohm’s law (Equation 4.7). The magnitude of this loss is determined by the size of Rohmic. (Curves calculated for Rohmic equal 0.50 Ω, 0.75 Ω, and 1.0 Ω, respectively.)

CHARACTERISTICS OF FUEL CELL CHARGE TRANSPORT RESISTANCE 125

where ASRohmic is the ASR of the fuel cell. Area-specific resistance accounts for the fact that fuel cell resistance scales with area, thus allowing fuel cells of different sizes to be compared. It is calculated by multiplying a fuel cell’s ohmic resistance Rohmic by its area:

ASRohmic = Afuel cell Rohmic (4.12)

Be careful, you must multiply resistance by area to get ASR, not divide! This calculation will probably seem unintuitive at first. Because a large fuel cell has so much more area to flow current through than a small fuel cell, its resistance is far lower. However, on a per-unit-area basis, their resistances should be about the same; therefore, the resistance of the large fuel cell must be multiplied by its area. This concept may be more understandable if you recall the original definition of resistance in Equation 4.6:

R = L A𝜎

(4.13)

Since resistance is inversely proportional to area, multiplication by area is necessary to get area-independent resistances. This point is reinforced by Example 4.2.

Example 4.2 Consider the two fuel cells illustrated in Figure 4.6. At a current density of 1 A∕cm2, calculate the ohmic voltage losses for both fuel cells. Which fuel cell incurs the larger ohmic voltage loss?

Fuel cell 1 A1 = 1 cm2

R1= 0.1 Ω

Fuel cell 2 A2 = 10 cm2

R2 = 0.02 Ω

Fuel cell 1 ASR R1A1= 0.1 Ω cm2

Fuel cell 2 ASR R2A2= 0.2 Ω cm2..

Figure 4.6. The importance of ASR is illustrated by these two fuel cells. Fuel cell 2 has lower total resistance than fuel cell 1 but yields a larger ohmic loss for a given current density. Fuel cell resistance is best compared using ASR rather than R.

Solution: There are two ways to solve this problem. To calculate voltage loss based on current density, we can either convert the resistances of the fuel cells to ASRs and then use Equation 4.11 (solution 1) or convert the current densities into currents and use Equation 4.6 (solution 2).

Solution 1: Calculating the ASRs for the two fuel cells gives

ASR1 = R1A1 = (0.1 Ω)(1 cm2) = 0.1 Ω ⋅ cm2

ASR2 = R2A2 = (0.02 Ω)(10 cm2) = 0.2 Ω ⋅ cm2 (4.14)

126 FUEL CELL CHARGE TRANSPORT

Then, the ohmic voltage losses for the two cells can be calculated using Equation 4.11:

𝜂1,ohmic = j(ASR1) = (1 A∕cm2)(0.1 Ω ⋅ cm2) = 0.1 V

𝜂2,ohmic = j(ASR2) = (1 A∕cm2)(0.2 Ω ⋅ cm2) = 0.2 V (4.15)

Solution 2: Converting current densities for the two fuel cells into currents gives

i1 = jA1 = (1 A∕cm2)(1 cm2) = 1 A

i2 = jA2 = (1 A∕cm2)(10 cm2) = 10 A (4.16)

Then, the ohmic voltage losses for the two cells can be calculated using Equation 4.6:

𝜂1,ohmic = i1(R1) = (1 A)(0.1 Ω) = 0.1 V

𝜂2,ohmic = i2(R2) = (10 A)(0.02 Ω) = 0.2 V (4.17)

In both solutions, the same answer is obtained; cell 2 incurs a greater voltage loss. Although the total resistance of cell 2 is lower than cell 1 (0.02 Ω versus 0.1 Ω), the ASR of cell 2 is higher than that of cell 1. Thus, on an area-normalized basis, cell 2 is actually more “resistive” than cell 1 and leads to poorer fuel cell performance.

4.3.2 Resistance Scales with Thickness

Referring again to Equation 4.6, it is apparent that resistance scales not only with the cross-sectional area of the conductor but also with the length (thickness) of the conductor. If we normalize resistance by using ASR, then

ASR = L 𝜎

(4.18)

The shorter the conductor length L, the lower the resistance. It is intuitive that a shorter path results in less resistance.

Ionic conductivity is orders of magnitude lower than the electronic conductivity of met- als, so minimizing the resistance of the fuel cell electrolyte is essential. Hence, we want the shortest path possible for ions between the anode and the cathode. Fuel cell electrolytes, therefore, are designed to be as thin as possible. Although reducing electrolyte thickness improves fuel cell performance, there are several practical issues that limit how thin the electrolyte can be made. The most important limitations are as follows:

• Mechanical Integrity. For solid electrolytes, the membrane cannot be made so thin that it risks breaking or develops pinholes. Membrane failure can result in catastrophic mixing of the fuel and oxidant!

CHARACTERISTICS OF FUEL CELL CHARGE TRANSPORT RESISTANCE 127

• Nonuniformities. Even mechanically sound, pinhole-free electrolytes may fail if the thickness varies considerably across the fuel cell. Thin electrolyte areas may become “hot spots” that are subject to rapid deterioration and failure.

• Shorting. Extremely thin electrolytes (solid or liquid) risk electrical shorting, espe- cially when the electrolyte thickness is on the same order of magnitude as the electrode roughness.

• Fuel Crossover. As the electrolyte thickness is reduced, the crossover of reactants may increase. This leads to an undesirable parasitic loss, which can eventually become so large that further thickness decreases are counterproductive.

• Contact Resistance. Part of the electrolyte resistance is associated with the interface between the electrolyte and the electrode. This “contact” resistance is independent of electrolyte thickness.

• Dielectric Breakdown. The ultimate physical limit to solid-electrolyte thickness is given by the electrolyte’s dielectric breakdown properties. This limit is reached when the electrolyte is made so thin that the electric field across the membrane exceeds the dielectric breakdown field for the material.

For most solid-electrolyte materials, the ultimate limit on thickness, as predicted by the dielectric breakdown field, is on the order of several nanometers. However, the other practical limitations listed above currently limit achievable thickness to about 10–100 𝜇m, depending on the electrolyte.

4.3.3 Fuel Cell Resistances Are Additive

As Figure 4.7 illustrates, the total ohmic resistance presented by a fuel cell is actually a combination of resistances coming from different components of the device. Depending on how much precision is needed, it is possible to assign individual resistances to the electrical interconnections, anode electrode, cathode electrode, anode catalyst layer, cathode catalyst layer, electrolyte, and so on. It is also possible to ascribe contact resistances associated with the interfaces between the various layers in the fuel cell (e.g., a flow structure/electrode con- tact resistance). Because the current produced by the fuel cell must flow serially through all of these regions, the total fuel cell resistance is simply the sum of all the individual resis- tance contributions. Unfortunately, it is experimentally very difficult to distinguish between all the various sources of resistance loss.

You might think that it should be a relatively easy experimental task to measure the resistance of each component in a fuel cell (e.g., the electrodes, the flow structures, the interconnections, the membrane) before assembling them together into a device. However, such measurements never completely reflect the true total resistance of a fuel cell device.

Variations in contact resistances, assembly processes, and operating conditions make total fuel cell resistance difficult to predict. These factors make fuel cell characterization extremely challenging, as discussed in Chapter 7, and emphasize the necessity of in situ fuel cell characterization. Despite the experimental difficulties involved in pinpointing all the sources of fuel cell resistance loss, the electrolyte yields the biggest resistance loss for most fuel cell devices.

128 FUEL CELL CHARGE TRANSPORT

CathodeElectrolyteAnode

Rinterconnect RinterconnectRanode Relectrolyte Rcathode

Figure 4.7. The total ohmic resistance presented by a fuel cell is actually a combination of resis- tances, each attributed to different components of the fuel cell. In this diagram, fuel cell resistance is divided into interconnect, anode, electrolyte, and cathode components. Since current flows serially through all components, total fuel cell resistance is given by the series sum of the individual resistance components.

4.3.4 lonic (Electrolyte) Resistance Usually Dominates

The best electrolytes employed in fuel cells have ionic conductivities of around 0.10 Ω−1 ⋅ cm−1. Even at a thickness of 50 𝜇m (very thin), this produces an ASR of 0.05 Ω ⋅ cm2. In contrast, a 50-𝜇m-thick porous carbon cloth electrode would have an ASR of less than 5 × 10−6 Ω ⋅ cm2. This example illustrates how electrolyte resistance usually dominates fuel cells.

Well-designed fuel cells have a total ASR in the range of 0.05–0.10 Ω ⋅ cm2, and elec- trolyte resistance usually accounts for most of the total. If electrolyte thickness cannot be reduced, decreasing ohmic loss depends on finding high-𝜎 ionic conductors. Unfortu- nately, developing satisfactory ionic conductors is challenging. The three most widely used electrolyte classes, discussed in Sections 4.5.1– 4.5.3, are aqueous, polymer, and ceramic electrolytes. The conductivity mechanisms and materials properties of these three elec- trolyte classes are quite different. Before we get to that discussion, however, it is helpful to develop a clear physical picture of conductivity in general terms.

4.4 PHYSICAL MEANING OF CONDUCTIVITY

Conductivity quantifies the ability of a material to permit the flow of charge when driven by an electric field. In other words, conductivity is a measure of how well a material accommo- dates charge transport. A material’s conductivity is influenced by two major factors: how many carriers are available to transport charge and the mobility of those carriers within the material. The following equation defines 𝜎 in those terms:

𝜎i = (|zi|F)ciui (4.19)

PHYSICAL MEANING OF CONDUCTIVITY 129

where ci represents the molar concentration of charge carriers (how many moles of carrier are available per unit volume) and ui is the mobility of the charge carriers within the mate- rial. The quantity |zi|F is necessary to convert charge carrier concentration from units of moles to units of coulombs. Here, zi is the charge number for the carrier (e.g., zi = +2 for Cu2+, zi = −1 for e–, etc.), the absolute-value function ensures that conductivity is always a positive number, and F is Faraday’s constant.

A material’s conductivity is therefore determined by the product of carrier concentration ci and carrier mobility ui. These properties are, in turn, set by the structure and conduc- tion mechanisms within the material. Up to this point, the charge transport equations we have learned apply equally well to both electronic and ionic conduction. Now, however, their paths will diverge. Because electronic and ionic conduction mechanisms are vastly different, electronic and ionic conductivities are also quite different.

CONDUCTIVITY AND MOBILITY

The difference between conductivity and mobility can be understood by an analogy. Pre- tend that we are studying the transport of people (in cars) down an interstate highway. Mobility describes how fast the cars are driving down the highway. Conductivity, how- ever, would also include information about how many cars are on the highway and how many people each car can hold. This analogy is not perfect but may help keep the two terms straight.

4.4.1 Electronic versus Ionic Conductors

Differences in the fundamental nature of electrons versus ions lead to differences in the mechanisms for electronic versus ionic conduction. Figure 4.8 schematically contrasts a typical electronic conductor (a metal) and a typical ionic conductor (a solid electrolyte).

Figure 4.8a illustrates the free-electron model of a metallic electron conductor. In this model, the valence electrons associated with the atoms of the metal become detached from the atomic lattice and are free to move about the metal. Meanwhile, the metal ions remain intact and immobile. The free valence electrons constitute a “sea” of mobile charges, which are able to move in response to an applied field.

By contrast, Figure 4.8b illustrates the hopping model of a solid-state ionic conductor. The crystalline lattice of this ion conductor consists of both positive and negative ions, all of which are fixed to specific crystallographic positions. Occasionally, defects such as miss- ing atoms (“vacancies”) or extra atoms (“interstitials”) will occur in the material. Charge transport is accomplished by the site-to-site “hopping” of these defects through the material.

The structural differences between the two kinds of conductors lead to dramatic differ- ences in carrier concentrations. In a metal, free electrons are populous, while carriers in a crystalline solid electrolyte are rare. The differences in the charge transport mechanisms, as illustrated in Figure 4.8, also lead to dramatic differences in carrier mobility. Combined, the differences in carrier concentration and carrier mobility lead to a very different picture for electron conductivity in a metal versus ion conductivity in a solid electrolyte. Let us take a closer look.

130 FUEL CELL CHARGE TRANSPORT

M+

M+ M+

M+

M+M+

M+

e–

e–e– e–

e–

e– e–

e–

e–e–

e–

CC+

C+

CC+

CC+ CC+ CC+

CC+

AA–

AA– AA– AA–

AA–

AA– AA–

Vacancy Interstitial

(a)

(b)

Figure 4.8. Illustration of charge transport mechanisms. (a) Electron transport in a free-electron metal. Valence electrons detach from immobile metal atom cores and move freely in response to an applied field. Their velocity is limited by scattering from the lattice. (b) Charge transport in this crystalline ionic conductor is accomplished by mobile anions, which “hop” from position to posi- tion within the lattice. The hopping process only occurs where lattice defects such as vacancies or interstitials are present.

4.4.2 Electron Conductivity in a Metal

For a simple electron conductor, such as a metal, the Drude model predicts that the mobility of free electrons in the metal will be limited by scattering (from phonons, lattice imperfec- tions, impurities, etc.):

u = q𝜏

m (4.20)

where 𝜏 gives the mean free time between scattering events, m is the mass of the electron (m = 9.11 × 10−31 kg), and q is the elementary electron charge in coulombs (q = 1.602 × 10−19 C).

Inserting the results for electron mobility (Equation 4.20) into the expression for con- ductivity (Equation 4.19) gives

𝜎 = |zeF|ceq𝜏

m (4.21)

Carrier concentration in a metal may be calculated from the density of free electrons. In general, each metal atom will contribute approximately one free electron. Atomic packing

PHYSICAL MEANING OF CONDUCTIVITY 131

densities are generally on the order of 1028 atoms/m3, which yields molar carrier concen- trations on the order of 104 mol/m3.

Inserting typical numbers into Equation 4.21 allows us to calculate ballpark electronic conductivity values. The charge number on an electron is, of course, –1(|ze| = 1). Typi- cal scattering times (in relatively pure metals) are 10−12–10–14 s. Using ce ≈ 104 mol∕m3 yields typical electron conductivities for metals in the range of 106–108 Ω–1 ⋅ cm–1).

4.4.3 Ion Conductivity in a Crystalline Solid Electrolyte

The conduction hopping process illustrated in Figure 4.8b for a solid ion conductor leads to a very different expression for mobility than that used for a metallic electron conductor. Ion mobility for the material in Figure 4.8b is dependent on the rate at which ions can hop from position to position within the lattice. This hopping rate, like the reaction rates studied in the previous chapter, is exponentially activated. The effectiveness of the hopping process is characterized by the material’s diffusivity D:

D = Doe−ΔGact∕(RT) (4.22)

where Do is a constant reflecting the attempt frequency of the hopping process, ΔGact is the activation barrier for the hopping process,R is the gas constant, and T is the temperature (K). The overall mobility of ions in the solid electrolyte is then given by

u = |zi|FD RT

(4.23)

Where |zi| is the charge number on the ion, F is Faraday’s constant, R is the gas constant, and T is the temperature (K).

Inserting the expression for ion mobility (Equation 4.23) into the equation for conduc- tivity (Equation 4.19) gives

𝜎 = c(ziF)2D

RT (4.24)

Carrier concentration in a crystalline electrolyte is controlled by the density of the mobile defect species. Most crystalline electrolytes conduct via a vacancy mechanism. These vacancies are intentionally introduced into the lattice by doping. Maximum effective vacancy doping levels are around 8–10%, leading to carrier concentrations of 102–103 mol∕m3.

Typical ion diffusivities are on the order of 10–8 m2∕s for liquid and polymer elec- trolytes at room temperature, and on the order of 10–11 m2∕s for ceramic electrolytes at 700–1000∘C. Typical ion carrier concentrations are 103–104 mol∕m3 for liquid electrolytes, 102–103 mol∕m3 for polymer electrolytes, and 102–103 mol∕m3 for ceramic electrolytes at 700–1000∘C. Inserting these values into Equation 4.24 yields ionic conductivity values of 10−4–102 Ω–1 ⋅ m−1 (10−6 − 100 Ω–1 ⋅ cm−1).

Note that solid-electrolyte ionic conductivity values are well below electronic conduc- tivity values for metals. As has been previously stated, ionic charge transport tends to be far more difficult than electronic charge transport. Therefore, much of the focus in fuel cell research is placed on finding better electrolytes.

132 FUEL CELL CHARGE TRANSPORT

4.5 REVIEW OF FUEL CELL ELECTROLYTE CLASSES

The search for better electrolytes has led to the development of three major candidate mate- rials classes for fuel cells: aqueous, polymer, and ceramic electrolytes. Regardless of the class, however, any fuel cell electrolyte must meet the following requirements:

• High ionic conductivity • Low electronic conductivity • High stability (in both oxidizing and reducing environments) • Low fuel crossover • Reasonable mechanical strength (if solid) • Ease of manufacturability

Other than the high-conductivity requirement, the electrolyte stability requirement is often the hardest to fulfill. It is difficult to find an electrolyte that is stable in both the highly reducing environment of the anode and the highly oxidizing environment of the cathode.

4.5.1 Ionic Conduction in Aqueous Electrolytes/Ionic Liquids

In this section, we discuss ionic conduction in aqueous electrolytes and ionic liquids. An aqueous electrolyte is a water-based solution containing dissolved ions that can transport charge. An ionic liquid is a material which is itself simultaneously liquid and ionic. Sodium chloride dissolved in water is an example of an aqueous electrolyte. Upon dissolution in water, the NaCl separates into mobile Na+ ions and mobile Cl– ions, which can transport charge by moving through the water solvent. Molten NaCl (when heated to high tempera- ture) is an example of an ionic liquid. Pure H3PO4 at 50

∘C is another example of an ionic liquid. At room temperature, H3PO4 is a somewhat waxy, white crystalline solid. However, when heated above 42∘C it becomes a viscous ionic liquid consisting of H+ ions, PO43– ions, and H3PO4 molecules.

Almost all aqueous/liquid electrolyte fuel cells use a matrix material to support or immo- bilize the electrolyte. The matrix generally accomplishes three tasks:

1. Provides mechanical strength to the electrolyte

2. Minimizes the distance between the electrodes while preventing shorts

3. Prevents crossover of reactant gases through the electrolyte

Reactant crossover, the last task on this list, is a particular problem for aqueous/liquid electrolytes (much more so than for solid electrolytes). In an unsupported liquid elec- trolyte, reactant gas crossover can be severe; in these situations, unbalanced-pressure or high-pressure operation is impossible. The use of a matrix material provides mechanical integrity and reduces gas crossover problems, while still permitting thin (0.1–1.0-mm) electrolytes.

Alkaline fuel cells use concentrated aqueous KOH electrolytes, while phosphoric acid fuel cells use either concentrated aqueous H3PO4 electrolytes or pure H3PO4 (an ionic liq- uid). Molten carbonate fuel cells use molten (K/Li)2CO3 immobilized in a supporting

REVIEW OF FUEL CELL ELECTROLYTE CLASSES 133

matrix. The (K/Li)2CO3 material melts at around 450 ∘C to become a liquid (“molten”)

electrolyte. (MCFCs must therefore obviously be operated above 450∘C.) Ionic conductivity in aqueous/liquid environments can best be approached using a driv-

ing force/frictional force balance model. In liquids, an ion will accelerate under the force of an electric field until frictional drag exactly counteracts the electric field force. The balance between the electric field and frictional drag determines the terminal velocity of the ion.

The electric field force, FE, is given by

FE = ziq dV dx

(4.25)

where zi is the charge number of the ion and q is the fundamental electron charge (1.6 × 10–19 C). Although we do not show the derivation here, the frictional drag force FD may be approximated from Stokes’s law as

FD = 6𝜋𝜇rv (4.26)

where 𝜇 is the viscosity of the liquid, r is the radius of the ion, and v is the velocity of the ion. Equating the two forces allows us to determine the mobility, ui, which is defined as the ratio between the applied electric field and the resulting ion velocity (because mobility is defined as a positive quantity, inclusion of the absolute value is again required):

ui = |||| 𝑣dV∕dx |||| = |zi|q6𝜋𝜇r (4.27)

Thus, mobility is determined by the ion size and the liquid viscosity. Intuitively, this expression makes sense: Bulky ions or highly viscous liquids should lead to lower mobili- ties, while nonviscous liquids and small ions should yield higher mobilities. The mobilities of a variety of ions in aqueous solution are given in Table 4.2. Note that in aqueous solu- tions the H+ ion tends to be hydrated by one or more water molecules. This ionic species is therefore better thought of as H3O

+ or H ⋅ (H2O)x +, where x represents the number of

water molecules “hydrating” the proton. Recall our expression for conductivity (Equation 4.19), which is repeated here for clarity:

𝜎i = (|zi|F)ciui (4.28) If the values of ion mobilities in Table 4.2 are inserted into this expression, the ionic

conductivity of various aqueous electrolytes may be calculated. Unfortunately, these

TABLE 4.2. Selected Ionic Mobilities at Infinite Dilution in Aqueous Solutions at 25∘C

Cation Mobility, u (cm2/V ⋅ s) Anion Mobility, u (cm2/V ⋅ s)

H+(H3O +) 3.63 × 10−3 OH− 2.05 × 10−3

K+ 7.62 × 10−4 Br− 8.13 × 10−4

Ag+ 6.40 × 10−4 I− 7.96 × 10−4

Na+ 5.19 × 10−4 Cl− 7.91 × 10−4

Li+ 4.01 × 10−4 HCO3− 4.61 × 10 −4

Source: From Ref. [6a].

134 FUEL CELL CHARGE TRANSPORT

calculations are only accurate for dilute aqueous solutions when the ion concentration is low. At high ion concentration (or for ionic liquids) strong electrical interactions between the ions make conductivity far more difficult to calculate. In general, the conductivity of highly concentrated aqueous solutions or pure ionic liquids will be much lower than that predicted by Equation 4.28. For example, the conductivity of pure H3PO4 is exper- imentally determined to be 0.1–1.0 Ω−1 ⋅ cm−1 (depending on the temperature), whereas Equation 4.28 predicts that the conductivity of pure H3PO4 should be approximately 18 Ω−1 ⋅ cm−1.

Table 4.2 does offer some other useful insights. For example, it explains why KOH is the electrolyte of choice in alkaline fuel cells. Besides being extremely inexpensive, KOH exhibits the highest ionic conductivity of any of the hydroxide compounds. (Compare the u value for K+ to other candidate hydroxide cations such as Na+ or Li+.) In alkaline fuel cells, fairly concentrated (30–65%) solutions of KOH are used, resulting in conductivities on the order of 0.1–0.5 Ω−1 ⋅ cm−1. How much would the conductivity be reduced if a far more dilute electrolyte was used? To get an answer, refer to Example 4.3, where the approx- imate conductivity of a 0.1 M KOH electrolyte solution is calculated using Equation 4.28.

Example 4.3 Calculate the approximate conductivity of a 0.1 M aqueous solution of KOH.

Solution: We use Equation 4.28 as our guide. Assuming that 0.1 M KOH completely dissolves into K+ ions and OH– ions (it does), the concentration of K+ and OH– will also be 0.1 M. Converting these concentrations to units of moles per cubic centimeter gives

cK+ = (0.1 mol∕L)(1 L∕1000 cm3) = 1 × 10−4 mol∕cm3

cOH− = (0.1 mol∕L)(1 L∕1000 cm3) = 1 × 10−4 mol∕cm3 (4.29)

The mobilities of K+ and OH– are given in Table 4.2. Inserting these numbers into Equation 4.28 yields

𝜎K+ = (1)(96, 485)(1 × 10−4 mol∕cm3)(7.62 × 10−4 cm2∕V ⋅ s) = 0.0073 Ω−1 ⋅ cm−1

𝜎OH− = (1)(96, 485)(1 × 10−4 mol∕cm3)(2.05 × 10−3 cm2∕V ⋅ s) = 0.0198 Ω−1 ⋅ cm−1

(4.30)

The total ionic conductivity of the electrolyte is then given by the sum of the cation and anion conductivities:

𝜎total = 𝜎K+ + 𝜎OH− = 0.0073 + 0.0198 = 0.0271 Ω−1 ⋅ cm−1 (4.31)

In reality, the conductivity of the 0.1 M KOH solution will likely be a little lower than this predicted value. Note that most of the conductivity is provided by the OH–

ion, rather than the K+ ion. This is due to the higher mobility of the OH– ion.

REVIEW OF FUEL CELL ELECTROLYTE CLASSES 135

4.5.2 Ionic Conduction in Polymer Electrolytes

In general, ionic transport in polymer electrolytes follows the exponential relationship described by Equations 4.22 and 4.24. By combining these two equations, we can obtain (see problem 4.11)

𝜎T = APEMe−Ea∕kT (4.32)

where APEM is a preexponential factor and Ea represents the activation energy (eV/atom) (Ea = ΔGact∕F, where F is Faraday’s constant). As this equation indicates, conductivity increases exponentially with increasing temperature. Most polymer and crystalline ion con- ductors obey this model quite well.

For a polymer to be a good ion conductor, at a minimum it should possess the following structural properties:

1. The presence of fixed charge sites

2. The presence of free volume (“open space”)

The fixed charge sites should be of opposite charge compared to the moving ions, ensur- ing that the net charge balance across the polymer is maintained. The fixed charge sites provide temporary centers where the moving ions can be accepted or released. In a poly- mer structure, maximizing the concentration of these charge sites is critical to ensuring high conductivity. However, excessive addition of ionically charged side chains will significantly degrade the mechanical stability of the polymer, making it unsuitable for fuel cell use.

Free volume correlates with the spatial organization of the polymer. In general, a typical polymer structure is not fully dense. Small-pore structures (or free volumes) will almost always exist. Free volume improves the ability of ions to move across the polymer. Increas- ing the polymer free volume increases the range of small-scale structural vibrations and motions within the polymer. These motions can result in the physical transfer of ions from site to site across the polymer. (See Figure 4.9.)

Because of these free-volume effects, polymer membranes exhibit relatively high ionic conductivities compared to other solid-state ion-conducting materials (such as ceramics).

Polymer free volume also leads to another well-known transport mechanism, known as the vehicle mechanism. In the vehicle mechanism, ions are transported through free-volume

–

– +

–

–– –

– +Charged site Ion Polymer chain

+ – –

– –

–

– –– –

Figure 4.9. Schematic of ion transport between polymer chains. Polymer segments can move or vibrate in the free volume, thus inducing physical transfer of ions from one charged site to another.

136 FUEL CELL CHARGE TRANSPORT

spaces by hitching a ride on certain free species (the “vehicles”) as these vehicles pass by. Water is a common vehicular species; as water molecules move through the free volumes in a polymer membrane, ions can go along for the ride. In this case, the conduction behavior of the ions in the polymer electrolyte is much like that in an aqueous electrolyte. Per- sulfonated polytetrafluoroethylene (PTFE)—more commonly known as Nafion—exhibits extremely high proton conductivity based on the vehicle mechanism. Since Nafion is the most popular and important electrolyte for PEMFC applications, we review its properties in the next section.

Ionic Transport in Nafion. Nafion has a backbone structure similar to polytetrafluo- roethylene (Teflon). However, unlike Teflon, Nafion includes sulfonic acid (SO3

–H+) func- tional groups. The Teflon backbone provides mechanical strength while the sulfonic acid (SO3

–H+) chains provide charge sites for proton transport. Figure 4.10 illustrates the struc- ture of Nafion.

It is believed that Nafion free volumes aggregate into interconnected nanometer-sized pores whose walls are lined by sulfonic acid (SO3

–H+) groups. In the presence of water, the protons (H+) in the pores form hydronium complexes (H3O

+) and detach from the sulfonic acid side chains. When sufficient water exists in the pores, the hydronium ions can transport in the aqueous phase. Under these circumstances, ionic conduction in Nafion is similar to conduction in liquid electrolytes (Section 4.5.1). As a bonus, the hydrophobic nature of the Teflon backbone further accelerates water transport through the membrane, since the hydrophobic pore surfaces tend to repel water. Because of these factors, Nafion exhibits proton conductivity comparable to that of a liquid electrolyte. To maintain this extraordinary conductivity, Nafion must be fully hydrated with liquid water. Usually, hydration is achieved by humidifying the fuel and oxidant gases provisioned to the fuel cell. In the following paragraphs, we review the key properties of Nafion in more detail.1

Nafion Absorbs Significant Amounts of Water. The pore structure in Nafion can hold significant amounts of water. In fact, Nafion can accommodate so much water that its volume will increase up to 22% when fully hydrated. (Strongly polar liquids, such as alcohols, can cause Nafion to swell up to 88%!) Since conductivity and water content are strongly related, determining water content is essential to determining the conductivity of the membrane. The water content λ in Nafion is defined as the ratio of the number of water molecules to the number of charged (SO3

–H+) sites. Experimental results suggest that λ can vary from almost 0 (for completely dehydrated Nafion) to 22 (for full saturation, under certain conditions). For fuel cells, experimental measurements have related the water con- tent in Nafion to the humidity condition of the fuel cell, as shown in Figure 4.11. Thus, if the humidity condition of the fuel cell is known, the water content in the membrane can be estimated. Humidity in Figure 4.11 is quantified by water vapor activity a𝑤 (essentially relative humidity):

a𝑤 = p𝑤 pSAT

(4.33)

1The Nafion model reviewed here was suggested by Springer et al. [8]

REVIEW OF FUEL CELL ELECTROLYTE CLASSES 137

H O H O SO 1nm 2 3 3

+ –

Polytetraflouroethylene (PTFE) Nafion

= =

F F

F F F F F F F F

F F F F F

F F

C C C C C C C C

n n n

O O

S H– +

(a)

(b)

Figure 4.10. (a) Chemical structure of Nafion. Nafion has a PTFE backbone for mechanical stability with sulfonic groups to promote proton conduction. (b) Schematic microscopic view of proton con- duction in Nafion. When hydrated, nanometer-sized pores swell and become largely interconnected. Protons bind with water molecules to form hydronium complexes. Sulfonic groups near the pore walls enable hydronium conduction.

where p𝑤 represents the actual partial pressure of water vapor in the system and pSAT rep- resents the saturation water vapor pressure for the system at the temperature of operation. The data in Figure 4.11 can be represented mathematically as

λ =

{ 0.043 + 17.18a𝑤 − 39.85a2𝑤 + 36.0a3𝑤 for 0 < a𝑤 ≤ 1 14 + 4

( a𝑤 − 1

) for 1 < a𝑤 ≤ 3 (4.34)

138 FUEL CELL CHARGE TRANSPORT

0 0.2 0.4 0.6 0.8 1 0

Water vapor activity (p w

/pSAT)

λ =

H 2O

/S O

3−

Figure 4.11. Water content versus water activity for Nafion 117 at 303 K (30∘C) according to Equation 4.34. Water vapor activity is defined as the ratio of the actual water vapor pressure (p𝑤) for the system compared to the saturation water vapor pressure (pSAT) for the system at the temper- ature of interest. Reprinted with permission from Ref. [8], Journal of the Electrochemical Society, 138: 2334, 1991. Copyright 1991 by the Electrochemical Society.

Equation 4.34 does not consider the effects of temperature; however, it is reasonably accurate for PEMFCs operating near 80∘C.

WATER VAPOR SATURATION PRESSURE

When the partial pressure of water vapor (p𝑤) within a gas stream reaches the water vapor saturation pressure pSAT for a given temperature, the water vapor will start to condense, generating water droplets. In other words, relative humidity is 100% when p𝑤 = pSAT. Importantly, pSAT is a strong function of temperature:

log10 pSAT = −2.1794 + 0.02953T − 9.1837 × 10−5T2 + 1.4454 × 10−7T3 (4.35)

where pSAT is given in bars (1 bar = 100,000 Pa) and T is the temperature in degrees Celsius. For example, if fully humidified air at 80∘C and 3 atm is provided to a fuel cell, the water vapor pressure is [9]

pSAT = 10−2.1794+0.02953×80−9.1837×10 −5×802+1.4454×10−7×803 = 0.4669 bar (4.36)

This gives the mole fraction of water in fully humidified air at 80∘C and 3 atm as 0.4669 bar/3 atm = 0.4669 bar/(3 × 1.0132501 bar) = 0.154 assuming an ideal gas.

REVIEW OF FUEL CELL ELECTROLYTE CLASSES 139

Under these same conditions, if the air is instead only partially humidified, such that the water mole fraction is 0.1, then the water vapor activity (or relative humidity) would be (again assuming an ideal gas)

a𝑤 = pH2O𝑤 pSAT

= xH2O × ptotal

xH2O,SAT × ptotal = 0.1

0.154 = 0.65 (4.37)

Nafion Conductivity Is Highly Dependent onWater Content. As previously men- tioned, conductivity and water content are strongly related in Nafion. Conductivity and temperature are also strongly related. In general, the proton conductivity of Nafion increases linearly with increasing water content and exponentially with increasing temperature, as shown by the experimental data in Figures 4.12 and 4.13. In equation form, these experi- mentally determined relationships may be summarized as

𝜎(T , λ) = 𝜎303K(λ) exp [ 1268

( 1 303

− 1 T

)] (4.38)

where 𝜎303K(λ) = 0.005193λ − 0.00326 (4.39)

where 𝜎 represents the conductivity (S/cm) of the membrane and T (K) is the temperature. Since the conductivity of Nafion can change locally depending on water content, the total

area-specific resistance of a membrane is found by integrating the local resistivity over the

0 5 10 15 20 25 0

0.02

0.04

0.06

0.08

0.1

0.12

λ = H 2 O/SO

σ (S

/c m

)

Figure 4.12. Ionic conductivity of Nafion versus water content λ according to Equations 4.38 and 4.39 at 303 K.

140 FUEL CELL CHARGE TRANSPORT

2.6 2.8 3 3.2 3.4 3.6 3.8 –1.3

–1.2

–1.1

–1

–0.9

–0.8

–0.7

–0.6

lo g(

σ )

[lo g(

S /c

m )]

1/T (x103 K)

100˚C 50˚C 0˚C

Figure 4.13. Ionic conductivity of Nafion versus temperature according to Equation 4.38 when λ = 22.

membrane thickness (tm) as

ASRm = ∫ tm

0 𝜌(z)dz = ∫

dz 𝜎[λ(z)]

(4.40)

Protons Drag Water with Them. Since conductivity in Nafion is dependent on water content, it is essential to know how water content varies across a Nafion membrane. During fuel cell operation, the water content across a Nafion membrane is generally not uniform. Water content varies across a Nafion membrane because of several factors. Perhaps most important is the fact that protons2 traveling through the pores of Nafion generally drag one or more water molecules along with them. This well-known phenomenon is called electro-osmotic drag. The degree to which proton movement causes water movement is quantified by the electro-osmotic drag coefficient ndrag, which is defined as the number of water molecules accompanying the movement of each proton (ndrag = nH2O∕H

+). Obvi- ously, how much water is dragged per proton depends on how much water exists in the Nafion membrane in the first place. It has been measured that ndrag = 2.5 ± 0.2 (between 30 and 50∘C) in fully hydrated Nafion (when λ = 22). When λ = 11, ndrag = ∼ 0.9. Com- monly, it is assumed that ndrag changes linearly with λ as

ndrag = nSATdrag λ

22 for 0 ≤ λ ≤ 22 (4.41)

2Actually, protons travel in the form of hydronium complexes as explained in the text. For simplicity, however, we use the term “proton” in these discussions. Also, it is more straightforward to define the electro-osmotic drag coefficient in terms of the number of water molecules per proton (rather than per hydronium, which contains a water molecule already).

REVIEW OF FUEL CELL ELECTROLYTE CLASSES 141

where nSATdrag ≈ 2.5. Knowledge of the electro-osmotic drag coefficient allows us to estimate the water drag flux from anode to cathode when a net current j flows through the PEMFC:

JH2O,drag = 2ndrag j

2F (4.42)

where J is the molar flux of water due to electro-osmotic drag (mol/cm2), j is the operating current density of the fuel cell (A/cm2), and the quantity 2F converts from current density to hydrogen flux. The factor of 2 in the front of the equation then converts from hydrogen flux to proton flux. As you will see in Chapter 6, the drag coefficient becomes very important in modeling the behavior of Nafion membranes in PEMFCs.

Back Diffusion of Water. In a PEMFC, electro-osmotic water drag moves water from the anode to the cathode. As this water builds up at the cathode, however, back diffu- sion occurs, resulting in the transport of water from the cathode back to the anode. This back-diffusion phenomenon occurs because the concentration of water at the cathode is generally far higher than the concentration of water at the anode (exacerbated by the fact that water is produced at the cathode by the electrochemical reaction). Back diffusion counterbalances the effects of electro-osmotic drag. Driven by the anode/cathode water concentration gradient, the water back-diffusion flux can be determined by

JH2O,back diffusion = − 𝜌dry

Mm Dλ

dλ dz

(4.43)

where 𝜌dry is the dry density (kg/m 3) of Nafion, Mm is the Nafion equivalent weight

(kg/mol), and z is the direction through the membrane thickness. The key factor in this equation is the diffusivity of water in the Nafion membrane (Dλ).

Unfortunately, Dλ is not constant but is a function of water content λ. Since the total water flux in Nafion is simply the addition of electro-osmotic drag and back diffusion, we have

JH2O = 2n SAT drag

2F λ

22 − 𝜌dry

Mm Dλ(λ)

dλ dz

(4.44)

This combined expression makes it explicitly clear that the water flux in Nafion is a complex function of λ. [We state the water diffusivity as Dλ(λ) in this equation to emphasize its dependency on water content.]

Summary. Based on the fuel cell operating conditions (humidity and current density), we can estimate the water content profile (λ(z)) in the membrane by using Equations 4.34 and 4.44. Once we have the water content profile, we can then calculate the ion conductivity of the membrane by using Equation 4.38. In this fashion, the ohmic losses in a PEMFC may be quantified. This procedure is demonstrated in Example 4.4. In Chapter 6 we will combine these equations with the other fuel cell loss terms to create a complete PEMFC model.

142 FUEL CELL CHARGE TRANSPORT

Example 4.4 Consider a hydrogen PEMFC powering an external load at 0.7 A/cm2. The activities of water vapor on the anode and cathode sides of the membrane are measured to be 0.8 and 1.0, respectively. The temperature of the fuel cell is 80∘C. If the Nafion membrane thickness is 0.125 mm, estimate the ohmic overvoltage loss across the membrane.

Solution: We can convert the water activity on the Nafion surfaces to water contents using Equation 4.34:

λA = 0.043 + 17.18 × 0.8 − 39.85 × 0.82 + 36.0 × 0.83 = 7.2

λC = 0.043 + 17.18 × 1.0 − 39.85 × 1.02 + 36.0 × 1.03 = 14.0 (4.45)

With these values as boundary conditions, we then solve Equation 4.44. In this equation, we have two unknowns, JH2O and λ. For convenience, we will set JH2O = 𝛼NH2 = 𝛼(j∕2F), where 𝛼 is an unknown that denotes the ratio of water flux to hydro- gen flux. After rearrangement, Equation 4.44 becomes

dλ dz

= (

2nSATdrag λ

22 − 𝛼

) jMm 2F𝜌dryDλ

(4.46)

EQUIVALENT WEIGHT

The equivalent weight of a species is defined by its atomic weight or formula weight divided by its valence:

Equivalent weight = atomic (formula) weight

valence (4.47)

Valence is defined by the number of electrons that the species can donate or accept. For example, hydrogen has a valence of 1 (H+). Oxygen has a valence of 2 (O2–). Thus, hydrogen has an equivalent weight of 1.008 g∕mol∕1 = 1.008 g∕mol and oxygen has an equivalent weight of 15.9994 g∕mol∕2 = 7.9997 g∕mol. In the case of sulfate radicals (SO4

2–), the formula weight is (1 × 32.06) + (4 × 15.9994) = 96.058 g∕mol. Thus, the equivalent weight is (96.058 g∕mol)∕2 = 48.029 g∕mol.

The sulfonic group (SO3 –H+) in Nafion has a valence of 1, since it can accept only

one proton. Thus, the equivalent weight of Nafion is equal to the average weight of the polymer chain structure that can accept one proton. This number is very useful since it facilitates the calculation of sulfonic charge (SO3

–) concentration in Nafion as

CSO−3 (mol∕m3) =

𝜌dry (kg∕m3) Mm (kg∕mol)

(4.48)

where 𝜌dry is the dry density of Nafion (kg/m 3) and Mm is the Nafion equivalent weight

(kg/mol).

REVIEW OF FUEL CELL ELECTROLYTE CLASSES 143

In a similar fashion, water content, λ (H2O∕SO3–), can be converted to water con- centration in Nafion as

CH2O(mol∕m 3) = λ

𝜌dry(kg∕m3) Mm(kg∕mol)

(4.49)

Typically, Nafion has an equivalent weight of around ∼ 1–1.1 kg∕mol and a dry den- sity of ∼ 1970 kg∕m3. Thus, the estimated charge density for Nafion would be

CSO−3 (mol∕m3) =

1970 kg∕m3

1 kg∕mol = 1970 mol∕m3 (4.50)

WATER DIFFUSIVITY IN NAFION

As emphasized above, water diffusivity in Nafion (Dλ) is a function of water content λ. Experimentally (using magnetic resonance techniques), this dependence has been mea- sured as

Dλ = exp [ 2416

( 1 303

− 1 T

)] × (2.563 − 0.33λ + 0.0264λ2 − 0.000671λ3) × 10−6

for λ > 4 (cm2∕s) (4.51)

The exponential part describes the temperature dependence, while the polynomial portion describes the λ dependence at the reference temperature of 303 K. This equation is only valid for λ > 4. For λ < 4, values extrapolated from Figure 4.14 (dotted line) should be used instead.

0 5 10 15 0

0.5

1.5

2.5

3.5

4 x 10−6

λ (H2O/SO3-)

W at

er d

iff us

iv ity

, D λ (

cm 2 /

Figure 4.14. Water diffusivity Dλ in Nafion versus water content λ at 303 K.

144 FUEL CELL CHARGE TRANSPORT

Even though this is an ordinary differential equation on λ, we may not solve it analyt- ically since Dλ is a function of λ. However, if we assume λ in the membrane changes from 7.2 to 14.0 according to the boundary conditions, we can see from Figure 4.14 that the water diffusivity is fairly constant over this range. If we assume an average value of λ = 10, we can estimate Dλ from Equation 4.51 as

Dλ = 10−6 exp [ 2416

( 1 303

− 1 353

)] × (2.563 − 0.33 × 10 + 0.0264 × 102 − 0.000671 × 103)

= 3.81 × 10−6 cm2∕s (4.52)

Now we can evaluate Equation 4.46, yielding the analytical solution

λ(z) 11𝛼 nSATdrag

+ C exp

[ jMmn

SAT drag

22 F 𝜌dryDλ z

] = 11𝛼

2.5

+ C exp

[ ( 0.7 A∕cm2

) × (1.0 kg∕mol) × 2.5

(22 × 96, 485 C∕mol) × (0.00197 kg∕cm3) × (3.81 cm2∕s) z

] = 4.4𝛼 + C exp(109.8z) (4.53)

where z is in centimeters and C is a constant to be determined from the boundary conditions. If we set the anode side as z = 0, we have λ(0) = 7.2 and λ(0.0125) = 14 from Equation 4.45. Accordingly, Equation 4.53 becomes

λ(z) = 4.4𝛼 + 2.30 exp(109.8z) where 𝛼 = 1.12 (4.54)

Now we know that about 1.12 water molecules are dragged per each hydrogen (or in other words, about 0.56 water molecules per proton). Figure 4.15a shows the result of how 𝜆 varies across the membrane in this example. At the start of the problem, we assumed a constant Dλ for λ in the range of 7.2–14. We can confirm that this assumption is reasonable from the results of Figure 4.15.

From Equations 4.38 and 4.54, we can determine the conductivity profile of the membrane:

𝜎(z) = {0.005193[4.4𝛼 + 2.30 exp(109.8z)] − 0.00326}

× exp [ 1268

( 1 303

− 1 353

)] = 0.0404 + 0.0216 exp(109.8z) (4.55)

REVIEW OF FUEL CELL ELECTROLYTE CLASSES 145

Figure 4.15b shows the result. Finally, we can determine the area-specific resis- tance of the membrane using Equation 4.40:

ASRm = ∫ tm

dz 𝜎[λ(z)]

= ∫ 0.0125

dz 0.0404 + 0.0216 exp(109.8z)

= 0.15 Ω ⋅ cm2

(4.56) Thus, the ohmic overvoltage due to the membrane resistance in this PEMFC is

approximately

Vohm = j × ASRm = (0.7 A∕cm2) × (0.15 Ω ⋅ cm2) = 0.105 V (4.57)

This section has focused exclusively on the details of Nafion. However, the conduc- tion properties and characteristics of other polymer electrolyte alternatives are discussed in Chapter 9 for the interested reader.

4.5.3 Ionic Conduction in Ceramic Electrolytes

This section explains the underlying physics of ion transport in SOFC electrolytes. As their name implies, SOFC electrolytes are solid, crystalline oxide materials that can conduct ions. The most popular SOFC electrolyte material is yttria-stabilized zirconia (YSZ). A typical YSZ electrolyte contains 8% yttria mixed with zirconia. What is the meaning of zirconia and yttria? Zirconia is related to the metal zirconium, and yttria derives its name from another metal, yttrium. Zirconia has the chemical composition ZrO2; it is the oxide of zirconium. By analogy, yttria, or Y2O3, is the oxide of yttrium. A mixture of zirconia and yttria is called yttria-stabilized zirconia because the yttria stabilizes the zirconia crystal structure in the cubic phase (where it is most conductive). Even more importantly, however, the yttria introduces high concentrations of oxygen vacancies into the zirconia crystal structure. This high oxygen vacancy concentration allows YSZ to exhibit high ion conductivity.

Adding yttria to zirconia introduces oxygen vacancies due to charge compensation effects. Pure ZrO2 forms an ionic lattice consisting of Zr

4+ ions and O2– ions, as shown in Figure 4.16a. Addition of Y3+ ions to this lattice upsets the charge balance. As shown in Figure 4.16b, for every two Y3+ ions taking the place of Zr4+ ions, one oxygen vacancy is created to maintain overall charge neutrality. The addition of 8% (molar) yttria to zirconia causes about 4% of the oxygen sites to be vacant. At elevated temperatures, these oxygen vacancies facilitate the transport of oxygen ions in the lattice, as shown in Figure 4.8b.

As discussed in Section 4.4, a material’s conductivity is determined by the combination of carrier concentration (c) and carrier mobility (u):

𝜎 = (|z|F)cu (4.58) In the case of YSZ, carrier concentration is determined by the strength of the yttria

doping. Because a vacancy is required for ionic motion to occur within the YSZ lattice, the

146 FUEL CELL CHARGE TRANSPORT

(b)

0 0.002 0.004 0.006 0.008 0.01 0.012 0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

Anode Membrane thickness(cm) Cathode

Lo ca

l c on

du ct

iv ity

( S

/c m

)

(a)

0 0.002 0.004 0.006 0.008 0.01 0.012 7

( H

O /S

O −

λ 2

3 )

Anode Membrane thickness(cm) Cathode

W at

er c

on te

Figure 4.15. Calculated properties of Nafion membrane for Example 4.4. (a) Water content profile across Nafion membrane. (b) Local conductivity profile across Nafion membrane.

REVIEW OF FUEL CELL ELECTROLYTE CLASSES 147

Y3+Zr 4+Zr4+ Zr4+

Zr4+Zr4+Zr4+

O2–2–O

Y3+ O2–O2–

Zr4+Zr4+ Zr4+ O2–O 2–

O2–O2– Zr4+

Zr4+

O2–

Zr4+ O2–

Zr 4+

Zr 4+Zr 4+Zr 4+

O2–O2–

O2–

Zr 4+Zr 4+ Zr 4+ O2–O2–

O2–O2– Zr 4+

Zr4+ O2–

O2–

Zr 4+ O2–

O2–

Vacancy

(a) (b)

Figure 4.16. View of the (110) plane in (a) pure ZrO2 and (b) YSZ. Charge compensation effects in YSZ lead to creation of oxygen vacancies. One oxygen vacancy is created for every two yttrium atoms doped into the lattice.

oxygen vacancies can be considered to be the ionic charge “carriers.” Increasing the yttria content will result in increased oxygen vacancy concentration, improving the conductivity. Unfortunately, however, there is an upper limit to doping. Above a certain dopant or vacancy concentration, defects start to interact with each other, reducing their ability to move. Above this concentration, further doping is counterproductive and conductivity actually decreases. Plots of conductivity versus dopant concentration show a maximum at the point where defect interaction or “association” commences. For YSZ, this maximum occurs at about 8% molar yttria concentration. (See Figure 4.17.)

1.6

1.7

1.8

1.9

2.1

2.2

2.3

2.4

6 7 8 9 10 11 12 13 14 15

%Y2O3

lo g(

σ T

) (Ω

–1 ·

cm –1

K )

Figure 4.17. YSZ conductivity versus %Y2O3 (molar basis) [10]; YSZ conductivity is displayed as σ(Ω–1 ⋅ cm–1)times T (K). In the next section, Figure 4.18 will clarify why it is convenient to multiply 𝜎 with T.

148 FUEL CELL CHARGE TRANSPORT

The complete expression for conductivity combines carrier concentration and carrier mobility, as described in Section 4.4.3:

𝜎 = c(zF) 2D

RT (4.59)

where carrier mobility is described by D, the diffusivity of the carrier in the crystal lattice. Diffusivity describes the ability of a carrier to move, or diffuse, from site to site within a crystal lattice. High diffusivities translate into high conductivities because the carriers are able to move quickly through the crystal. The atomic origins and physical explana- tion behind diffusivity will be detailed in forthcoming sections. For now, however, it is sufficient to know that carrier diffusivity in SOFC electrolytes is exponentially tempera- ture dependent:

D = D0e−ΔGact∕(RT) (4.60)

where D0 is a constant (cm 2/s), ΔGact is the activation barrier for the diffusion process

(J/mol), R is the gas constant, and T is the temperature (K). Combining Equations 4.59 and 4.61 provides a complete expression for conductivity in SOFC electrolytes:

𝜎 = c(zF)2D0e−ΔGact∕(RT)

RT (4.61)

INTRINSIC CARRIERS VERSUS EXTRINSIC CARRIERS

In YSZ and most other SOFC electrolytes, dopants are used to intentionally create high vacancy (or other charge carrier) concentrations. These carriers are known as extrinsic carriers because their presence is extrinsically created by intentional doping. However, any crystal, even an undoped one, will have at least some natural carrier population. These natural charge carriers are referred to as intrinsic carriers because they occur intrinsically due to the natural energetics of the crystal. Intrinsic carriers exist because no crystal is perfect (unless it is at absolute zero). All crystals will contain “mistakes” such as vacancies that can act as charge carriers for conduction. These mistakes are actually energetically favorable, because they increase the entropy of the crystal. (Recall Section 2.1.4.) For the case of vacancies, an energy balance may be developed that considers the enthalpy cost to create the vacancies versus the entropy benefit they deliver. Solving for this balance results in the following expression for intrinsic vacancy concentration as a function of temperature in an ionic crystal:

xV ≈ e−Δh𝑣∕(2kT) (4.62)

where xV represents the fractional vacancy concentration (expressed as the fraction of lattice sites of the species of interest that are vacant), Δh𝑣 is the formation enthalpy for

REVIEW OF FUEL CELL ELECTROLYTE CLASSES 149

the vacancy in electron-volts (in other words, the enthalpy cost to “create” a vacancy), k is Boltzmann’s constant, and T is the temperature in Kelvin. This expression states that the intrinsic concentration of vacancies within a crystal increases exponentially with temperature. However, since Δh𝑣 is typically on the order of 1 eV or larger, intrinsic vacancy concentrations are generally quite low, even at high temperatures. At 800∘C, the intrinsic vacancy concentration in pure ZrO2 is around 0.001, or about one vacancy per 1000 sites. Compare this to extrinsically doped crystal structures, which can attain vacancy concentrations as high as 0.1, or about one vacancy per 10 sites.

This equation can be further refined depending on whether the charge carriers are extrin- sic or intrinsic:

• For extrinsic carriers, c is determined by the doping chemistry of the electrolyte. In this case, c is a constant and Equation 4.62 can be used as is.

• For intrinsic carriers, c is exponentially dependent on temperature, and Equation 4.62 must be modified as follows:

𝜎 = csites(zF)2D0e−Δh𝑣∕(2kT)e−ΔGact∕(RT)

RT (4.63)

where csites stands for the concentration of lattice sites for the species of interest in the material (moles of sites/cm3).

Almost all useful fuel cell electrolyte materials are purposely doped to increase the number of charge carriers, and therefore the concentration of intrinsic carriers is usually insignificant compared to the concentration of extrinsic carriers (see text box on previous page). Thus, Equation 4.62 is far more important than Equation 4.63 for describing ionic conduction in practical electrolytes. Equation 4.62 is often simplified to a pseudo-empirical expression by lumping the various preexponential terms into a single factor, yielding

𝜎T = ASOFCe−ΔGact∕RT (4.64)

Similarly to Equation 4.32, the term ΔGact∕RT can instead be written as Ea∕kT , yielding

𝜎T = ASOFCe−Ea∕kT (4.65)

Experimental observations confirm the relationship described by Equation 4.64 (or 4.65).

Figure 4.18 shows experimental plots of log(𝜎T) versus 1∕T for both YSZ and gadolinia-doped ceria (GDC, another candidate SOFC electrolyte). The multiplication of 𝜎 with T ensures that the slopes in these plots are indicative of the activation energy for ion migration, ΔGact. The size of ΔGact is often critical for determining the conductivity

150 FUEL CELL CHARGE TRANSPORT

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

–1

–2

–3

–4

1000/T K–1

Gd-doped ceria Y-stabilized zirconia

G =0.60eVactΔ

G =0.89eVactΔlo g(

σ T

) (Ω

–1 ·

cm –1

K )

Figure 4.18. Conductivity of YSZ and GDC electrolytes versus temperature.

of SOFC electrolytes. Typically, its value ranges between about 50,000 and 120,000 J/mol (0.5–1.2 eV).

Further details on specific fuel cell electrolyte materials properties, including a more in-depth discussion on YSZ and GDC, are provided in Chapter 9.

CALCULATING EXTRINSIC DEFECT CONCENTRATIONS IN CRYSTALLINE CERAMIC MATERIALS

As was pointed out earlier in this chapter, almost all useful ceramic fuel cell electrolyte materials are purposely doped to increase the number of charge carriers, and therefore extrinsically created carriers dominate the conduction process. In order to calculate the concentration of the extrinsically created charge carriers (c), which is needed in Equation 4.62, information about the material composition, the doping concentration, and the crystal structure or density is required.

As an example, consider the classic case of 8YSZ, which is zirconia doped with 8 mol% yttria. As shown in Figure 4.16, for every 2 Y that are substituted into the ZrO2 lattice, one oxygen vacancy is created. These extrinsically created oxygen vacan- cies become the source of ionic conduction in this material. To create 8YSZ, 8 mol % Y2O3 is combined with 92 mol % ZrO2. The chemical formula of 8YSZ can therefore be represented as 0.92(ZrO2) + 0.08(Y2O3) = Zr0.92Y0.16O2.08. Because of the 2-to-1 relationship between Y dopants and the created oxygen vacancies, the number of oxygen vacancies can be explicitly shown by writing the formula as Zr0.92Y0.16O2.08V0.08. One

REVIEW OF FUEL CELL ELECTROLYTE CLASSES 151

mole of this material will therefore contain 0.08 mol of oxygen vacancies. The fraction of oxygen sites that are vacant, xv, is 0.08∕2.16 = 0.037. This vacancy fraction can be converted into a vacancy concentration (cv, units of vacancies/cm

3) by applying knowl- edge about the molecular weight and density of the material or by applying knowledge about the molar volume of the material.

If the density of the material is known, this information can be used to convert molar vacancy fraction to vacancy concentration as follows:

c𝑣 = x𝑣co = x𝑣 no

V (4.66)

where co is the concentration of oxygen sites in the material (mol/cm 3), no is the moles of

oxygen atoms per mole of material, and V is the molar volume of the material (cm3/mol). The molar volume can be calculated from the molecular weight (M, g/mol) and the den- sity (𝜌, g∕cm3) as

V = M 𝜌

(4.67)

For 8YSZ, 𝜌 = 6.15 g∕cm3 and M = (91.22 g∕mol × 0.92 + 88.9 g∕mol × 0.16 + 16 g∕mol × 2.08) = 131.4 g∕mol. Thus

V = 131.4 g∕mol 6.15 g∕cm3

= 21.4 cm3 (4.68)

c𝑣 = 0.037 vacancies∕O site

( 2.16 mol O sites∕mol YSZ(

21.4 cm3∕mol YSZ ) )

= 0.0037 mol vacancies∕cm3 (4.69)

If the lattice constant and crystal structure of the material are known, this information can be used to convert vacancy fraction to vacancy concentration in an analogous fash- ion. In this case, the molar volume can be calculated from the unit cell information. For example, 8YSZ has the cubic (fluorite-type) structure with a lattice constant a = 5.15 Å and a total of four ZrO2 formula units per unit cell (e.g., four cations and eight anions). Based on this information the molar volume can be estimated as

V = (5.15A) 3 × (6.022 × 1023)

4 = 20.5 cm3 (4.70)

which is reasonably close to the density-based value calculated from Equation 4.68. From this point, the vacancy concentration, cv, can be calculated as before using Equation 4.69.

152 FUEL CELL CHARGE TRANSPORT

4.5.4 Mixed Ionic–Electronic Conductors

So far, this chapter has focused almost exclusively on pure ionic conductors. These are materials that conduct charged ionic species but do not conduct electrons. Beyond the tra- ditional classes of pure ionic conductors and pure electronic conductors, however, there are also interesting classes of materials that can conduct both ions and electrons. These mate- rials are known as “mixed ionic–electronic conductors” (MIECs) or, more simply, “mixed conductors.”

Many doped metal oxide ceramic materials exhibit both electronic and ionic conduc- tivity. This is because doping can introduce both ionic defects (like oxygen vacancies) and electronic defects (like free electrons or free holes). Both the ionic and electronic defects can then “wander” through the material, leading to simultaneous ionic and elec- tronic conductivity. If an oxide material is a mixed conductor, it is unsuitable for use as a fuel cell electrolyte (since the electronic conductivity would essentially “short” the fuel cell). However, MIECs are extremely attractive for SOFC electrode structures, because they can dramatically increase electrochemical reactivity and thereby improve fuel cell performance.

Why do MIECs increase electrochemical activity? As you may recall from Chapter 3 (Section 3.11), fuel cell reactions can only occur where the electrolyte, electrode, and gas phases are all in contact. This requirement is expressed by the concept of the “triple-phase zone,” which refers to regions or points where the gas pores, electrode, and electrolyte phases converge (see Figure 3.14). In order to maximize the number of these three-phase zones, most fuel cell electrode–electrolyte interfaces employ a highly nanostructured geom- etry with significant intermixing, or blending, of the electrode and electrolyte phases (along with gas porosity). However, another strategy to increase the number of reaction zones is to employ a mixed-conductor electrode. Because a MIEC conducts both ions and elec- trons, it can simultaneously provide both the ionic species and the electrons needed for an electrochemical reaction. In this case, only one additional phase (the gas phase) is needed for electrochemical reaction. Thus, fuel cell reactions can occur anywhere along the entire surface of the MIEC where it is in contact with the gas phase. Figure 4.19 schematically illustrates the difference between a standard fuel cell electrode (Figure 4.19a) and a MIEC electrode (Figure 4.19b).

As you can imagine, MIECs are scientifically fascinating materials. Most MIECs are ceramic materials and are therefore employed in SOFC electrodes—particularly as cathode electrode materials. In contrast, there is very little research on MIECs for low-temperature PEMFCs, but perhaps this will be an interesting area for future work. The prototypical MIEC is (La,Sr)MnO3 (LSM). LSM is used as the cathode electrode in many SOFC designs. In LSM, Sr2+ is substituted for La3+ as a dopant in order to create oxygen vacancies and holes. Due to the charge difference between La3+ and Sr2+, either oxygen vacancies or electron holes must be created to maintain charge neutrality, as illustrated by the following defect reactions:

Oxygen vacancy formation: 2Oxo → 2Sr ′ La + V

− o

Electron hole formation: null → 2Sr′La + 2h ⋅

MORE ON DIFFUSIVITY AND CONDUCTIVITY (OPTIONAL) 153

e–e–

O2 O2 O2

O2–O 2– O2–

Electrolyte Electrolyte

MIEC Electrode: Entire surface is active for reaction

Standard Electrode: Only TPBs are active for reaction

(a) (b)

Figure 4.19. A standard SOFC cathode electrode (a) versus a mixed ionic–electronic conducting (MIEC) SOFC cathode electrode (b).

In the first reaction, one oxygen vacancy (V ⋅⋅o ) is formed for every two Sr 2+ dopant substi-

tutions. This process is identical to the vacancy creation process in YSZ (see Section 4.5.3). In the second reaction, two holes (h⋅) are formed for every two Sr2+ dopant substitutions. Under typical SOFC conditions, hole conduction in LSM is dominant compared to oxygen vacancy conduction. Therefore, LSM is only a marginal MIEC (i.e., for all intents and pur- poses it is almost exclusively a p-type electronic conductor). Nevertheless, its remarkable stability and compatibility with other SOFC materials make it a popular choice in many SOFC designs.

Significant recent research has been conducted to develop better MIEC materials, and there are several other La-based perovskites that show increased ionic conductivity, and therefore better mixed-conduction behavior, compared to LSM. These materials include (La,Sr)(Co,Mn)O3, (La,Sr)FeO3, and (La,Sr)CoO3. These materials tend to provide much higher ionic conductivity compared to LSM and therefore function as true mixed ionic–electronic conductors. Unfortunately, these materials also tend to be less stable than LSM and have therefore proven difficult to deploy in functional SOFC designs. Nevertheless, the electrochemical benefits of MIEC electrodes are substantial, and therefore MIEC development remains an extremely intriguing area of research. Further details on these materials are provided for the interested reader in Chapter 9.

4.6 MORE ON DIFFUSIVITY AND CONDUCTIVITY (OPTIONAL)

In this optional section, we develop an atomistic picture to explore conductivity and diffusivity in more detail. We find that for conductors where charge transport involves a “hopping”-type mechanism, conductivity and diffusivity are intimately related. Diffusivity measures the intrinsic rate of this hopping process. Conductivity incorporates how this

154 FUEL CELL CHARGE TRANSPORT

hopping process is modified by the presence of an electric field driving force. Diffusivity is therefore actually the more fundamental parameter.

Diffusivity is a more fundamental parameter of atomic motion because even in the absence of any driving force, hopping of ions from site to site within the lattice still occurs at a rate that is characterized by the diffusivity. Of course, without a driving force, the net movement of ions is zero, but they are still exchanging lattice sites with one another. This is another example of a dynamic equilibrium; compare it to the exchange current density phenomenon that we learned about in Chapter 3.

4.6.1 Atomistic Origins of Diffusivity

Using the schematic in Figure 4.20b, we can derive an atomistic picture of diffusivity. The atoms in this figure are arranged in a series of parallel atomic planes. We would like to calculate the net flux (net movement) of gray atoms from left to right across the imaginary plane labeled A in Figure 4.20 (which lies between two real atomic planes in the material). Examining atomic plane 1 in the figure, we assume that the flux of gray atoms hopping in the forward direction (and therefore through plane A) is simply determined by the number

(b)

(a)

A (c )1 2(c )

ΔX

JA+

JA–

Distance (x)

Jnet

C on

ce nt

ra tio

n of

ay a

to m

Figure 4.20. (a) Macroscopic picture of diffusion. (b) Atomistic view of diffusion. The net flux of gray atoms across an imaginary plane A in this crystalline lattice is given by the flux of gray atoms hopping from plane 1 to plane 2 minus the flux of gray atoms hopping from plane 2 to plane 1. Since there are more gray atoms on plane 1 than plane 2, there is a net flux of gray atoms from plane 1 to plane 2. This net flux will be proportional to the concentration difference of gray atoms between the two planes.

MORE ON DIFFUSIVITY AND CONDUCTIVITY (OPTIONAL) 155

(concentration) of gray atoms available to hop times the hopping rate:

JA+ = 1 2 𝑣c1Δx (4.71)

where JA+ is the forward flux through plane A (mol/cm 2 ⋅ s), v is the hopping rate (s–1),

c1 is the volume concentration (mol/cm 3) of gray atoms in plane 1, Δx (cm) is the atomic

spacing required to convert volume concentration to planar concentration (mol/cm2), and the 1/2 accounts for the fact that on average only half of the jumps will be “forward” jumps. (On average, half of the jumps will be to the left, half of the jumps will be to the right.)

Similarly, the flux of gray atoms hopping from plane 2 backward through plane A will be given by

JA− = 1 2 𝑣c2Δx (4.72)

where JA− is the backward flux through plane A and c2 is the volume concentration (mol/cm3) of gray atoms in plane 2. The net flux of gray atoms across plane A is therefore given by the difference between the forward and backward fluxes through plane A:

Jnet = 1 2 𝑣Δx(c1 − c2) (4.73)

We would like to make this expression look like our familiar equation for diffusion: J = −D(dc∕dx) We can express Equation 4.73 in terms of a concentration gradient as

Jnet = − 1 2 𝑣(Δx)2

(c1 − c2) Δx

= − 1 2 𝑣(Δx)2 Δc

Δx

= − 1 2 𝑣(Δx)2 dc

dx (for small x) (4.74)

Comparison with the traditional diffusion equation J = −D(dc∕dx) allows us to identify what we call the diffusivity as

D = 1 2 𝑣(Δx)2 (4.75)

We therefore recognize that the diffusivity embodies information about the intrinsic hop- ping rate for atoms in the material (v) and information about the atomic length scale (jump distance) associated with the material.

As mentioned previously, the hopping rate embodied by v is exponentially activated. Consider Figure 4.21b, which shows the free-energy curve encountered by an atom as it hops from one lattice site to a neighboring lattice site. Because the two lattice sites are essentially equivalent, in the absence of a driving force a hopping atom will possess the same free energy in its initial and final positions. However, an activation barrier impedes the motion of the atom as it hops between positions. We might associate this energy barrier with the displacements that the atom causes as it squeezes through the crystal lattice between lattice sites. (See Figure 4.21a, which shows a physical picture of the hopping process.)

156 FUEL CELL CHARGE TRANSPORT

Distance

∆Gact

C+

A–

F re

e en

er gy

(a)

(b)

C+ C+

C+

Figure 4.21. Atomistic view of hopping process. (a) Physical picture of the hopping process. As the anion (A−) hops from its original lattice site to an adjacent, vacant lattice site, it must squeeze through a tight spot in the crystal lattice. (b) Free-energy picture of the hopping process. The tight spot in the crystal lattice represents an energy barrier for the hopping process.

In a treatment analogous to the reaction rate theory developed in the previous chapter, we can write the hopping rate as

𝑣 = 𝑣0e−ΔGact∕(RT) (4.76)

where ΔGact is the activation barrier for the hopping process and v0 is the jump attempt frequency.

Based on this activated model for diffusion, we can then write a complete expression for the diffusivity as

D = 1 2 (Δx)2𝑣0e−ΔGact∕(RT) (4.77)

or, lumping all the preexponential constants into a D0 term.

D = D0e−ΔGact∕(RT) (4.78)

4.6.2 Relationship between Conductivity and Diffusivity (1)

To understand how conductivity relates to diffusivity, we take a look at how an applied electric field will affect the hopping probabilities for diffusion. Consider Figure 4.22, which shows the effect of a linear voltage gradient on the activation barrier for the hopping process. From this picture, it is clear that the activation barrier for a “forward” hop is

MORE ON DIFFUSIVITY AND CONDUCTIVITY (OPTIONAL) 157

Distance

∆G’act

F re

e en

er gy

dV zF∆x

2 1

x∆ 2 1

x∆

dV zF

Voltage gradient

Figure 4.22. Effect of linear voltage gradient on activation barrier for hopping. The linear variation in voltage with distance causes a linear drop in free energy with distance. This reduces the forward activation barrier (ΔG′act < ΔGact). Two adjacent lattice sites are separated by Δx; therefore, the total free-energy drop between them is given by zFΔx(dV∕dx). If the activation barrier occurs halfway between the two lattice sites, ΔGact will be decreased by

2 zFΔx(dV∕dx). [In other words, ΔG′act =

ΔGact − 1

2 zFΔx(dV∕dx).]

reduced by 1 2 zFΔx(dV∕dx) while the activation barrier for the “reverse” hop is increased

by 1 2 zFΔx(dV∕dx). (We are assuming that the activated state occurs exactly halfway

between the two lattice positions, or in other words that 𝛼 = 1 2 .) The forward-(𝑣1) and

reverse-(𝑣2) hopping-rate expressions are therefore

𝑣1 = 𝑣0 exp − [ ΔGact −

1 2 zFΔx (dV∕dx)

] RT

𝑣2 = 𝑣0 exp − [ ΔGact +

1 2 zFΔx (dV∕dx)

] RT

(4.79)

This voltage gradient modification to the activation barrier turns out to be small. In fact,

1 2

zF RT

ΔxdV dx

≪ 1

158 FUEL CELL CHARGE TRANSPORT

so we can use the approximation ex ≈ 1 + x for the second term in the exponentials. This allows us to rewrite the hopping rate expressions as

𝑣1 ≈ 𝑣0e−ΔGact∕(RT) (

1 + 1 2

zF RT

ΔxdV dx

) 𝑣2 ≈ 𝑣0e−ΔGact∕(RT)

( 1 − 1

zF RT

ΔxdV dx

) (4.80) Proceeding as before, we can then write the net flux across an imaginary plane A in a

material as Jnet = JA+ − JA− =

1 2 Δx(c1𝑣1 − c2𝑣2) (4.81)

Since we are interested in conductivity this time, we would like to consider a flux that is driven purely by the potential gradient. In other words, we want to get rid of any effects of a concentration gradient by saying that c1 = c2 = c. Making this modification and inserting the formulas for v1 and v2 give

Jnet = 1 2 Δx𝑣0e−ΔGact∕(RT)

(czF RT

ΔxdV dx

) = 1

2 (Δx)2𝑣0e−ΔGact∕(RT)

(czF RT

dV dx

) (4.82) Recognizing the first group of terms as our diffusion coefficient D, we thus have

Jnet = czFD RT

dV dx

(4.83)

Comparing this to the conduction equation

J = 𝜎 zF

dV dx

we see that 𝜎 and D are related by

𝜎 = c(zF) 2D

RT (4.84)

For conductors that rely on a diffusive hopping-based charge transport mechanism, this important result relates the observed conductivity of the material to the atomistic diffusivity of the charge carriers. This equation is our key for understanding the atomistic underpin- nings of ionic conductivity in crystalline materials.

4.6.3 Relationship between Diffusivity and Conductivity (2)

Recall from Section 2.4.4 that the introduction of the electrochemical potential gave us an alternate way to understand the Nernst equation. In a similar fashion, looking at charge

MORE ON DIFFUSIVITY AND CONDUCTIVITY (OPTIONAL) 159

transport from the perspective of the electrochemical potential gives us an alternate way to understand the relationship between conductivity and diffusivity. Recall the definition of the electrochemical potential (Equation 2.99):

𝜇i = 𝜇0i + RT ln ai + ziF𝜙i

If we assume that activity is purely related to concentration (ai = ci∕c0), then the elec- trochemical potential can be written as

𝜇i = 𝜇0i + RT ln ci c0

+ ziF𝜙i (4.85)

The charge transport flux due to a gradient in the electrochemical potential will include both the flux contributions due to the concentration gradient and the flux contributions due to the potential gradient:

Ji = −Mi𝜇 𝜕𝜇

𝜕x = −Mi𝜇

( RT

d [ ln ( ci∕c0

)] dx

+ ziF dV dx

) (4.86)

The concentration term in the natural logarithm can be processed by remembering the chain rule of differentiation:

d[ln(ci∕c0)] dx

= c 0

d(ci∕c0) dx

= 1 ci

dci dx

(4.87)

Therefore, the total charge transport flux due to an electrochemical potential gradient is really made up of two fluxes, one driven by a concentration gradient and one driven by a voltage gradient:

Ji = − Mi𝜇RT

dci dx

−Mi𝜇ziF dV dx

(4.88)

Comparing the concentration gradient term in this equation to our previous expression for diffusion allows us to identify Mi𝜇 in terms of diffusivity:

Mi𝜇RT

ci = D

Mi𝜇 = Dci RT

(4.89)

Comparing the voltage gradient term in this expression to our previous expression for conduction allows us to identify 𝜎 in terms of diffusivity:

Mi𝜇zF = 𝜎|z|F , where 𝜎 = ci(zF)2DRT (4.90)

160 FUEL CELL CHARGE TRANSPORT

By using the electrochemical potential, we arrive at the same result as before. Inter- estingly, we did not have to make any assumptions about the mechanism of the transport process this time. Thus, we see that the relationship between diffusivity and conductivity is completely general. (In other words, it does not just apply to hopping mechanisms.) The conductivity and diffusivity of a material are related because the fundamental driving forces for diffusion and conduction are related via the electrochemical potential.

4.7 WHY ELECTRICAL DRIVING FORCES DOMINATE CHARGE TRANSPORT (OPTIONAL)

Our relationship between conductivity and diffusivity allows us to explain why electrical driving forces dominate charge transport.

In metallic electron conductors, the extremely high background concentration of free electrons means that electron concentration is basically invariant across the conductor. This means that there are no gradients in electron chemical potential across the conductor. Additionally, since metal conductors are solid materials, pressure gradients do not exist. Therefore, we find that electron conduction in metals is driven only by voltage gradients.

What about for ion conductors? Like the metallic conductors, most fuel cell ion con- ductors are also solid state, therefore pressure gradients do not exist. (Even in fuel cells that employ liquid electrolytes, the electrolyte is usually so thin that convection does not contribute significantly). Similarly, the background concentration of ionic charge carriers is also usually large, so that significant concentration gradients do not arise. However, even if large concentration gradients were to arise, we find that the “effective strength” of a voltage gradient driving force is far greater than the effective strength of a concentration gradient driving force. To illustrate this point, let’s compare the charge flux generated by a concen- tration gradient to the charge flux generated by a voltage gradient. The charge flux generated by a concentration gradient (jc) is given by

jc = zFD dc dx

(4.91)

The charge flux generated by a voltage gradient (j𝑣) is given by

j𝑣 = 𝜎 dV dx

(4.92)

Note that the quantity zF is required to convert moles in the diffusion equation into charge in coulombs. As we have learned, 𝜎 and D are related by

𝜎 = c(zF) 2D

RT (4.93)

The maximum possible sustainable charge flux due to a concentration gradient across a material is

jc = zFD c0 L

(4.94)

QUANTUM MECHANICS–BASED SIMULATION OF ION CONDUCTION IN OXIDE ELECTROLYTES (OPTIONAL) 161

where L is the thickness of the material and c0 is the bulk concentration of charge carri- ers. The voltage, V, that would be required to produce an equivalent charge flux can be calculated from

j𝑣 = jc c0(zF)2D

RT V L

= zFD c0 L

(4.95)

Solving for V gives

V = RT zF

(4.96)

At room temperature, for z = 1, RT∕zF = 0.0257 V. Therefore a voltage drop of 25.7 mV across the thickness of the material accomplishes the same thing as the maximum possible chemical driving force available from concentration effects. Effectively, the quantity RT∕zF sets the strength of the electric driving force relative to the chemical (concentration) driving force. Because RT∕zF is small (for the fuel cell temperature range of interest), fuel cell charge transport is dominated by electrical driving forces rather than chemical potential driving forces.

4.8 QUANTUM MECHANICS–BASED SIMULATION OF ION CONDUCTION IN OXIDE ELECTROLYTES (OPTIONAL)

In the previous sections, we have discussed the atomistic mechanisms of conduction and diffusion. In particular, you have learned that diffusion (and hence conduction) in crystalline oxide electrolytes occurs by a hopping process and that the rate of this hopping process is determined by the size of the energy barrier for motion, ΔGact. In general, materials with a lower barrier height will yield higher ionic diffusivities and hence higher ionic conduc- tivities. This is exemplified in Figure 4.18 where GDC displays higher ionic conductivity than YSZ (especially at lower temperatures) due to a smaller ΔGact. The quest for new solid-oxide electrolyte materials has therefore focused on creating materials with higher concentrations of mobile defects and lower activation barriers.

New electrolyte development, like new catalyst development, is largely a trial-and-error process. Researchers first develop new candidate materials and then screen them for high ionic conductivity and stability. Recently, however, the same quantum mechanics tech- niques that have been developed to help identify new catalyst materials (recall Chapter 3.12) are also being applied to identify new oxide electrolyte materials. The basic idea is that quantum mechanics techniques can be used to directly calculate the size of activation barriers associated with atomic motion through a crystalline lattice. Based on these cal- culated barrier heights, the conductivity of potential new electrolyte materials can then be theoretically predicted.

Consider a quantum simulation approach applied to YSZ. In YSZ the diffusing species are oxide ions, which must jump from an occupied site in the lattice to an adjacent (unoc- cupied) “vacancy.” The height of the barrier associated with this jump depends on the exact nature and symmetry of all the other atoms in the nearby vicinity. The exact neighborhood

162 FUEL CELL CHARGE TRANSPORT

surrounding a single atom in the lattice can vary significantly—in fact, a detailed analysis reveals that there are 42 different atomic configurations that an oxide ion may encounter when jumping into a neighboring vacancy in YSZ [11]! (And this analysis considers only nearest neighbors and next-nearest neighbors.) The barrier heights for each of these 42 dif- ferent atomic configurations will be different because the local environment associated with each of these configurations is different. These barrier heights can be calculated based on approximations to the Schrödinger equation (as discussed in Appendix D), which allows the determination of the energy “landscape” for a system of atoms at zero degrees Kelvin. The barrier height associated with moving an atom into a vacancy is calculated by deter- mining the energy of the entire atomic configuration in a step-by-step fashion as the oxide ion moves into the vacancy. Figure 4.23 shows the concept of this barrier height calcula- tion, performed step by step by considering atomic rearrangements, applied to one of the 42 possible configurations in YSZ. Once this process has been completed for the first con- figuration, it must then be repeated for the other 41 atomic configurations—a laborious and time-consuming process!

After calculating each of the 42 possible barrier heights associated with moving an atom from its lattice position to an open vacancy, the next step is to employ the methods of statistical thermodynamics to calculate the overall macroscopic diffusivity. Statistical thermodynamics teaches us that barriers with lower height can be more easily overcome than those with a higher barrier height. Thus, the macroscopic diffusivity will largely be dominated by the atomic configurations that occur most frequently and that have the low- est barrier heights. Diffusion processes are typically simulated using kinetic Monte Carlo (KMC) techniques, which assume that all atoms move randomly, but that the probability of a successful move depends exponentially on the barrier height as we discussed in Section 4.5.3. In KMC methods, the rate of successful atomic jumps is proportional to a random number multiplied with an exponential Boltzmann factor that contains the barrier height for diffusion. By simulating hundreds of thousands (if not millions) of individual atomic jumps using this KMC technique, the averaged “macroscopic” diffusivity for a material can be estimated. This diffusivity information can then be used to predict the performance of new ion conductors or help in understanding the behavior of current ion conductors.

R el

at iv

e en

er gy

Migration path

ΔE m

Figure 4.23. Illustration of the migration energy barrier. The middle point corresponds to the saddle where the oxygen ion and two cations such as zirconia align in the same plane before the oxide ion continues its path forward creating a vacancy in the location where it started.

CHAPTER SUMMARY 163

Experiment

KMC

mole % Y2O3

2.4

2.3

2.2

2.1

2.0

1.9

1.8

1.7

1.6 6 8 10 12 14 16

lo g

D /D

–4.2

–4.3

–4.4

–4.5

–4.6

–4.7

–4.8

–4.9

lo g(

σ T

) (Ω

–1 ·

cm –1

K )

Figure 4.24. Logarithmic plot of conductivity times T versus mol% Y2O3 in YSZ comparing exper- iment (open squares) and calculation (closed circles).

As an example of the power provided by this combined quantum–KMC technique, Figure 4.24 compares experimental measurements and theoretical predictions for the conductivity of YSZ as a function of yttria dopant concentration. As discussed in Section 4.5.3, adding excessive amounts of yttria to zirconia will actually decrease ionic conduc- tivity because defects begin to interact with one another, reducing their ability to move. This subtle effect is captured beautifully by the combined quantum–KMC simulation approach.

4.9 CHAPTER SUMMARY

• Charge transport in fuel cells is predominantly driven by a voltage gradient. This charge transport process is known as conduction.

• The voltage that is expended to drive conductive charge transport represents a loss to fuel cell performance. Known as the ohmic overvoltage, this loss generally obeys Ohm’s law of conduction, V = iR, where R is the ohmic resistance of the fuel cell.

• Fuel cell ohmic resistance includes the resistance from the electrodes, electrolyte, interconnects, and so on. However, it is usually dominated by the electrolyte resis- tance.

• Resistance scales with conductor area A, thickness L, and conductivity σ: R = L∕𝜎A. • Because resistance scales with area, area-specific fuel cell resistances (ASRs) are

computed to make comparisons between different-size fuel cells possible (ASR = A × R).

• Because resistance scales with thickness, fuel cell electrolytes are made as thin as possible.

164 FUEL CELL CHARGE TRANSPORT

• Because resistance scales with conductivity, developing high-conductivity electrode and electrolyte materials is critical.

• Conductivity is determined by carrier concentration and carrier mobility: 𝜎i = (|zi|F)ciui.

• Metals and ion conductors show vastly different structures and conduction mecha- nisms, leading to vastly different conductivities.

• Ion conductivity even in good electrolytes is generally four to eight orders of magni- tude lower than electron conductivity in metals.

• In addition to having high ionic conductivity, electrolytes must be stable in both highly reducing and highly oxidizing environments. This can be a significant challenge.

• The three major electrolyte classes employed in fuel cells are (1) liquid, (2) polymer, and (3) ceramic electrolytes.

• Mobility (and hence conductivity) in aqueous electrolytes is determined by the bal- ance between ion acceleration under an electric field and frictional drag due to fluid viscosity. In general, the smaller the ion and the greater its charge, the higher the mobility.

• Conductivity in Nafion (a polymer electrolyte) is dominated by water content. High water content leads to high conductivity. Nafion conductivity may be determined by modeling the water content in the membrane.

• Conductivity in ceramic electrolytes is controlled by defects (“mistakes”) in the crys- tal lattice. Natural (intrinsic) defect concentrations are generally low, so higher (extrin- sic) defect concentrations are usually introduced into the lattice on purpose via doping.

• Mixed ionic and electronic conductors (MIECs) conduct both electrons and ions. They are useful for SOFC electrodes, where simultaneous conduction of electrons and ions enables improved reactivity by extending three-phase boundaries into two-phase reac- tion zones.

• (Optional section) At the atomistic level, we find that conductivity is determined by a more basic parameter known as diffusivity D. Diffusivity expresses the intrinsic rate of movement of atoms within a material.

• (Optional section) By examining an atomistic picture of diffusion and conduction, we can explicitly relate diffusivity and conductivity: 𝜎 = c(zF)2D∕(RT).

• (Optional section) Using the relationship between conductivity and diffusivity, we can understand why voltage driving forces (conduction) dominate charge transport.

CHAPTER EXERCISES

Review Questions

4.1 Why does charge transport result in a voltage loss in fuel cells?

4.2 If a fuel cell’s area is increased 10-fold and its resistance is decreased 9-fold, will the ohmic losses in the fuel cell increase or decrease (for a given current density, all else being equal)?

CHAPTER EXERCISES 165

4.3 What are the two main factors that determine a material’s conductivity?

4.4 Why are the electron conductivities of metals so much larger than the ion conductiv- ities of electrolytes?

4.5 List at least four important requirements for a candidate fuel cell electrolyte. Which requirement (other than high conductivity) is often the hardest to fulfill?

Calculations

4.6 Redraw Figure 4.4c for a SOFC, where O2– is the mobile charge carrier in the elec- trolyte. Is there any change in the figure?

4.7 Draw a fuel cell voltage profile similar to those shown in Figure 4.4 that simultane- ously shows the effects of both activation losses and ohmic losses.

4.8 Given that fuel cell voltages are typically around 1 V or less, what would be the abso- lute minimum possible functional electrolyte thickness for a SOFC if the dielectric breakdown strength of the electrolyte is 108 V/m?

4.9 In Section 4.3.2, we discussed how fuel cell electrolyte resistance scales with thick- ness (in general as L∕𝜎). Several practical factors were listed that limit the useful range of electrolyte thickness. Fuel crossover was stated to cause an undesirable parasitic loss which can eventually become so large that further thickness decreases are coun- terproductive! In other words, at a given current density, an optimal electrolyte thick- ness may exist, and reducing the electrolyte thickness below this optimal value will actually increase the total fuel cell losses. We would like to model this phenomenon. Assume that the leak current jleak across an electrolyte gives rise to an additional fuel cell loss of the following form: 𝜂leak = A ln jleak. Furthermore, assume that jleak varies inversely with electrolyte thickness L as jleak = B∕L. For a given current density j determine the optimal electrolyte thickness that minimizes 𝜂ohmic + 𝜂leak.

4.10 A 5-cm2 fuel cell has Relec = 0.01 Ω and 𝜎electrolyte = 0.10 Ω−1 ⋅ cm−1. If the elec- trolyte is 100 𝜇m thick, predict the ohmic voltage losses for this fuel cell at j = 50 mA∕cm2.

4.11 Derive Equation 4.32 using Equations 4.22 and 4.24.

4.12 Consider a PEMFC operating at 0.8 A/cm2 and 70∘C. Hydrogen gas at 90∘C and 80% relative humidity is provided to the fuel cell at the rate of 8 A. The fuel cell area is 8 cm2 and the drag ratio of water molecules to hydrogen, α, is 0.8. Find the water activity of the hydrogen exhaust. Assume that p = 1atm and that the hydrogen exhaust exits at the fuel cell temperature, 70∘C.

4.13 Consider two H2–O2 PEMFCs powering an external load at 1 A/cm 2. The fuel cells

are running with differently humidified gases: (a) aW,anode= 1.0, aW,cathode= 0.5; (b) aW,anode= 0.5, aW,cathode= 1.0. Estimate the ohmic overpotential for both fuel cells if they are both running at 80∘C. Assume that they both employ a 125-𝜇m-thick Nafion electrolyte. Based on your results, discuss the relative effects of humidity at the anode versus the cathode.

166 FUEL CELL CHARGE TRANSPORT

4.14 (a) Calculate the diffusion coefficient for oxygen ions in a pure ZrO2 electrolyte at T = 1000∘C given ΔGact = 100 kJ∕mol, 𝑣0 = 1013 Hz. ZrO2 has a cubic unit cell with a lattice constant a = 5 Å and contains four Zr atoms and eight O atoms. Assume that the oxygen–oxygen “jump”distance Δx = 1

2 a.

(b) Calculate the intrinsic carrier concentration in the electrolyte given Δh𝑣 = 1 eV. (Assume vacancies are the dominant carrier.)

4.15 You have determined the resistance of a 100-𝜇m-thick, 1.0-cm2-area YSZ electrolyte sample to be 47.7 Ω at T = 700 K and 0.680 Ω at T = 1000 K. Calculate D0 and ΔGact for this electrolyte material given that the material is doped with 8% molar Y2O3. Recall from problem 4.14 that pure ZrO2 has a cubic unit cell with a lattice constant of 5 Å and contains four Zr atoms and eight O atoms. Assume that the lattice constant does not change with doping.

4.16 Which of the following is a correct statement for the water behavior in a Nafion-based PEMFC operating on dry H2/dry air at room temperature:

(a) Both electro-osmotic drag and backdiffusion move water from the anode to the cathode.

(b) Both electro-osmotic drag and backdiffusion move water from the cathode to the anode

(c) Electro-osmotic drag moves water from the cathode to the anode while backdif- fusion moves water from the anode to the cathode

(d) Electro-osmotic drag moves water from the anode to the cathode while backdif- fusion moves water from the cathode to the anode

4.17 A solid-oxide fuel cell electrolyte has ASR = 0.20 Ω ⋅ cm2 at T = 726.85∘C and ASR = 0.05 Ω ⋅ cm2 at T = 926.85∘C. What is the activation energy (ΔGact) for conduction in this electrolyte material?

CHAPTER 5

FUEL CELL MASS TRANSPORT

As discussed in the introductory chapter, to produce electricity, a fuel cell must be continually supplied with fuel and oxidant. At the same time, products must be continu- ously removed so as to avoid “strangling” the cell. The process of supplying reactants and removing products is termed fuel cell mass transport. As you will learn, this seemingly simple task can turn out to be quite complicated.

In the previous chapters, you learned about the electrochemical reaction process (Chapter 3) and the charge transport process (Chapter 4). Mass transport represents the last major fuel cell process to be discussed. After completing this chapter, you will have all the basic tools you need to understand fuel cell operation.

In this chapter, we will concentrate on the movement of reactants and products within a fuel cell. The previous chapter (on charge transport) has already introduced you to many of the fundamental equations that govern the transport of matter from one location to another. Indeed, ionic charge transport is actually just a special subset of mass transport where the mass being transported consists of charged ions. We now deal with the transport of uncharged species, thus distinguishing this chapter from the last chapter. Uncharged species are unaffected by voltage gradients and so must instead rely on convective and diffusive forces for movement. Furthermore, we are concerned mostly with gas-phase transport (and occasionally liquid-phase transport). Contrast this to the mostly solid-phase ionic transport discussed in the previous chapter.

Why are we so interested in fuel cell mass transport? The answer is because poor mass transport leads to significant fuel cell performance losses. To understand why poor mass transport can lead to a performance loss, remember that fuel cell performance is deter- mined by the reactant and product concentrations within the catalyst layer, not at the fuel cell inlet. Thus, reactant depletion (or product accumulation) within the catalyst layer will adversely affect performance. This loss in performance is called a fuel cell “concentration”

167

168 FUEL CELL MASS TRANSPORT

loss or mass transport loss. Concentration loss is minimized by careful optimization of mass transport in the fuel cell electrodes and fuel cell flow structures.

5.1 TRANSPORT IN ELECTRODE VERSUS FLOW STRUCTURE

This chapter is divided into two major parts: one part on mass transport in fuel cell elec- trodes and a second part on mass transport in fuel cell flow structures. Why do we make this distinction, and what is the difference between them?

The difference between the two domains is one of length scale. More importantly, how- ever, this difference in length scale leads to a difference in transport mechanism. For fuel cell flow structures, dimensions are generally on the millimeter or centimeter scale. Flow patterns typically consist of geometrically well-defined channel arrays that are amenable to the laws of fluid mechanics. Gas transport in these channels is dominated by fluid flow and convection. In contrast, fuel cell electrodes exhibit structure and porosity on the micrometer and nanometer length scale. The tortuous, sheltering geometry of these electrodes insu- lates gas molecules from the convective forces present in the flow channel. Sheltered from convective flow, gas transport within the electrodes is dominated by diffusion.

CONVECTION VERSUS DIFFUSION

It is important to understand the difference between convection and diffusion. Convec- tion refers to the transport of a species by bulk motion of a fluid (under the action of a mechanical force). Diffusion refers to the transport of a species due to a gradient in concentration. Figure 5.1 illustrates the difference between the two transport modes. Interestingly (and importantly for fuel cells), convection turns out to be far more “effec- tive” at transporting species than diffusion. For example, at STP, the maximum likely diffusive O2 flux across a 500-μm-thick porous electrode is ≈ 4 × 10–5mol∕(cm2 ⋅ s). This flux could instead be provided by 0.01 m/s (or less) convective flow of O2.

(a) (b)

Figure 5.1. Convection versus diffusion. (a) Convective fluid transport in this system moves material from the upper tank to the lower tank. (b) A concentration gradient between white and gray particles results in net diffusive transport of gray particles to the left and white particles to the right.

TRANSPORT IN ELECTRODE VERSUS FLOW STRUCTURE 169

Where do the convective forces that dominate transport in the flow channels come from? They are imposed by the user (us) who forces fuel or oxidant through the fuel cell at a given rate. The pressure (driving force) required to push a given rate of fuel or oxidant through fuel cell flow channels may be calculated using fluid dynamics. High flow rates can ensure good distribution of reactants (and effective removal of products) across a fuel cell but may require unacceptably high driving pressures or lead to other problems.

Where do the concentration gradients that dominate diffusive transport in the electrode come from? They develop due to species consumption/production within the catalyst layer. As Figure 5.2 illustrates, a fuel cell anode operating at high current density is consuming H2 molecules at a voracious rate. This leads to a depletion of H2 in the vicinity of the catalyst layer, extending out into the electrode. The resulting concentration gradient provides the driving force for the diffusive transport of H2 from the electrode to the reaction zones.

The “dividing line,” or boundary between convective-dominated flow and diffusive- dominated flow, often occurs where the fuel cell gas channel and porous electrode meet. Within the flow channel, convection serves to keep the gas stream well mixed, so that

Anode Electrolyte Cathode

H2 O2

Anode electrode

Flow channel

C on

ce nt

ra tio

Distance

c0H2

c*H2 Flow channel Electrode

Diffusion layer

H+

Figure 5.2. Schematic of diffusion layer that develops at the anode of an operating H2–O2 fuel cell. Consumption of H2 gas at the anode–electrolyte interface results in a depletion of H2 within the electrode. The concentration of H2 gas falls from its bulk value (c

0 H2

) at the flow channel to a much lower value (c∗H2 ) at the catalyst layer. The magnitude of the H2 gas velocity in the flow channel is schematically illustrated by the size of the flow arrows. Near the channel–electrode interface, the H2 gas velocity drops toward zero, marking the start of the diffusion layer.

170 FUEL CELL MASS TRANSPORT

concentration gradients do not occur. However, due to frictional effects, the velocity of the moving gas stream tends toward zero at the electrode–channel boundary (as shown in Figure 5.2). In the absence of convective mixing, concentration gradients are then able to form within the stagnant gas of the electrode. We call this stagnant gas region the diffusion layer, since it is the region where diffusion dominates mass transport. Because the demar- cation line where convective transport ends and diffusive transport begins is necessarily fuzzy, the exact thickness of the diffusion layer is often hard to define. Furthermore, it can change depending on the flow conditions, flow channel geometry, or electrode structure. For example, at very low gas velocities, the diffusion layer may stretch out into the middle of the flow channels. In contrast, at extremely high gas velocities convective mixing may penetrate into the electrode itself, causing the diffusion layer to retreat.

In the following two major sections of this chapter, we will first treat mass transport within the electrode using diffusion. Then, we will treat mass transport within the flow structure using fluid dynamics techniques.

5.2 TRANSPORT IN ELECTRODE: DIFFUSIVE TRANSPORT

In this section, we examine mass transport within the fuel cell electrodes. Technically, we are really treating mass transport within the diffusion layer, but for the purposes of this discussion, we assume that the electrode thickness coincides with the diffusion layer thickness. For most flow situations, this is a reasonable assumption. As mentioned previously, high flow velocities or unusual flow patterns can decrease the diffusion layer; calculating the true diffusion layer thickness in these situations requires sophisticated models. Likewise, low-flow velocities can increase the diffusion layer but again require treatment by sophisticated models.

5.2.1 Electrochemical Reaction Drives Diffusion

For most flow scenarios, the mass transport situation within the fuel cell electrode is similar to that shown in Figure 5.3. As illustrated in this figure, an electrochemical reaction on one side of an electrode and convective mixing on the other side of the electrode set up concen- tration gradients, leading to diffusive transport across the electrode. From this figure, you can see that the electrochemical reaction leads to reactant depletion (and product accumula- tion) at the catalyst layer. In other words, c∗R < c

0 R and c

∗ P > c

0 P, where c

∗ R, c

∗ P represent the

catalyst layer reactant and product concentrations, respectively, and c0R, c 0 P represent

the bulk (flow channel) reactant and product concentrations, respectively. This reactant depletion (and product accumulation) affects fuel cell performance in two ways, which will now briefly be described:

1. Nernstian Losses. The reversible fuel cell voltage will decrease as predicted by the Nernst equation since the reactant concentration at the catalyst layer is decreased relative to the bulk concentration, and the product concentration at the catalyst layer is increased relative to the bulk concentration.

TRANSPORT IN ELECTRODE: DIFFUSIVE TRANSPORT 171

Anode electrodeFlow channel

C on

ce nt

ra tio

c*R

c0R

Catalyst layer Electrolyte

Flow structure

c*P

c0P

Reactants (R) In

Products (P) Out

Reaction in catalyst layer consumes R, generates P

j rxn

Figure 5.3. Schematic of mass transport situation within a typical fuel cell electrode. Convective mixing of reactants and products in the flow channel establishes constant bulk species concentrations outside the diffusion layer (c0R and c

0 P). The consumption/generation of species (at a rate given by jrxn)

within the catalyst layer leads to reactant depletion and product accumulation (c∗R < c 0 R and c

∗ P > c

0 P).

Across the diffusion layer, a reactant concentration gradient is established between c0R and c ∗ R, while

a product concentration gradient is established between c0P and c ∗ P.

2. Reaction Losses. The reaction rate (activation) losses will be increased because the reactant concentration at the catalyst layer is decreased relative to the bulk concen- tration, and the product concentration at the catalyst layer is increased relative to the bulk concentration.

The combination of these two loss effects is what we collectively refer to as the fuel cell’s concentration (or mass transport) loss. To determine the size of the concentration loss, it is essential to determine exactly how much the catalyst layer reactant and product concentrations differ from their bulk values. How do we make this determination? Let’s see if we can come up with an answer by taking a closer look at the diffusion process occurring inside a fuel cell electrode.

Consider the fuel cell electrode depicted in Figure 5.4. Imagine that at some time t = 0 this fuel cell is “turned on” and it begins producing electricity at a fixed current density j. Initially, the reactant and product concentrations everywhere in this fuel cell are constant (they are given by c0R and c

0 P). As soon as the fuel cell begins producing current,

however, the electrochemical reaction leads to depletion of reactants (and accumulation

172 FUEL CELL MASS TRANSPORT

C on

ce nt

ra tio

c*R

c0R

c*P

c0P

Anode electrodeFlow channel Catalyst

layer

∞t ∞t

1t 2t

3t 2t

0=t

Figure 5.4. Time dependence of reactant and product concentration profiles at fuel cell electrode. The fuel cell begins producing current at time t = 0. Starting from constant initial values (c0R and c0P), the reactant and product concentration profiles evolve with increasing time, as shown for t1 < t2 < t3. Eventually the profiles approach a steady-state balance (indicated by the dark solid lines) where concentration varies (approximately) linearly with distance across the diffusion layer. At steady state, the diffusion flux down these linear concentration gradients exactly balances the reaction flux at the catalyst layer.

of products) at the catalyst layer. Reactants begin to diffuse toward the catalyst layer from the surrounding area, while products begin to diffuse away from the catalyst layer. Over time, the reactant and product concentration profiles will evolve as shown in the figure. Eventually, a steady-state situation will be reached as indicated by the dark lines. At steady-state, the reactant and product concentration profiles drop linearly (at least in approximation) with distance across the electrode (diffusion layer). Furthermore, the flux of reactants and products down these concentration gradients will exactly match the consumption/depletion rate of reactants and products at the catalyst layer. (This should make intuitive sense: At steady state, the rate of consumption must equal the rate of supply.) Mathematically,

j = nF Jdiff (5.1)

where j is the fuel cell’s operating current density (remember, the current density is a measure of the electrochemical reaction rate) and Jdiff is the diffusion flux of reactants to the catalyst layer (or the diffusion flux of products away from the catalyst layer). The now familiar quantity nF is, of course, required to convert the molar diffusion flux into the units of current density.

TRANSPORT IN ELECTRODE: DIFFUSIVE TRANSPORT 173

CALCULATING NOMINAL DIFFUSIVITY

The gas diffusion of a species i depends not only on the properties of i but also on the properties of the species j through which i is diffusing. For this reason, binary gas diffusion coefficients are typically written as Dij, where i is the diffusing species and j is the species through which the diffusion is occurring. For a binary system of two gases, Dij is a strong function of temperature, pressure, and the molecular weights of species i and j. At low pressures, nominal diffusivity can be estimated from the following equation based on the kinetic theory of gases [12]:

p ⋅ Dij = a

( T√ TciTcj

)b (pcipcj)1∕3(TciTcj)5∕12

( 1 Mi

+ 1 Mj

)1∕2 (5.2)

where p is the total pressure (atm), Dij is the binary diffusion coefficient (cm 2/s), and

T is the temperature (K); Mi, Mj are the molecular weights (g/mol) of species i and j, and Tci, Tcj, pci, pcj are the critical temperatures and pressures of species i and j. Table 5.1 summarizes Tc and pc values for some useful gases. The final parameters in Equation 5.2 are a and b. Typically, one can use a = 2.745 × 10−4 and b = 1.823 for pairs of non- polar gases, such as H2, O2, and N2. For pairs involving H2O (polar) and a nonpolar gas, one can use a = 3.640 × 10−4 and b = 2.334. Other equations to estimate diffusivity can be found in the literature.

TABLE 5.1. Critical Properties of Gases

Substance Molecular Weight (g/mol) Tc(K) pc (atm)

H2 2.016 33.3 12.80 Air 28.964 132.4 37.0 N2 28.013 126.2 33.5 O2 31.999 154.4 49.7 CO 28.010 132.9 34.5 CO2 44.010 304.2 72.8 H2O 18.015 647.3 217.5

Source: From Ref. [12].

CALCULATING EFFECTIVE DIFFUSIVITY

In porous structures, the gas molecules tend to be impeded by the pore walls as they diffuse. The diffusion flux should therefore be corrected to account for the effects of such blockage. Usually this is accomplished by employing a modified or effective diffusivity. According to the Bruggemann correction, the effective diffusivity in a porous structure can be expressed as [13]

Deffij = 𝜀 1.5Dij (5.3)

174 FUEL CELL MASS TRANSPORT

where 𝜀 stands for the porosity of the porous structure. Porosity represents the ratio of pore volume to total volume. Usually, fuel cell electrodes have porosities of around 0.4, which means 40% of the total electrode volume is occupied by pores. In open space, porosity is 1 and Deffij = Dij. Often, Equation 5.3 is modified to include tortuosity 𝜏 as

Deffij = 𝜀 𝜏Dij (5.4)

Tortuosity describes the additional impedance to diffusion caused by a tortuous or convoluted flow path. Highly “mazelike” or meandering pore structures yield high tor- tuosity values. It is known that tortuosity can vary from 1.5 to 10, depending on pore structure configuration. At high temperatures, however, a different correlation for effec- tive diffusivity proves more accurate [14]:

Deffij = Dij 𝜀

𝜏 (5.5)

The diffusion flux, Jdiff, can be calculated using the diffusion equation. Recall from the previous chapter (Table 4.1) that diffusive transport may be described by

Jdiff = −D dc dx

(5.6)

For the steady-state situation shown in Figure 5.4, this equation becomes (written for the flux of a diffusing reactant)

Jdiff = −Deff c∗R − c

0 R

𝛿 (5.7)

where c∗R is the catalyst layer reactant concentration, c 0 R is the bulk (flow channel) reactant

concentration, δ is the electrode (diffusion layer) thickness, and Deff is the effective reac- tant diffusivity within the catalyst layer. (The “effective” diffusivity will be lower than the “nominal” diffusivity due to the complex structure and tortuosity of the electrode. For more on calculating nominal and effective diffusivity, refer to the text box above.) By combining Equations 5.1 and 5.7, we can then solve for the reactant concentration in the catalyst layer:

j = nFDeff c∗R − c

0 R

𝛿 (5.8)

c∗R = c 0 R −

j𝛿

nFDeff (5.9)

What this equation says is that the reactant concentration in the catalyst layer (c∗R) is less than the bulk concentration c0R by an amount that depends on j, δ, and D

eff. As j increases, the reactant depletion effect intensifies. Thus, the higher the current density, the worse the concentration losses. However, these concentration losses can be mitigated if the diffusion layer thickness, δ, is reduced or the effective diffusivity Deff is increased.

TRANSPORT IN ELECTRODE: DIFFUSIVE TRANSPORT 175

5.2.2 Limiting Current Density

It is interesting to consider the situation when the reactant concentration in the catalyst layer drops all the way to zero. This represents the limiting case for mass transport. The fuel cell can never sustain a higher current density than that which causes the reactant concentration to fall to zero. We call this current density the limiting current density of the fuel cell. The limiting current density (jL) can be calculated from Equation 5.8 by setting c∗R = 0:

jL = nFDeff c0R 𝛿

(5.10)

Fuel cell mass transport design strategies focus on increasing the limiting current density. These design strategies include the following:

1. Ensuring a high c0R (by designing good flow structures that evenly distribute reactants)

2. Ensuring that Deff is large and δ is small (by carefully optimizing fuel cell operating conditions, electrode structure, and diffusion layer thickness)

Typical values are about 100–300 μm for δ and 10–2 cm2/s for Deff. Therefore, typical limiting current densities are on the order of 1–10 A/cm2. This mass transport effect rep- resents the ultimate limit for fuel cells; a fuel cell will never be able to produce a higher current density than that determined by its limiting current density. (Note, however, that other fuel cell losses, for example, ohmic and activation losses, may reduce the fuel cell voltage to zero well before the limiting current density is ever reached.)

While the limiting current density defines the ultimate fuel cell mass transport limit, concentration losses still occur at lower current densities as well. Recall from Section 5.2.1 that concentration differences in the catalyst layer affect fuel cell performance in two ways: first, by decreasing the Nernst (thermodynamic) voltage and, second, by increasing the acti- vation (reaction rate) loss. We will now examine both of these effects in detail. Surprisingly, we will find that both lead to the same result. This result, when generalized, is what we will refer to as the fuel cell’s “concentration” overvoltage, 𝜂conc.

LIMITING CURRENT DENSITIES AT ANODES AND CATHODES

In general, a limiting current density can be calculated for each reactant species in a fuel cell. For example, in an H2–O2 fuel cell, a jL value can be calculated for both the anode (based on H2) and the cathode (based on O2). In both cases, care must be taken to cor- rectly match the reactant species considered with the correct value for n in Equation 5.10. For the case of H2, 1 mol H2 will provide 2e

–, and hence n = 2. However, for the case of O2, 1 mol O2 will consume 4e

–, and hence n = 4. For most fuel cells, only jL for oxy- gen is considered when determining mass transfer losses. Mass transfer limitations due to oxygen transport are typically much more severe than for hydrogen. This is because air (rather than pure oxygen) is typically used and O2 diffuses more slowly than H2.

176 FUEL CELL MASS TRANSPORT

For the sake of clarity and simplicity, we will consider only reactant depletion effects when developing our concentration overvoltage expressions in the following sections. These expressions can be developed in an analogous manner if the product accumulation effects are considered instead.

5.2.3 Concentration Affects Nernst Voltage

The first way that concentration affects fuel cell performance is through the Nernst equation. This is because the real reversible thermodynamic voltage of a fuel cell is determined by the reactant and product concentrations at the reaction sites, not at the fuel cell inlet. From Chapter 2, recall the form of the Nernst equation (Equation 2.89):

E = E0 − RT nF

ln Πa𝑣iproducts Πa𝑣ireactants

(5.11)

For simplicity, we will consider a fuel cell with a single reactant species. As mentioned previously, we will neglect the product accumulation effects in this treatment. We retain our notation from the previous sections: c∗R = catalyst layer reactant concentration, c

0 R = bulk

reactant concentration. We would like to calculate the incremental voltage loss due to reactant depletion in the

catalyst layer (we will call this 𝜂conc). In other words, we would like to calculate how much the Nernst potential changes when using c∗R values instead of c

0 R values:

𝜂conc,Nernst = E0Nernst − E ∗ Nernst

( E0 − RT

nF ln

c0R

) − ( E0 − RT

nF ln

1 c∗R

)

= RT nF

ln c0R c∗R

(5.12)

where E0Nernst is the Nernst voltage using c 0 values and E∗Nernst is the Nernst voltage using

c∗ values. Recall that c0R can be described in terms of the limiting current density (from Equation 5.10),

c0R = jL𝛿

nFDeff (5.13)

and that c∗R can be described in terms of the diffusion Equation 5.9,

c∗R = c 0 R −

j𝛿

nFDeff

= jL𝛿

nFDeff −

j𝛿

nFDeff

(5.14)

TRANSPORT IN ELECTRODE: DIFFUSIVE TRANSPORT 177

Thus, the ratio c0R ∕c ∗ R can be written as

c0R c∗R

= jL𝛿∕nFDeff

jL𝛿∕nFDeff − j𝛿∕nFDeff

= jL

jL − j (5.15)

Substituting this result into our expression for 𝜂conc provides the final result:

𝜂conc,Nernst = RT nF

ln jL

jL − j (5.16)

Note that this expression is valid only for j < jL (j should never be greater than jL anyway). For j << jL, this expression implies that the concentration loss 𝜂conc will be minor; however, as j → jL, 𝜂conc increases sharply.

5.2.4 Concentration Affects Reaction Rate

The second way that concentration affects fuel cell performance is through the reaction kinetics. This is because the reaction kinetics also depend on the reactant and product con- centrations at the reaction sites. Recall from Chapter 3 that the reaction kinetics may be described by the Butler–Volmer equation 3.33:

j = j 00

( c∗R c0∗R

e𝛼nF𝜂act∕(RT) − c∗P c0∗P

e−(1−𝛼)nF𝜂act∕(RT) )

(5.17)

where c∗R and c ∗ P are arbitrary concentrations and j

0 0 is measured at the reference reactant

and product concentration values c0∗R and c 0∗ P . (Note that c

0∗ R and c

0∗ P , which are the reference

reactant and product concentration values, may be different from c0R and c 0 P, the reactant and

product bulk concentration values in our fuel cell.) We are concerned primarily with the high-current-density region, since this is where the

concentration effects become most pronounced. At high current density, the second term in the Butler–Volmer equation drops out and the expression simplifies to

j = j 00

( c∗R c0∗R

e𝛼nF𝜂act∕(RT) )

(5.18)

Written in terms of the activation overvoltage, this becomes

𝜂act = RT 𝛼nF

ln jc0∗R j 00 c

∗ R

(5.19)

As in the previous section, we would like to calculate the incremental voltage loss due to reactant depletion in the catalyst layer (which we will again call 𝜂conc). In other words,

178 FUEL CELL MASS TRANSPORT

we would like to calculate how much the activation overvoltage changes when using c∗R values instead of c0R values (keeping in mind that c

0∗ R and c

0 R are different):

𝜂conc, BV = 𝜂∗act − 𝜂 0 act

( RT 𝛼nF

ln jc0∗R j 00 c

∗ R

) −

( RT 𝛼nF

ln jc0∗R j 00 c

0 R

)

= RT 𝛼nF

ln c0R c∗R

(5.20)

where 𝜂0act is the activation loss using c 0 values and 𝜂∗act is the activation loss using c

∗

values. As before, we can then write the ratio c0R∕c ∗ R as

c0R c∗R

= jL

jL − j (5.21)

Substituting this result into our expression for 𝜂conc provides almost the same final result as before:

𝜂conc, BV = RT 𝛼nF

ln jL

jL − j (5.22)

This result differs from our previous expression for the concentration loss (Equation 5.16) only by a factor of 𝛼. Because the two effects are virtually identical, we can generalize the total concentration loss as follows:

𝜂conc = 𝜂conc,Nerst + 𝜂conc,BV = (RT nF

)( 1 + 1

𝛼

) ln

jL jL − j

(5.23)

Written in the most general form, this becomes

𝜂conc = c ln jL

jL − j (5.24)

where c is a constant.

5.2.5 Concentration Loss Explained on the j–V Curve

In this section, we explore in more detail how concentration losses affect the fuel cell j–V curve. According to Equations 5.12 and 5.20, the difference between the reactant concen- tration at the catalyst surface (c∗R) versus the bulk (c

0 R) causes the concentration loss. The

more severe the depletion of concentration within the catalyst layer (in other words, the smaller c∗R), the greater the concentration loss.

Let’s first consider the “Nernstian” concentration losses. Equation 5.12 tells us that reactant concentration depletion causes a drop in the Nernst potential. The effect of this Nernstian concentration loss can be directly illustrated on a fuel cell j–V curve, as shown in Figure 5.5.

TRANSPORT IN ELECTRODE: DIFFUSIVE TRANSPORT 179

C el

l v ol

ta ge

( V

)

Current density (A/cm2)

Theoretical EMF or ideal voltage

E’

A’

V’

A ηact ηconc, Nernstηconc, Nernstηact+

Figure 5.5. Concentration loss due to Nernstian effects. When the fuel cell operates at a current density j, the surface concentration decreases below the bulk value due to reactant consumption. Accordingly, the ideal voltage drops by an amount given by 𝜂conc,Nernst from E to E

′. (For now, we do not consider the additional activation losses due to concentration depletion, and so the activation loss curve (𝜂act) is simply translated from A to A

′).

The ideal voltage curve E and the activation loss A in Figure 5.5 represent the perfor- mance of a fuel cell when the concentration in the catalyst layer is exactly the same as the bulk concentration (zero depletion). Typically, this zero-depletion condition only occurs at zero current density. As soon as the fuel cell begins to generate current, reactant con- sumption leads to a decrease in reactant concentration at the catalyst surface. Because the Nernst voltage depends on the reactant concentration at the catalyst surface, a decrease in the reactant concentration within the catalyst layer causes a commensurate decrease in the ideal Nernstian voltage for the fuel cell. This new Nernstian voltage curve is shown by E′ in Figure 5.5. The difference betweenE andE′ represents the concentration loss obtained from Equation 5.12. For now, we ignore the impact of concentration losses on the activation loss curve. However, you should recognize that even though we are ignoring the effect of con- centration on activation losses, the activation loss curve must still be translated downward from A to A′ because E has been translated downward to E′. The increased Nernstian losses due to reactant depletion, shown by the shift from E to E′, therefore, causes a commensurate shift in the fuel cell voltage curve from V to V′.

Although it was ignored in Figure 5.5, let’s now consider the impact of concentration losses on the activation loss curve. As described in Equation 5.20, reactant depletion at the catalyst layer causes an increase in the activation loss. As shown in Figure 5.6, this causes a shift in the activation loss curve from A′ to A∗. The difference between A′ and A∗ represents the activation loss obtained from Equation 5.20 (based on Butler–Volmer kinetics). This loss is marked as 𝜂conc.BV in Figure 5.6. The combined losses due to 𝜂conc,Nernst and 𝜂conc.BV, therefore, lower the overall fuel cell voltage to V

∗, which captures both the concentration-induced Nernstian and activation losses.

The dotted line in Figure 5.6 represents the j–V performance behavior of a fuel cell considering both the activation loss and concentration loss at the same time. As the current density increases, the reactant concentration at the catalyst layer decreases commensurately. Accordingly, the concentration loss is especially severe in the high-current-density region of the j–V curve.

180 FUEL CELL MASS TRANSPORT

C el

l v ol

ta ge

( V

) Current density (A/cm2)

ηact ηconc, Nernst

Theoretical EMF or ideal voltage

ηact + ηconc, Nernst

ηact + ηconc, Nernst + ηconc, BV

E’

A’

Figure 5.6. Concentration loss due to Nernstian effects and activation effects. The new activa- tion curve A∗ accounts for additional kinetic losses due to the decreasing catalyst surface con- centration with increasing current density. The difference between A∗ and A′ represents this concentration-induced concentration loss (𝜂conc.BV).

5.2.6 Summary of Fuel Cell Concentration Loss

In the previous sections, we have seen how species depletion/accumulation in the cata- lyst layer leads to fuel cell performance loss. This performance loss, called the fuel cell concentration loss (or mass transport loss), may be described by the general form

𝜂conc = c ln jL

jL − j (5.25)

where c, a constant, might have the approximate form

c = RT nF

( 1 + 1

𝛼

) (5.26)

Interestingly, real fuel cell behavior often exhibits an effective c value, which is much larger than that predicted by Equation 5.26 above. Therefore, in many cases, c is obtained empirically. Noting the discrepancy between “actual” values for c and the value predicted by the theoretical treatment provided in this text, S.B. Beale has provided a more general treatment for mass transfer losses in fuel cells. Based on this treatment, Beale suggests that the following formula should be used to calculate concentration losses:

𝜂conc = RT 𝛼nF

ln (1 + rB

1 + B

) where B is a generalized mass transfer driving force.1

1Students interested in applying this more generalized mass transfer analysis are encouraged to consult S.B. Beale, Calculation procedure for mass transfer in fuel cells, Journal of Power Sources, 128:185–192, 2004.

TRANSPORT IN ELECTRODE: DIFFUSIVE TRANSPORT 181

1.2

C el

l v ol

ta ge

( V

)

Current density (A/cm2)

0.5

Theoretical EMF or ideal voltage Concentration loss

2.01.0

jL = 1.0 A/cm2

jL = 2.0 A/cm2

jL = 1.5 A/cm2

Figure 5.7. Effect of concentration loss on fuel cell performance. Concentration effects in the catalyst layer contribute to a characteristic drop in fuel cell operating voltage as determined by Equation 5.25. The shape of this loss is determined by c and jL. (Curves calculated for jL = 1, 1.5, 2 A∕cm2, respec- tively, while c was held constant; c was fixed at 0.0388 V using Equation 5.26 with T = 300 K, n = 2, α = 0.5.)

Figure 5.7 shows the effect of concentration loss on the j–V behavior of a fuel cell. The curves in this figure were generated for various values of jL (1, 1.5, and 2 A/cm

2, respectively) while c was held constant (c = 0.0388 V using T = 300K, n = 2, 𝛼 = 0.5). As the curves clearly indicate, the concentration loss only significantly affects fuel cell performance at high current density (when j approaches jL). Although the con- centration loss appears mainly at high current density, its effect is abrupt and severe. The onset of significant concentration loss marks the practical limit of a fuel cell’s operating range.

As shown in Figure 5.7, increasing jL can greatly extend a fuel cell’s potential operating range; therefore mass transport design is an active area of current fuel cell research. Recall how jL is defined:

jL = nFDeff c0R 𝛿

(5.27)

As previously discussed, this equation shows that the limiting current density depends on Deff, c0R, and δ, where D

eff and δ are mostly determined by the electrode. Many constraints exist on electrode design, so it is often difficult to optimize the elec-

trode solely for its mass transport properties. Instead, the flow structure often provides the best opportunities for mass transport optimization. Flow structure design affects the limit- ing current density because it determines c0R, the bulk concentration of reactant (or product) in the flow channel. It is important to realize that c0R is not constant within fuel cell flow channels. (We wish that it was!) Instead, c0R decreases with distance along a fuel cell flow channel because the reactants are being consumed. The best flow structure designs mini- mize this gas depletion effect so that c0R is consistently high across an entire fuel cell device. As we will learn in the next section, maintaining a consistent, high c0R value is often the best way to minimize the concentration losses in a fuel cell.

182 FUEL CELL MASS TRANSPORT

Example 5.1 Consider a fuel cell operating at 80∘C. In the cathode, humidified air at 1.0 atm is supplied with a water vapor mole fraction of 0.2. (a) Calculate the limiting current density, jL, for this cathode assuming that the diffusivity of oxygen in humid air at this temperature is 0.1 cm2/s and that the cathode is 500 μm thick and 40% porous. (b) Calculate the concentration overpotential (ηconc) experienced by this fuel cell if it is operating at a current density of 2.0 A/cm2. Assume 𝛼 = 0.5 and c = 0.1 V.

Solution: The limiting current density is given by Equation 5.27, repeated here for convenience:

jL = nFDeff c0R 𝛿

Most of the terms in this expression are provided by the problem statement. How- ever, it is necessary to calculate c0R. Gas concentrations can be calculated from gas partial pressures using the ideal gas law:

c0R = n0R V

= P0R RT

From the problem statement, our cathode is supplied with humid air at 1.0 atm total pressure, with a water vapor mole fraction of 0.2. Air is 78% nitrogen and 21% oxygen. However, in this case, our air is “diluted” by 20% with water vapor, so 78% of the remaining 80% is nitrogen, and 21% of the remaining 80% is oxygen. In other words, the partial pressure of oxygen is 0.8 × 0.21 = 0.168. Inserting this value into the ideal gas law gives

c0O2 = P0O2 RT

= 0.168 × (101,300 Pa∕atm) (8.314 J∕mol ⋅ K) × (353 K)

= 5.8 mol∕m3 = 5.8 × 10-6 mol∕cm3

Be careful when evaluating ideal gas law expressions!!! To avoid units problems, SI units should be used for all quantities (for example, the pressure must be converted to pascals). If SI quantities are used, the resulting concentration will have units of mol/m3.

Using Equation 5.3 and the quantities given in the problem statement, the effective diffusivity of oxygen in the cathode of the fuel cell can be calculated as

DeffO2,N2 = 𝜀 1.5DO2,N2 = (0.4

1.5)(0.2 cm2∕s) = 0.0506 cm2∕s

Finally, applying these results to the expression for jL yields

jL = nFDeff c0R 𝛿

= 4(96, 485 C∕mol)(0.0506 cm2∕s) 5.8 × 10−6 mol∕cm3

0.05 cm

= 2.26 A∕cm2

TRANSPORT IN FLOW STRUCTURES: CONVECTIVE TRANSPORT 183

Note than n = 4 is used here since we are calculating jL for the cathode (oxygen). Limiting current densities on the order of 1–10 A∕cm2 are typical for most fuel cells. Limiting current density calculations are generally straightforward; however, units are always a source of trouble. Take care when evaluating these expressions!

The concentration overpotential can be calculated by applying Equation 5.25:

𝜂conc = c ln jL

jL − j = 0.1 ln

[ 2.26 A∕cm2

2.26 A∕cm2 − 2 A∕cm2

] = 0.22 V

5.3 TRANSPORT IN FLOW STRUCTURES: CONVECTIVE TRANSPORT

Fuel cell flow structures are designed to distribute reactants across a fuel cell. Perhaps the simplest “flow structure” you could imagine would be a single-chamber structure. To make a single-chamber flow structure, we could encapsulate the entire fuel cell anode in a single compartment, then introduce H2 gas into one corner. Unfortunately, this single-chamber design would lead to poor fuel cell performance. The H2 would tend to stagnate inside the chamber, leading to poor reactant distribution and high mass transport losses.

In real fuel cells, mass transport losses are minimized by employing intricate flow struc- tures containing many small flow channels. Compared to a single-chamber design, a design employing many small flow channels keeps the reactants constantly flowing across the fuel cell, encouraging uniform convection, mixing, and homogeneous reactant distribu- tion. Small-flow-channel designs also provide more contact points across the surface of the electrode from which the fuel cell electrical current can be harvested.

To make a fuel cell flow structure, the flow channel design is typically stamped, etched, or machined into a flow field plate. The channels (there can be dozens or even hundreds of them) often snake, spiral, and twist across the flow field plate from a gas inlet at one cor- ner to a gas outlet at another corner. Analyzing convective gas transport in these complex real-world flow structures is only really possible with numerical methods. A common tech- nique is to use a computer simulation tool known as computational fluid dynamics (CFD) modeling, which will be overviewed in Chapter 6 and will be discussed in more detail in Chapter 13. Without using CFD, however, a basic analysis of simple flow scenarios is still possible. This kind of basic analysis, which relies on the principles of fluid mechanics, can still yield great insight into fuel cell mass transport and flow structure design. Therefore, the rest of this chapter focuses on applying fluid mechanical principles to simplified convection in fuel cell flow channels. We begin with a brief review of fluid mechanics.

5.3.1 Fluid Mechanics Review

It is important to realize that when we talk about “fluid” in the context of fuel cell mass transport, we are usually talking about a gas. In the science of fluid mechanics, fluid does

184 FUEL CELL MASS TRANSPORT

not have to mean liquid. A gas is a fluid. We use fluid mechanics to set up the rules governing how gases flow through fuel cell flow channels.

The nature of fluid flow in confined channels is characterized by an important dimen- sionless number known as the Reynolds number, Re:

Re = 𝜌VL 𝜇

= VL 𝜈

(5.28)

where V is the characteristic velocity of the flow (m/s), L is the characteristic length scale of the flow (m), 𝜌 is the fluid density (kg/m3), μ is the fluid viscosity (kg/m ⋅ s or N ⋅ s/m2), and v is the kinematic viscosity (m2/s). (The kinematic viscosity is the ratio of 𝜇 over 𝜌.) Physically, the Reynolds number describes the ratio of inertial forces to viscous forces in dynamic flow. Regardless of fluid type, flow velocity, or geometry, flows with the same Reynolds number show similar viscous behavior.

All fluids have a characteristic viscosity. Viscosity measures the resistance to fluid flow. On the microscopic scale, viscosity measures how easily molecules slide past one another when driven by a shear force. It can therefore be thought of as a measure of internal fluid “friction.” Mathematically, viscosity relates shear stress 𝜏xy to strain rate �̇�xy. For simple fluids such as water and gases, the relationship between shear stress and strain rate is linear:2

𝜏xy = 2𝜇�̇�xy = 2𝜇 ⋅ 1 2

( 𝜕u 𝜕y

+ 𝜕𝑣 𝜕x

) (5.29)

where u is the fluid velocity (m/s) in the x direction and v is the fluid velocity (m/s) in the y direction.

Considering the microscopic origin of viscosity, it is not surprising that 𝜇 is strongly tem- perature dependent. Viscosity increases with increasing temperature for gases. For dilute gases, the temperature dependence of viscosity can be approximated either by a simple power law,

𝜇

𝜇0 ≈ (

T T0

)n (5.30)

or by Sutherland’s law using the kinetic theory of gases [15],

𝜇

𝜇0 ≈ (

T T0

)1.5 T0 + S T + S

(5.31)

In these equations, n, μ0, T0, and S can be obtained from experiments or kinetic theory. For most gases of interest, the viscosity values obtained from these equations give less than 3% error over a wide range of temperatures (0–1000∘C). Table 5.2 summarizes values for common gases relevant to fuel cells.

2Fluids obeying this equation are called Newtonian fluids.

TRANSPORT IN FLOW STRUCTURES: CONVECTIVE TRANSPORT 185

TABLE 5.2. Parameters for Viscosity Calculation

Gas 𝜇0 (10 −6 kg∕m ⋅ s) T0 (K) n S

Air 17.16 273 0.666 111

CO2 13.7 273 0.79 222

CO 16.57 273 0.71 136

N2 16.63 273 0.67 107

O2 19.19 273 0.69 139

H2 8.411 273 0.68 47

H2O (vapor) 11.2 350 1.15 1064

Source: From Ref. [16].

FLOW BETWEEN PLATES

Assume that a fluid is present between two parallel plates where the lower plate is fixed and the upper plate moves to the right at a steady velocity V, as shown in Figure 5.8. Since the plate only moves in the x direction, u = V and 𝑣 = 0. Equation 5.29 for this case becomes

𝜏xy = 2𝜇 ⋅ 1 2

( 𝜕u 𝜕y

+ 𝜕u 𝜕x

) = 𝜇 ⋅ 𝜕u

𝜕y = const (5.32)

u(y)

0 u = O

u = V

Figure 5.8. Fluid flow between two parallel plates.

Here, 𝜏 is constant since the system is in steady state with no acceleration or pressure variation. By solving Equation 5.32, we can obtain the velocity profile in the y direction, u(y), assuming u(0) = 0 and u(H) = V (where H is the distance between the plates):

u(y) = V y

H and 𝜏 = 𝜇 ⋅ V

H (5.33)

To obtain Equation 5.33, we made the critical assumption that u(0) = 0 and u(H) = V . In other words, we assumed that the fluid velocity was the same as the plate velocity at both of the fluid/plate boundaries. This is the most widely assumed boundary condition for fluid flow, and it is generally a good assumption. In generalized form, this assumption can be stated as

Vfluid = Vsolid (5.34)

186 FUEL CELL MASS TRANSPORT

where V is a vector. This assumption is commonly called the no-slip condition. In certain cases, slip boundary conditions must instead be used. Situations where slip boundary conditions must be used include gas flow in microchannels or gas flow at extremely low pressures. Such scenarios are generally not relevant to fuel cells.

Viscosity is also pressure dependent, increasing slowly with increasing pressure. Fuel cells rarely operate at gas pressures higher than 5 atm. At these low pressures, the “low-density limit” for viscosity applies, and the pressure effects on viscosity can be safely ignored. Thus, viscosity pressure effects will not be considered in this text.

Fuel cell gas streams are rarely composed of a single species. Instead, we usually deal with gas mixtures (e.g., O2 and N2). The following semiempirical expression provides a good approximation for the viscosity of a gas mixture [17]:

𝜇mix = N∑ i=1

xi𝜇i∑N j=1

xjΦij (5.35)

where Φij is a dimensionless number obtained from

Φij = 1√ 8

( 1 +

Mi Mj

)−1∕2[ 1 +

( 𝜇i

𝜇j

)1∕2(Mi Mj

)1∕4]2 (5.36)

where N is the total number of species in the mixture, xi, xj are the mole fractions of species i and j, and Mi, Mj are the molecular weights (kg/mol) of species i and j.

Under most conditions, gas flow in fuel cell flow channels is fairly smooth, or laminar. At extremely high flow rates, gas flow can become turbulent instead. The difference between laminar and turbulent flow is illustrated in Figure 5.9. Turbulent flow is extremely rare in

(a)

(b)

Particle injector

Flow

Figure 5.9. (a) Laminar versus (b) turbulent flow.

TRANSPORT IN FLOW STRUCTURES: CONVECTIVE TRANSPORT 187

fuel cell flow channels. The boundary between laminar and turbulent flow is determined by the Reynolds number, Re. In circular pipes, for example, laminar flow occurs when Re ≤ 2000, while turbulent flow occurs for Re ≥ 3000.

Example 5.2 Consider a fuel cell operating at 80∘C. In the cathode, humidified air at 1 atm is supplied with a water vapor mole fraction of 0.2. If the fuel cell employs circular channels with a diameter of 1 mm, find the maximum tolerable air velocity that still ensures laminar flow. Solution: Using Equation 5.30 and Table 5.2, we can determine the viscosity of each gas component in the humidified air stream. For example, the viscosity of N2 may be calculated as follows:

𝜇N2|80∘C = 𝜇0 (

T T0

)n = 16.63 × 10−6

(353.15 273

)0.67 19.76 × 10−6 kg∕m ⋅ s (5.37)

Similarly, we can obtain μO2|80∘C = 22.92 × 10−6 kg∕m ⋅ s and μH2O|80∘C = 11.32 × 10−6 kg∕m ⋅ s.

To calculate the total viscosity of the mixture using Equation 5.36, we first assem- ble the following parameters:

Species Mole Fraction, xi

Molecular Weight, Mi

Viscosity, μi (10−6 kg∕m ⋅ s)

1. N2 0.8 × 0.79 = 0.632 28.02 19.76 2. O2 0.8 × 0.21 = 0.168 32.00 22.92 3. H2O 0.200 18.02 11.32

Then, we can use Equation 5.36 to produce the following:

Species i Species j Mi∕Mj μi∕μj Φij xjΦij 3∑ j=1

xjΦij

1. N2 1. N2 1.000 1.000 1.000 0.632 2. O2 0.876 0.862 0.930 0.156 1.059 3. H2O 1.555 1.746 1.356 0.271

2. O2 1. N2 1.142 1.160 1.079 0.682 2. O2 1.000 1.000 1.000 0.168 1.146 3. H2O 1.776 2.025 1.482 0.296

3. H2O 1. N2 0.643 0.573 0.776 0.491 2. O2 0.563 0.494 0.732 0.123 0.814 3. H2O 1.000 1.000 1.000 0.200

188 FUEL CELL MASS TRANSPORT

Finally, Equation 5.35 gives the mixture viscosity:

𝜇mix = (0.632 × 19.76

1.059 + 0.168 × 22.92

1.146 + 0.200 × 11.32

0.814

) × 10−6

= 17.93 × 10−6 kg∕m ⋅ s

The molecular weight of the mixture is given by

Mmix = N∑ i=1

xiMi = 0.632 × 28.02 + 0.168 × 32.00 + 0.200 × 18.02

= 26.69g∕mol

Then, the density of the mixture can be obtained using the ideal gas law:

𝜌 = p

RT∕Mmix = 101,325Pa

8.314J∕mol ⋅ K 0.02669kg∕mol

(273.15 + 80) = 0.921kg∕m3 (5.38)

Roughly, laminar flow holds for Re ≤ 2000; thus,

Vmax = Remax𝜇mix

𝜌L =

2000 × (17.93 × 10−6 kg∕m ⋅ s) (0.921kg∕m3) × (0.001m)

= 38.03 m∕s (5.39)

This is very fast flow considering the channel is only 1 mm in diameter.

5.3.2 Mass Transport in Flow Channels

Pressure Drop in Flow Channels. Figure 5.10 illustrates (in 2D) the typical mass transport situation in a fuel cell flow channel. In this diagram, we have a gas moving from left to right through the flow channel at a mean velocity u. A pressure difference between the inlet (pin) and the outlet (pout) drives the fluid flow. Increasing the pressure drop between the inlet and the outlet will increase the mean gas velocity in the channel, improving convection.

For circular flow channels, the relationship between pressure drop and mean gas velocity may be calculated from the relation

dx = 4

D 𝜏𝑤 (5.40)

TRANSPORT IN FLOW STRUCTURES: CONVECTIVE TRANSPORT 189

Inlet Outlet

D h

u u

Pout

T W

Membrane

ElectrodeDiffusionJD

Convection transfer at surface JC

Figure 5.10. Schematic of 2D mass transport in fuel cell flow channel.

where dp∕dx is the pressure gradient, D is the flow channel diameter, and the mean wall shear stress 𝜏𝑤 may be calculated from a nondimensionalized number called the friction factor, f:

f = 𝜏𝑤

1∕2𝜌u2 (5.41)

where 𝜌 is the fluid density (kg/m3) and u is the mean flow velocity (m/s). It is found that regardless of channel size or flow velocity, f ⋅ Re = 16 for laminar flow in circular channels. Furthermore, for circular channels

Re = 𝜌uD 𝜇

(5.42)

Thus, by combining Equations 5.40, 5.41, and 5.42 and the fact that f ⋅ Re = 16, pressure drop and mean gas velocity may be related:

dx = 32u

D2 (5.43)

Unfortunately, most fuel cell flow channels are rectangular instead of circular. For rect- angular channels, Equation 5.43 cannot be used. For rectangular channels, we must use a “hydraulic diameter” to compute the effective Reynolds number compared to a circular channel:

Reh = 𝜌uDh 𝜇

(5.44)

where Dh =

4A P

= 4 × cross-sectional area perimeter

(5.45)

For circular channels, Dh = D. Hence, Dh can be thought of as the “effective” diameter of a noncircular channel.

190 FUEL CELL MASS TRANSPORT

0 0.2 0.4 0.6 0.8 1

b/a

e Rf

Rectangle

Circle

Figure 5.11. Friction factors of circular and rectangular channels.

For rectangular channels, the relationship between Reh and f is also more complex than for circular channels. It can be approximated as [18]

fReh = 24(1 − 1.355𝛼∗ + 1.9467𝛼∗2 − 1.7012𝛼∗3 + 0.9564𝛼∗4 − 0.2537𝛼∗5) (5.46)

where α∗ is the aspect ratio of the channel cross section: α∗ = b∕a, where 2a and 2b are the lengths of the channel sides. Equation 5.46 is plotted as a function of α∗ in Figure 5.11.

By determining τ𝑤 for a rectangular channel from Equations 5.41, 5.44, and 5.46, the pressure gradient can then be determined using Equation 5.40 (making sure that Dh is used in place of D).

Example 5.3 Fluid is flowing at a velocity of 1 m/s through a 1-mm-wide, 2-mm-high, 20-cm-long rectangular channel. Find the pressure drop in the channel if the viscosity of the flowing fluid is 17.9 × 10–6 kg∕m ⋅ s.

Solution: We know

dx = 4

Dh 𝜏𝑤 =

4 Dh

f 1 2 𝜌u2

= 4 Dh

f Reh Reh

1 2 𝜌u2 = 4

fReh𝜇

𝜌uDh

1 2 𝜌u2

= 2 D2h

f Reh𝜇u (5.47)

TRANSPORT IN FLOW STRUCTURES: CONVECTIVE TRANSPORT 191

Assume α∗ = b∕a = 1∕2, and from Equation 5.46

f Reh = 24(1 − 1.3553 ⋅ 0.5 + 1.9467 ⋅ 0.52 − 1.7012 ⋅ 0.53

+ 0.9564 ⋅ 0.54 − 0.2537 ⋅ 0.55) = 15.56 (5.48)

Using

Dh = 4 × (1 × 2) 2 × (1 + 2)

= 1.33mm = 0.00133m

Equation 5.47 gives

dx = 2

(0.00133 m)2 15.56 × 17.9 × 10−6 kg∕m ⋅ s × 1 m∕s = 315Pa∕m (5.49)

Thus the pressure drop is

Pdrop = L × dp

dx = 0.2 m × 315 Pa∕m = 63 Pa (5.50)

Convective Mass Transport from Flow Channels to the Electrode. As shown in Figure 5.10, although gas is flowing in the x direction from left to right along the flow channel, convective mass transport can also occur in the y direction from the flow channel into (or out of) the electrode. This type of convective mass transport occurs when the density of a species i is different at the electrode surface and the flow channel bulk. For example, in a fuel cell cathode, water is produced at the electrode. The local density of water at the electrode surface will be greater than the density of water in the flow channel bulk, leading to convective mass transport of water away from the electrode surface.

Mathematically, the mass flux due to this form of convective mass transfer may be estimated by

JC,i = hm(𝜌i,s − 𝜌i) (5.51)

where JC,i is the convective mass flux (kg/m 2 ⋅ s), 𝜌i,s is the density (kg/m

3) of species i at the electrode surface, 𝜌i is the mean density (kg/m

3) of species i in the bulk fluid, and hm is the mass transfer convection coefficient (m/s). The value of hm is dependent on the channel geometry, the physical properties of species i and j, and the wall conditions.

Commonly, hm can be found from a nondimensional number called the Sherwood (or Nusselt) number:3

hm = Sh Dij Dh

(5.52)

3The Nusselt number applies to convective heat transport problems. Due to the similarity between heat and mass transport, both numbers are essentially the same.

192 FUEL CELL MASS TRANSPORT

TABLE 5.3. Sherwood Numbers for Laminar Flows in Circular, Rectangular, and Three-Sided Closed Rectangular Ducts

Cross Section 𝛼 = 0.2 𝛼 = 0.4 𝛼 = 0.7 𝛼 = 1.0 𝛼 = 2.0 𝛼 = 2.5 𝛼 = 5.0 𝛼 = 10.0

ShD 4.36

ShF 3.66

ShD 4.80 3.67 3.08 2.97 3.38 3.67 4.80 5.86

ShF 5.74 4.47 3.75 3.61 4.12 4.47 5.74 6.79

ShD 0.83 1.42 2.02 2.44 3.19 3.39 3.91 4.27

ShF 0.96 1.60 2.26 2.71 3.54 3.78 4.41 4.85

Note: Channel aspect ratio 𝛼 = b∕a, where b and a are channel dimensions. Source: From Ref. [19].

where Sh is the Sherwood number, Dh is the hydraulic diameter, and Dij is the binary dif- fusion coefficient for species i and j. The Sherwood number depends on channel geometry. Table 5.3 summarizes some values of Sh for geometries commonly encountered in fuel cell flow channels. In most cases, only one wall in a rectangular channel of a fuel cell participates in convective mass transport (the third case represented in the table). The table distinguishes between two different Sherwood numbers: ShD values apply when density ρi is uniform along a channel; ShF values apply when flux JC,i is uniform along a channel. If neither density nor flux is uniform along the channel, Equations 5.51 and 5.52 should not be used.

5.3.3 Gas Is Depleted along Flow Channel

Since either hydrogen (anode) or air (cathode) is consumed continuously along a fuel cell flow channel, these reactants tend to become depleted, especially near the outlet. Depletion poses an adverse effect on fuel cell performance, since concentration losses increase as the reactant concentrations decrease.

In this section, we will develop a simple 2D mass transport model for a fuel cell cathode. We will use this model to determine how the oxygen density (concentration) decreases along the flow channel using a macroscale mass flux balance.

Consider the simple half PEMFC geometry shown in Figure 5.12. Pure oxygen flows from left to right along the flow channel depicted in this diagram from the fuel cell inlet to the fuel cell outlet. As the gas travels from left to right along the flow channel, it is also being consumed. The y-direction flux JO2 |y=E represents the oxygen gas that is removed from the flow channel by convective mass transport into the gas diffusion layer. This oxygen gas then diffuses to the catalyst layer where it reacts to produce the fuel cell current.

For this simple model, we assume the flow channel has a square crosssection. We also make a few additional simplifying assumptions:

1. The catalyst layer is infinitely thin.4

2. Water exists only in the vapor form.

4This is a fairly good approximation since real catalyst layers are very thin (∼10 μm) compared to gas diffusion layers (100–350 μm) in PEMFCs.

TRANSPORT IN FLOW STRUCTURES: CONVECTIVE TRANSPORT 193

cathode flow channel

cathode catalyst layer

gas diffusion layer

electrolyte

OutletInlet

uin

JO2| y=E CONV

JO2| y=E DIFF

JO2| y=C RXN

ρ O2 ρ

Figure 5.12. Schematic of a 2D fuel cell transport model including diffusion and convection.

3. Diffusive mass transport dominates in the diffusion layer. Furthermore, only y-direction diffusion is considered.

4. Convection dominates in the flow channel.

The current density produced by the fuel cell will vary along the x direction because the concentration of oxygen varies along the x direction. We denote the local current density produced by the fuel cell at position X as j(X). From Faraday’s law, if the fuel cell is pro- ducing a current density j(X) at location X, then the mass flux of oxygen it is consuming is given by

ĴO2 |rxnx=X,y=C = MO2 j(X)4F (5.53) where ĴO2 is the oxygen mass flux (kg/cm

2⋅s), y = C denotes the catalyst layer (where the reaction to produce electricity takes place), and MO2 is the molecular weight (kg/mol) of oxygen.

The oxygen flux consumed by the electrochemical reaction must be provided by dif- fusion in the gas diffusion layer. As you have previously seen, diffusive mass transport is described by Fick’s law:

ĴO2 |diffx=X,y=E = −DeffO2 𝜌O2 |x=X,y=C − 𝜌O2 |x=X,y=EHE (5.54) where HE is the thickness of the diffusion layer. In this equation, we have converted the molar concentrations normally seen in Fick’s law into mass concentrations (density 𝜌 is effectively a “mass concentration”). The flux ĴO2 is therefore a mass flux rather than a molar flux.

The oxygen flux due to diffusive transport through the gas diffusion layer is provided by convective mass transport between the flow channel and the gas diffusion layer surface

194 FUEL CELL MASS TRANSPORT

(represented in the diagram by ĴO2 |convy=E ). Recall from Equation 5.51 that this convective mass transport process can be described by

ĴO2 |convx=X,y=E = −hm(𝜌O2 |x=X,y=E − 𝜌O2 |x=X,y=channel) (5.55) where hm is the convection mass transfer coefficient and 𝜌O2 is the average density of oxygen in the flow channel. To maintain flux balance, the oxygen fluxes in Equations 5.53, 5.54, and 5.55 must be the same (steady-state condition). In other words,

ĴO2 |rxnx=X,y=C = ĴO2 |diffx=X,y=E = ĴO2 |convx=X,y=E (5.56) Thus, we can obtain the following relations:

Now, we couple the y-direction mass transport of oxygen to the x-direction mass trans- port of oxygen in the flow channel by considering the overall flux balance in the control volume (dotted box) in Figure 5.12. Oxygen is entering into this control volume from the left and leaving to the right. The difference between the amount of oxygen entering on the left and the amount of oxygen leaving on the right yields the amount of oxygen that is leaving out the top into the gas diffusion layer. Mathematically,

uinHC𝜌O2 |x=0,y=channel amount of gas

entering from left

− uinHC𝜌O2 |x=0,y=channel amount of gas

leaving from right

= ∫ X

( ĴO2 |convy=E) dx

amount of gas

leaving out the top

(5.60)

Equation 5.57 then allows us to relate the gas leaving out the top of the control volume to the current density produced by the fuel cell:

∫ X

0 (ĴO2 |convy=E )dx = ∫ X0 MO2 j(x)4F dx (5.61)

Remember, we are seeking an expression for the x-direction oxygen profile at the cat- alyst layer. (In other words, we want to find 𝜌O2 |x=X,y=C.) Starting with Equation 5.58, 𝜌O2 |x=X,y=C may be determined by plugging in Equations 5.59, 5.60, and 5.61. This yields

𝜌O2 |x=X,y=C = 𝜌O2 |x=0,y=channel − MO24F ( j (X) hm

+ HEj(X) Deff

+ ∫ X

j(x) uinHC

) (5.62)

TRANSPORT IN FLOW STRUCTURES: CONVECTIVE TRANSPORT 195

Distance from inlet (cm)

D en

si ty

o f o

xy ge

n a

t c at

al ys

t l ay

er (

kg /m

3 )

0 2 4 6 8 10 12 14 16 18 20 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Figure 5.13. Oxygen density profile predicted from Equation 5.65 for the following case: elec- trode porosity 𝜀 = 0.4, inlet gas pressure p = 2 atm, model temperature T = 80∘C, current density j = 1 A∕cm2, inlet gas velocity uin = 10 cm/s, channel height HC = 0.1 cm, electrode thickness HE = 0.035 cm, and the Sherwood number ShF = 2.71.

For an exact solution, Equation 5.62 can then be solved in combination with the Tafel equation. However, to avoid mathematical complication, we assume that the current density j is constant along the x direction. (This assumption is not quite true. The oxygen concentra- tion changes along the x direction, and thus the local current density will also change. Even for fairly substantial oxygen concentration changes, however, the current density effect will be minor. For instance, the oxygen concentration changes shown in Figure 5.13, where 𝜌O2 decreases by more than a factor of 4 at the outlet compared to the inlet, would result in only a 20% decrease in local current density at the outlet compared to the inlet.) Using the constant-current-density assumption, Equation 5.62 becomes

𝜌O2 |x=X,y=C = 𝜌O2 |x=0,y=channel −MO2 j4F (

1 hm

+ HE DeffO2

+ X uinHC

) (5.63)

Using Equation 5.52, we can determine hm based on the constant-flux Sherwood number ShF for the flow channel:

hm = ShFDO2 HC

(5.64)

Plugging this result into Equation 5.63 yields a final expression for the oxygen profile:

𝜌O2 |x=X,y=C = 𝜌O2 |x=0,y=channel −MO2 j4F (

HC ShFDO2

+ HE DeffO2

+ X uinHC

) (5.65)

196 FUEL CELL MASS TRANSPORT

Equation 5.65 tells us that oxygen density decreases linearly as X increases.5 In other words, the oxygen concentration is depleted linearly as the gas moves along the channel. The three terms in the parentheses represent the effects of channel size HC, diffusion layer thickness HE, and inlet flow velocity uin in that order. Supplying more oxygen (increasing uin) improves mass transport, thus increasing the oxygen density at the catalyst layer. Sim- ilarly, decreasing the diffusion layer thickness HE also increases the oxygen density at the catalyst layer. The effect of channel size HC is a little tricky to calculate, since HC appears in both the first and third terms in the parentheses. However, if we assume that the total gas supply to the fuel cell (either by volume or mass) is constant, we have

Ntotal = uinHC = const (5.66)

Thus, if the oxygen supply rate is constant, uinHC in the last term is fixed. In this case, decreasing the channel sizeHC will increase the oxygen density. An example oxygen profile prediction given by Equation 5.65 is displayed in Figure 5.13.

5.3.4 Flow Structure Design

Flow StructureMaterials. In the most general terms, the flow structure serves two main purposes: (1) it supplies the reactant gases and removes the reaction products and (2) it harvests the electrical current generated by the fuel cell. In spite of these seemingly simple tasks, flow structures are subject to a challenging set of materials selection criteria [20]:

• High electrical conductivity • High corrosion resistance • High chemical compatibility • High thermal conductivity • High gas tightness • High mechanical strength • Low weight and volume • Ease of manufacturability • Cost-effectiveness

The most commonly used material for low-temperature fuel cell flow plates is graphite. Graphite satisfies most of the criteria discussed above except for (1) ease of manufactura- bility, (2) cost, and (3) high mechanical strength. These criteria are not fulfilled because of costly machining requirements and the intrinsic brittleness of the material. Surprisingly, the machining of graphite is so expensive that graphite plates can comprise up to half the cost of a fuel cell system [21]. Alternatives to graphite include corrosion-resistant metals

5See problem 5.8.

TRANSPORT IN FLOW STRUCTURES: CONVECTIVE TRANSPORT 197

such as stainless steel [22, 23]. In general, metal plates offer less expensive fabrication and higher mechanical strength compared to graphite plates. Thin metal flow plates can significantly reduce the volume and weight of a fuel cell system. One critical issue con- cerning metal plates is the formation of surface metal oxides. Even a thin metal oxide layer will increase the contact resistance between the flow plate and the electrode, resulting in degraded fuel cell performance [23–25]. This problem has been partially overcome by the use of corrosion-resistant surface coatings [24, 25], although the long-term stability of such coatings needs improvement.

Flow plates in high-temperature fuel cells are made from ceramics such as lanthanum chromite (for high temperatures) or ferritic stainless steel (for moderate temperatures). These materials are discussed in more detail in Chapter 9. In SOFCs and MCFCs, flow plate stability and durability are critical, since the high operating temperature facilitates degrada- tion. Also, any thermal mismatch between the plate material and the electrode material will be a source of serious mechanical stress during thermal cycles. Thus, the thermal proper- ties of the flow plate should be carefully matched to the rest of the fuel cell system. Certain SOFC designs, such as tubular SOFCs, do not require flow plates and avoid the issue of high-temperature sealing. These designs are discussed in Chapter 10.

Flow Structure Patterns. As was previously mentioned, flow plates contain dozens or even hundreds of fine channels (or “grooves”) to distribute the gas flow over the surface of the fuel cell. The shape, size, and pattern of flow channels can significantly affect the performance of a fuel cell. Choosing the right flow pattern is especially critical for PEMFCs. In PEMFCs, flow field design efforts often focus on the water removal capability of the cathode side. Poorly designed flow field plates leave certain regions flooded with liquid water, thus blocking gas access and reducing the output current of the cell. Such blocked regions not only reduce performance but can actually cause irreversible damage to the fuel cell. This is because cell polarity can be locally reversed in gas-starved regions, leading to corrosion and material degradation [26].

Although a wide variety of flow patterns are employed by research groups and develop- ers, most fall under three basic flow pattern archetypes (see Figure 5.14):

1. Parallel flow

2. Serpentine flow

3. Interdigitated flow

Parallel Flow. In a parallel configuration, flow evenly enters each straight channel and exits through the outlet. (See Figure 5.15a.) A significant advantage of the parallel pattern is the low overall pressure drop between gas inlet and outlet. However, when the width of the flow field is relatively large, flow distribution in each channel may not be uniform. This causes water buildup in certain channel areas, leading to increased mass transfer losses (and a corresponding current density decrease). Several fuel cell developers (e.g., Ballard, Honda) employ this channel type in their PEMFC fuel cell systems.

198 FUEL CELL MASS TRANSPORT

Figure 5.14. Major flow channel geometries: (a) parallel, (b) serpentine, (c) parallel–serpentine, (d) interdigitated. Flow channel geometries seek to provide homogeneous distribution of reactants across an electrode surface while minimizing pressure drop losses and maximizing water removal capability.

Figure 5.15. Gas transport modes in various flow channel geometries. Each channel type induces a different convective transport scheme in the electrode.

CHAPTER SUMMARY 199

Serpentine Flow. This is the most common geometry found in fuel cell prototypes. The advantage of the serpentine pattern lies in the water removal capability. Only one flow path exists in the pattern, so liquid water is forced to exit the channel. (See Figure 5.15b.) Unfortunately, in large-area cells, a serpentine design leads to a large pressure drop. Several variations of the serpentine design have been investigated, such as the parallel–serpentine configuration. This hybrid design, combining the advantages of serpentine and parallel pat- terns, is famously used in Ballard PEMFC stacks.

Interdigitated Flow. The interdigitated design promotes forced convection of the reactant gases through the gas diffusion layer. (See Figure 5.15c). Subject to much recent attention, research shows that this design provides far better water management, leading to improved mass transport [27]. The forced convection through the gas diffusion layer leads to signif- icant pressure drop losses. However, there is evidence that this major disadvantage might be partially overcome by employing extremely small rib spacing [28].

In addition to the channel pattern, channel shape and size can also significantly affect performance [24], [28–31] . These parameters are best explored using computer numerical simulations. One such simulation technique, known as CFD modeling, will be discussed in the forthcoming chapter and again in Chapter 13.

5.4 CHAPTER SUMMARY

• Mass transport governs the supply and removal of reactants and products in a fuel cell.

• Poor mass transport leads to a loss in fuel cell performance due to reactant depletion (or product clogging) effects.

• Mass transport in fuel cell electrodes is typically dominated by diffusion. Mass trans- port in fuel cell flow structures is typically dominated by convection.

• Convection refers to the transport of a species by the bulk motion of a fluid. Diffusion refers to the transport of a species due to a gradient in concentration.

• Diffusive transport limitations in the electrode lead to a limiting current density jL. The limiting current density corresponds to the point where the reactant concentration falls to zero in the fuel cell catalyst layer. A fuel cell can never sustain a current density higher than jL.

• Reactant depletion affects both the Nernstian cell voltage and the kinetic reaction rate. Depletion leads to a similar loss in both cases. This “concentration loss” can be generalized as 𝜂conc = c ln[jL∕(jL − j)] where c is a constant that depends on the geometry and mass transport properties of the fuel cell.

• Concentration losses are most effectively minimized by careful consideration of the convective transport situation in the fuel cell flow channels.

• Convection in fuel cell flow channels is characterized by the Reynolds number Re, a nondimensional parameter that characterizes the viscous behavior of the flow. Usually, gas flow in fuel cells is laminar.

200 FUEL CELL MASS TRANSPORT

• Viscosity μ characterizes the resistance of a fluid to flow. Viscosity can be thought of as a measure of the “internal” friction in the fluid.

• The viscosity of a gas mixture is dependent on the temperature and composition of the mixture.

• A pressure difference is required to drive gas flow through a channel. • The pressure drop in a flow channel is mainly caused by friction between fluid and the

channel walls. This friction is quantified by wall shear stress 𝜏𝑤. Pressure drops can be determined using the friction factor f, which is dependent on the Reynolds number and channel geometry.

• Although gases in fuel cell flow channels move along the flow channel, they can also be transported between the flow channel and the electrode. This is known as convec- tive mass transport. Convective mass transport is characterized by a convective mass transfer coefficient hm, which may be calculated from the Sherwood number, Sh.

• A simple 2D fuel cell mass transport model can be constructed to show how reactant gases are depleted in a flow channel from the inlet to the outlet. In general, increas- ing the gas flow velocity, decreasing the channel size, or decreasing the diffusion layer thickness will improve the mass transport situation along the length of the flow channel.

• Choice of the flow field pattern significantly affects the size of the mass transport losses. Due to the liquid water formation in the cathode, PEMFCs require flow fields with high water removal capability.

• Serpentine or parallel–serpentine designs are the most commonly used flow field types. They provide a decent compromise between pressure drop and water removal capability.

CHAPTER EXERCISES

Review Questions

5.1 Everything else being equal, would the concentration losses in a fuel cell using “syn- thetic air” (21% oxygen, 79% helium) be higher or lower than the concentration losses in a fuel cell using real air (≈ 21% oxygen, ≈ 79% nitrogen)? Defend your answer.

5.2 Discuss why cathode flow channel design is less important for SOFCs than for PEMFCs. Hint: Consider the typical operating temperature of a SOFC and its effect on jL.

5.3 Discuss the factors that determine jL. List at least three ways to increase jL.

Calculations

5.4 Using Equation 5.10, calculate the limiting current density for a fuel cell cathode running on air at STP. Assume only O2 and N2 and ignore the presence of water vapor. Assume that the diffusion layer is 500 μm thick and has a porosity of 40%.

CHAPTER EXERCISES 201

5.5 Generate a series of plots similar to the ones shown in Figure 5.7 but for different values of c, while holding jL constant at 2.0 A/cm

2. Generate plots for c values of 0.1, 0.05, and 0.01, respectively.

5.6 Consider a fuel cell operating at 800∘C, 1 atm. In the cathode, humidified air is sup- plied with the mole fraction of water vapor equal to 0.1. If the fuel cell employs circular flow channels with a diameter of 1 mm, find the maximum velocity of air that can be used while still maintaining laminar flow. Compare your result to Example 5.2.

5.7 Estimate the maximum fuel cell area that can be operated at 1 A/cm2, under the con- dition from Example 5.2. Assume a stoichiometric number of 2. Assume that the fuel cell is made of a single straight flow channel. Discuss why channel flow in fuel cells is almost always considered to be laminar.

5.8 Plot the oxygen distribution along the channel (the x direction at the catalyst layer) for the fuel cell flow model developed in Section 5.3.3, assuming uin = 1m∕s, HC = 1 mm, and an operating temperature of 80∘C. Estimate DO2,H2O and D

eff O2,H2O

, using

Equations 5.2 and 5.3 assuming 𝜀 = 0.4 and p = 1atm. (Use the same HE, Sh, and j as in the Section 5.3.3 example.)

5.9 Following a procedure similar to that illustrated by the model developed in Section 5.3.3, derive an equation for the water vapor density distribution along a fuel cell flow channel (at the catalyst layer).

5.10 Find the oxygen density distribution along the channel (at the catalyst layer) for the fuel cell model developed in Section 5.3.3, assuming constant voltage but not con- stant current. Hint: Use the Tafel equation to set up an ordinary differential equation for j(X).

5.11 Consider a direct methanol fuel cell at T = 95∘C. Calculate jL at the anode assuming 1M methanol fuel supply, given Deff = 10–5 cm2∕s and δ = 150 μm.

(a) 386 A/cm2

(b) 0.386 A/cm2

(d) 0.00852 A/cm2

5.12 Everything else being equal, the limiting current density (maximum current density produced) for a direct methanol fuel cell (DMFC) operated on 5M methanol instead of 1M methanol:

(a) will decrease

(b) will increase

(d) cannot be determined

5.13 True or False: If both temperature and pressure are increased by the same relative proportions, the limiting current density, jL, will increase.

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CHAPTER 6

FUEL CELL MODELING

In the last four chapters we have acquired the necessary tools to describe the basic operation of a fuel cell. Now it is time to complete the picture. In this chapter, we will put all those tools together to build a complete fuel cell model. Our model will include thermodynamics (Chapter 2), reaction kinetics (Chapter 3), charge transport (Chapter 4), and mass transport (Chapter 5). Do not worry if putting all these things together sounds intimidating. In fact, it is surprisingly simple! You will be amazed at the predictive power provided by even a modest fuel cell model. Furthermore, modeling offers a great opportunity to see how the material we have learned in the last four chapters fits together into a cohesive unit.

After discussing the big picture in the context of a simple fuel cell model, we will delve into the details of several more sophisticated modeling approaches. One example is a flux balance-based approach, which we use to model both a PEMFC and an SOFC. Still more complex is the CFD approach to fuel cell modeling. Computational fluid dynamics modeling allows the detailed interactions between flow structure geometry, fluid dynam- ics, multiphase flow, and electrochemical reaction to be simulated numerically. These more sophisticated modeling techniques can provide predictive capability and may one day allow fuel cell designers to better optimize fuel cells computationally before ever testing them in the laboratory.

6.1 PUTTING IT ALL TOGETHER: A BASIC FUEL CELL MODEL

If you recall from the first chapter of this book, we noted that the real voltage output of a fuel cell could be written by starting with the thermodynamically predicted voltage and then subtracting the various overvoltage losses:

V = Ethermo − 𝜂act − 𝜂ohmic − 𝜂conc (6.1)

203

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204 FUEL CELL MODELING

where

V = operating voltage of fuel cell Ethermo = thermodynamically predicted voltage of fuel cell 𝜂act = activation losses due to reaction kinetics

𝜂ohmic = ohmic losses from ionic and electronic resistance 𝜂conc = concentration losses due to mass transport

In the last four chapters, we determined basic expressions for each of the quantities in Equation 6.1. For example, in Chapters 3 we learned how the activation loss 𝜂act could be described by the Butler–Volmer equation (or the simpler Tafel equation). We were even able to draw a graph, which showed the effect of the activation loss on fuel cell perfor- mance. In Chapters 4 and 5, we were able to draw graphs describing the effects of charge transport and mass transport on fuel cell performance. As Equation 6.1 illustrates, overall fuel cell performance is simply given by the combined effects of all these various losses. Pictorially, the concept is illustrated in Figure 6.1. By starting with the thermodynamically predicted fuel cell voltage and then graphically subtracting out the losses from activation, ohmic resistance, and concentration effects, we are left with the net fuel cell performance. Mathematically (using the simplest expressions developed in Chapters 3–5 for 𝜂act, 𝜂ohmic, and nconc), the net fuel cell j–V behavior can be written as

V = Ethermo − (aA + bA ln j) − (aC + bC ln j) − (jASRohmic) − ( c ln

jL jL − j

) (6.2)

where

𝜂act = (aA + bA ln j) + (aC + bC ln j): activation losses from both anode (A) and the cathode (C) based on natural logarithm version of the Tafel Equation 3.41

𝜂ohmic = jASRohmic: ohmic resistance loss based on current density and ASR (see Equation 4.11)

𝜂conc = c ln jL jL−j

: combined fuel cell concentration loss based on Equation 5.25, where c is an empirical constant

Because we use the Tafel approximation for the fuel cell kinetics, this model is only valid when j >> j0. For detailed modeling of the low-current-density region, the full form of the Butler–Volmer equation is required.

In its most general form, the simple model represented by Equation 6.2 has seven “fitting constants”: aA, aC, bA, bC, c, ASRohmic, and jL. However, for H2–O2 fuel cells, the anode kinetic losses can often be neglected compared to the cathode kinetic losses (eliminating aA and bA). Also, if the “first-principles” values of a, b, and c are used, we know that they are really related to the two more fundamental constants α and j0. In the extremely streamlined case, then, as few as four parameters (𝛼C, j0,C, ASRohmic, and jL) are required.

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PUTTING IT ALL TOGETHER: A BASIC FUEL CELL MODEL 205

C el

l v ol

ta ge

( V

)

Current density (A/cm2)

Concentration loss (Chapter 5)

C el

l v ol

ta ge

( V

) Current density (A/cm2)

Net fuel cell performance

C el

l v ol

ta ge

( V

)

Current density (A/cm2)

C el

l v ol

ta ge

( V

)

Current density (A/cm2)

Activation loss (Chapter 3)

C el

l v ol

ta ge

( V

)

Current density (A/cm2)

Ohmic loss (Chapter 4)Reversible voltage (Chapter 2)

Figure 6.1. Pictorial summary of major factors that contribute to fuel cell performance. The over- all fuel cell j–V performance can be determined by starting from the ideal thermodynamic fuel cell voltage and subtracting out the losses from activation, conduction, and concentration.

Theoretical EMF or Ideal voltage

C el

l v ol

ta ge

( V

)

Measured current density (A/cm2) 1.00.0

j leak

No leakage loss

FC with leakage loss

Figure 6.2. Pictorial illustration of the effect of a leakage current loss on overall fuel cell perfor- mance. A leakage current effectively “offsets” a fuel cell’s j–V curve, as shown by the dotted curve in the figure. This has a significant effect on the open-circuit voltage of the fuel cell (y-axis intercept), which is reduced below its thermodynamically predicted value.

In reality, we find that one additional term is usually needed to reflect the true behavior of most fuel cell systems. This additional term, jleak, is associated with the parasitic loss from current leakage, gas crossover, and unwanted side reaction. In almost all fuel cell systems, some current is lost due to these parasitic processes. You might recall that we have already talked a little bit about gas crossover in previous chapters. The net effect of this parasitic current loss is to offset the fuel cell’s operating current by an amount given by jleak. In other words, the fuel cell has to produce extra current to compensate for the current that is lost due to parasitic effects. Pictorially, this loss effect is illustrated in Figure 6.2.

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206 FUEL CELL MODELING

Mathematically, jgross = j + jleak (6.3)

where jgross is the gross current produced at the fuel cell electrodes, jleak is the parasitic current that is wasted, and j is the actual fuel cell operating current that we can measure and use. In our fuel cell model, 𝜂act and 𝜂conc should be based on jgross since the reaction kinetics and species concentrations are affected by the leakage current. In general, however, 𝜂ohmic should be based on j, since only the operating current of the fuel cell is actually conducted through the cell. (The leakage current is wasted by side reactions or non-electrochemical reactions at the electrodes and does not give rise to real current flow across the cell.) Thus, we can rewrite our fuel cell model in the following final form:

V = Ethermo − [aA + bA ln(j + jleak)] − [aC + bC ln(j + jleak)]

−(jASRohmic) −

( c ln

jL jL −

( j + jleak

)) (6.4) The most noticeable effect of leakage current is to reduce a fuel cell’s open-circuit volt-

age below its thermodynamically predicted value. At high current density, the limiting current density will also be reduced by the leakage current. However, at midrange current densities, the leakage current effects tend to be minor or insignificant. Careful inspection of the two curves in Figure 6.2 illustrates this effect.

The simple fuel cell model described by Equation 6.4 can be used for virtually unlim- ited numbers of “what-if ” scenarios. For example, the model can be used to contrast the j–V behavior of a typical low-temperature (e.g., polymer electrolyte membrane) fuel cell versus a typical high-temperature (e.g., solid oxide) fuel cell. In a typical H2–O2 PEMFC, activation losses are significant due to the low reaction temperature, but ohmic losses are relatively small due to the high conductivity of the polymer electrolyte. In contrast, ohmic losses tend to dominate H2–O2 SOFC performance while the activation losses are minor due to the high reaction temperature.

Typical parameters for H2–O2 PEMFCs and SOFCs are summarized in Table 6.1. Using these parameters as inputs into our simple model (Equation 6.4) produces the contrasting j–V behaviors shown in Figure 6.3. The large j0 values in the SOFC model require the use of the full Butler–Volmer equation for 𝜂act. Alternatively, since j0 is so large in the SOFC, the small 𝜂act approximation of the Butler–Volmer equation can be successfully used. (Recall from Equation 3.38 that this approximation gives 𝜂act ≈ [(RTj)∕(nFj0)].)

6.2 A 1D FUEL CELL MODEL

Having discussed a simple fuel cell model in the previous section, we now introduce a more sophisticated 1D model for SOFCs and PEMFCs. This model is based on the flux balance concept. Flux balance allows us to keep track of all the species that flow in, out, and through a fuel cell. Flux-balance-based models are popular in the fuel cell literature. The model that we will develop in this section is really just a simplified version of the popular literature models developed in the last decade [8, 32–37].

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A 1D FUEL CELL MODEL 207

TABLE 6.1. Summary of Typical Parameters for Low-Temperature PEMFC versus High-Temperature SOFC

Parameter Typical Value for PEMFC Typical Value for SOFC

Temperature 350 K 1000 K

Ethermo 1.22 V 1.06 V

j0(H2) 0.10 A∕cm2 10 A/cm2

j0(O2) 10 −4 A∕cm2 0.10 A/cm2

α(H2) 0.50 0.50

α(O2) 0.30 0.30

ASRohmic 0.01Ω ⋅ cm2 0.04Ω ⋅ cm2

jleak 10 −2 A∕cm2 10−2 A∕cm2

jL 2 A∕cm2 2 A∕cm2

c 0.10 V 0.10 V

Theoretical EMF or ideal voltage

Current density (A/cm2)

C el

l v ol

ta ge

( V

) Theoretical EMF or ideal voltage

Current density (A/cm2)

C el

l v ol

ta ge

( V

)

Typical PEMFC Typical SOFC

0.2

0.4

0.6

0.8

1.2

0 0.5 1 1.5 2 0

0.2

0.4

0.6

0.8

1.2

0 0.5 1 1.5 2

Figure 6.3. Comparison of our simple model results for a typical PEMFC versus a typical SOFC. As shown by the shape of the curves, the PEMFC benefits from a higher thermodynamic voltage but suffers from larger kinetic losses. SOFC performance is dominated by ohmic and concentration losses. The input parameters used to generate these model results are summarized in Table 6.1.

Flux-balance-based models are suited to both PEMFCs and SOFCs. Generally PEM- FCs are more difficult to model because water can be transported through the membrane, complicating the flux balance. Also, in PEMFCs, water is present as a liquid. Liquid water is far more difficult to model than water vapor. Remember that in SOFCs all the reactants and products exist as gases (including water); this makes the modeling easier. However, SOFC modeling can be complicated by other issues such as nonisothermal behavior and thermal-expansion-induced mechanical stress. While these issues can be integrated into a structural SOFC model, the complexity swiftly becomes daunting. In the present models, therefore, we will focus only on fuel cell species transport. By keeping track of species concentration profiles inside a model fuel cell, we can extract electrochemical losses and the j–V curve.

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208 FUEL CELL MODELING

6.2.1 Flux Balance in Fuel Cells

A 1D flux balance fuel cell model starts as a very careful bookkeeping exercise. To generate an accurate model, the fluxes of all chemical species going into, out of, and through the fuel cell must be detailed. Figure 6.4 illustrates the high-level flux detail needed in our 1D fuel cell model. In this diagram, individual fluxes are numbered consecutively. While the exact meaning of each flux term is unimportant for now, this diagram essentially allows us to keep track of the H2O and H2 flowing into/out of the anode, the H2O, N2, and O2 flowing into/out of the cathode, and the H2O and H

+ (for PEMFC) or O2– (for SOFC) flowing across the electrolyte membrane.

The fluxes in Figure 6.4 can be related to one another using the principle of flux balance. Flux balance expresses the idea that what comes in must go out. In fuel cells, all fluxes can be related to a single characteristic flux—the current density, or charge flux of the fuel cell. Here is an example of how the current density (flux 14 in Figure 6.4a) can be related to the other fluxes in a PEMFC. Based on an examination of the fluxes in Figure 6.4a, we can write

flux14 = flux5 = flux1 − flux4 = flux8 − flux13 (6.5)

In other words, the current density produced by the fuel cell must equal the proton flux across the electrolyte, which must equal the hydrogen flux into the anode catalyst layer, which must equal the oxygen flux into the cathode catalyst layer. Mathematically,

2F =

JH+

2 = JAH2 = 2J

C O2

= SCH2O (6.6)

where j, F, and J stand for current density (A∕cm2), Faraday’s constant (96,484 C∕mol), and molar flux (mol∕s ⋅ cm2), respectively; JAH2 stands for the net flux of H2 in the anode (in other words, the flux of hydrogen coming in minus the flux of hydrogen going out). Since the net hydrogen flux is the difference between what goes in and what goes out, it represents hydrogen that is consumed inside the fuel cell by the reaction. Likewise, JCO2

stands for the

net flux of oxygen at the cathode. Also, note that the water generation rate SCH2O (mol/s ⋅ cm2)

at the cathode is equal to the net hydrogen flux. (For each mole of hydrogen that is con- sumed, 1 mol of water will be produced.)

In an analogous manner, the following water flux balance must also be satisfied:

flux2 − flux3 anode

= flux6 − flux7 membrane

= flux12 − flux9 − flux5 cathode

(6.7)

In other words, the net water flux into the anode catalyst layer must be equal to the net water flux across the electrolyte (given by the balance between the electro-osmotic drag and back-diffusion water fluxes), which must be equal to the net water flux out of the cathode catalyst layer. Note that the water generation at the cathode (flux 5) also must be included for correct flux balance. Mathematically,

JAH2O = J M H2O

= JCH2O − j

2F (6.8)

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A 1D FUEL CELL MODEL 209

(a)

(b)

Anode Electrolyte Cathode

Flow structure

Porous electrode

H2 O2 H2O

H2O N2

O2 H2O N2

H2 H2O

10 9

13 4

a b c d z

Convection

Diffusion

Electro-osmotic

Electronic

Ionic

drag

conduction

H2 2H+ + 2e-

2H+ + 2e- + --O2 H2O

1 2

Anode Electrolyte Cathode

Flow structure Porous electrode

H2 O2

H2O N2

N2 H2

H2O

9 4

a b c dz

Convection

Diffusion

Electronic

Ionic conduction

conduction

H2 + O2- H2O + 2e-

--O2 + 2e- O2- 1 2

5 +

Figure 6.4. Flux details for (a) 1D PEMFC model and (b) 1D SOFC model. (a) In a PEMFC, water (H2O) and protons (H

+) transport through the electrolyte. (b) In a SOFC, oxygen ions (O2–) transport through the electrolyte.

where JAH2O , JMH2O

, and JCH2O represent the net flux into the anode catalyst layer, across the

electrolyte, and out of the cathode catalyst layer, respectively, and j∕2F represents the water generation rate at the cathode due to electrochemical reaction.

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210 FUEL CELL MODELING

For convenience (see Example 4.4), we introduce an unknown, α, which represents the ratio between the water flux across the membrane and the charge flux across the membrane:

𝛼 = JMH2O j∕2F

(6.9)

Using Equation 6.9, we can write Equation 6.8 in terms of j and 𝛼:

JCH2O = j

2F (1 + 𝛼) (6.10)

Now, by combining Equations 6.6, 6.8, 6.9, and 6.10, all the fluxes in the fuel cell may be connected together through j and 𝛼:

2F =

JM H+

2 = JAH2 = 2J

C O2

= JAH2O 𝛼

= JMH2O 𝛼

= JCH2O 1 + 𝛼

(6.11)

This is the master flux balance equation for our PEMFC model. The flux balance princi- ple captured by this equation relates to what are known as the conservation laws. To arrive at Equation 6.11, we have used the laws of mass conservation, species conservation, and charge conservation.

In an analogous manner, we can set up a flux balance equation for a SOFC as shown in Figure 6.4b:

2F = JM

O2− = JAH2 = 2J

C O2

= −JAH2O (6.12)

The overall flux balance for a SOFC is simpler than that for a PEMFC since only oxygen ions (O2–) are transported through the electrolyte. Since a SOFC generates water at the anode, the water flux at the anode is equal to the current density. Also, the water flux at the cathode will be zero.

When we set up the governing equations for the anode, membrane, and cathode of our fuel cell models, they will all be connected by Equation 6.11 (for a PEMFC) or 6.12 (for a SOFC). Current density j is usually the known quantity in the flux balance. Solving our model equations as a function of j will provide detailed information on the oxygen con- centration in the cathode catalyst layer and the water (or O2–) concentration profile in the electrolyte membrane. From this information, we can calculate the activation and ohmic overvoltages for the fuel cell, allowing us to determine the operating voltage.

6.2.2 Simplifying Assumptions

Possessing a flux balance for the species in the fuel cell, it is almost time to write equations describing how the species move and interact inside the fuel cell. These equations are called governing equations. If we wanted to include all the possible processes occurring inside our fuel cell, we would have to write governing equations for all the items listed in Table 6.2. Modeling all of these different phenomena for all these different species in all these different

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T A B L E 6. 2.

D es cr ip ti on

of F ul lP

E M F C (o r SO

F C ,i n it al ic s) M od

D om

ai ns

C on

ve ct

io n

D if

fu si

on C

on du

ct io

n E

le ct

ro ch

em ic

al R

ea ct

io n

A no

Fl ow

ch an

ne ls

(1 )H

2 ,

H 2 O

(g ),

H 2 O

(l) (2 )H

2 ,

H 2 O

(g ),

H 2 O

(l) (3

) e−

—

(1 )H

2 , H

2 O

(g )

(2 )H

2 , H

2 O

(g )

(3 )e

− —

E le

ct ro

de (1 )H

2 ,

H 2 O

(g ),

H 2 O

(l) (6 )H

2 ,

H 2 O

(g ),

H 2 O

(l) (3 )e

− —

(1 )H

2 , H

2 O

(g )

H 2 , H

2 O

(g )

(3 ,5 )e

− , O

2− (5 )H

2 + O 2 − →

H 2 O + 2 e−

C at

al ys

t (1 )H

2 ,

H 2 O

(g ),

H 2 O

(l) (5 )H

2 ,

H 2 O

(g ),

H 2 O

(l) (3

,5 )e

− ,

H +

(4 )H

2 →

2H + +

2e −

(1 )H

2 , H

2 O

(g )

(5 )H

2 , H

2 O

(g )

(3 ,5 )e

− , O

2− H 2 + O 2 − →

H 2 O + 2 e−

E le

ct ro

ly te

— (6 )H

2 O

(l) (6

) H

+ ,H

2 O

(l )a

—

— —

O 2−

—

C at

ho de

C at

al ys

t (1 )N

2 ,

O 2 ,

H 2 O

(g ),

H 2 O

(l) (5 )N

2 ,

O 2 ,

H 2 O

(g ),

H 2 O

(l) (3

,5 )e

− ,

H +

(6 )2

H + +

1 2 O

2 +

2e − →

H 2 O

(l)

(1 )N

2 , O

2 (5 )N

2 , O

2 (3

,5 )e

− , O

2− 1 2 O 2 + 2 e−

→ O 2 −

E le

ct ro

de (1 )N

2 ,

O 2 ,

H 2 O

(g ),

H 2 O

(l) (6 )N

2 ,

O 2 ,

H 2 O

(g ),

H 2 O

(l) (3 )e

− —

(1 )N

2 , O

2 N

2 , O

2 (3

,5 )e

− , O

2− (5 )

1 2 O 2 + 2 e−

→ O 2 −

Fl ow

ch an

ne ls

(1 )N

2 ,

O 2 ,

H 2 O

(g )

(2 )N

2 ,

O 2 ,

H 2 O

(g ),

H 2 O

(l) (3 )e

− —

(1 )N

2 , O

2 (2 )N

2 , O

2 (3 )e

− —

N ot e:

Si x

ke y

as su

m pt

io ns

,n um

be re

d 1–

6 in

pa re

nt he

se s,

le ad

to th

e si

m pl

ifi ed

m od

el sh

ow n

in Ta

bl e

6. 3.

a To

be pr

ec is

e, th

is w

at er

tr an

sp or

tp he

no m

en on

is du

e to

el ec

tr o-

os m

ot ic

dr ag

(s ee

C ha

pt er

4) .F

or co

nv en

ie nc

e, it

ha s

be en

ca te

go ri

ze d

as co

nd uc

tio n

du e

to its

cl os

e re

la tio

ns hi

p w

ith pr

ot on

co nd

uc tio

211

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212 FUEL CELL MODELING

domains would be daunting. Fortunately, by making the following simplifying assumptions, most of the items in Table 6.2 can be ignored in our current model:

1. Convective transport is ignored. Except for special cases, it is extremely difficult to obtain an analytical solution for convection. Convection is typically the dominant mass transport phenomena in fuel cells. However, since our model is a 1D model, we can safely ignore convection. As Figures 6.4 indicates, convective transport is mostly along the y-axis, but in our 1D model we consider transport only along the z-axis.

2. Diffusive transport in the flow channels is ignored. In the flow channels, diffusion is far less dominant than convection. Since we are already ignoring convection, dif- fusion in flow channels can be ignored, too. (We will not ignore diffusion in the electrodes, however.)

3. We assume that all the ohmic losses come from the electrolyte membrane. For most fuel cells, this is a reasonable assumption, because the ohmic losses from ionic con- duction in the electrolyte tend to dominate the other ohmic losses. (See Chapter 4.) This assumption means that we can ignore any conduction phenomena occurring in the electrode, catalyst layer, and flow channels.

4. We ignore the anode reaction kinetics. In H2–O2 fuel cells, the anode activation losses are usually much smaller than the cathode activation losses since oxygen reduction is the most sluggish process. (See Chapter 3.) We assume that the kinetic losses in our fuel cell model are determined by the oxygen concentration at the cathode catalyst layer (see the following text box).

5. We assume that the catalyst layers are extremely thin or act as “interfaces” (no thickness). With this assumption, we can ignore all convection, diffusion, and conduction processes in the catalyst layer, focusing instead only on the reaction kinetics. This is a reasonable assumption for most PEMFCs since the catalyst layer is extremely thin (∼10𝜇m) compared to the electrode (100–350 μm). In most SOFCs, however, the catalyst layer and electrode form a single unified body. Ionic conduction and electrochemical reactions may happen throughout the entire thickness of the electrode. Usually, however, reactions are localized to a very thin region of the catalyst/electrode bordering the electrolyte. In this case, our assumption is still justified.

6. The last and fairly bold assumption we make is that water exists only as water vapor. For SOFCs, this assumption is justified; only water vapor will exist at typical SOFC operating temperatures. In PEMFCs, however, we would expect both water vapor and liquid water to be present. Unfortunately, however, it is difficult to model the com- bined transport of a liquid and gas mixture. (Combined liquid–gas transport models are known as two-phase flow models. Developing a two-phase flow model for PEM- FCs is currently an area of active research.) By ignoring two-phase flow, we will introduce significant error into our PEMFC cathode water distribution results. This will affect the cathode overvoltage results, making our model less realistic. The depar- ture from reality is most pronounced at high current density, when significant amounts of liquid water are produced at the cathode. In real fuel cells, this leads to flooding, a phenomenon that our model cannot capture.

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A 1D FUEL CELL MODEL 213

TABLE 6.3. Description of Simplified PEMFC (or SOFC, in italics) Model

Domains Convection Diffusion Conduction Electrochemical Reaction

Anode

Flow channels — — — —

Electrode — H2, H2O(g) — —

— H2, H2O(g) — —

Catalyst — — — —

— — — H2 + O2− → H2O(g) + 2e−

Electrolyte — H2O(g) H +, H2O(g) —

O2−

Cathode

Catalyst — — — 2H+ + 1 2 O2 + 2e− → H2O(g)

— — — 1 2 O2 + 2e− → O2−

Electrode — N2, O2, H2O(g) — —

— N2, O2 — —

Flow channels — — — —

Notes: The items to be modeled in this table are described by governing equations, which are developed in the next section.

The simplifying assumptions listed above significantly reduce our modeling require- ments, as shown in Table 6.3.

SOFC STRUCTURE AFFECTS MODELING ASSUMPTIONS

In anode-supported SOFC structures, several of the modeling assumptions listed above prove problematic. Because the components in a SOFC are quite brittle, the anode electrode, the cathode electrode, or the electrolyte must be made thick enough to act as a support. Thus, three potential types of SOFC structures exist—anode-supported, cathode-supported, and electrolyte-supported SOFCs (see Chapter 9 for more details). When modeling anode-supported SOFC structures, the assumptions listed above cannot be used. For example, we may not ignore anodic reaction losses for anode-supported SOFCs. This is because hydrogen diffusion limitations in thick anode structures can lead to severe mass transport constraints and therefore high anodic reaction losses despite fast anode reaction kinetics. The assumptions described above in the text should be used only for cathode- and electrolyte-supported SOFCs.

6.2.3 Governing Equations

We must now assign reasonable governing equations for each domain in Table 6.3. Actu- ally, we have already learned all the required governing equations in previous chapters.

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214 FUEL CELL MODELING

By solving these governing equations, we can determine how the concentrations of H2, O2, H2O, and N2 vary across our fuel cell (in the z direction). From these concentration profiles, we can then calculate the mass transport overvoltage 𝜂conc, activation overvoltage 𝜂act, and ohmic overvoltage 𝜂ohmic at different current density levels j. With this information, we are then able to construct a j–V curve.

Electrode Layer. We start by writing the governing equations for the electrodes. In the electrodes, we must model diffusion processes for H2, O2, H2O, and N2. We start with a modified form of the basic diffusion model that was described by Equation 5.7:

Ji = −pDeffij RT

dxi dz

(6.13)

where xi stands for the mole fraction of species i and p is the total gas pressure (Pa) at the electrode, which satisfies pi = pxi. This equation is more convenient than Equation 5.7 because it is based on gas pressures instead of concentrations. It can be derived directly from Equation 5.7 by using the ideal gas law (pi = ciRT). Recall how the effective diffusivity Deffij is obtained using Equations 5.2–5.5 based on the measured/assumed porosity of the electrode structure.

Equation 6.13 is sufficient to describe diffusion processes involving two gas species. At PEMFC cathodes, however, three gas species are typically present (N2, O2, and H2O). In such cases, we need to apply a multicomponent diffusion model such as the Maxwell–Stefan equation. However, since there is no N2 diffusion flux in fuel cells (no generation or con- sumption of N2), we will simply ignore the nitrogen flux. This sacrifices model accuracy but allows us to use a simple binary diffusion model based on the oxygen and water fluxes only. Students interested in employing the more accurate multicomponent models are directed to the explanatory text box below.

DIFFUSION MODELS FOR FUEL CELLS

Binary Diffusion Model

In simple cases, the rate of diffusion is directly proportional to a gradient in concentration (as explained in Chapter 5):

Ji = −Dij dci dz

(6.14)

This equation is called Fick’s law of binary diffusion. It works well for binary systems where only two species (i and j) are involved in diffusion. A good example of a binary system is a stream of humidified hydrogen. In a mixture of hydrogen and water vapor, the only possible diffusion processes are hydrogen diffusion (species i) in water vapor (species j) or vice versa. The binary diffusivity Dij can be calculated using Equation 5.2. Fick’s law of binary diffusion also works when species j diffuses in species i; in this case

Jj = −Dij dcj dz

(6.15)

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A 1D FUEL CELL MODEL 215

From the definition of the diffusion flux, the relationship Ji + Jj = 0 always holds, which results in Dij = Dji. (See problem 6.5.)

Maxwell–Stefan Model

Multicomponent diffusion applies when three or more species are involved in a diffu- sion process. At low density, multicomponent gas diffusion can be approximated by the Maxwell–Stefan equation [38]:

dxi dz

= RT ∑ j≠i

xiJj − xjJi pDeffij

(6.16)

This equation allows us to calculate the z-profile of a species i by summing the effects due to the interactions with the j other species making up the mixture. In this equation, xi and xj stand for the mole fractions of species i and j, Ji and Jj stand for the molar fluxes of species i and j (mol∕m2 ⋅ s), R is the gas constant (J∕mol ⋅ K), T is the temperature (K), p is the total gas pressure (Pa), and Deffij is the effective binary diffusivity (m

2∕s). Even though we do not use the Maxwell–Stefan model in our text due to mathematical complication, you may find it useful in more sophisticated models [8].

Electrolyte. Having used the diffusion equations to describe gas transport in the elec- trodes, we now write the governing equations for species transport in the electrolyte. The governing equation we use depends on whether we are modeling a SOFC or a PEMFC.

For SOFCs, we only need to worry about the O2– flux across the electrolyte. From our flux balance (Equation 6.12) we can relate the O2– flux to the current density:

JM O2−

= j

2F (6.17)

Then, we can determine the ohmic voltage loss from Equation 4.11:

𝜂ohmic = j(ASRohmic) = j ( tM

𝜎

) (6.18)

where tM is the thickness of the electrolyte. To calculate the electrolyte conductivity σ, we use Equation 4.64:

𝜎 = ASOFCe

−ΔGact∕(RT)

T (6.19)

where ASOFC (K∕Ω ⋅ cm) and ΔGact (J∕mol) are usually obtained from experiment. For PEMFCs, we know the proton flux from Equation 6.11. In addition to the proton

flux, however, we also need to consider the water flux in the electrolyte. Water causes the electrolyte conductivity to vary spatially. Therefore, we need to be able to calculate the water profile in the electrolyte. In a Nafion membrane, two water fluxes exist: back diffusion

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216 FUEL CELL MODELING

and electro-osmotic drag. Revisiting Equation 4.44, we can account for both of these fluxes, resulting in the following combined water flux balance within the membrane:

JMH2O = 2n SAT drag

2F 𝜆

22 − 𝜌dry

Mm D𝜆

d𝜆 dz

(6.20)

Keep in mind that the water content λ in this equation is not constant, but a function of z [λ = λ(z)]. By obtaining the water profile λ(z), we can estimate the resistance of the electrolyte. A detailed explanation and an example of this process have been provided in Section 4.5.2.

Catalyst. The governing equations for the catalyst are quite straightforward. As discussed previously, we consider only the cathode reaction kinetics. Since the oxygen partial pressure at the cathode is the dominant factor in determining the cathodic overvoltage, we can use the simplified form of the Butler–Volmer equation from Section 5.2.4 (Equation 5.19):

𝜂cathode = RT 4𝛼F

ln jc0O2 j0cO2

(6.21)

Here, the 4 in the denominator represents the electron transfer number for an oxygen molecule. For an ideal gas (p = cRT), the above equation becomes

𝜂cathode = RT 4𝛼F

ln j

j0pCxO2 (6.22)

where pC is the total pressure at the cathode and xO2 is the oxygen mole fraction at the cathode catalyst layer. Note that we use atm as the unit of pressure p and the reference pressure p0, which is 1 atm, disappears.

6.2.4 Examples

Having developed simplified governing equations for our 1D fuel cell model in the previous sections, we are now ready to introduce a few examples, showing how we can obtain j–V curve predictions from our model for both a SOFC and a PEMFC.

SOFC Model Example. For the 1D SOFC example, we will use Figure 6.4b for our model. From Equation 6.13, we can describe H2 and H2O transport in the anode as

JAH2 = −pADeffH2,H2O

dxH2 dz

JAH2O = −pADeffH2,H2O

dxH2O dz

(6.23)

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A 1D FUEL CELL MODEL 217

Using Equation 6.12, we can relate JAH2 and J A H2O

to the fuel cell current density j. When we integrate Equation 6.23, however, we need to provide boundary conditions. Fortunately, we know (or can impose) the values of xH2 and xH2O at the fuel cell inlet (interface “a” in Figure 6.4b). These inlet values serve as our boundary conditions. Solving Equation 6.23 gives linear profiles for the hydrogen and water concentrations in the anode:

xH2 (z) = xH2 |a − z jRT2FpADeffH2,H2O xH2O(z) = xH2O|a + z jRT2FpADeffH2,H2O (6.24)

Solving for the hydrogen and water concentrations at the anode–membrane interface (interface “b” in Figure 6.4b) yields

xH2 |b = xH2 |a − tA jRT2FpADeffH2,H2O xH2O|b = xH2O|a + tA jRT2FpADeffH2,H2O (6.25)

where tA represents anode thickness. Following a similar procedure, we can also obtain the oxygen profile at the cathode and hence the oxygen concentration at the cathode catalyst layer:

xO2 |c = xO2 |d − tC jRT4FpCDeffO2,N2 (6.26) Note that we ignore the nitrogen profile since the nitrogen flux is zero (nitrogen is neither

produced nor consumed in the fuel cell). Having determined the oxygen concentration at the cathode catalyst layer, we can combine Equations 6.26 and 6.22 to calculate the cathode overpotential:

𝜂cathode = RT 4𝛼F

⎡⎢⎢⎢⎣ j

j0pC { xO2 |d − tCjRT∕(4FpCDeffO2,N2)}

⎤⎥⎥⎥⎦ (6.27) Because we account for the oxygen concentration in this equation, we are effectively accounting for both the activation losses and the concentration losses at the same time. All that remains, then, is to calculate the ohmic losses. From Equations 6.18 and 6.19, we can calculate the ohmic loss as

𝜂ohmic = j(ASRohmic) = j tM

𝜎 = j t

MT ASOFCe−ΔGact∕(RT)

(6.28)

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218 FUEL CELL MODELING

TABLE 6.4. Physical Properties of SOFC Used in Example

Physical Properties Values

Thermodynamic voltage, Ethermo(V) 1.0

Temperature, T(K) 1073

Hydrogen inlet mole fraction, xH2 |a 0.95 Oxygen inlet mole fraction, xO2 |d 0.21 Cathode pressure, pC(atm) 1

Anode pressure, pA(atm) 1

Effective hydrogen (or water) diffusivity, DeffH2 ,H2O(m 2/s) 1 × 10−4

Effective oxygen diffusivity, DeffO2 ,N2 (m 2/s) 2 × 10−5

Transfer coefficient, α 0.5

Exchange current density, j0(A∕cm2) 0.1

Electrolyte constant, ASOFC(K∕Ω ⋅ m) 9 × 107

Electrolyte activation energy, ΔGact(kJ∕mol) 100

Electrolyte thickness, tM(μm) 20

Anode thickness, tA(μm) 50

Cathode thickness tC(μm) 800

Gas constant, R (J∕mol ⋅ K) 8.314

Faraday constant, F (C∕mol) 96,485

Finally, we obtain the fuel cell voltage as

V = Ethermo − 𝜂ohmic − 𝜂cathode

= Ethermo − j tMT

ASOFCe−ΔGact∕(RT) − RT

4𝛼F ln

⎡⎢⎢⎢⎣ j

j0pC { xO2 |d − tC [jRT∕(4FpCDeffO2,N2)]}

⎤⎥⎥⎥⎦ (6.29)

where Ethermo is the thermodynamically predicted fuel cell voltage. We now apply Equation 6.29 to predict the performance of a realistic SOFC. For

example, consider the parameter values and conditions shown in Table 6.4. We compute the output voltage for this SOFC at a current density of 500 mA/cm2:

𝜂ohmic = 0.5A∕cm2 (

104cm2

) (0.00002m)(1073K)

(9 × 107 K ⋅Ω-1 ⋅ m-1)e−(100,000 J∕mol)∕(8.314 J∕mol⋅K×1073 K)

= (0.5A∕cm2)(0.176Ω cm2) = 0.088V (6.30)

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A 1D FUEL CELL MODEL 219

𝜂cathode = (8.314J∕mol ⋅ K) (1073K)

4 × 0.5 × 96485C∕mol ln

⎡⎢⎢⎢⎣ 0.5A∕cm2

0.1A∕cm2 ⋅ 1atm × 101300 Pa∕atm

× 1

0.210 − 0.0008m 5000A∕m2 × 8.314J∕ (mol ⋅ K) × 1073K

(4 × 96,485C∕mol) × 101,325Pa × 0.00002m2∕s

⎤⎥⎥⎥⎥⎦ = 0.158V (6.31)

V = 1.0V − 0.088V − 0.158V = 0.754V (6.32)

By iteratively following this procedure over a range of current densities, we can easily construct a complete j–V curve. Figure 6.5 presents the complete j–V curve for this example.

PEMFCModel Example. Now we will explore the PEMFC model shown in Figure 6.4a. Just as in a SOFC anode, we must account for hydrogen and water in the PEMFC anode. From Equation 6.13, we obtain the model equations:

JAH2 = −pADeffH2,H2O

dxH2 dz

JAH2O = −pADeffH2,H2O

dxH2O dz

(6.33)

Figure 6.5. The j–V curve of 1D SOFC model from simplified governing equations. The activation overvoltage is prominent at low current density while the ohmic overvoltage is dominant throughout the entire range of current density. The concentration overvoltage increases sharply at high current density.

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220 FUEL CELL MODELING

These equations look exactly like the SOFC anode Equations 6.23. One significant and important difference, however, is that JAH2O

is unknown in our PEMFC model since we do not know α in the flux balance equation 6.11. Using this flux balance information, where α is an unknown, the above equations have the following solutions:

xH2 (z) = xH2 |a − z jRT2FpADeffH2,H2O (6.34) xH2O(z) = xH2O|a − z 𝛼∗jRT2FpADeffH2,H2O (6.35)

Note that we add an asterisk to the unknown α to avoid confusion with the transfer coefficient (which is also represented by α). From the above equations, we can calculate the hydrogen and water concentrations at the anode–membrane interface (interface “b” in Figure 6.4a):

xH2 |b = xH2 |a − tA jRT2FpADeffH2,H2O (6.36) xH2O|b = xH2O|a − tA 𝛼∗jRT2FpADeffH2,H2O (6.37)

In a similar manner, we can obtain the oxygen and water concentrations at the cathode– membrane interface “c”:

xO2 |c = xO2 |d − tC jRT2FpCDeffO2,H2O (6.38) xH2O|c = xH2O|d + tC (1 + 𝛼∗)jRT2FpCDeffO2,H2O (6.39)

As before, we have ignored the nitrogen flux to simplify the model. Similarly to the anode solution, the cathode solution also contains the unknown α∗. Just as in the SOFC model, once we obtain the oxygen concentration at interface “c,” we can calculate the cathodic overpotential via Equation 6.27.

The biggest challenge of our PEMFC model is to find the ohmic overpotential. The critical issue is to obtain the water profile in the membrane, since the water profile lets us calculate the membrane resistance. We can obtain the water profile in the membrane along with the unknown 𝛼* by solving the membrane water flux equation 6.20. Equations 6.37 and 6.39 serve as our boundary conditions.

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A 1D FUEL CELL MODEL 221

The solution to Equation 6.20 has been previously worked out in Chapter 4 (see Equation 4.53):

𝜆(z) = 11𝛼 ∗

nSATdrag + C exp

( jMmn

SAT drag

22F𝜌dryD𝜆 z

) = 11𝛼

∗

2.5

+C exp

( j ( A∕cm2

) × 1.0kg∕mol × 2.5

22 × 96,485C∕mol × 0.00197kg∕cm3 × D𝜆(cm2∕s) z(cm)

)

= 4.4𝛼∗ + C exp

( 0.000598 ⋅ j

( A∕cm2

) ⋅ z(cm)

D𝜆(cm2∕s)

) (6.40)

Using this equation, we can obtain the water content λ at the anode–membrane interface “b” and the cathode–membrane interface “c” as

𝜆|b = 𝜆(0) = 4.4𝛼∗ + C (6.41) 𝜆|c = 𝜆(tM) = 4.4𝛼∗ + C exp(0.000598 ⋅ j (A∕cm2) ⋅ tM(cm)D𝜆(cm2∕s)

) (6.42)

where tM represents the membrane thickness. So far, we have two unknowns: C in the above equation and α∗ from Equations 6.37 and 6.39. To make further progress, we need to relate the water fluxes in Equations 6.37 and 6.39 to the water contents in Equations 6.41 and 6.42.

As explained in Section 4.5.2, the Nafion water content is a nonlinear function of the sur- rounding water vapor pressure. As it is quite complicated to solve these nonlinear equations, we introduce two more simplifying assumptions:

1. Water content in the Nafion membrane increases linearly with water activity. Thus, we use the following linearized form of Equation 4.34:

𝜆 = 14aW for 0 < aW ≤ 1 (6.43) 𝜆 = 10 + 4aW for 1 < aW ≤ 3 (6.44)

This piecewise equation linearly approximates the real water content versus water activity behavior shown in Figure 4.11.

2. Water diffusivity in Nafion is constant. This is a fairly reasonable assumption, since the water diffusivity does not change much over most water content ranges.

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222 FUEL CELL MODELING

TABLE 6.5. Physical Properties of PEMFC Used in Example

Physical Properties Values

Thermodynamic voltage, Ethermo (V) 1.0

Operating current density, j (A∕cm2) 0.5

Temperature, T(K) 343

Vapor saturation pressure, pSAT (atm) 0.307

Hydrogen mole fraction, xH2 0.9

Oxygen mole fraction, xO2 0.19

Cathode water mole fraction, xH2O 0.1

Cathode pressure, pC (atm) 3

Anode pressure, pA (atm) 3

Effective hydrogen (or water) diffusivity, DeffH2 ,H2O (cm 2/s) 0.149

Effective oxygen (or water) diffusivity, DeffO2 ,H2O (cm 2/s) 0.0295

Water diffusivity in Nafion, Dλ (cm2∕s) 3.81 × 10−6

Transfer coefficient, α 0.5

Exchange current density, j0 (A∕cm2) 0.0001

Electrolyte thickness, tM(μm) 125

Anode thickness, tA(μm) 350

Cathode thickness tC(μm) 350

Gas constant, R (J∕mol ⋅ K) 8.314

Faraday constant, F (C∕mol) 96,485

Since a𝑤|b = pAxH2O|b∕pSAT, combining Equations 6.43 and 6.37 gives 𝜆|b = 14a𝑤|b = 14 pApSAT

( xH2O|a − tA 𝛼∗jRT2FpADeffH2,H2O

) (6.45)

Similarly, combining Equations 6.39 and 6.44 for the cathode side yields

𝜆|c = 10 + 4a𝑤|c = 10 + 4 pCpSAT ( xH2O|d + tC (1 + 𝛼∗) jRT2FpCDeffO2,H2O

) (6.46)

In the above two equations, we have assumed that a𝑤 < 1 for “b” and a𝑤 > 1 for “c.” At “b,” water is consumed to provide water flux to Nafion, and at “c,” water is gener- ated. Since water is depleted at “b” and produced at “c,” the water activity assumptions are reasonable.

Using the system of equations that we have set up, we will now work a practical example. Consider the specific fuel cell properties listed in Table 6.5. Incorporating these properties

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A 1D FUEL CELL MODEL 223

into Equations 6.45 and 6.46 gives

𝜆|b = 14 3atm0.307atm × (

0.1 − 0.00035m 𝛼∗ × 0.5A∕0.0001m2 ⋅ 8.314J∕molK × 343K

(2 × 96,485C∕mol) (3 × 101,325Pa)(0.149 × 0.0001m2∕s)

) = 13.68 − 0.781𝛼∗ (6.47)

𝜆|c = 10 + 4 3atm0.307atm × (

0.1 + 0.00035m (1 + 𝛼∗) × 0.5A∕0.0001m2 ⋅ 8.314J∕mol ⋅ K × 343K

(2 × 96,485C∕mol)(3 × 101,325Pa)(0.0295 × 0.0001m2∕s)

) = 15.04 − 1.127𝛼∗ (6.48)

and Equations 6.41 and 6.42 become

𝜆|b = 𝜆(0) = 4.4𝛼∗ + C (6.49) 𝜆|c = 4.4𝛼∗ + C exp(0.000598 × 0.5A∕cm2 × 0.0125cm

3.81 × 10−6

) = 4.4𝛼∗ + 2.667C (6.50)

Now, we can equate Equation 6.47 with Equation 6.49 and Equation 6.48 with Equation 6.50 to find 𝛼 = 2.034 and C = 3.141.

From Equations 4.38 and 6.40, we can then determine the conductivity profile of the membrane:

𝜎(z) = {

0.005193

[ 4.4𝛼 + Cexp

( 0.000598 × 0.5

3.81 × 10−6 z

)] − 0.00326

} × exp

[ 1268

( 1 303

− 1 343

)] = 0.0704 + 0.0266 exp(78.48z) (6.51)

Finally, we can determine the resistance of the membrane using Equation 4.40:

ASRm = ∫ tm

dz 𝜎(z)

= ∫ 0.0125

dz 0.0704 + 0.0266 exp(78.48z)

= 0.109Ω ⋅ cm2 (6.52)

Thus, the ohmic overvoltage due to the membrane resistance in this PEMFC is approximately

𝜂ohmic = j × ASRm = 0.5 A∕cm2 × 0.109Ω ⋅ cm2 = 0.0505V (6.53)

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224 FUEL CELL MODELING

Figure 6.6. The j–V curve of 1D PEMFC model from simplified governing equations. Please notice the sharp drop of the voltage near zero current density due to large activation overvoltage (typi- cal behavior for PEMFC). The gradual change of slope of the j–V curve after 1 A/cm2 represents the increase of the ohmic resistance in the proton exchange membrane due to the water depletion. Remember (from Chapter 4) that the electro-osmotic drag of water increases with current density, which reduces the water content in the membrane. In the previous 1D SOFC example, the concen- tration overvoltage was clearly observed at high current density due to the thick cathode (800 μm in Table 6.4). In this example, the concentration overvoltage is not observable since the thickness of the cathode is small (350 μm in Table 6.5).

and we can compute the cathodic overvoltage using Equation 6.27 as

𝜂cathode = (8.314J∕mol ⋅ K)(343K) 4 × 0.5 × 96485C∕mol

⎡⎢⎢⎢⎣ 0.5A∕cm2

0.0001A∕cm2 × 3atm × 101300Pa∕atm

× 1( 0.19 − 0.00035m

5000A∕m2 × 8.314J∕mol ⋅ K × 343K (4 × 96,485C∕mol) (3 × 101,325Pa)(0.0295 × 0.0001m2∕s)

) ⎤⎥⎥⎥⎥⎦

= 0.135 V (6.54)

Finally, we find the fuel cell voltage as

V = 1.0V − 0.0505V − 0.135V = 0.810V (6.55)

Figure 6.6 shows the complete j–V curve of this 1D PEMFC model.

Gas Depletion Effects: Modifying the 1D SOFC Model. So far in our example models, we have assumed an infinite supply of hydrogen and oxygen at the fuel cell inlets. Physically, this is represented by assigning constant mole fractions for the species at bound- aries “a” and “d” in Figure 6.4b. Now, however, we will consider a more realistic case where oxygen can be depleted at these boundaries depending on the relative rates of oxy- gen supply and consumption. For simplicity, we illustrate this modification with our SOFC model, although a similar modification could also be applied to the PEMFC model. Also,

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A 1D FUEL CELL MODEL 225

we consider only oxygen depletion effects. Hydrogen depletion is not considered since our model ignores the anodic overvoltage losses in the first place. At the cathode outlet (boundary “d”) we may derive the expression

xO2 |d = JCO2,outletJCO2,outlet + JCN2,outlet (6.56) where the denominator represents the total species flux at the fuel cell cathode outlet. This equation simply tells us that the oxygen mole fraction at the boundary is given by the ratio of the outlet oxygen flux to the total outlet gas flux. As oxygen is consumed in the fuel cell, the mole fraction of oxygen will decrease at “d.” Although we fix the inlet flux values in our model, the outlet flux will change according to usage of oxygen (which in turn corresponds to the operating current density).

We will now replace JCO2,outlet and JCN2,outlet

with known values. From the SOFC flux balance Equation 6.12, we know that

JCO2,outlet = J C O2,inlet

− JCO2 = J C O2,inlet

− j

4F (6.57)

Commonly, in fuel cell operation, the oxygen inlet flux JCO2,inlet (and the hydrogen inlet

flux) are regulated according to the stoichiometric number. The concept of a stoichiomet- ric number is briefly introduced in the text box that follows. From the definition of the stoichiometric number,

JCO2,inlet = 𝜆O2J C O2

(6.58)

Plugging the above equation into Equation 6.57 allows us to solve for JCO2,outlet in terms

the stoichiometric number:

JCO2,outlet = (𝜆O2 − 1)J C O2

= (𝜆O2 − 1) j

4F (6.59)

Finding JCN2,outlet is easier. Since there is no nitrogen consumption,

JCN2,outlet = J C N2,inlet

= 𝜔JCO2,inlet = 𝜔𝜆O2J C O2

= 𝜔𝜆O2 j

4F (6.60)

where 𝜔 represents the molar ratio of nitrogen to oxygen in air (typically 𝜔 = 0.79∕0.21 = 3.76).

Now, we plug Equations 6.59 and 6.60 into Equation 6.56 and solve for xO2 |d: xO2 |d = (𝜆O2 − 1)[j∕(4F)](𝜆O2 − 1)[j∕(4F)] + 𝜔𝜆O2 [j∕(4F)]

= 𝜆O2 − 1

(1 + 𝜔)𝜆O2 − 1 (6.61)

When 𝜆O2 = 1, Equation 6.61 tells us that xO2 |d = 0 , since all the oxygen is consumed in the fuel cell.

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226 FUEL CELL MODELING

STOICHIOMETRIC NUMBER

As described in Section 2.5.2, it is common to operate a fuel cell at a certain stoichio- metric number to maximize fuel cell efficiency. The stoichiometric number 𝜆 reflects the rate at which a reactant is provided to a fuel cell relative to the rate at which it is consumed. For example, 𝜆 = 2 means that twice as much reactant as needed is being provided to a fuel cell. Choosing an optimal 𝜆 is a delicate task. A large 𝜆 is waste- ful, resulting in parasitic power consumption due to higher pumping losses and/or lost fuel. As 𝜆 decreases toward 1, however, reactant depletion effects become more severe. Obviously, two stoichiometric numbers must be specified in fuel cells—one for hydro- gen and one for oxygen. For our SOFC model, we can define the hydrogen and oxygen stoichiometric number based on the ratios of the inlet to consumption fluxes:

𝜆H2 = JH2,inlet

JAH2

𝜆O2 = JO2,inlet

JCO2

(6.62)

We can incorporate gas depletion effects into our SOFC model by simply plugging Equation 6.61 into 6.29, giving us the following final model equation:

V = Ethermo − 𝜂ohmic − 𝜂cathode

= Ethermo − j tMT

ASOFCe−ΔGact∕(RT)

− RT 4𝛼F

⎡⎢⎢⎢⎢⎢⎣ j

j0pC

( 𝜆O2 − 1

(1 + 𝜔) 𝜆O2 − 1 − tC

jRT

4FpCDeffO2,N2

) ⎤⎥⎥⎥⎥⎥⎦

(6.63)

Using the same table of fuel cell parameters as in the previous SOFC example with 𝜆O2 = 1.5 and j = 500 mA∕cm2, this modified model gives

ncathode = (8.314J∕mol ⋅ K)(1073K)

4 ⋅ 0.5 ⋅ 96485C∕mol ln

⎡⎢⎢⎢⎣ 0.5A∕cm2

0.1A∕cm2 ⋅ 1atm × 101300Pa∕atm

× 1( 1.5 − 1

(1 + 3.76) 1.5 − 1 − 0.0008m

5000A∕m2 × 8.314J∕mol ⋅ K × 1073K (4 × 96,485C∕mol)(101,325Pa)(0.00002m2∕s)

) ⎤⎥⎥⎥⎥⎦

= 0.228V (6.64)

V = 1.0V − 0.088V − 0.228V = 0.684V (6.65)

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FUEL CELL MODELS BASED ON COMPUTATIONAL FLUID DYNAMICS (OPTIONAL) 227

Figure 6.7. The j–V curve of 1D SOFC model considering stoichiometry number. Two curves repre- sent cases where the oxygen stoichiometries are 1.5 (example case in the text) and 5, respectively. The behavior of the concentration overvoltage is quite different from Figure 6.5 where no stoichiometry effect was considered. The example in Figure 6.5 considered only the diffusion limit at the cathode. In other words, the oxygen stoichiometry number was assumed to be infinitely large. In this example, the concentration overvoltage is much larger and limiting current density is greatly reduced.

Note how we obtain a much higher cathodic overvoltage compared to the first example. This is because the low 𝜆O2 value (𝜆O2 = 1.5) causes significant gas depletion effects (xO2 |d = 0.21 in the first example versus xO2 |d = 0.0814 in the current example).

Figure 6.7 shows the complete j–V curve of this modified SOFC model.

6.2.5 Additional Considerations

As additional levels of detail are introduced, fuel cell modeling quickly becomes more diffi- cult. For the case of the 1D model, recall how we made a series of simplifying assumptions in Section 6.2.2 to keep the system manageable. By relaxing some of these assumptions, a more accurate fuel cell model can be generated. However, this accuracy comes at the cost of greatly increased complexity.

Ambitious fuel cell models may incorporate thermal or mechanical effects. Thermal fuel cell modeling is extremely difficult. Numerous heat flows must be considered, including convective heat transfer via the fuel and air streams, conductive heat transfer through the fuel cell structures, heat absorption/release from phase changes of water, entropy losses from the electrochemical reaction, and heating due to the various overvoltages. Mechanical modeling is likewise challenging.

In most cases, these issues are implemented using sophisticated computer software pro- grams based on numerical methods. In the next section, we introduce a fuel model based on CFD, which includes most of the issues we ignored earlier in this chapter.

6.3 FUEL CELL MODELS BASED ON COMPUTATIONAL FLUID DYNAMICS (OPTIONAL)

Computational fluid dynamics modeling is a broad field of research. The intricacies of the field are beyond the scope of this chapter. Our purpose here is to only briefly introduce

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228 FUEL CELL MODELING

Air inlet

Hydrogen inlet

Air outlet

Figure 6.8. Isometric view of serpentine flow channel fuel cell model (500 μm channel feature size). Since no repetitive unit exists, the entire physical domain is modeled.

the subject. In this section, we will use CFD to simulate a PEMFC with serpentine flow channels. Rather than discuss the detailed governing equations and theory behind CFD, we instead present this serpentine flow channel example to illustrate the utility, advantages, and limitations of the CFD technique. For those students interested in the details of CFD modeling, further discussion may be found in Chapter 13.

Figure 6.8 shows a CFD model of our example serpentine channel fuel cell. The complex flow geometry embodied by this fuel cell would be difficult, if not impossible, to model analytically. Fortunately, it is quite amenable to computer-based numerical modeling. Referring to Figure 6.8, note that this fuel cell employs a single serpentine channel pattern for both the anode and cathode flow structures. The cathode structure (air side) is located on top and the anode structure (hydrogen side) is on the bottom. Inlet and outlet gas locations are marked on the figure. Table 6.6 summarizes the major physical properties used in this fuel cell model.

Figure 6.9 shows the j–V curve obtained from the CFD model. This j–V curve does not look much different from the curves obtained by simple analytical fuel cell models. In addi- tion to this j–V curve, however, our CFD model permits us to investigate and visualize the effects of geometry. This is where the true power of CFD becomes apparent. For example, we can use our CFD model to examine the oxygen distribution across the serpentine channel pattern as shown in Figures 6.10 and 6.11. Figure 6.10 shows a cross-sectional cut across the center of the serpentine pattern. The cathode side is on the top. As the air is introduced from the inlet on the left and travels to the outlet on the right, note how the oxygen concentration gradually drops. As a result, fuel cell performance is inhomogeneous. Less current is pro- duced near the outlet as the oxygen stream becomes depleted. Figure 6.11 illustrates how the channel rib structures also cause oxygen depletion. The channel ribs block the diffusion flux, leading to local “dead zones.” Our CFD model provides performance enhancement hints. For example, a multichannel design and/or narrower ribs might alleviate the oxygen depletion problems.

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FUEL CELL MODELS BASED ON COMPUTATIONAL FLUID DYNAMICS (OPTIONAL) 229

TABLE 6.6. Physical Properties Used in CFD Fuel Cell Model

Properties Values

Fuel cell area 14 × 14 mm

Electrode thickness, tg 0.25 mm

Catalyst thickness, tc 0.05 mm

Membrane thickness, tm 0.125 mm

Flow channel width, 𝑤f 0.5 mm

Flow channel height, tf 0.5 mm

Rib width, 𝑤r 0.5 mm

Relative humidity of inlet gases 100%

Temperature, T 50∘C

Hydrogen inlet flow rate 1.8 A/cm2 equivalent

Air inlet flow rate 1.9 A/cm2 equivalent

Outlet pressure 1 atm

Note: The gas flow rates are expressed in terms of equivalent current density.

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.2

1.1

1.2

200 400 600 8000 100 300 500 700 900

V ol

ta ge

( V

)

Current density(mA/cm2)

Figure 6.9. Cell j–V curves for serpentine flow channel model. Activation, ohmic, and concentration losses are clearly observed.

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230 FUEL CELL MODELING

O2 0214B

0.2

0.1

0 0

Figure 6.10. Oxygen concentration in cathode at 0.8 V overvoltage. This cross-sectional cut across the center of the serpentine pattern illustrates how the oxygen concentration in the flow channel slowly decreases from inlet to outlet. (see color insert)

O2 0214B

0.2

Air Outlet

Air Inlet

0.1

0 0

Figure 6.11. Oxygen concentration in cathode at 0.8 V overvoltage. The plan view shows the oxygen concentration profile across the cathode surface. Low oxygen concentration is observed under the channel ribs due to the blockage of oxygen flux. (see color insert)

In a 1D or 2D fuel cell model, these geometric effects are difficult to observe. The visu- alization tools provided by CFD modeling provide a highly intuitive way to understand and explore geometric effects in fuel cells. CFD models are especially useful when experimen- tal investigation is difficult or impractical. When used in combination with experimentation, CFD models can add significant speed and power to the fuel cell design process. To learn more, see Chapter 13 for further detailed information on CFD-based fuel cell modeling.

6.4 CHAPTER SUMMARY

Fuel cell models are used to understand and predict fuel cell behavior. Simple models can be used to understand basic trends (e.g., what happens when temperature increases or pressure decreases). Sophisticated models can be used as design guides (e.g., to answer

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CHAPTER EXERCISES 231

questions such as what happens when the diffusion layer thickness is reduced from 500 to 100 μm). All fuel cell models incorporate assumptions. When interpreting model results, major assumptions and limitations must be taken into account.

• There are three major fuel cell losses: activation losses (𝜂act), ohmic losses (𝜂ohmic), and concentration losses (𝜂conc).

• A simple fuel cell model can be developed by starting with the thermodynamic fuel cell voltage and then deducting the three major loss terms: V = Ethermo − 𝜂act − 𝜂ohmic − 𝜂conc.

• To accurately reflect the behavior of most fuel cells, an additional loss term, known as the leakage loss jleak, must be introduced.

• The leakage loss jleak is associated with the parasitic loss due to current leakage, gas crossover, unwanted side reaction, and so on. The net effect of this parasitic current loss is to offset a fuel cell’s operating current to the left by an amount given by jleak. This has the effect of reducing a fuel cell’s open-circuit voltage below the thermody- namically predicted value.

• The basic fuel cell model requires four parameters. Two parameters (𝛼 and j0) describe the kinetic losses, one parameter (ASRohmic) describes the ohmic losses, and one parameter (jL) describes the concentration losses.

• A wide variety of different fuel cell behaviors can be explored by varying only a few basic parameters.

• All models include assumptions. The number and type of assumptions determine the complexity and accuracy of the model.

• More sophisticated fuel cell models use conservation laws and governing equations to relate fuel cell behavior to basic physical principles.

• The governing equations of a fuel cell model are related to one another by flux balance and conservation laws. Proper boundary conditions are required to generate solutions.

• In a SOFC, proper model assumptions can be significantly impacted by the electrode and electrolyte geometry.

• In PEMFCs, proper modeling of water distribution is critical. • The CFD fuel cell models use numerical methods to simulate fuel cell behavior. Com-

putational fluid dynamics modeling permits detailed investigation and visualization of electrochemical and transport phenomena. It is especially useful when experimental investigation is difficult or impractical and illustrates tremendous promise and power as a fuel cell design tool.

CHAPTER EXERCISES

Review Questions

6.1 Match the following five scenarios to the five corresponding hypothetical j–V curves in Figure 6.12:

(a) A SOFC limited by an extremely high electrolyte resistance

(b) A PEMFC suffering from a large leakage current loss

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232 FUEL CELL MODELING

C el

l v ol

ta ge

( V

)

C el

l v ol

ta ge

( V

)

C el

l v ol

ta ge

( V

)

C el

l v ol

ta ge

( V

)

C el

l v ol

ta ge

( V

)

Current density (A/cm2) Current density (A/cm2)

Current density (A/cm2)Current density (A/cm2) Current density (A/cm2)

(a) (b) (c )

(d ) (e)

Figure 6.12. Curves for problem 6.1.

(d) A PEMFC with an extremely low ohmic resistance

(e) A SOFC suffering from reactant starvation

6.2 From an efficiency standpoint, which fuel cell in Figure 6.3 would be more desirable, the PEMFC or the SOFC?

Calculations

6.3 This problem estimates the effect of jleak on the open-circuit voltage of a fuel cell. Assume a simple fuel cell model that depends only on the activation losses at the cathode (i.e., do not include the effects of ohmic or concentration losses). For a typ- ical pure H2–O2 PEMFC cathode, assume n = 2, j0 ≈ 10−3 A∕cm2, and 𝛼 ≈ 0.3. Using these values, determine the approximate drop in open-circuit voltage caused by a leakage current jleak = 10 mA∕cm2 (assume STP). Hint: To solve this question properly, carefully consider which approximation of the Butler–Volmer equation you should use. Cross-check your final answer with the approximation assumptions.

6.4 This problem has several parts. By following each part, you will develop a simple fuel cell model similar to the one discussed in the text.

(a) Calculate Ethermo for a PEMFC running on atmospheric pressure H2 and atmo- spheric air at 330 K.

(b) Calculate ac and bc (the constants for the natural log form of the Tafel equation for the cathode of this PEMFC) if j0 = 10−3 A∕cm2, n = 2, and 𝛼 = 0.5.

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CHAPTER EXERCISES 233

(c) Calculate ASRohmic if the membrane has a conductivity of 0.1 Ω−1 ⋅ cm−1 and a thickness of 100 μm. Assume that there are no other contributions to cell resistance.

(d) Calculate the effective binary diffusion coefficient for O2 in air in the cathode electrode. Neglect the effect of water vapor (consider only O2 and N2) and assume the cathode electrode has a porosity of 20%.

(e) Calculate the limiting current density in the cathode given δ = 500 μm. (f) To complete your model, assume c (the geometric constant in the concentration

loss equation) has a value of 0.10 V. Assume jleak = 5 mA∕cm2. Neglect all anode effects. Using some type of software package, plot the j–V and power density curves for your model.

(g) What is the maximum power density for your simulated fuel cell? At what current density does the power density maximum occur?

(h) Assuming 90% fuel utilization, what is the total efficiency of your simulated fuel cell at the maximum power density point?

6.5 Show that Dji = Dij using the fact that Ji + Jj = 0 and xi + xj = 1. Hint: Use the equation Ji = 𝜌Dij(dxi∕dz).

6.6 Show that the Maxwell–Stefan equation 6.16 satisfies x1 + x2 + ⋅ ⋅ ⋅ + xN = 1.

6.7 (a) Plot the complete j–V curve for the 1D SOFC model example (without the gas depletion modification) in the text (Section 6.2.4).

(b) Plot the ohmic overvoltage and cathodic overvoltage versus current density. Find the limiting current density from the j–V curve.

6.8 (a) Plot the complete j–V curve for the 1D SOFC model example in the text assum- ing that all the properties are unchanged as shown in Table 6.4 except that the operating temperature is now 873 K.

(b) Plot the ohmic overvoltage and the cathodic overvoltage versus current density. Compare your results with problem 6.7. Which overvoltage (ohmic or cathodic) shows a larger change?

6.9 (a) Using the 1D SOFC model, plot the j–V curve of an electrolyte-supported SOFC that has a 200-μm-thick electrolyte, a 50-μm-thick cathode, and a 50-μm-thick anode. Ignore the anodic overpotential and use the properties provided in Table 6.4.

(b) Repeat the process in (a) assuming that the fuel cell operating temperature is 873 K. Explain why an electrolyte-supported SOFC may not be suitable for lower temperature operation.

6.10 In the text, our 1D SOFC model did not incorporate anodic overvoltage. In this prob- lem, we consider it.

(a) Using a linear approximation of Butler–Volmer equations for the anode as

j = j0 p

2𝛼F RT

𝜂act (6.66)

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234 FUEL CELL MODELING

show that the anodic overvoltage can be modeled as

𝜂anode = RT 2𝛼F

j0pA

( xH2

||a − tA jRT2FpADeffH2,H2O ) (6.67)

(b) Based on the information from Table 6.4, plot the anodic and cathodic overvolt- ages for this model SOFC. (Assume that the Table 6.4 j0 and α values apply to the SOFC cathode. For the SOFC anode, use j0,A = 10 A∕cm2 and αA = 0.5.)

6.11 (a) Plot the j–V curve for an anode-supported SOFC that has a 1000-μm-thick anode and a 50-μm-thick cathode. Consider both the anodic and the cathodic overvolt- ages by using Equations 6.27 and 6.67. Use the properties provided in Table 6.4. (Assume that the Table 6.4 j0 and α values apply to the SOFC cathode. For the SOFC anode, use j0,A = 10 A∕cm2 and αA = 0.5.)

(b) Plot the anodic overvoltage and cathodic overvoltage for this fuel cell.

(c) Find the limiting current for each overvoltage curve. Which electrode shows more loss? Explain the consequence of ignoring the anodic overpotential in an anode-supported SOFC.

6.12 (a) Plot the complete j–V curve for the 1D PEMFC model example from the text (Section 6.2.4.2).

(b) Plot the ohmic overvoltage versus current density. Is the curve linear? If not, explain why.

6.13 (a) Plot the complete j–V curve for the final 1D SOFC example in the text, where oxygen gas depletion effects are considered. Assume the oxygen stoichiometric number is 1.2.

(b) Assume that this fuel cell employs an air pump that consumes 10% of the fuel cell power to deliver an oxygen stoichiometric number of 1.2. When the stoichio- metric number is set to 2.0, the pump consumes 20% of fuel cell power. Ignore all other sources of parasitic load. Which operation mode provides more power? Discuss your answer by carefully calculating the power density curves for each of the two operating modes.

6.14 Assume that a solid-oxide fuel cell’s j–V curve may be approximated by a “sideways parabola” with an equation given by V = 0.5(4 – j)1∕2 (valid only for j > 0, V > 0), where j is the current density (A∕cm2) and V is the operating voltage (V). (a) What is the open-circuit voltage (OCV) for this fuel cell?

(b) What is the limiting current density (jL) for this fuel cell?

(d) What is the maximum power that this fuel cell can produce, and at what current density does the maximum power point occur?

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CHAPTER EXERCISES 235

(e) Draw both the j–V curve and the j–P curves for this fuel cell. Be careful to label all axes, include units, and designate important points. In particular, indicate VOCV, jL, Pmax and the current density associated with Pmax on your curves.

6.15 Flooding can be a serious issue in low-temperature PEMFCs. Consider a H2∕air PEMFC at room temperature and atmospheric pressure:

(a) Calculate DeffO2,air at the cathode of this fuel cell given DO2,air = 0.2 cm

2∕s, porosity 𝜀 = 0.4, and tortuosity τ = 2.5.

(b) Calculate jL at the cathode of this fuel cell given δ = 200μm. Remember that the fuel cell cathode is supplied with air at STP.

(c) Liquid water flooding will affect mass transport by reducing the porosity of the electrode and increasing the tortuosity. Assuming that cathode flooding reduces 𝜀 to 0.26 and increases τ to 3, calculate DeffO2,air for this “flooded” fuel cell cathode.

(d) Calculate jL for this “flooded” fuel cell cathode.

(e) At j = 0.50 A∕cm2, calculate 𝜂conc for the “unflooded” fuel cell cathode and 𝜂conc for the “flooded” fuel cell cathode. (Assume c = 0.10 V.)

(f) How many times larger is 𝜂conc for the “flooded” fuel cell cathode versus the “unflooded” fuel cell cathode?

6.16 Consider a pure H2–O2 fuel cell at T = 80∘C and Pcathode = Panode = 1 atm: (a) Calculate the ideal thermodynamic voltage for this fuel cell given E0 = 1.23 V

and ΔSrxn = –163 J∕K ⋅ molH2 (remember E0 is given for STP conditions; assume liquid water product).

(b) At j = 1 A∕cm2, calculate 𝜂act for the cathode given α = 0.3, n = 4, and j0 = 10–3 A∕cm2. Check any assumptions/simplifications made.

(c) Calculate jL at the cathode given Deff = 10–2 cm2∕s and δ = 150μm. (d) At j = 1 A∕cm2, calculate 𝜂conc at the cathode. (Assume c = 0.10 V.) (e) We now pressurize the fuel cell cathode to 10 atm (but the anode pressure remains

1 atm). Calculate the new thermodynamic voltage for this situation.

(f) At j = 1 A∕cm2, calculate 𝜂act for the pressurized cathode given α = 0.3, n = 4, and j0 = 10–3 A∕cm2. Keep in mind that j0 is given for 1 atm pressure conditions and thus 𝜂act will need to be corrected for the new cathode pressure. Check any assumptions/simplifications made.

(g) Calculate jL for the pressurized fuel cell cathode, again assuming Deff = 10–2 cm2∕s and δ = 150μm.

(h) At j = 1 A/cm2, calculate 𝜂conc at the pressurized fuel cell cathode (again, assume c = 0.10 V). How much total voltage boost is gained when operating at j = 1A∕cm2 by pressurizing the fuel cell cathode to 10 atm?

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CHAPTER 7

FUEL CELL CHARACTERIZATION

Characterization techniques permit the quantitative comparison of fuel cell systems, distinguishing good fuel cell designs from poor ones. The most effective characterization techniques also indicate why a fuel cell performs well or poorly. Answering these “why” questions requires sophisticated testing techniques that can pinpoint performance bottlenecks. In other words, the best characterization techniques discriminate between the various sources of loss within a fuel cell: fuel crossover, activation, ohmic, and concentration losses.

As mentioned in previous chapters, in situ testing is critically necessary. Usually, the performance of a fuel cell system cannot be determined simply by summing the perfor- mance of its individual components. Besides the losses due to the components themselves, the interfaces between components often contribute significantly to the total losses in a fuel cell system. Therefore, it is important to characterize all aspects of a fuel cell, while it is assembled and running under realistic operating conditions.

In this chapter, the most popular and effective fuel cell characterization techniques are introduced and discussed. We focus on in situ electrical characterization techniques because these techniques provide a wealth of information about operational fuel cell behavior. In spite of our emphasis on in situ testing, there are many useful ex situ characterization techniques that can supplement or accentuate the information provided by in situ testing. Therefore, some of these techniques are also discussed.

237

238 FUEL CELL CHARACTERIZATION

7.1 WHAT DO WE WANT TO CHARACTERIZE?

We start this chapter with a list of the various fuel cell properties we might want to characterize:

• Overall performance ( j–V curve, power density) • Kinetic properties (𝜂act, j0, α, electrochemically active surface area) • Ohmic properties (Rohmic, electrolyte conductivity, contact resistances, electrode

resistances, interconnect resistances)

• Mass transport properties ( jL, Deff, pressure losses, reactant/product homogeneity) • Parasitic losses ( jleak, side reactions, fuel crossover) • Electrode structure (porosity, tortuosity, conductivity) • Catalyst structure (thickness, porosity, catalyst loading, particle size, electro-

chemically active surface area, catalyst utilization, triple-phase boundaries, ionic conductivity, electrical conductivity)

• Flow structure (pressure drop, gas distribution, conductivity) • Heat generation/heat balance • Lifetime issues (lifetime testing, degradation, cycling, startup/shutdown, failure,

corrosion, fatigue)

This list is certainly not comprehensive. Nevertheless, it gives a sense of the literally dozens, if not hundreds, of properties, effects, and issues that contribute to the overall per- formance and behavior of a fuel cell. Some of these play just a minor role, while others can have a huge effect. How do we know on which properties to focus? Which ones are most important to characterize? Essentially, the answers to these questions depend on your interests, your goals, and your desired level of detail.

In this chapter, we will focus our efforts on just a few of the most widely used charac- terization techniques. We organize our goals with a reminder of the two main reasons to characterize fuel cells:

1. To separate good fuel cells from bad fuel cells

2. To understand why a given fuel cell performs the way it does

Separating the good from the bad is fairly straightforward. This separation is usually obtained by measuring j–V performance; the fuel cell that delivers the highest voltage at the current density of interest wins. Of course, fuel cell j–V performance can change dramat- ically depending on factors like the operating conditions and testing procedures. To ensure that j–V performance comparisons are fair, identical operating conditions, testing proce- dures and device histories must be applied. In addition, j–V performance is the ultimate “acid test” for new fuel cell innovations. For example, say you develop a marvelous new ultrahigh-conductivity electrolyte or an incredible new fuel cell catalyst. This is great—but until you put your material into a working fuel cell and show that it delivers high perfor- mance, the scientific community will reserve their applause.

OVERVIEW OF CHARACTERIZATION TECHNIQUES 239

It is considerably more difficult to understand why a given fuel cell performs the way it does. Generally, the best way to tackle this problem is to think about a fuel cell’s perfor- mance in terms of the various major loss categories: activation loss, ohmic loss, concen- tration loss, and leakage loss. If we can somehow determine the relative sizes of each of these losses, then we are closer to understanding our fuel cell’s problems. For example, if we find that concentration losses are killing performance, then a redesigned flow structure might solve the problem. In another instance, testing may reveal that our fuel cell has an abnormally large ohmic resistance. In this case, we probably want to check the electrolyte, the electrical contacts, the conductive coatings, or the electrical interconnects.

As these examples illustrate, diagnostic fuel cell testing needs to be able to separate the various fuel cell losses, 𝜂act, 𝜂ohmic, and 𝜂conc. In the ideal case, characterization techniques should even determine the underlying fundamental properties of the fuel cell, such as j0, α, σelectrolyte, and Deff.

In the next several sections, we work toward this characterization goal. Starting with basic fuel cell tests that give overall quantitative information about fuel cell performance, we then move to more sophisticated characterization techniques that distinguish between various fuel cell losses. With refinement and care, some of these tests can even be used to determine such fundamental properties as j0 or D

eff.

7.2 OVERVIEW OF CHARACTERIZATION TECHNIQUES

We divide fuel cell characterization techniques into two types:

1. Electrochemical Characterization Techniques (In Situ). These techniques use the electrochemical variables of voltage, current, and time to characterize the performance of fuel cell devices under operating conditions.

2. Ex Situ Characterization Techniques. These techniques characterize the detailed structure or properties of the individual components composing the fuel cell, but generally only components removed from the fuel cell environment in an unassembled, nonfunctional form.

Within the area of in situ electrochemical characterization, we discuss four major methods:

1. Current–Voltage (j–V) Measurement. The most ubiquitous fuel cell characterization technique, a j–V measurement provides an overall quantitative evaluation of fuel cell performance and fuel cell power density.

2. Current Interrupt Measurement. This method separates the contributions to fuel cell performance into ohmic and nonohmic processes. Versatile, straightforward, and fast, current interrupt can be used even for high-power fuel cell systems and is easily implemented in parallel with j–V curve measurements.

3. Electrochemical Impedance Spectroscopy (EIS). This more sophisticated technique can distinguish between ohmic, activation, and concentration losses. However, the

240 FUEL CELL CHARACTERIZATION

results may be difficult to interpret. In addition, EIS is relatively time consuming, and it is difficult to implement for high-power fuel cell systems.

4. Cyclic Voltammetry (CV). This is another sophisticated technique that provides insight into fuel cell reaction kinetics. Like EIS, CV can be time consuming and results may be difficult to interpret. It may require specialized modification of the fuel cell under test and/or use of additional test gases such as argon or nitrogen.

In the area of ex situ characterization, we discuss the following methods:

1. Porosity Determination. Effective fuel cell electrode and catalyst structures must have well-controlled porosity. Several characterization techniques determine the porosity of sample structures, although many of them are destructive tests. More sophisticated techniques even produce approximate pore size distributions.

2. Brunauer–Emmett–Teller (BET) Surface Area Measurement. Fuel cell performance critically depends on the use of extremely high surface area catalysts. Some elec- trochemical techniques yield approximate surface area values; however, the BET method allows highly accurate ex situ surface area determinations for virtually any type of sample.

3. Gas Permeability. Even highly porous fuel cell electrodes may not be very gas permeable if the pores do not lead anywhere. Understanding mass transport in fuel cell electrodes therefore requires permeability measurements in addition to porosity determination. While fuel cell electrodes and catalyst layers should be highly permeable, electrolytes should be gas tight. Gas permeability testing of electrolytes is critical to the validation of ultrathin membranes, where gas leaks can prove catastrophic.

4. Structure Determinations. A wide variety of microscopy and diffraction techniques are used to investigate the structure of fuel cell materials. By structure, we mean grain size, crystal structure, orientation, morphology, and so on. This determination is especially critical when new catalysts, electrodes, or electrolytes are being developed or when new processing methods are used.

5. Chemical Determinations. In addition to characterizing physical structure, charac- terizing the chemical composition of fuel cell materials is also critical. Fortunately, many techniques are available for chemical composition and analysis. Often, the hardest part is deciding which technique is best for a given situation.

7.3 IN SITU ELECTROCHEMICAL CHARACTERIZATION TECHNIQUES

In the following section, we detail the most commonly used in situ electrochemical charac- terization techniques. All in situ electrochemical fuel cell characterization techniques rely on the measurement of current and voltage. Of course, these tests often involve the variation of other variables besides current and voltage. For example, we may want to vary tempera- ture, gas pressure, gas flow rate, or humidity. In all these cases, we are trying to answer the following question: What effect does a given variable have on fuel cell current and voltage? Current and voltage are the “end indicators” of fuel cell performance.

IN SITU ELECTROCHEMICAL CHARACTERIZATION TECHNIQUES 241

7.3.1 Fundamental Electrochemical Variables: Voltage, Current, and Time

In an electrochemical experiment, the three fundamental variables are voltage (V), current (i), and time (t). We can measure or control the voltage of our system, we can measure or control the current of our system, and we can do either as a function of time. That’s it. From an electrical characterization standpoint, there is nothing else we can do. Furthermore, since current and voltage are intimately related in a fuel cell, we cannot independently vary both of them at the same time. If we choose to control voltage, then the electrochemistry of our system sets the current. If we instead choose to control current, then the electrochemistry of our system sets the voltage. Because of this interdependence between current and voltage, there are really only two fundamental types of electrochemical characterization techniques: potentiostatic techniques and galvanostatic techniques:

1. Potentiostatic Techniques. The voltage of a system is controlled by the user and the resulting current response is measured. “Static” is an unfortunate historical mis- nomer. Potentiostatic techniques can either be steady state (where the control voltage is constant in time) or dynamic (where the control voltage varies with time).

2. Galvanostatic Techniques. The current of a system is controlled by the user and the resulting voltage response is measured. Galvanostatic techniques can also be steady state (where the control current is constant in time) or dynamic (where the control current varies with time).

Both potentiostatic and galvanostatic techniques can be applied to fuel cells. For example, fuel cell j–V curves are generally acquired using steady-state potentiostatic or galvanostatic measurements. In fact, at steady state, it does not matter whether a potentiostatic or galvanostatic measurement is used to record a fuel cell’s j–V curve—the measurements represent two sides of the same coin. In the steady-state condition, a potentiostatic and a galvanostatic measurement of a system made at the same point will yield the identical result. In other words, if a steady-state galvanostatic measurement of a fuel cell yields 0.5 V at an imposed current of 1.0 A, then the steady-state potentiostatic measurement of the same fuel cell should yield a current of 1.0 A at an imposed voltage of 0.5 V.

For short time periods or under non-steady-state conditions, potentiostatic and galvanos- tatic measurements may deviate from one another. Often, this deviation is because a system has not had enough time to relax to its steady-state condition. Actually, deviations from the steady state due to slow relaxation processes can be exploited to help understand fuel cell behavior. This is where the more sophisticated dynamic techniques enter in. One technique that exploits the dynamic behavior of a fuel cell is known as the current interrupt measure- ment. We will briefly contrast the difference between a true steady-state j–V measurement and a current interrupt measurement:

• Steady-State j–V Measurement. The current of the fuel cell is held fixed in time and the steady-state value of the fuel cell voltage is recorded after a long equilibration time. Or, the voltage of the fuel cell is held fixed in time and the steady-state value of the fuel cell current is recorded after a long equilibration time.

242 FUEL CELL CHARACTERIZATION

• Current Interrupt Measurement. A current is abruptly imposed (or withdrawn) at time t = 0, and the system voltage’s resulting time-dependent approach to steady state is measured.

While time-invariant techniques can give useful information about the steady-state prop- erties of fuel cells, it is the dynamic (time-variant) techniques that give truly powerful insight into the various loss components that contribute to performance. In addition to current interrupt, two other powerful dynamic techniques, cyclic voltammetry and elec- trochemical impedance spectroscopy, are also detailed in this chapter. We briefly compare these two dynamic techniques:

• Cyclic Voltammetry. In this dynamic technique, the voltage applied to a system is swept linearly with time back and forth across a voltage window of interest. The resulting cyclic current response is measured as a function of time but is plotted as a function of the cyclic voltage sweep.

• Electrochemical Impedance Spectroscopy. In this dynamic technique a sinusoidal per- turbation (usually a voltage perturbation) is applied to a system and the amplitude and phase shift of the resulting current response are measured. Measurements can be con- ducted over a wide range of frequencies, resulting in the construction of an impedance spectrum.

All of these techniques require a basic fuel cell testing platform and some standard electrochemical measurement equipment. Therefore, before going into further detail on the techniques themselves, we will take a brief look at the basic fuel cell test station requirements.

7.3.2 Basic Fuel Cell Test Station Requirements

Figure 7.1a illustrates a basic test station used for in situ fuel cell characterization mea- surements. This diagram is specifically for a PEMFC; a similar setup for a SOFC is shown in Figure 7.1b. Since fuel cell performance strongly depends on the operating conditions, a good test setup must allow flexible control over the operating pressures, temperatures, humidity levels, and flow rates of the reactant gases.

Mass flow controllers, pressure gauges, and temperature sensors allow the operating conditions of the fuel cell to be continually monitored during testing. Electrochemical measurement equipment, usually including a potentiostat/galvanostat and an impedance analyzer, is attached to the fuel cell. These measurement devices have at least two leads; one connects to the fuel cell cathode, while the other connects to the fuel cell anode. Often a third lead is provided for a reference electrode. Most commercially available potentiostats can perform a wide range of potentiostatic/galvanostatic experiments, including j–V curve measurements, current interrupt, and cyclic voltammetry. Electrochemical impedance spec- troscopy often requires a dedicated impedance analyzer or an add-on unit in addition to the potentiostat.

Compared to PEMFCs, SOFCs require a more elaborate test station (see Figure 7.1b). This is primarily due to the fact that SOFCs run at substantially higher temperatures and

IN SITU ELECTROCHEMICAL CHARACTERIZATION TECHNIQUES 243

0.00 sccm

Exhaust

Fuel cell

Pressure gauges

Electric measurement

Mass flow controller

50.0 C

Heater

Humidifier

(a)

(b)

0.00 sccm

Exhaust

Pressure gauges

Mass flow controller

Tube furnace

Fuel cell

Temperature sensors

Power supply

Optional humidifiers (necessary for proton conducting ceramic electrolytes)

0.00 sccm

50.0 C

Gas line heaters

Power source/temperature controller with feedback

loop

Electric measurement

Figure 7.1. (a) Typical PEMFC test station. Pressures, temperatures, humidity levels, and flow rates of gases are controlled. (b) Typical SOFC test station. Compared to the PEMFC test station, the SOFC test station is more elaborate due to the challenges associated with working at high temperatures.

244 FUEL CELL CHARACTERIZATION

are often supplied with hydrocarbon fuels rather than simple hydrogen. Accordingly, the fuel cell in a SOFC test station needs to reside inside a furnace with precise temperature control over a wide temperature range. Working at elevated temperatures presents special challenges, particularly in providing robust seals, electrical leads, and connections to/from the fuel cell. Accurately monitoring the fuel cell conditions (such as temperature, pres- sure, and gas compositions) while at elevated temperatures is also challenging. Designing a proper test station gets even more complicated when considering that SOFCs are frequently intended for use with hydrocarbon fuels. Such fuels tend to crack at elevated temperatures and leave undesirable carbon coatings behind. Methods for removing, burning, or con- trolling these carbon residues become essential in fuel cell test stations operating at high temperatures with hydrocarbon fuels. SOFC testing brings unique experimental require- ments and constraints but also brings unique opportunities by broadening the range of fuels that can be explored relative to a PEMFC.

With a complete fuel cell test station like the ones shown in Figure 7.1, there are literally dozens of possible characterization experiments that can be conducted. One of the first measurements you will probably want to take is a j–V curve.

7.3.3 Current–Voltage Measurement

As previously introduced, the performance of a fuel cell is best summarized by its current–voltage response, or j–V curve (recall Figure 1.11). The j–V curve shows the voltage output of the fuel cell for a given current density loading. High-performance fuel cells will exhibit less loss and therefore a higher voltage for a given current load. Fuel cell j–V curves are usually measured with a potentiostat/galvanostat system. This system draws a fixed current from the fuel cell and measures the corresponding output voltage. By slowly stepping the current demand, the entire j–V response of the fuel cell can be determined.

In taking j–V curve measurements of fuel cells, the following important points must be considered:

• Steady state must be ensured. • The test conditions should be carefully controlled and documented.

These points will now be addressed.

Steady State. Reliable j–V curve measurements require a steady-state system. Steady state means that the voltage and current readings do not change with time. When current is demanded from a fuel cell, the voltage of the cell drops to reflect the higher losses associated with producing current. However, this voltage drop is not instantaneous. Instead, it can take seconds, minutes, or even hours for the voltage to relax all the way to a steady-state value. This delay is due to subtle changes, such as temperature changes and reactant concentration changes that take time to propagate through the fuel cell. Usually, the larger the fuel cell, the slower the approach to steady state. It is not unusual for a large automotive or residential fuel cell stack to require 30 min to reach steady state after an abrupt current or voltage change. Current or voltage measurements recorded before a fuel cell reaches steady state will be artificially high or artificially low.

IN SITU ELECTROCHEMICAL CHARACTERIZATION TECHNIQUES 245

For large fuel cell systems, j–V curve testing can be a tedious, time-consuming process. Often, measurements are made galvanostatically: The fuel cell is subjected to a given cur- rent load, and the voltage response is monitored until it no longer changes significantly in time. This voltage is recorded. Then, the current load is increased to a new predetermined value and the procedure is repeated. Frequently, time constraints only permit 10–20 points along the fuel cell’s j–V curve to be acquired. While the data are coarse, it is generally sufficient to outline the fuel cell’s performance.

For small fuel cell systems, slow-scan j–V curve measurements can be acquired. In a slow-scan galvanostatic measurement, the current demanded from the fuel cell is gradually scanned in time from zero to some predetermined limit. The voltage of the fuel cell will continuously drop as the current is ramped. The resulting graph of current versus voltage represents a pseudo-steady-state version of the fuel cell’s j–V curve if the current scan is slow enough. The question is, how does one know if the current scan is sufficiently slow? The answer is found by conducting a series of j–V measurements at several different scan speeds. If the scan speed is too fast, the j–V curve will be artificially high. If decreasing the scan speed no longer affects the j–V curve, the speed is sufficiently slow.

Test Conditions. Test conditions will dramatically affect fuel cell performance. There- fore, care must be taken to fully document measurement operating conditions, testing pro- cedures, device histories, and so on. A “bad” PEMFC operating at 80∘C on humidified oxygen and hydrogen gases under 5 atmpressure may show better j–V curve performance than a “good” PEMFC operating at 30∘C on dry air and dilute hydrogen at atmospheric pressure. However, if the two fuel cells are tested under identical conditions, the truly good fuel cell will become apparent.

The most important testing conditions to document are now briefly discussed:

• Warm-up. To ensure that a fuel cell system is well equilibrated, it is customary to conduct a standardized warm-up procedure prior to cell characterization. A typical warm-up procedure might involve operating the cell at a fixed current load for 30–60 min prior to testing. Failure to properly warm up a fuel cell system can result in highly nonstationary (non-steady-state) behavior.

• Temperature. It is important to document and maintain a constant fuel cell temperature during measurement. Both the gas inlet and exit temperatures should be measured as well as the temperature of the fuel cell itself. Sophisticated techniques even allow temperature distributions across a fuel cell device to be monitored in real time. In general, increased temperature will improve performance due to improved kinetics and conduction processes. (For PEMFCs, this is only true up to about 80∘C, above which membrane drying becomes an issue.)

• Pressure. Gas pressures are generally monitored at both the fuel cell inlets and out- lets. This allows the internal pressure of the fuel cell to be determined as well as the pressure drop within the cell. Increased cell pressure will improve performance. (However, increasing the pressure requires additional energy “input” from compres- sors, fans, etc.)

• Flow Rate. Flow rates are generally set using mass flow controllers. During a j–V test, there are two main ways to handle reactant flow rates. In the first method, flow rates

246 FUEL CELL CHARACTERIZATION

are held constant during the entire test at a flow rate that is sufficiently high so that even at the largest current densities there is sufficient supply. This method is known as the fixed-flow-rate condition. In the second method, flow rates are adjusted stoi- chiometrically with the current so that the ratio between reactant supply and current consumption is always fixed. This method is known as the fixed-stoichiometry con- dition. Fair j–V curve comparisons should be done using the same flow rate method. Increased flow usually improves performance. (For PEMFCs, increasing the flow rate of extremely humid or extremely dry gases can upset the water balance in the fuel cell and actually decrease performance.)

• Compression Force. For most fuel cell assemblies, there is an optimal cell compres- sion force, which leads to best performance; thus, cell compression force should be noted and monitored. Cells with lower compression forces can suffer increased ohmic loss, while cells with higher compression forces can suffer increased pressure or con- centration losses.

Interpreting j–V Curve Measurements. Generally, j–V curve measurements are used to quantitatively describe the overall performance of a fuel cell system. At first glance, it appears impossible to individually separate the various loss contributions (e.g., activation, ohmic, concentration losses) from the j–V curve. Nevertheless, careful data analysis can sometimes permit approximate activation losses to be isolated using the Tafel equation (at least in PEMFCs).

In PEMFCs at low current densities, the ohmic loss is usually small compared to the acti- vation loss. Thus, the ohmic loss can be ignored and the approximate activation loss can be calculated directly from the data. If plotted on a log scale, the low-current-density j–V curve regimen shows linear behavior, as expected from the Tafel equation 3.41. The transfer coef- ficient and the exchange current density can be obtained by fitting a line through the data. The line can be extended throughout the j–V curve, allowing the approximate activation loss contribution to be identified at each current density. Figure 7.2 briefly illustrates the process.

7.3.4 Electrochemical Impedance Spectroscopy

While the j–V curve provides general quantification of fuel cell performance, a more sophis- ticated test is required to accurately differentiate between all the major sources of loss in a fuel cell. Electrochemical impedance spectroscopy is the most widely used technique for distinguishing the different losses.

EIS Basics. Like resistance, impedance is a measure of the ability of a system to impede the flow of electrical current. Unlike resistance, impedance can deal with time- or frequency-dependent phenomena. Recall how we define resistance R from Ohm’s law as the ratio between voltage and current:

R = V i

(7.1)

IN SITU ELECTROCHEMICAL CHARACTERIZATION TECHNIQUES 247

0 0.2 0.4 0.6 0.8 1 0

0.2

0.4

0.6

0.8

(a) (b)

(c)

10–2 10–1 100 0

0.2

0.4

0.6

0.8

10–2 10–1 0

0.2

0.4

0.6

0.8

IV curve Tafel fitting

Activation loss

Ohmic and concentration loss

V ol

ta ge

( V

)

V ol

ta ge

( V

)

V ol

ta ge

( V

)

Current density (A/cm2)

Current density (A/cm2) Current density (A/cm2)

IV curve Activation loss

Figure 7.2. (a) Typical log-scaled j–V curve. The activation loss contribution is plotted by the dashed line. (b) The low-current-density regimen of the j–V curve shows linear behavior on a log scale. Fitting this line to the Tafel equation gives the transfer coefficient and the exchange current density. (c) Activation loss is plotted throughout the j–V curve. The difference between the activation loss and the j–V curve represents the sum of ohmic and concentration losses.

In an analogous manner, impedance Z is given by the ratio between a time-dependent voltage and a time-dependent current:

Z = V(t) i(t)

(7.2)

Impedance measurements are usually made by applying a small sinusoidal voltage per- turbation, V(t) = V0 cos(𝑤t), and monitoring the system’s resultant current response, i(t) = i0 cos(𝑤t). In these expressions, V(t) and i(t) are the potential and current at time t, V0, and i0 are the amplitudes of the voltage and current signals, and 𝑤 is the radial frequency.

248 FUEL CELL CHARACTERIZATION

C ur

re nt

( A

)

V ol

ta ge

( V

)

Phase shift (ϕ)

Figure 7.3. A sinusoidal voltage perturbation and resulting sinusoidal current response. The current response will possess the same period (frequency) as the voltage perturbation but will generally be phase shifted by an amount 𝜙.

The relationship between radial frequency 𝑤 (expressed in radians per second) and fre- quency f (expressed in hertz) is

𝑤 = 2𝜋 f (7.3)

In general, the current response of a system may be shifted in phase compared to the voltage perturbation. This phase shift effect is described by 𝜙. A graphical representation of the relationship between a sinusoidal voltage perturbation and a phase-shifted current response is shown in Figure 7.3 (for a linear system).

Following Equation 7.2, we can write the sinusoidal impedance response of a system as

Z = V0 cos(𝑤t)

i0 cos(𝑤t − 𝜙) = Z0

cos(𝑤t) cos(𝑤t − 𝜙)

(7.4)

Alternatively, we can use complex notation to write the impedance response of a system in terms of a real and an imaginary component:

Z = V0e

j𝑤t

i0e(j𝑤t−j𝜙) = Z0ej𝜙 = Z0(cos𝜙 + j sin𝜙) (7.5)

The impedance of a system can therefore be expressed in terms of an impedance mag- nitude Z0 and a phase shift 𝜙, or in terms of a real component (Zreal = Z0 cos𝜙) and an imaginary component (Zimag = Z0 sinϕj). Note that j in these expressions represents the imaginary number (j =

√ −1), not the current density! Typically, impedance data are plotted

in terms of the real and imaginary components of impedance (Zreal on the x-axis and –Zimag on the y-axis). Such graphical representations of impedance data are known as Nyquist plots.

IN SITU ELECTROCHEMICAL CHARACTERIZATION TECHNIQUES 249

Current (A)

C el

l v ol

ta ge

( V

)

Small-signal voltage

perturbation Probe pseudolinear portion of i--V curve

Yields small- signal current response

Figure 7.4. Application of a small-signal voltage perturbation confines the impedance measurement to a pseudolinear portion of a fuel cell’s i–V curve.

Because impedance measurements are made at dozens or even hundreds of different fre- quencies, Nyquist plots generally summarize the impedance behavior of a system over many orders of magnitude in frequency.

System linearity is required for facile impedance analysis. In a linear system, doubling the current will double the voltage. Obviously, electrochemical systems are not linear. (Con- sider Butler–Volmer kinetics, which predicts an exponential relationship between voltage and current.) We circumvent this problem by using small-signal voltage perturbations in our impedance measurements. As Figure 7.4 illustrates, if we sample a small enough portion of a cell’s i–V curve, it will appear linear. In normal EIS practice, a 1–20-mV alternating current (AC) signal is applied to the cell. This signal is generally small enough to confine us to a pseudolinear segment of the cell’s i–V curve.

EIS and Fuel Cells. Before we get into the details of impedance theory, we will present a brief example illustrating the power of EIS for fuel cell characterization. Consider a hypo- thetical fuel cell that suffers from three loss effects:

1. Anode activation loss 2. Ohmic electrolyte loss 3. Cathode activation loss

Figure 7.5 shows what the EIS Nyquist plot for this fuel cell might look like. Don’t worry about understanding this spectrum yet. The key thing to note is that two semicircu- lar peaks are visible in the plot. For the hypothetical fuel cell in this example, the size of these two semicircles can be attributed to the magnitude of the two (anode and cathode) activation losses. Looking more closely at the diagram, you will see that the three x-axis intercepts defined by the semicircles mark off three impedance regions, which are denoted by ZΩ, ZfA, and ZfC. The size of these three impedances correspond to the relative size of

250 FUEL CELL CHARACTERIZATION

Cathode activation

losses

Ohmic losses

Anode activation

losses

0 0

ZΩ ZΩ + ZfA ZΩ + ZfA + Zfc

Figure 7.5. Example Nyquist plot from a hypothetical fuel cell. The three regions marked on the impedance plot are attributed to the ohmic, anode activation, and cathode activation losses. The rela- tive size of the three regions provides information about the relative magnitude of the three losses in this fuel cell.

𝜂ohmic, 𝜂act,anode, and 𝜂act,cathode in our fuel cell. Thus, in this hypothetical EIS example, it is clear that the cathode activation loss dominates the fuel cell’s performance, while the ohmic and anode activation losses are small.

How were we able to generate this spectrum using EIS and how could we assign the various intercepts in the spectrum to the various loss processes in the fuel cell? This requires a discussion on impedance theory and equivalent circuit modeling.

EIS and Equivalent Circuit Modeling. The processes that occur inside a fuel cell can be modeled using circuit elements. For example, we can assign groups of resistors and capacitors to describe the behavior of electrochemical reaction kinetics, ohmic conduction processes, and even mass transport. Such circuit-based representations of fuel cell behavior are known as equivalent circuit models. If we measure a fuel cell’s impedance spectrum and compare it to a good equivalent circuit model, it is then possible to extract information about the reaction kinetics, ohmic conduction processes, mass transport, and other properties.

We now introduce the common circuit elements used to describe fuel cell behavior. We will then build a sample equivalent circuit model of a fuel cell using these circuit elements for illustration. We begin with the ohmic conduction processes.

Ohmic Resistance. The equivalent circuit representation of an ohmic conduction process is rather straightforward; it is a simple resistor!

ZΩ = RΩ (7.6)

IN SITU ELECTROCHEMICAL CHARACTERIZATION TECHNIQUES 251

–Z imag

Z real

R0 0

Figure 7.6. Circuit diagram and Nyquist plot for a simple resistor. The impedance of a resistor is a single point of value R on the real impedance axis (x-axis). The impedance of a resistor is independent of frequency.

As was mentioned previously, impedance data are generally plotted on a Nyquist dia- gram. Recall from the complex definition of impedance that the impedance of a system can be represented in terms of its real component Z0 cos𝜙 and its imaginary component ( jZ0 sin𝜙):

Z = Z0 cos𝜙 + jZ0 sin𝜙 (7.7)

A Nyquist diagram plots the real component of impedance versus the imaginary com- ponent of impedance (actually, the negative of the imaginary component of impedance) over a range of frequencies. For the case of a simple resistor, the imaginary component of resistance is zero, 𝜙 is zero, and the impedance does not change with frequency. The Nyquist plot for a resistor is therefore a single point on the real axis (x-axis) with value R. The equivalent circuit and corresponding Nyquist diagram of a simple resistor are given in Figure 7.6.

Electrochemical Reaction. The equivalent circuit representation of an electrochemical reaction is more complicated. Figure 7.7 depicts the typical electrochemical reaction inter- face. As illustrated in this figure, the impedance behavior of the reaction interface can be modeled as a parallel combination of a resistor and a capacitor (Rf and Cdl). Here, Rf , the Faradaic resistance, models the kinetics of the electrochemical reaction, while Cdl, the double-layer capacitance, reflects the capacitive nature of the interface. We will briefly discuss both Cdl and Rf .

The easiest to visualize is Cdl. As Figure 7.7 illustrates, during an electrochemical reac- tion, a significant separation of charge occurs across the reaction interface, with electron accumulation in the electrode matched by ion accumulation in the electrolyte. The charge separation causes the interface to behave like a capacitor. The strength of this capacitive behavior is reflected in the size of Cdl. For a perfectly smooth electrode–electrolyte interface, Cdl is typically on the order of 30 μF∕cm2 interfacial area. However, with high-surface-area fuel cell electrodes, Cdl can be orders of magnitude larger.

252 FUEL CELL CHARACTERIZATION

Figure 7.7. Physical representation and proposed equivalent circuit model of an electrochemical reaction interface. The impedance behavior of an electrochemical reaction interface can be modeled as a parallel combination of a capacitor and a resistor. The capacitor (Cdl) describes the charge separa- tion between ions and electrons across the interface. The resistor (Rf ) describes the kinetic resistance of the electrochemical reaction process.

The impedance response of a capacitor is purely imaginary. The equation relating voltage and current for a capacitor is

i = C dV dt

(7.8)

For a sinusoidal voltage perturbation (V = V0ej𝑤t), this gives

i(t) = C d(V0ej𝑤t)

dt = C(j𝑤)V0ej𝑤t (7.9)

which yields an impedance of

Z = V(t) i(t)

= V0e

j𝑤t

C(j𝑤)V0ej𝑤t = 1

j𝑤C (7.10)

If this capacitor is placed in series with a resistor, the net impedance will be given by the sum of the impedances of the two elements. In other words, series impedances, like series resistances, are additive:

Zseries = Z1 + Z2 (7.11)

For a capacitor and resistor in series, the net impedance would be

Z = R + 1 j𝑤C

(7.12)

The equivalent circuit diagram and corresponding Nyquist impedance plot of the resistor–capacitor series combination is shown in Figure 7.8. One drawback of the Nyquist

IN SITU ELECTROCHEMICAL CHARACTERIZATION TECHNIQUES 253

–Z imag

Z real

R0 0

D ec

re as

in g

Figure 7.8. Circuit diagram and Nyquist plot for a series RC. The impedance is a vertical line that increases with decreasing w. The real component of the impedance is given by the value of the resis- tor. As frequency decreases, the imaginary component of the impedance (as given by the capacitor) dominates the response of the circuit.

plot is that you cannot tell what frequency was used to record each point. In Figure 7.8, we mitigate this disadvantage by noting the general frequency trend for reference.

For the case of the reaction interface shown in Figure 7.7, the capacitor and resistor are in parallel rather than in series. Before we talk about parallel impedances, however, we will discuss the Faradaic resistance, Rf , in more detail.

To understand how the reaction process can be modeled by Rf , recall the Tafel simplifi- cation of reaction kinetics (Equation 3.40):

𝜂act = − RT 𝛼nF

ln i0 + RT 𝛼nF

ln i (7.13)

Note that we have replaced current density j by raw current i to facilitate the impedance calculation. For a small-signal sinusoidal perturbation, the impedance response Z = V(t)∕i(t) can be approximated as Z = dV∕di. (In other words, the impedance is the instantaneous slope of the i–V response at the point of interest.) Thus, the impedance of a Tafel-like kinetic process may be calculated as

Zf = d𝜂 di

= RT 𝛼nF

1 i

(7.14)

Substituting i = i0e𝛼nF𝜂act∕(RT) into this expression yields

Zf = Rf = ( RT 𝛼nF

) 1 i0e𝛼nF𝜂act∕(RT)

(7.15)

Notice that Zf has no imaginary component and therefore can be represented as a pure resistor (Zf = Rf ). The size of Rf depends on the kinetics of the electrochemical reaction. A high Rf indicates a highly resistive electrochemical reaction. A large i0 or a large activation overvoltage (𝜂act) will decrease Rf , decreasing the kinetic resistance of the reaction.

As was previously mentioned, the total impedance of our electrochemical interface model is given by the parallel combination of the capacitive double-layer impedance and

254 FUEL CELL CHARACTERIZATION

the resistive Faradaic impedance. Just like combining parallel resistances, the parallel combination of two impedance elements is given by

1 Zparallel

= 1 Z1

+ 1 Z2

(7.16)

For our case, this becomes 1 Z

= 1 Rf

+ j𝑤Cdl (7.17)

Thus Z = 1

1∕Rf + j𝑤Cdl (7.18)

The equivalent circuit and corresponding Nyquist diagram of this reaction interface model is given in Figure 7.9. Note that the impedance shows a characteristic semicircular response. The leftmost point on the diagram corresponds to the highest frequency; fre- quency then steadily decreases as we progress from left to right across the diagram. In most electrochemical systems, the real component of impedance will almost always increase (or remain constant) with decreasing frequency.

The high-frequency intercept of the semicircle in Figure 7.9 is zero, while the low-frequency intercept is Rf . Thus, the diameter of the semicircle provides information about the size of the activation resistance. A fuel cell with highly facile reaction kinetics will show a small impedance loop. In contrast, a blocking electrode (one where Rf → ∞ because the electrode “blocks” the electrochemical reaction) shows an impedance response similar to the pure capacitor in Figure 7.8. Examination of the limits in Equation 7.18 for 𝑤→ ∞ and 𝑤 → 0 confirms these observations. At intermediate frequencies, the impedance response contains both real and imaginary components. The frequency at the apex of the semicircle is given by the RC time constant of the interface: 𝑤 = 1∕(Rf Cdl). From this value, Cdl may be determined.

–Zimag

Z real

Cdl

Decreasing ω

Rf0 0

ω = 1/R Cf dl

Figure 7.9. Circuit diagram and Nyquist plot for a parallel RC. This semicircular impedance response is typical of an electrochemical reaction interface. The high-frequency intercept of the semicircle is zero, while the low-frequency intercept of the impedance semicircle is Rf . The diameter of the semicircle (Rf ) gives information about the reaction kinetics of the electrochemical interface. A small loop indicates facile reaction kinetics while a large loop indicates sluggish reaction kinetics.

IN SITU ELECTROCHEMICAL CHARACTERIZATION TECHNIQUES 255

The impedance behavior illustrated in Figure 7.9 can be understood intuitively by examining the RC circuit model. At extremely high frequencies, capacitors act as short circuits; at extremely low frequencies, capacitors act as open circuits. Thus, at high frequency, the current can be completely shunted through the capacitor and the effective impedance of the model is zero. In contrast, at extremely low frequencies, all of the current is forced to flow through the resistor and the effective impedance of the model is given by the impedance of the resistor. For intermediate frequencies, the situation is somewhere in between, and the impedance response of the model will have both resistive and capacitive elements.

Mass Transport. Mass transport in fuel cells can be modeled by Warburg circuit elements. Time does not permit the derivation of Warburg elements here. However, they are based on (and can be derived from) diffusion processes. The impedance of an “infinite” Warburg element (used for an infinitely thick diffusion layer) is given by the equation

Z = 𝜎i√ 𝑤 (1 − j) (7.19)

where σi in this equation is the Warburg coefficient for a species i (not the conductivity) and is defined as

𝜎i = RT

(niF)2A √

( 1

c0i √

) (7.20)

where A is the electrode area, c0i is the bulk concentration of species i, and Di is the diffusion coefficient of species i. Thus, σi characterizes the effectiveness of transporting species i to or away from a reaction interface. If species i is abundant (c0i is large) and diffusion is fast (Di, is high), then σi will be small and the impedance due to mass transport of species i will be negligible. On the other hand, if the species concentration is low and diffusion is slow, σi will be large and the impedance due to mass transport can become significant. Note from Equation 7.19 that the Warburg impedance also depends on the frequency of the potential perturbation. At high frequencies the Warburg impedance is small since diffusing reactants do not have to move very far. However, at low frequencies the reactants must diffuse farther, thereby increasing the Warburg impedance.

The equivalent circuit and corresponding Nyquist diagram of the infinite Warburg impedance element are given in Figure 7.10. Note that the infinite Warburg impedance shows a characteristic increasing linear response with decreasing ω. The infinite Warburg impedance appears as a diagonal line with a slope of 1.

The infinite Warburg impedance is only valid if the diffusion layer is infinitely thick. In fuel cells, this is rarely the case. As we learned in Chapter 5, convective mixing in fuel cell flow structures usually restricts the diffusion layer to the thickness of the electrode. For such situations, the impedance at lower frequencies no longer obeys the infinite Warburg equation. In these cases, it is better to use a porous bounded Warburg model (also called the “O” diffusion element), which has the form

Z = 𝜎i√ 𝑤 (1 − j) tanh

( 𝛿

√ j𝑤

) (7.21)

256 FUEL CELL CHARACTERIZATION

0 0

Slope = 1.0De

cr ea

sin g

–Z imag

Zreal

Figure 7.10. Circuit diagram and Nyquist plot for a Warburg element used to model diffusion pro- cesses. The impedance response is a diagonal line with a slope of 1. Impedance increases from left to right with decreasing frequency.

–Z imag

Z real

0 0

Slope = 1.0

Decreasing ω

Follows infinite Warburg for ω > 10Di/δ

i i D

Z 2

= δσ

Figure 7.11. Circuit diagram and Nyquist plot for a porous bounded Warburg element, which is used to model finite diffusion processes (with diffusion occurring through a fixed diffusion layer thickness from an inexhaustible bulk supply of reactants). This situation is typical in fuel cell systems. At high frequency, the porous bounded Warburg impedance response mirrors the behavior of an infinite Warburg; at low frequency, it returns toward the real impedance axis. (This makes intuitive sense: A finite diffusion layer thickness should yield finite real impedance.) The low-frequency real axis impedance intercept yields information about the diffusion layer thickness.

where δ is the diffusion layer thickness. As shown in Figure 7.11, at high frequencies or cases where δ is large, the porous bounded Warburg impedance converges to the infinite Warburg behavior. However, at low frequencies or for small diffusion layers, the porous bounded Warburg impedance loops back toward the real axis.

We have now assembled enough tools to describe basic fuel cell processes using equiv- alent circuit elements. The equivalent circuit elements that we have developed (as well as a few others) are summarized in Table 7.1.

IN SITU ELECTROCHEMICAL CHARACTERIZATION TECHNIQUES 257

TABLE 7.1. Impedance Summary of Common Equivalent Circuit Elements

Circuit Element Impedance

Resistor R

Capacitor 1∕j𝑤C

Constant-phase element 1∕[Q(j𝑤)α]

Inductor j𝑤L

Infinite Warburg (𝜎i∕ √ 𝑤)(1 − j)

Finite (porous bounded) Warburg (𝜎i∕ √ 𝑤)(1 − j) tanh(δ

√ j𝑤∕Di)

Series impedance elements Zseries = Z1 + Z2 Parallel impedance elements 1∕Zparallel = 1∕Z1 + 1∕Z2

Simple Equivalent Circuit Fuel Cell Model. We now construct a simple equivalent circuit model for a complete fuel cell using the elements described previously. We assume that our fuel cell suffers from the following loss processes:

1. Anode activation 2. Cathode activation 3. Cathode mass transfer 4. Ohmic loss

For simplicity, we assume that the cathode mass transfer process can be modeled with an infinite Warburg impedance element. Also, we assume that the anode kinetics are fast com- pared to the cathode activation kinetics. The physical picture, equivalent circuit model, and corresponding Nyquist plot for our fuel cell are shown in Figure 7.12. The Nyquist plot was generated using the equivalent circuit values given in Table 7.2. Note how the impedance response of this fuel cell model is given by a combination of the impedance behaviors from each individual element in our circuit! The Nyquist plot shows two semicircles followed by a diagonal line. The high-frequency (far left), real-axis intercept corresponds to the ohmic resistance of our fuel cell model. The first loop corresponds to the RC model of the anode activation kinetics while the second loop corresponds to the RC model of the cathode acti- vation kinetics. The diameter of the first loop gives Rf for the anode while the diameter of the second loop gives Rf for the cathode. Note how the cathode loop is significantly larger than the anode loop. This visually indicates that the cathode activation losses are signifi- cantly greater than the anode activation losses. From the Rf values, the kinetics of the anode and cathode reactions can be extracted using Equation 7.15. Fitting the Cdl values gives an indication of the effective surface area of the fuel cell electrodes.

The diagonal line at low frequencies is due to mass transport as modeled by the infinite Warburg impedance. From the frequency–impedance data of this line, the mass transport properties of the fuel cell can be extracted. If a porous bounded Warburg is used instead, a diffusion layer thickness could also be extracted.

258 FUEL CELL CHARACTERIZATION

Figure 7.12. Physical picture, circuit diagram, and Nyquist plot for a simple fuel cell impedance model. The equivalent circuit for this fuel cell consists of two parallel RC elements to model the anode and cathode activation kinetics, an infinite Warburg element to simulate cathode mass transfer effects, and an ohmic resistor to simulate the ohmic losses. While schematically shown in the electrolyte region, the ohmic resistor models the ohmic losses arising from all parts of the fuel cell (electrolyte, electrodes, etc.). The impedance response shown in the Nyquist plot is based on the circuit element values given in Table 7.2. Each circuit element contributes to the shape of the Nyquist plot, as indicated in the diagram. The ohmic resistor determines the high-frequency impedance intercept. The small semicircle is due to the anode RC element, while the large semicircle is due to the cathode RC element. The low-frequency diagonal line comes from the infinite Warburg element.

TABLE 7.2. Summary of Values Used to Generate Nyquist Plot in Figure

Fuel Cell Process Circuit Element Value

Ohmic resistance RΩ 10 mΩ

Anode Faradaic resistance Rf ,A 5 mΩ

Anode double-layer capacitance Cdl,A 3 mF

Cathode Faradaic resistance Rf ,C 100 mΩ

Cathode double-layer capacitance Cdl,C 30 mF

Cathode Warburg coefficient σ 15 mΩs1∕2

IN SITU ELECTROCHEMICAL CHARACTERIZATION TECHNIQUES 259

(a) (b)

–Z imag

Z real0 0

–Z

Z real

imag

0 0

Figure 7.13. In H2–O2 fuel cells the cathode impedance is often significantly larger than the anode impedance. In these cases, the cathode impedance can mask the impedance of the anode, as shown to varying degrees in (a) and (b). This masking (or “merging”) also occurs if the RC time constants for the anode and cathode reactions overlap. If Rf for the anode is extremely small, the RC time constant for the anode may correspond to frequencies that are beyond the limits of most impedance hardware. (EIS is usually limited to f < 1 MHz.) In these cases, the anode impedance may be unmeasurable.

For clarity in this example, we deliberately chose RC values for the anode and cathode that allowed the two semicircles to be distinguished from one another. In many real fuel cells, however, the RC loop for the cathode overwhelms the RC loop for the anode, as shown in Figure 7.13.

To fully understand fuel cell behavior, it is essential to measure the impedance response at several different points along a fuel cell’s i–V curve. The impedance behavior of a fuel cell will change along the i–V curve, depending on which loss processes are dominant. Figure 7.14 gives several illustrative examples. At low currents, the activation kinetics dom- inate and Rf is large, while the mass transport effects can be neglected. In these situations, an impedance response similar to that shown in Figure 7.14a is typical. At higher currents (higher activation overvoltages), Rf decreases since the activation kinetics improve with increasing 𝜂act (refer to Equation 7.15). Thus, the activation impedance loop decreases, as shown in Figure 7.14b. A decreasing impedance loop with increasing activation overvolt- age is indicative of an activated electrochemical reaction. At high currents, mass transport effects occur and the impedance response may look something like Figure 7.14c.

While the power of EIS is considerable, the technique is complex and fraught with pit- falls. Caution! There be dragons here! Due to time and space limitations, this EIS overview is not comprehensive. Interested readers who plan to use EIS for fuel cell characterization are highly encouraged to consult the extensive literature on EIS beforehand [39–41].

Example 7.1 Assume that point a on the i–V curve in Figure 7.14 corresponds to i = 0.25 A and V = 0.77 V. Assume that point b on the i–V curve corresponds to i = 1.0 A and V = 0.62 V. From the EIS data in Figure 7.14, calculate nohmic and nact at points a and b on the fuel cell i–V curve. Assume that only ohmic and activation losses contribute to fuel cell performance. If the activation losses are wholly due to the cathode, calculate i0 and α for the cathode based on your 𝜂act values (T = 300 K, n = 2, and Ethermo = 1.2 V).

260 FUEL CELL CHARACTERIZATION

Solution: At point a, i = 0.25 A, Rohmic = 0.10 Ω, and 𝜂tot = 1.2 V – 0.77 V = 0.43 V. Thus

𝜂ohmic = iRohmic = (0.25A)(0.10Ω) = 0.025V

𝜂act = 𝜂tot − 𝜂ohmic = 0.43V − 0.025V = 0.405V (7.22)

Note: It is not appropriate to write 𝜂act = iRf since Rf changes as a function of i. Thus, the best we can do is infer the activation loss by subtracting the ohmic loss from the total loss. At point b, i = 1.0 A, Rohmic = 0.10 Ω, and 𝜂tot = 1.2 V – 0.62 V = 0.58 V. Thus

𝜂ohmic = iRohmic = (1.0A)(0.10Ω) = 0.10V

𝜂act = 𝜂tot − 𝜂ohmic = 0.58V − 0.1V = 0.48V (7.23)

Note that Rf decreases at point b, but the total activation loss still increases slightly (from 0.405 to 0.48 V). This is expected; the total activation loss increases with increasing current, but the “effective resistance” of the activation process decreases. We can fit the EIS data from a and b to Equation 7.13 to extract j0 and α:

For point b:

𝜂act = − ( RT 𝛼nF

) ln i0 +

( RT 𝛼nF

) ln i

(7.24)

0.48V = − ( RT 𝛼nF

) ln i0

Substitution into a similar equation for point a allows us to solve for α: For point a:

𝜂act = − ( RT 𝛼nF

) ln i0 +

( RT 𝛼nF

) ln i

0.405V = 0.48V + ( RT 𝛼nF

) ln 0.25 (7.25)

𝛼 = 0.239 for T = 300K, n = 2

Substituting α back into the equation for point b yields i0:

0.48 V = − (

(8.314) (300) (0.239)(2)(96400)

) ln i0

i0 = 1.4 × 10−4A (7.26)

If we knew the area of the fuel cell, we could then calculate the more fundamental properties ASRohmic and j0 from Rohmic and i0.

IN SITU ELECTROCHEMICAL CHARACTERIZATION TECHNIQUES 261

Zreal

–Z imag

0.10Ω 0

1.1Ω

–Z imag

0.10Ω 0

0.30Ω

–Z imag

0.10Ω 0

0.40Ω Zreal

Current (A)

C el

l v ol

ta ge

( V

)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5

a b

(a) (b) (c)

Figure 7.14. EIS characterization of a fuel cell requires impedance measurements at several different points along an i–V curve. The impedance response will change depending on the operating voltage. (a) At low current, the activation kinetics dominate and Rf is large, while the mass transport effects can be neglected. (b) At intermediate current (higher activation overvoltages), the activation loops decrease since Rf decreases with increasing 𝜂act. (Refer to Equation 7.15.) (c) At high current, the activation loops may continue to decrease, but the mass transport effects begin to intercede, resulting in the diagonal Warburg response at low frequency.

7.3.5 Current Interrupt Measurement

The current interrupt method can provide some of the same information provided by EIS. While not as accurate or as detailed as an impedance experiment, current interrupt has several major advantages compared to impedance:

• Current interrupt is extremely fast. • Current interrupt generally requires simpler measurement hardware. • Current interrupt can be implemented on high-power fuel cell systems. (Such systems

are generally not amenable to EIS.)

• Current interrupt can be conducted in parallel with a j–V curve measurement.

262 FUEL CELL CHARACTERIZATION

(a)

(b)

(c)

RΩ Cdl

R f ZW

0.5

0.0 t C ur

re nt

( A

) V

ol ta

ge (

V )

0.6

0.7

1.0

Figure 7.15. (a) Simplified equivalent circuit of a fuel cell system. The RC components from the anode and cathode have been consolidated into a single branch. (b) Hypothetical current interrupt profile applied to the circuit in (a). In this example, an original steady-state current load of 500 mA is abruptly zeroed. (c) Hypothetical time response of fuel cell voltage when the current interrupt in (b) is applied to the system. The instantaneous rebound in the voltage is associated with the pure ohmic losses in the system. The time-dependent voltage rebound is associated with the activation and mass transport losses in the system.

For these reasons, current interrupt has found wide acceptance in the fuel cell research community, especially for characterization of large fuel cells (e.g., residential or vehicular fuel cell stacks).

The basic idea behind the current interrupt technique is illustrated in Figure 7.15. When a constant-current load on a fuel cell system is abruptly interrupted, the resulting time-dependent voltage response will be representative of the capacitive and resistive behaviors of the various components in the fuel cell. The same equivalent circuit models that were used to analyze the impedance behavior of fuel cells may be used to understand the current interrupt behavior of fuel cells.

For example, consider the simple equivalent circuit fuel cell model shown in Figure 7.15a. If the current flowing through this cell is abruptly interrupted, as shown

IN SITU ELECTROCHEMICAL CHARACTERIZATION TECHNIQUES 263

in Figure 7.15b, the corresponding voltage–time response will resemble Figure 7.15c. Interruption of the current causes an immediate rebound in the voltage, followed by an additional, time-dependent rebound in the voltage. The immediate voltage rebound is associated with the ohmic resistance of the fuel cell. The time-dependent rebound is associated with the much slower reaction and mass transport processes.

The voltage rebound process can be understood via the circuit diagram in Figure 7.15a. As the circuit diagram illustrates, the reaction and masstransport processes are modeled by time-dependent RC and Warburg elements. Due to their capacitive nature, the voltage across these elements recovers over a period of time. The recovery time for the RC element can be approximated by its RC time constant. Because the voltage rebound across the resistor is immediate while the voltage rebound across the RC/Warburg element is time dependent, the voltage–time response can be used to separate the two contributions. Example 7.2 illustrates this technique.

Example 7.2 Calculate 𝜂ohmic and Rohmic from the current interrupt data in Figure 7.15.

Solution: In Figure 7.15, when the fuel cell is held under 500-mA current load, the steady-state voltage is 0.60 V. When the current is abruptly zeroed, the cell voltage instantaneously rises to 0.70 V. We associate this instantaneous rebound in the cell voltage with the ohmic processes in the fuel cell. Therefore, the fuel cell must have been experiencing an ohmic loss of 100 mV at the 500 mA current load point:

𝜂ohmic = 0.70 V − 0.60 V = 0.10 V (at i = 500 mA) (7.27)

The ohmic resistance may be calculated from 𝜂ohmic and the current:

Rohmic = 𝜂ohmic

i = 0.10 V

0.50 A = 0.2 Ω (7.28)

After a long relaxation time, the fuel cell’s voltage recovers to a final value of around 1.0 V. Thus, the activation and concentration losses in this fuel cell must amount to about 0.30 V at a 500-mA current load (1.0 V – 0.70 V = 0.30 V).

To get accurate results from the current interrupt technique, the current should be interrupted sharply and cleanly (on the order of microseconds to milliseconds), and a fast oscilloscope should be employed to record the voltage response. Current interrupt is often implemented in parallel with i–V curve measurements. It is especially useful for determining the ohmic component of fuel cell loss at each measurement point on the fuel cell i–V curve. Typically, after a fuel cell i–V data point is recorded, a current interrupt measurement is then made to determine RΩ at that point. Then, the i–V measurement procedure is stepped to the next current level and the voltage is allowed to equilibrate to the steady state. In this way, the i–V curve information is collected along with detailed ohmicloss information from each point. The ohmicloss portion of the i–V curve data can then be removed; such curves are called “iR-free” or “iR-corrected” i–V curves. When fit to

264 FUEL CELL CHARACTERIZATION

the Tafel equation, these iR-corrected curves allow the activation and concentration losses to be separated. The result is a nearly complete quantification of the ohmic, activation, and concentration losses associated with the fuel cell.

7.3.6 Cyclic Voltammetry

Cyclic voltammetry is typically used to characterize fuel cell catalyst activity in more detail. In a standard CV measurement, the potential of a system is swept back and forth between two voltage limits while the current response is measured. The voltage sweep is gener- ally linear with time, and the plot of the resulting current versus voltage is called a cyclic voltammogram. An illustration of a typical CV waveform is provided in Figure 7.16.

In fuel cells, CV measurements can be used to determine in situ catalyst activity by using a special “hydrogen pump mode” configuration. In this mode, argon gas is passed through the cathode instead of oxygen, while the anode is supplied with hydrogen. The CV measurement is performed by sweeping the voltage of the system between about 0 and 1 V with respect to the anode. An example of a hydrogen pump mode cyclic voltammogram from a fuel cell is shown in Figure 7.17. When the potential increases from 0 V, a current begins to flow. (See Figure 7.17.) There are two contributions to this current. One contribu- tion is constant—a simple, capacitive charging current that flows in response to the linearly changing voltage. The second current response is nonlinear and corresponds to a hydrogen adsorption reaction occurring on the electrochemically active cathode catalyst surface. As the voltage increases further, this reaction current reaches a peak and then falls off as the entire catalyst surface becomes fully saturated with hydrogen. The active catalyst surface area can be obtained by quantifying the total charge (Qh) provided by hydrogen adsorp- tion on the catalyst surface. The total charge essentially corresponds to the area under the hydrogen adsorption reaction peak in the CV after converting the potential axis to time and

(a) (b)

Voltage

Time

Voltage

Current

V1 V2

Figure 7.16. Schematic of a (CV) waveform and typical resulting current response. (a) In a CV experiment, the voltage is swept linearly back and forth between two voltage limits (denoted V1 and V2 on the diagram). (b) The resulting current is plotted as a function of voltage. When the voltage sweeps past a potential corresponding to an active electrochemical reaction, the current response will spike. After this initial spike, the current will drop off as most of the readily available reactants are consumed. On the reverse voltage scan, the reverse electrochemical reaction (with a corresponding reverse current direction) may be observed. The shape and size of the peaks give information about the relative rates of reaction and diffusion in the system.

EX SITU CHARACTERIZATION TECHNIQUES 265

Potential (mV vs. hydrogen anode)

C ur

re nt

( μ A

)

0 300 600 900 1200 1500 1800

200

150

100

–50

–100

–150 –200

Figure 7.17. Fuel cell CV curve. The peaks marked Qh and Q ′ h represent the hydrogen adsorption

and desorption peaks on the platinum fuel cell catalyst surface, respectively. The gray rectangular area between the two peaks denotes the approximate contribution from the capacitive charging current. The active catalyst surface area can be calculated from the area under the Qh or Q

′ h peak (recognizing that

the voltage axis can be converted to a time axis if the scan rate of the experiment is known).

making sure to exclude the capacitive charging current contribution. Instead of using hydro- gen absorption to probe electrochemically active surface area, CO can also be used (at least for pure Pt catalysts) since it reversibly saturates a Pt catalyst surface in a similar way.

An active catalyst area coefficient Ac may be calculated that represents the ratio of the measured active catalyst surface area compared to the active surface area of an atomically smooth catalyst electrode of the same size:

Ac = measured active catalyst surface area

geometric surface area =

Qh QmAgeometric

(7.29)

where Qm is the adsorption charge for an atomically smooth catalyst surface, generally accepted to be 210 μC∕cm2 for a smooth platinum surface.

As noted before, a highly porous, well-made fuel cell electrode may have an active sur- face area that is orders of magnitude larger than its geometric area. This effect is expressed through Ac.

7.4 EX SITU CHARACTERIZATION TECHNIQUES

While the direct in situ electrical characterization techniques are the most popular methods used to study fuel cell behavior, indirect ex situ characterization techniques can provide enormous additional insight into fuel cell performance. Most ex situ techniques focus on evaluating the physical or chemical structure of fuel cell components in an effort to identify which elements most significantly impact fuel cell performance. Pore structure, catalyst surface area, electrode/electrolyte microstructure, and electrode/electrolyte chemistry are among the most important characteristics to evaluate.

266 FUEL CELL CHARACTERIZATION

7.4.1 Porosity Determination

The porosity 𝜙 of a material is defined as the ratio of void space to the total volume of the material. To be effective, fuel cell electrodes and catalyst layers must exhibit substantial porosity. Furthermore, this pore space should be interconnected and open to the surface. Porosity determination is accomplished in several ways. First, if the density of a porous sample (ρs) can be determined by measuring its mass and volume, and the bulk density of the material used to make the sample is also known (ρb), then the porosity may be calculated as

𝜙 = 1 − 𝜌s

𝜌b (7.30)

For fuel cells, however, effective porosity is more important than total porosity. Effective porosity counts only the pore space that is interconnected and open to the surface. (In other words, dead pores are ignored.) Effective porosity can be determined using volume infil- tration techniques. For example, the total volume of a porous sample is first determined by immersing the sample in a liquid that does not enter the pores. For example, at low pressure, mercury will not infiltrate pore spaces due to surface tension effects. Then, the sample may be inserted into a container of known volume that contains an inert gas. The gas pressure in the container is noted, then a second evacuated chamber of known volume is connected to the system and the new system pressure is noted. Using the ideal gas law, the volume of open pores in the sample may be obtained and thus the effective porosity.

Pore size distributions may be obtained from mercury porosimetry. In this method, the porous sample is placed into a chamber, which is then evacuated. Mercury is then injected into the porous sample, first at extremely low pressure and then at steadily increasing pres- sures. The volume of mercury taken up at each pressure is noted. Mercury will enter a pore of radius r only when the pressure p in the chamber is

p ≥ 2𝛾 r

cos 𝜃 (7.31)

where γ is the surface tension of mercury and θ is the contact angle of mercury. Fitting this equation to the experimental mercury uptake pressure data allows approximate pore size distribution curves to be calculated.

7.4.2 BET Surface Area Determination

As discussed many times, the most effective fuel cell catalyst layers have extremely high real surface areas. Surface area determination, therefore, represents an important char- acterization tool. As you learned for CV, the electrochemically active surface area can be determined from specialized in situ electrochemical measurements. Additionally, the double-layer capacitance Cdl in impedance measurements may be used to roughly estimate surface areas based on the fact that a smooth reaction interface should have a capacitance of about 30 μF∕cm2. However, for the most accurate surface area determination, an ex situ technique known as the Brunauer–Emmett–Teller (BET) method is employed.

EX SITU CHARACTERIZATION TECHNIQUES 267

The BET method makes use of the fact that a fine layer of an inert gas like nitrogen, argon, or krypton will absorb on a sample surface at extremely low temperatures. In a typical experiment, a dry sample is evacuated of all gas and cooled to 77 K, the temperature of liquid nitrogen. A layer of inert gas will physically adhere to the sample surface, lowering the pressure in the analysis chamber. From the measured absorption isotherm of the experiment, the surface area of the sample can be calculated.

7.4.3 Gas Permeability

High surface area and high porosity accomplish nothing if the fuel cell electrode and catalyst structure exhibit low permeability. Permeability measures the ease with which gases move through a material. Even highly porous materials can have low permeability if most of their pores are closed or fail to interconnect. Fuel cell electrodes and catalyst layers should have high permeabilities. On the other hand, fuel cell electrolytes need to be gas tight. Permeability K is determined by measuring the volume of gas (ΔV) that passes through a sample in a given period of time (Δt) when driven by a given pressure drop (Δp = p1 − p2):

K = I Δp

− ΔV Δt

2p2 (p1 + p2)Δp

(7.32)

where I is a constant.

7.4.4 Structure Determinations

Significant information about microstructure, porosity, pore size distribution, and inter- connectedness is gleaned from microscopy. Optical microscopy (OM), scanning electron microscopy (SEM), transmission electron microscopy (TEM), and atomic force microscopy (AFM) are invaluable characterization techniques. Specific quantitative structural informa- tion can be provided from x-ray diffraction (XRD) measurements, which provide crystal structure, orientation, and chemical compound information. This information is extremely important when developing new electrode, catalyst, or electrolyte materials. Furthermore, XRD peak broadening measurements can provide information about particle size (in cat- alyst powder samples) or grain size (in bulk crystalline samples). Combined with TEM, XRD allows structural, chemical, and powder size distribution determinations for catalyst particles as small as 10 Å.

7.4.5 Chemical Determinations

When developing new catalyst, electrode, or electrolyte materials, it is always important to know what you have. Therefore, chemical determinations of composition, phase, bond- ing, or spatial distribution are just as important as structural determinations. For chemical determinations, TEM and XRD prove invaluable. In addition, other techniques like Auger electron spectroscopy (AES), x-ray photoelectron spectroscopy (XPS), and secondary-ion

268 FUEL CELL CHARACTERIZATION

mass spectrometry (SIMS) can provide useful information. While it is beyond the scope of this book to describe the advantages and disadvantages of these techniques, the interested reader is invited to consult the literature available on the subject.

7.5 CHAPTER SUMMARY

This chapter discussed many of the major techniques used to characterize fuel cells. We have seen that fuel cell characterization has two major goals: (1) to quantitatively separate good fuel cell designs from bad fuel cell designs and (2) to understand why fuel cell designs are good or bad.

• In situ electrical characterization techniques make use of the three fundamental elec- trochemical variables (voltage, current, and time) to probe fuel cell behavior.

• Ex situ characterization techniques focus on correlating the structure (porosity, grain size, morphology, surface area, etc.) or the chemistry (composition, phase, spatial distribution) of fuel cell components to fuel cell performance.

• The major in situ electrical characterization techniques are (1) j–V curve measure- ment, (2) electrochemical impedance spectroscopy (EIS), (3) current interrupt, and (4) cyclic voltammetry (CV).

• A careful j–V curve measurement yields the steady-state performance of a fuel cell under well-documented conditions. A fuel cell’s j–V performance is sensitive to the measurement procedure and test conditions. Fuel cell j–V curves can only be fairly compared if they are acquired using similar measurement procedures and testing conditions.

• Current interrupt, EIS, and CV measurements utilize the non-steady-state (dynamic) behavior of fuel cells to distinguish between the major processes that contribute to fuel cell performance.

• Current interrupt distinguishes ohmic and nonohmic fuel cell processes. The immedi- ate voltage rise after an abrupt current interruption is associated with ohmic processes, while the time-dependent voltage rise is associated with activation and mass transport processes. Current interrupt is fast and relatively easy to implement. It is especially attractive for high-power systems.

• In EIS, the impedance of a fuel cell system is measured over many orders of magnitude in frequency. A Nyquist plot of the resulting impedance data can be fit to an equivalent circuit model of the fuel cell. From this fit, the ohmic, activation, and mass transport losses in the fuel cell can often be resolved separately. Electrochemical impedance spectroscopy can be slow and requires sophisticated hardware. It is difficult to imple- ment for high-power systems.

• While the subject of impedance is complex (no pun intended), you should become familiar with the equivalent circuit models of common fuel cell components and the resulting impedance responses that these models produce.

CHAPTER EXERCISES 269

• In a standard CV measurement, the potential of a system is swept back and forth between two voltage limits while the current response is measured. In general, CV measurements are used to determine in situ catalyst activity, although they may also be used for detailed reaction kinetics analysis.

• Some of the more popular ex situ characterization techniques include porosity anal- ysis, surface area determination, permeability measurement, inspection microscopy (OM, SEM, TEM, AFM), and chemical analysis (XRD, AES, XPS, SIMS).

CHAPTER EXERCISES

Review Questions

7.1 What are the two main goals of fuel cell characterization?

7.2 List at least three major operation variables that can affect fuel cell performance (e.g., temperature). For each, provide what you believe is the most important equation that describes how fuel cell performance is affected by the variable in question.

7.3 Discuss the relative advantages and disadvantages of EIS versus current interrupt mea- surement.

7.4 A fuel cell’s j–V curve is acquired at two different scan rates: 1 and 100 mA∕s. (a) Which scan rate will result in better apparent performance? (Assume the scans

were acquired with increasing current starting at zero current.)

(b) Which portion of the j–V curve (low current density, moderate current density, high current density) will be most affected by the change in scan rate and why?

7.5 (a) Draw a schematic EIS curve for a fuel cell with one blocking electrode (repre- sented by a series RC) and one activated electrode (represented by a parallel RC). Assume that the RC product for the parallel RC is much smaller than the RC product for the series RC.

(b) Draw a schematic EIS curve for the scenario above if the RC product for the parallel RC is much greater than the RC product for the series RC.

(c) Draw a schematic EIS curve for a fuel cell modeled by two parallel RC elements, an ohmic resistance component, and a porous bounded Warburg element. Assume that the time constants of the two parallel RC elements are separated by at least two orders of magnitude.

7.6 Sketch an example material structure that has high porosity but low permeability.

Calculations

7.7 In Example 7.1 we calculated 𝜂ohmic, 𝜂act, i0, and α from the i–V and EIS data in points a and b of Figure 7.14. In this problem, calculate 𝜂ohmic, 𝜂act, i0, and α from the i–V and EIS data in points b and c of Figure 7.14. Assume that point c on the i–V

270 FUEL CELL CHARACTERIZATION

curve corresponds to i = 2.2 A and V = 0.45 V. Assume that the activation losses are wholly due to the cathode and T = 300 K and n = 2.

7.8 Calculate the approximate active platinum catalyst area coefficient from the CV curve in Figure 7.17 assuming that it was acquired from a 0.1 × 0.1-cm2 test electrode at a scan rate of 10 mV∕s.

7.9 True or False: Assuming that a fuel cell may be modeled by a simple parallel RC circuit, if the fuel cell resistance increases and the capacitance remains constant, the fuel cell current output will take a longer amount of time to transiently respond to an abrupt change in voltage.

7.10 True or False: In electrochemical impedance spectroscopy (EIS), the Warburg ele- ment is usually used to model the Butler–Volmer reaction kinetics response of a fuel cell.

7.11 From an electrochemical impedance spectroscopy (EIS) experiment, you determine that ηact = 0.2V at j = 0.5 A∕cm2 for the cathode of a PEMFC and that j0 = 1 × 10–3 A∕cm2. All else being equal, and assuming simple Tafel-type reaction kinetics, what would ηact for the cathode of this fuel cell be at j = 1 A∕cm2?

PART II

FUEL CELL TECHNOLOGY

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CHAPTER 8

OVERVIEW OF FUEL CELL TYPES

8.1 INTRODUCTION

As described in the first chapter of this book, there are five major types of fuel cells, differ- entiated from one another on the basis of their electrolyte:

1. Phosphoric acid fuel cell (PAFC)

2. Polymer electrolyte membrane fuel cell (PEMFC)

3. Alkaline fuel cell (AFC)

4. Molten carbonate fuel cell (MCFC)

5. Solid-oxide fuel cell (SOFC)

Many of the discussions and examples in the first part of this book focused on the PEMFC and the SOFC. Of all the fuel cell types, the PEMFC and the SOFC appear well positioned to deliver on the promise of the technology. Still, the other fuel cell classes have unique advantages, properties, and histories that make a succinct overview worthwhile. In the following sections we briefly discuss each of the five major fuel cell types. We will also briefly introduce a diverse set of exciting “nonstandard” fuel cell types and related electrochemical devices, which defy conventional classification. These include direct liquid-fueled fuel cells (such as direct methanol, direct formic acid, and direct borohydride fuel cells), biological fuel cells, membraneless fuel cells, metal–air cells, single-chamber SOFCs, direct-flame SOFCs, liquid-tin anode SOFCs, protonic ceramic fuel cells, reversible fuel cell/electrolyzers, and redox flow batteries. We conclude the chapter with a summary of the relative merits of each of the primary fuel cell types.

273

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274 OVERVIEW OF FUEL CELL TYPES

8.2 PHOSPHORIC ACID FUEL CELL

In the PAFC, liquid H3PO4 electrolyte (either pure or highly concentrated) is contained in a thin SiC matrix between two porous graphite electrodes coated with a platinum catalyst. Hydrogen is used as the fuel and air or oxygen may be used as the oxidant. The anode and cathode reactions are

Anode: H2 → 2H + + 2e−

Cathode: 1 2 O2 + 2H+ + 2e− → H2O

(8.1)

A schematic of a PAFC is provided in Figure 8.1. Figure 8.2 gives a photograph of a 200-kW stationary power commercial PAFC system. Pure phosphoric acid solidifies at 42∘C. Therefore, PAFCs must be operated above this temperature. Because freeze–thaw cycles can cause serious stress issues, commissioned PAFCs are usually maintained at operating temperature. Optimal performance occurs at temperatures of 180–210∘C. Above 210∘C, H3PO4 undergoes an unfavorable phase transition, which renders it unsuitable as an electrolyte. The SiC matrix provides mechanical strength to the electrolyte, keeps the two electrodes separated, and minimizes reactant gas crossover. During operation, H3PO4 must be continually replenished because it gradually evaporates to the environment (especially during higher-temperature operation). Electrical efficiencies of PAFC units are ≈ 40%with combined heat and power units achieving ≈ 70%.

Because PAFCs employ platinum catalysts, they are susceptible to carbon monoxide and sulfur poisoning at the anode. This is not an issue when running on pure hydrogen but can be important when running on reformed or impure feedstocks. Susceptibility depends on temperature; because the PAFC operates at higher temperatures than the PEMFC, it exhibits somewhat greater tolerance. Carbon monoxide tolerance at the anode can be as high as 0.5–1.5%, depending on the exact conditions. Sulfur tolerance in the anode, where it is typically present as H2S, is around 50 ppm (parts per million).

H3PO4 in SiC matrix

Porous graphite cathode

H+

H2 2H + + 2e-

+ 2e- + 2H+ H2O

Porous graphite anode

Pt/C catalyst

--O2 1 2

Figure 8.1. Schematic of H2–O2 PAFC. The phosphoric acid electrolyte is immobilized within a porous SiC matrix. Porous graphitic electrodes coated with a Pt catalyst mixture are used for both the anode and the cathode. Water is produced at the cathode.

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POLYMER ELECTROLYTE MEMBRANE FUEL CELL 275

Figure 8.2. Photograph of PureCell™ 200 power system, a commercial 200-kW PAFC. The unit includes a reformer, which processes natural gas into H2 for fuel. This system provides clean, reliable power at a range of locations from a New York City police station to a major postal facility in Alaska to a credit-card processing system facility in Nebraska to a science center in Japan. It also can provide heat for the building.

PAFC Advantages • Mature technology • Excellent reliability/long-term performance • Electrolyte is relatively low cost

PAFC Disadvantages • Expensive platinum catalyst • Susceptible to CO and S poisoning • Electrolyte is a corrosive liquid that must be replenished during operation

8.3 POLYMER ELECTROLYTE MEMBRANE FUEL CELL

The PEMFC is constructed from a proton-conducting polymer electrolyte membrane, usu- ally a perfluorinated sulfonic acid polymer. Because the polymer membrane is a proton conductor, the anode and cathode reactions in the PEMFC (like the PAFC) are

Anode: H2 → 2H + + 2e−

Cathode: 1 2 O2 + 2H+ + 2e− → H2O

(8.2)

A schematic diagram of a PEMFC is provided in Figure 8.3. Figure 8.4 gives a photo- graph of the system layout of a Hyundai ix35 fuel cell vehicle powered by PEMFCs.

The polymer membrane employed in PEMFCs is thin (20–200 μm), flexible, and trans- parent. It is coated on either side with a thin layer of platinum-based catalyst and porous

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276 OVERVIEW OF FUEL CELL TYPES

Porous

Polymer electrolyte

carbon cathode

H+

H2 2H + + 2e-

+ 2e- + 2H+ H2O

Porous carbon anode

Pt/C catalyst

--O2 1 2

Figure 8.3. Schematic of H2–O2 PEMFC. Porous carbon electrodes (often made from carbon paper or carbon cloth) are used for both the anode and the cathode. The electrodes are coated with a Pt catalyst mixture. Water is produced at the cathode.

carbon electrode support material. This electrode–catalyst–membrane–catalyst–electrode sandwich structure is referred to as a membrane electrode assembly (MEA). The entire MEA is less than 1 mm thick. Because the polymer membrane must be hydrated with liq- uid water to maintain adequate conductivity (see Chapter 4), the operating temperature of the PEMFC is limited to 90∘C or lower. Because of the low operating temperature, platinum-basedmaterials are the only practical catalysts currently available.While H2 is the fuel of choice, for low-power (< 1-kW) portable applications, liquid fuels such as methanol and formic acid are also being considered. One such liquid fuel solution, the direct methanol fuel cell (DMFC), is a PEMFC that directly oxidizesmethanol (CH3OH) to provide electric- ity. The DMFC is under extensive investigation at this time (2016). Some researchers assign these alternative-fuel PEMFCs their own fuel cell class. Later in this chapter, Section 8.7.1 provides additional information on the DMFC.

The PEMFC currently exhibits the highest power density of all the fuel cell types (500–2500 mW/cm2). It also provides the best fast-start and on–off cycling characteris- tics. For these reasons, it is well suited for portable power and transport applications. Fuel cell development at most of the major car companies is almost exclusively focused on the PEMFC.

PEMFC Advantages

• Highest power density of all the fuel cell classes • Good start–stop capabilities • Low-temperature operation makes it suitable for portable applications

PEMFC Disadvantages

• Uses expensive platinum catalyst • Polymer membrane and ancillary components are expensive • Active water management is often required • Very poor CO and S tolerance

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POLYMER ELECTROLYTE MEMBRANE FUEL CELL 277

(a)

(b)

Figure 8.4. (a) Rendering of the 2015 Hyundai ix35 fuel cell car power train. The PEMFC stack gen- erates 100 kW of electricity. The 24–kW Li-ion battery delivers a high rate of electrical energy to the motor during startup and acceleration and stores electricity recovered during braking. The drive train consists of a motor, transmission, and drive shaft, with the AC induction motor producing 100 kW maximum power and 302.8 N ⋅ m maximum torque. The inverter converts DC electric power from the fuel cell stack to AC electrical power for motive force. The high-pressure hydrogen tank can store 5.64 kg of hydrogen at 700 atm. The fuel economy of the vehicle is 0.95 kg H2 / 100 km, which means the car can travel 594 km with a full tank of hydrogen. Maximum speed of the car is 160 km/h and 0–100 km acceleration takes 12.5 s. (b) The Hyundai ix35 fuel cell car undergoing cold- and hot-weather testing. Beside the durability issue during the lifetime of the vehicle operation, PEMFCs face several other big challenges for automotive application. These include cold-start operation and cooling of the fuel cell stack. The water in the fuel cell stack and systemwill freeze under cold weather after the vehicle turns off. When turned on, a “frozen” fuel cell will not operate normally until the ice in the fuel cell melts. Through clever design and control of fuel cell systems, a state-of-the-art fuel cell engine can start even at –25∘C. Cooling of the fuel cell stack is also a big challenge. Since the ideal operating temperature of the PEMFC is around 80∘C, hot weather (∼45∘C) easily overloads the fuel cell cooling system because all the heat generated by the 100-kW fuel cell must be rejected by the cooling system even if the temperature difference is only 35∘C! Thus, as automotive manu- facturers continue to test out their fuel cell cars in exotic mountain or desert locations, they aren’t just having fun, they’re performing serious research! (Images courtesy of Hyundai Motor Company). (see color insert)

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278 OVERVIEW OF FUEL CELL TYPES

8.4 ALKALINE FUEL CELL

The AFC employs an aqueous potassium hydroxide electrolyte. In contrast to acidic fuel cells where H+ is transmitted from the anode to the cathode, in an alkaline fuel cell OH– is conducted from the cathode to the anode. The anode and cathode reactions are therefore

Anode: H2 + 2OH− → 2H2O + 2e−

Cathode: 1 2 O2 + 2e− + H2O → 2OH−

(8.3)

Thus, water is consumed at the cathode of an AFC while it is produced (twice as fast) at the anode. If the excess water is not removed from the system, it can dilute the KOH elec- trolyte, leading to performance degradation. A schematic diagram of an AFC is provided in Figure 8.5. Figure 8.6 gives a photograph of an AFC fuel cell unit that was used on the NASA Apollo missions.

For reasons that are still poorly understood, the cathode activation overvoltage in an AFC is significantly less than in an acidic fuel cell of similar temperature. Also, many more metal-based catalysts are stable in an alkaline environment. Thus, under some conditions, nickel (rather than platinum) catalysts can be used as the cathode catalyst. Because the ORR kinetics proceed much more rapidly in an alkaline medium than in an acidic medium, AFCs can achieve operating voltages as high as 0.875 V. Remember that a high operating voltage leads to high efficiency—an important point if fuel is at a premium.

Depending on the concentration of KOH in the electrolyte, the AFC can operate at tem- peratures between 60 and 250∘C. Alkaline fuel cells require pure hydrogen and pure oxygen as fuel and oxidant because they cannot tolerate even atmospheric levels of carbon dioxide. The presence of CO2 in an AFC degrades the KOH electrolyte as follows:

2OH− + CO2 → CO32− + H2O (8.4)

e- Pt/C or Ni catalyst

Porous carbon anode

Porous carbon cathode

Aqueous KOH electrolyte

OH- H2O

H2 + 2OH - 2H2O + 2e-

+ 2e- + H2O 2OH -

--O2 1 2

Figure 8.5. Schematic of an H2–O2 AFC. Porous carbon or nickel electrodes are used for both the anode and the cathode. Either Pt or nonprecious metal catalyst alternatives can be used. Water is produced at the anode and consumed at the cathode; therefore, the water must be extracted from the anode waste stream or recycled through the electrolyte, using electrolyte recirculation.

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ALKALINE FUEL CELL 279

Figure 8.6. Photograph of United Technologies Corporation(UTC) AFC. These fuel cell units sup- plied the primary electric power for the Apollo space missions. The units were rated to 1.5 kW with a peak power capability of 2.2 kW, weighed 250 lb, and were fueled with cryogenic H2 and O2. Fuel cell performance during the Apollo missions was exemplary. Over 10,000 h of operation were accumulated in 18 missions without an in-flight incident.

Over time, the concentration of OH– in the electrolyte declines. Additionally, K2CO3 can begin to precipitate out of the electrolyte (due to its lower solubility), leading to sig- nificant problems. These issues can be partially mitigated by the use of CO2 scrubbers and the continual resupply of fresh KOH electrolyte. However, both solutions entail significant additional cost and equipment.

Due to these limitations, the AFC is not economically viable for most terrestrial power applications. However, the AFC demonstrates impressively high efficiencies and power densities, leading to an established application in the aerospace industry. Alkaline fuel cells were employed on the Apollo missions as well as on the Space Shuttle orbiters.

Recently, a number of solid-polymer based alkaline electrolytemembranematerials have been developed that partially mitigate the CO2 instability issue associated with AFC oper- ation. Thus, a number of research initiatives are now reexamining the AFC for portable terrestrial applications.

AFC Advantages • Improved cathode performance • Potential for nonprecious metal catalysts • Low materials costs, extremely low cost electrolyte

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280 OVERVIEW OF FUEL CELL TYPES

AFC Disadvantages • Must use pure H2–O2 • KOH electrolyte may need occasional replenishment • Must remove water from anode

8.5 MOLTEN CARBONATE FUEL CELL

The electrolyte in the MCFC is a molten mixture of alkali carbonates, Li2CO3 and K2CO3, immobilized in a LiO–AlO2 matrix. The carbonate ion, CO3

2−, acts as the mobile charge carrier in the MCFC. The anode and cathode reactions are, therefore,

Anode: H2 + CO32− → CO2 + H2O + 2e−

Cathode: 1 2 O2 + CO2 + 2e− → CO2−3

(8.5)

In the MCFC, CO2 is produced at the anode and consumed at the cathode. Therefore, MCFC systems must extract the CO2 from the anode and recirculate it to the cathode. (This situation contrasts with the AFC, where CO2 must be excluded from the cathode.) The CO2 recycling process is actually less complicated than one might suppose. Typically, the waste stream from the anode is fed to a burner, where the excess fuel combusts. The resulting mix- ture of steam and CO2 is then mixed with fresh air and supplied to the cathode. The heat released at the combustor preheats the reactant air, thus improving the efficiency and main- taining the operating temperature of the MCFC. A schematic diagram of a MCFC is pro- vided in Figure 8.7. Figure 8.8 gives a photograph of a 2.5-MW pressurized MCFC system.

Molten carbonate in ceramic matrix

Porous nickel oxide

CO3 2-

Porous nickel/ chrome

CO2

+ CO2 + 2e- CO3 2-

H2 + CO3 2- CO2 + H2O + 2e-

--O2 1 2

Figure 8.7. Schematic of H2–O2 MCFC. The molten carbonate electrolyte is immobilized in a ceramic matrix. Nickel-based electrodes provide corrosion resistance, electrical conductivity, and cat- alytic activity. The CO2 must be recycled from the anode to the cathode to sustain MCFC operation since CO3

2– ions are otherwise depleted. Water is produced at the anode.

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MOLTEN CARBONATE FUEL CELL 281

(a)

(b)

Medium Temperature Water Supply

Medium Temperature Water Return

Distribution Panel

Electricity Supply

Ex-PLC

Fuel Supply

Fuel Supply Device Water Supply Pump

Water Supply

Hot Water Supply Pump

60˚C

120˚C

District Heating

End User

Heat Exchanger

2. 5

~ 2

.8 M

C om

m un

ic at

io n

MBOP

EBOP

STACK

Electricity Supply

Ex-PLC

Fuel Supply

Fuel Supply Device Water Supply Pump

Water Supply

Hot Water Supply Pump

H ea

t S up

pl y

60˚C

120˚C

District Heating

End User

Heat Exchanger

2. 5

~ 2

.8 M

C om

m un

ic at

io n

MBOP

EBOP

STACK

Figure 8.8. Photograph of a 2.5-MW MCFC system. (a) The system can power roughly 3500 indi- vidual homes using liquefied natural gas, biogas, or synthesized natural gas as fuel. The footprint of the system is 500 m2. (b) The system is composed of a fuel cell stack, an MBOP (Mechanical Balance of Plant), and an EBOP (Electrical Balance of Plant). The functions of the MBOP include treatment, preheating, and humidification of the fuel, air, and process water. The system supplies heated water for neighborhood or industrial use via the waste heat from the fuel cell. Through the EBOP, the fuel cell is connected to the electric grid, thereby providing electricity to the end user. The EBOP includes a DC/AC converter, power metering, switching equipment, and a voltage transformer. (Images courtesy of POSCO Energy.)

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282 OVERVIEW OF FUEL CELL TYPES

The electrodes in a typical MCFC are nickel based; the anode usually consists of a nickel/chromium alloy while the cathode consists of a lithiated nickel oxide. At both elec- trodes, the nickel provides catalytic activity and conductivity. At the anode, the chromium additions maintain the high porosity and surface area of the electrode structure. At the cathode, the lithiated nickel oxide minimizes nickel dissolution, which could otherwise adversely affect fuel cell performance.

The relatively high operating temperature (650∘C) of theMCFC provides fuel flexibility. TheMCFC can run on hydrogen, simple hydrocarbons (like methane), and simple alcohols. Carbon monoxide tolerance is not an issue for MCFCs; rather than acting as a poison, CO acts as a fuel!

Due to stresses created by the freeze–thaw cycle of the electrolyte during startup/ shut- down cycles, the MCFC is best suited for stationary, continuous power applications. The electrical efficiency of a typical MCFC unit is near 50%. In combined heat and power appli- cations, efficiencies could reach close to 90%.

MCFC Advantages • Fuel flexibility • Nonprecious metal catalyst • High-quality waste heat for cogeneration applications

MCFC Disadvantages • Must implement CO2 recycling • Corrosive, molten electrolyte • Degradation/lifetime issues • Relatively expensive materials

8.6 SOLID-OXIDE FUEL CELL

The SOFC employs a solid ceramic electrolyte. The most popular SOFC electrolyte mate- rial is yttria-stabilized zirconia (YSZ), which is an oxygen ion (oxygen vacancy) conductor. Since O2– is the mobile conductor in this case, the anode and cathode reactions are

Anode: H2 + O2− → H2O + 2e−

Cathode: 1 2 O2 + 2e− → O2−

(8.6)

In an SOFC, water is produced at the anode, rather than at the cathode, as in a PEMFC. A schematic of an SOFC is provided in Figure 8.9. Figure 8.10 is a photograph of an SOFC prototype.

The anode and cathode materials in an SOFC are different. The fuel electrode must be able to withstand the highly reducing high-temperature environment of the anode, while the air electrode must be able to withstand the highly oxidizing high-temperature

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SOLID-OXIDE FUEL CELL 283

Solid ceramic

electrolyte

Porous mixed conducting

oxide

- -

Porous nickel/YSZ

cermet H2 + O 2 H2O + 2e

+ 2e O2- --O2 1 2

Figure 8.9. Schematic of H2–O2 SOFC. The ceramic electrolyte is solid state. A nickel–YSZ cer- met anode and a mixed conducting ceramic cathode provide the required thermal, mechanical, and catalytic properties at high SOFC operating temperatures. Water is produced at the anode.

Figure 8.10. Photograph of Siemens-Westinghouse 220-kW hybrid SOFC/micro gas-turbine sys- tem. This system was delivered to Southern California Edison in May 2000.

environment of the cathode. The most common material for the anode electrode in the SOFC is a nickel–YSZ cermet (a cermet is a mixture of ceramic and metal). Nickel provides conductivity and catalytic activity. The YSZ adds ion conductivity, thermal expansion compatibility, and mechanical stability and maintains the high porosity and sur- face area of the anode structure. The cathode electrode is usually a mixed ion-conducting

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284 OVERVIEW OF FUEL CELL TYPES

and electronically conducting (MIEC) ceramic material. Typical cathode materials include strontium-doped lanthanum manganite (LSM), lanthanum–strontium ferrite (LSF), lanthanum–strontium cobaltite (LSC), and lanthanum–strontium cobaltite ferrite (LSCF). These materials show good oxidation resistance and high catalytic activity in the cathode environment.

The operating temperature of the SOFC is currently between 600 and 1000∘C. The high operating temperature provides both challenges and advantages. The challenges include stack hardware, sealing, and cell interconnect issues. High temperature makes the materi- als requirements, mechanical issues, reliability concerns, and thermal expansion matching tasks more difficult. Advantages include fuel flexibility, high efficiency, and the ability to employ cogeneration schemes using the high-quality waste heat that is generated. The elec- trical efficiency of the SOFC is about 50–60%; in combined heat and power applications, efficiencies could reach 90%.

An intermediate-temperature (400–700∘C) SOFC design could remove most of the dis- advantages associated with high-temperature operation while maintaining the most signif- icant SOFC benefits. Such SOFCs could employ much cheaper sealing technologies and robust, inexpensive metal (rather than ceramic) stack components. At the same time, these SOFCs could still provide reasonably high efficiency and fuel flexibility. However, there are still many fundamental problems that need to be solved before the routine operation of lower temperature SOFCs can be achieved.

SOFC Advantages • Fuel flexibility • Nonprecious metal catalyst • High-quality waste heat for cogeneration applications • Solid electrolyte • Relatively high power density

SOFC Disadvantages • Significant high-temperature materials issues • Sealing issues • Relatively expensive components/fabrication

8.7 OTHER FUEL CELLS

Fuel cells are a wonderfully rich and varied technology. Although we have so far classified fuel cells into five standard, or “classic,” types in this chapter, there are many other fuel cells that represent variants of the standard types or do not easily fall into the typical clas- sification. These nonstandard fuel cell types include direct liquid-fueled fuel cells (such as direct methanol, direct formic acid, and direct borohydride fuel cells), biological fuel cells, membraneless fuel cells, metal–air cells, single-chamber SOFCs, direct flame SOFCs, and liquid-tin anode SOFCs. This section briefly discusses these diverse and exciting nonstan- dard fuel cell technologies.

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OTHER FUEL CELLS 285

8.7.1 Direct Liquid-Fueled Fuel Cells

Direct liquid-fueled fuel cells produce electricity directly from liquid fuels such as methanol, ethanol, formic acid, or borohydride solutions. In these cells, a liquid fuel is supplied directly to the fuel cell, where it is electrochemically oxidized to H2O and other products, and electricity is generated. Direct operation on liquid fuels is attractive because of the exceptionally high-energy density and convenience of liquid fuels. For low-temperature fuel cells (PEMFC), only relatively simple liquid fuels such as lower alcohols (methanol, ethanol), formic acid, or borohydride solutions (which release hydrogen gas in the presence of fuel cell catalysts) can be used. Even with these relatively simple fuels, electrochemical reactivity is significantly more sluggish than for hydrogen, and therefore direct liquid fuel cells tend to exhibit poor power density and poor efficiency.

The prototypical direct liquid fuel cell is the direct methanol fuel cell (DMFC). The methanol electro-oxidation reaction in acidic electrolytes, such as the PEMFC environment, is

CH3OH + H2O → CO2 + 6H+ + 6e−

A fuel cell running on methanol requires water as an additional reactant at the anode. It produces CO2 at the anode as a waste product.

DMFCs have been widely investigated as portable power sources to replace recharge- able batteries due to the high energy density of methanol fuel. Figure 8.11 shows recent prototypes of portable DMFCs.

As with the H2–O2 PEMFC, the best catalysts for low-temperature methanol fuel cells are Pt based. Unfortunately, the j0 values for the methanol reaction are quite low, result- ing in large activation overvoltage losses at the anode and the cathode. The low exchange current density reflects the complexity of the methanol oxidation reaction. The reaction occurs by many individual steps, several of which can lead to the formation of undesirable intermediates, including CO, which acts as a poison.

Carbon monoxide tolerance is provided by alloying the Pt catalyst with a secondary component such as Ru, Sn, W, or Re. Ruthenium is considered to be most effective at providing tolerance. It creates an adsorption site capable of forming OHads species. These OHads species react with the bound CO species to produce CO2, thereby removing the poi- son. Current DMFCs exhibit power densities of about 30–100 mW/cm2. (Compare this to H2 PEMFC power densities of 500–2500 mW/cm

2.) In addition to the considerable acti- vation overvoltage losses at the anode, DMFCs suffer from significant methanol crossover through the electrolyte.

In order to overcome some of these shortcomings, researchers are also investigating alkaline-based direct methanol and direct ethanol fuel cells. In an alkaline environment, the methanol electro-oxidation reaction is

CH3OH + 6OH− → CO2 + 5H2O + 6e−

While at the cathode, the oxygen reduction proceeds as

3 2 O2 + 3H2O + 6e− → 6OH−

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286 OVERVIEW OF FUEL CELL TYPES

(a)

(b)

Figure 8.11. Recent example prototype portable DMFC systems. (a) A 20-WDMFC notebook com- puter charger can directly power a notebook or recharge the battery in the computer to extend the operating time. The methanol cartridge (the small box detached from the 540-cm3 main unit at the back of the computer) stores 130cm3 of pure methanol fuel. The system provides up to 160 Wh with an overall energy density of 230 Wh/L. (b) A 2-W prototype DMFC system can charge a cell phone in 2 h using a 10-cm3 methanol fuel cartridge. The system occupies roughly 150 cm3.

The methanol electro-oxidation and oxygen reduction kinetics are considerably improved in alkaline environments compared to acidic environments, leading to much better performance. Furthermore, there are a number of non-platinum catalysts such as nickel, silver, and various chevrel phase chalcogenides (which contain molybdenum, usually with selenium) that show excellent potential for alkaline-based direct alcohol fuel cells. However, as was discussed previously in this chapter, the switch from an acidic PEMFC fuel cell to an alkaline fuel cell also brings new concerns and challenges, including issues with CO2 degradation of the electrolyte.

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OTHER FUEL CELLS 287

In addition to the direct methanol and direct ethanol fuel cell, other liquid-fueled fuel cells have also been developed. One of them is the direct formic acid fuel cell [42]. Formic acid (HCOOH), like methanol, is a liquid at room temperature and can be directly used in a fuel cell, obviating the need for complicated external reforming. The reactions in a direct formic acid fuel cell are

Anode: HCOOH → CO2 + 2H+ + 2e−

Cathode: 1 2 O2 + 2H+ + 2e− → H2O

Overall: HCOOH + 1 2 O2 → CO2 + H2O

Like direct methanol fuel cells, most direct formic acid fuel cells are based on PEMFC technologies, although different anode catalysts (often Pd-based rather than Pt-based) are typically used.

Another liquid-fueled fuel cell is the direct borohydride fuel cell. The direct borohy- dride fuel cell uses a solution of sodium borohydride (NaBH4) or, alternatively, ammonium borohydride (NH4BH4) for fuel. Direct borohydride fuel cells are based on alkaline fuel cell technology because the borohydride fuel itself is highly alkaline. The waste product generated by the direct sodium borohydride cell (NaBO2 = borax) protects the cell from CO2 poisoning, thus obviating the CO2 concerns associated with most alkaline fuel cell arrangements. The reactions in a direct sodium borohydride fuel cell are

Cathode: 2O2 + 4H2O + 8e− → 8OH−

Anode: NaBH4 + 8OH− → NaBO2 + 6H2O + 8e−

Overall: NaBH4 + 2O2 → NaBO2 + 2H2O

The theoretical open cell voltage of the direct sodium borohydride fuel cell is 1.64 V (at STP). The sodium borohydride fuel has an extremely high energy density in its dry (pow- dered) form. However, it is typically mixed with water and KOH to create a liquid solution, which is delivered to the cell, lowering the energy density of the fuel but improving the ease of implementation. The borohydride fuel cell is unique because a solid waste product, NaBO2 (borax), is created. Borax is a common detergent and soap additive and is relatively nontoxic. It is also soluble in a water–KOH mixture and thus can be dissolved and flushed from the cell by the circulating fuel stream.

It must be understood that direct liquid-fueled fuel cells are very different from liquid-fueled reformer + fuel cell systems. In a liquid-fueled reformer + fuel cell system, liquid fuel is first supplied to a reformer, which generates H2 and CO2, and then the H2 is passed on to a conventional fuel cell to produce electricity and H2O. For example, an indirect sodium borohydride fuel cell can be created by feeding NaBH4 to a catalytic reactor, generating H2, which can then be supplied to a conventional PEMFC. In this approach, the reformer reaction would be

Reformer Reaction: NaBH4 + 2H2O → NaBO2 + 4H2

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288 OVERVIEW OF FUEL CELL TYPES

The hydrogen can then be used in a conventional H2/O2 PEMFC fuel cell according to

Anode: H2 → 2H + + 2e−

Cathode: 1∕2O2 + 2H+ + 2e− → H2O

Both the direct approach and the reformer approach have strengths and weaknesses. These trade-offs are discussed in more detail in Chapter 10, Section 10.3.2.

8.7.2 Biological Fuel Cells

Biological fuel cells are fuel cell devices that use living cells, biological catalysts, microor- ganisms, and/or enzymes to convert chemical energy (often contained in the form of lower alcohols, such as methanol, or simple sugars, such as glucose) into electrical energy [43]. In these fuel cells, power is generated directly from a biofuel by the catalytic activity of the bacteria or enzymes.

Like any other fuel cell, a functioning biological fuel cell must have an anode elec- trode and a cathode electrode separated by an electrolyte membrane. In most biological fuel cells, the anode is supplied with fuel (typically glucose, although demonstrations have even used “wastewater”) in the absence of oxygen. Under anaerobic (zero-oxygen) condi- tions, microorganisms at the anode can oxidize sugars to produce carbon dioxide, protons, and electrons, as shown in the following equation:

C12H22O11 + 13H2O → 12CO2 + 48H+ + 48e−

These electrons can then be harvested either directly (this is called the mediator-free approach) or indirectly by using inorganic mediators (this is called the mediator approach).

The mediator-free approach uses bacteria that contain electrochemically active redox enzymes such as cytochromes on their outer membrane [44]. Examples of such bacteria include Shewanella putrefaciens and Aeromonas hydrophila. These bacteria can deliver the electrons that they have generated by the oxidation of fuel directly to a metallic (or graphitic) electrode.

The mediator approach uses redox active dye molecules (such as thionine, methyl blue, humic acid, or neutral red), which can exist in both oxidized and reduced states, to mediate electron liberation in the anode compartment. In most biological fuel cells, a mediator approach is required because very few bacteria or microorganisms will yield their electrons directly to a metallic or graphitic electrode. Instead, an intermediate species is needed to “steal” electrons from the bacteria that generate them and then shuttle them to the anode electrode.

As in any other fuel cell, electrons collected in the anode compartment are sent through an external circuit (doing useful work), before recombining in the cathode with protons (delivered across the electrolyte membrane) and an oxidizing species to complete the cir- cuit. The oxidant at the cathode can be oxygen, in which case the cathode of a biological fuel cell can look very similar to any other fuel cell. However, because the introduction of large volumes of circulating gas to the cathode often proves difficult in biological fuel cell experiments, many researchers choose to introduce high concentrations of a strong chemical oxidizing agent into the cathode as a surrogate to oxygen.

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OTHER FUEL CELLS 289

8.7.3 Membraneless Fuel Cells

The membraneless fuel cell exploits laminar flow in microfluidic channels to create a fuel cell that does not require an electrolyte membrane. Figure 8.12 provides a schematic example of a membraneless fuel cell design. As shown in the schematic, a Y-shaped microfluidic channel is used to merge two liquid streams, one containing an oxidant/electrolyte solution, the other containing a fuel/electrolyte solution. The oxidant and fuel streams flow in a laminar regime and therefore do not mix [45]. As in any other fuel cell, fuel is oxidized at the anode electrode, releasing protons and electrons. Electrons are sent through an external circuit, producing useful work, while the protons transport across the laminar flow stream to the cathode, where they react with electrons and oxidant, completing the circuit. Microscale channel width dimensions are needed to ensure laminar flow conditions. The laminar flow condition prevents fuel and oxidant streams from mixing turbulently. However, fuel and oxidant will be depleted near the electrodes and will begin

Fuel/electrolyte solution

O2/electrolyte

solution

C athode

A node

Depletion zone

Diffusion (mixing) zone

Figure 8.12. A membraneless fuel cell design based on a Y-shaped microfluidic channel configu- ration that places fuel and oxidant streams into diffusional contact without mixing. The left-hand oxygenated electrolyte stream passes over a cathode electrode, while the right-hand fuel-saturated electrolyte stream passes over the anode. Protons can transport across the stream, but fuel and oxy- gen do not mix because of the laminar flow. However, fuel and oxidant will be depleted near the electrodes and will begin to mix in the center region by diffusion; these two processes set a maximum effective length for the fuel cell (typically micrometers to millimeters).

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290 OVERVIEW OF FUEL CELL TYPES

to mix in the center region by diffusion; these two processes set a maximum effective length for the fuel cell (typically micrometers to millimeters).

Membraneless fuel cells are extremely simple and can be very compact. This makes them potentially intriguing for microscale power source applications. However, membrane- less fuel cells generally exhibit low power densities and poor efficiencies. Furthermore, operation requires constant flow of both an oxygenated cathode electrolyte stream and a fuel-enriched anode electrolyte stream. Without the implementation of an effective fluid recycling stream, the reservoir volumes required for these two streams would likely render membraneless fuel cells impractical for most applications.

8.7.4 Metal–Air Cells

The metal–air cell is essentially halfway between a fuel cell and a battery. The anode “fuel” in ametal–air cell is a solid (powdered), highly reactivemetal. Thus, like batteries, metal–air cells have a limited life and are depleted once this solid-metal fuel is expended. However, unlike batteries, metal–air cells have an oxygen-breathing cathode rather than a second metal electrode. In this sense, they are similar to fuel cells. Metal–air cells exploit the high electrochemical reactivity between the anode metal and oxygen from air to produce electricity. Because air is used at the cathode instead of a second heavy metal electrode, metal–air cells can achievemuch higher energy density thanmost batteries. However, power densities tend to be modest, so metal–air cells are best used for low-current/low-power applications.

Typical metal–air cells use zinc, aluminum, or magnesium as fuel. Figure 8.13 schemati- cally illustrates the operating principle of a zinc–air cell. The zinc–air cell consists of a zinc anode, an aqueous alkaline electrolyte (typically KOH), and a highly porous, electrically conductive, air-breathing cathode (typically carbon-based with oxygen reduction catalysts).

Zinc anode

KOH electrolyte (in porous matrix)

Porous cathode

Insulating gasket

Figure 8.13. Schematic diagram of a zinc–air cell. A zinc metal anode and a porous air-breathing cathode are separated by a porous, KOH electrolyte saturated membrane. Oxygen from the air reacts with the zinc metal anode to create ZnO, producing electricity in the process. The anode and cathode electrodes are typically housed inside a two-piece coin-cell arrangement, with electrical isolation and sealing provided by an insulating ring gasket.

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OTHER FUEL CELLS 291

At the anode, zinc is oxidized by OH– ions in the electrolyte to create ZnO and electrons are liberated:

AnodeReaction: Zn + 2OH− → ZnO + H2O + 2e− (8.7)

The electrons pass through an external circuit, providing useful work, before recombin- ing at the cathode with O2 and water to produce fresh OH

– ions:

CathodeReaction: 1 2 O2 + H2O + 2e− → 2OH− (8.8)

The OH– ions transport back through the electrolyte, thus completing the circuit:

Overall Reaction: 1 2 O2 + Zn → ZnO (8.9)

The cell voltage of a zinc–air cell is typically 1.4 V, slightly higher than an H2/O2 fuel cell. The zinc–air cell produces electrical power until the zinc anode is depleted or until so much ZnO has built up that access to fresh zinc is blocked. Metal–air technology is currently used for a number of low-power applications, including batteries for hearing aids and high-capacity (at low current) batteries for long-lifetime sensor applications.

8.7.5 Single-Chamber SOFC

The single-chamber SOFC is a type of solid-oxide fuel cell that is designed to operate in a single chamber where both the fuel and air are supplied in combination [46]. An example of a single-chamber SOFC design is illustrated in Figure 8.14. Successful operation of a single-chamber SOFC requires highly selective anode and cathode electrodes: An anode material must be chosen that only oxidizes fuel (and ignores the oxygen), while a cathode material must be chosen that only reduces oxygen (and ignores the fuel). If a material like platinum is used in a single-chamber SOFC, no electricity will be produced, since plat- inum catalyzes both the oxidation of fuel and the reduction of oxygen. Platinum electrodes will simply cause the fuel + air mixture to burn. Other common SOFC electrode materials, such as Ni–YSZ (the common SOFC anode material) and LSM (the common SOFC cath- ode materials) also cannot be used in single-chamber SOFCs because they are not selective enough. However, several highly selective cathode and anode materials have been devel- oped, permitting the demonstration of actual working single-chamber SOFC prototypes. Typical selective electrode materials include Ni–GDC (GDC = gadolinium-doped ceria) cermets for the anode and Sm0.5Sr0.5CoO3-x (SSC) for the cathode.

Single-chamber SOFC designs offer several compelling advantages. Single-chamber designs are simple and require no high-temperature seals. The electrolyte no longer needs to be gastight (it must only electrically separate the anode and cathode electrodes), sig- nificantly relaxing electrolyte fabrication requirements. Size reduction/miniaturization is facilitated by the intrinsic simplicity of single-chamber design and reduced gas manifolding requirements. However, single-chamber SOFCs also impose several serious limitations.

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292 OVERVIEW OF FUEL CELL TYPES

Electrolyte

Selective anode

Selective cathode

Fuel + Air

Fuel

Figure 8.14. Operating principle of the single-chamber SOFC. The single-chamber SOFC employs a selective anode that only reacts with fuel species and a selective cathode that only reacts with oxygen. Because of this selectivity, both fuel and air can be simultaneously introduced into a single chamber, greatly simplifying fuel cell design and sealing.

The risk of fuel–air mixture explosions is significant. Therefore, most single-chamber SOFC designs are operated on very dilute (typically < 4%) fuel mixtures, decreasing per- formance. Electrode materials are never 100% selective, and parasitic non-electrochemical reactions will therefore reduce fuel utilization and decrease efficiency. In spite of these limitations, because single-chamber SOFCs offer compelling design simplifications, they remain an intriguing area of continuing research.

8.7.6 Direct Flame SOFC

The direct flame SOFC concept [47], illustrated in Figure 8.15, is based on the combination of a combustion flame with an open (“no-chamber”) solid oxide fuel cell. In the direct flame SOFC, a fuel-rich flame is placed a few millimeters away from the anode. The fuel-rich flame provides partially oxidized/reformed fuel species to the anode, while at the same time providing the heat required for SOFC operation. The cathode is freely exposed to ambient air. As long as the cell is somewhat larger than the flame, no sealing is required and the device can be operated in a no-chamber configuration.

The direct flame SOFC offers a number of intriguing advantages. First, the system is fuel flexible. Because intermediate flame species are similar for all kinds of hydrocarbons, the cell can be operated on virtually any carbon-based fuel. Second, the cell is remarkably simple. The anode is simply held in the exhaust gases close to a fuel-rich flame, while the cathode breathes ambient air. The system is thermally self-sustained and there are no seal- ing requirements. Finally, system start-up is rapid—typically within seconds depending on the thermal mass of the fuel cell. Disadvantages include low-efficiency, low-power density,

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OTHER FUEL CELLS 293

Partially combusted products (e.g., H2, CO)

Air

Anode

Electrolyte

Cathode

Burner

Flame: combustion chemistry and heat release

Figure 8.15. Schematic illustration of the direct flame fuel cell. A direct flame fuel cell is designed to operate in a “zero-chamber” mode, where the anode side is exposed to a flame combustion source, which provides both heat and partially combusted fuel species, while the cathode faces the ambient air.

and issues with coking (depending on the flame chemistry) and thermal shock (due to rapid thermal cycles). Nevertheless, the direct flame fuel cell might have interesting applications in emergency or recreational activities. Imagine, for example, a direct flame fuel cell pro- ducing electric power from your campfire!

8.7.7 Liquid-Tin Anode SOFC

The liquid-tin anode solid-oxide fuel cell (LTA-SOFC), developed by CellTech Power of Westborough, Massachusetts [48], is an intriguing variant on the SOFC that allows the direct conversion of almost any carbonaceous fuel. The primary advantage of the LTA-SOFC is its remarkable ability to run on almost any fuel—including biomass, JP8 (a sulfur-rich military logistics fuel), coal, woodchips, even plastic bags! The LTA-SOFC uses conventional SOFC electrolytes and cathodes but employs an anode based on liquid tin. The liquid-tin anode is the key feature of the LTA-SOFC. It allows direct oxidation of almost any carbon-containing fuel without reforming or other fuel processing. Furthermore, the liquid-tin anode is surprisingly durable—it is not harmed by coking, and it is not poisoned by sulfur (sulfur instead can be used as a fuel).

The basic operation of the LTA-SOFC is illustrated in Figure 8.16. The LTA-SOFC works by using a Sn/SnO2 redox couple to oxidize fuel species. At the anode–electrolyte interface, the liquid tin is oxidized to SnO2. The SnO2 is then transported to the anode–fuel interface, where it is reduced in the presence of fuel, back to Sn. This reduction process is apparently facile and versatile, as many different fuel species can be reduced, includ- ing S (reduced to SO2), C and CO (reduced to CO2), hydrogen (reduced to H2O), and

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294 OVERVIEW OF FUEL CELL TYPES

Porous anode separator

Liquid tin anode

YSZ electrolyte Porous

cathode

O2-

SnO

H2 or hydro- carbon fuel

H2O, CO2, SO2

e-e-

+ 2e- O2-

O2 (from air)

SnO + Fuel Sn + H2O, CO2, SO2, etc.

Sn + O2- SnO + 2e-

--O2 1 2

Figure 8.16. Operating principle of the liquid-tin anode SOFC (LTA-SOFC). Based on a conven- tional SOFC electrolyte and cathode, the LTA-SOFC employs liquid tin for the anode, enabling direct utilization of virtually any hydrocarbon species. The liquid tin functions as a reaction “intermediate” by undergoing a redox cycle, converting to SnO2 at the liquid tin–YSZ interface, then reducing back to Sn at the liquid tin–fuel interface.

hydrocarbons (reduced to CO2 + H2O). While the LTA-SOFC is still under preliminary development, it appears to be an extremely attractive technology for fuel-flexible power gen- eration applications. Currently, the required operation temperature is quite high (> 900∘C), power densities remain low, and lifetime/durability issues must be investigated further.

8.7.8 Protonic Ceramic Fuel Cells

Recently, protonic ceramic fuel cells (PCFCs) have become of great interest in the fuel cell research community. PCFCs are based on solid-state ion-conducting oxide electrolytes. However, unlike SOFCs, which are based on oxygen-ion-conducting ceramic electrolytes, PCFCs are based on proton-conducting ceramic electrolytes. PCFCs share many character- istics in common with SOFCs. They operate at relatively high temperatures (usually greater than 500∘C), they can enable operation on non-hydrogen fuels, and they are generally made from relatively inexpensive oxide materials (requiring little or no precious metal catalysts). Like PEMFCs, however, PCFCs produce water at the cathode. This means that the anode fuel is not diluted by product water gas, enabling potential gains in cell operating voltage and efficiency. This stands in contrast to SOFCs, where water is produced at the anode and consequently dilutes the anode fuel stream.

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OTHER FUEL CELLS 295

The most common proton-conducting ceramic electrolytes include acceptor-doped per- ovskite compositions based on BaZrO3 and BaCeO3. Like SOFC electrolytes, PCFC elec- trolytes require elevated temperatures to facilitate ion conduction, since the hopping process associated with ionic motion in these materials has a relatively high activation energy. How- ever, because protons are lighter and are generally more weakly bound than oxygen ions in these materials, reasonable ionic conductivity can be achieved in PCFC electrolytes at much lower temperatures than in SOFC electrolytes. Thus, researchers are currently design- ing and studying PCFC devices that can operate at temperatures as low as 350∘C! PCFC electrolyte materials are further described in Chapter 9 of this textbook.

Currently, the greatest limitation to PCFCs is the development of new electrode mate- rials (especially new cathode materials) that work at lower temperatures and are com- patible with PCFC electrolytes. Early PCFCs used the same electrode materials devel- oped for SOFCs, but this generally resulted in poor performance. It has become clear that oxygen-ion-conducting electrode materials designed for SOFCs operating at 800–1000∘C generally provide poor performance when matched with proton-conducting electrolytes operating at 500∘C. A variety of new, mixed protonic-and-electronic conducting oxide materials are currently being developed as electrodes for PCFCs.

8.7.9 Solid-Acid Fuel Cells

Solid-acid fuel cells (SAFCs) use a solid proton-conducting electrolyte based on an inor- ganic acid salt (“a solid acid”). Chemically, solid acids can be thought of as in-between normal salts and normal acids. For example, if sulfuric acid (H2SO4) is reacted with cesium sulfate (Cs2SO4) salt, the solid acid CsHSO4 is produced:

1 2 H2SO4 +

1 2 Cs2SO4 → CsHSO4 (8.10)

CsHSO4 is the prototypical solid acid used in most SAFCs. At room temperature, the structure of most solid acids like CsHSO4 is highly ordered and crystalline. Under these conditions, they are poor ionic conductors. However, at slightly higher temperatures (typ- ically between 50 and 150∘C) they undergo a “superprotonic phase transition” where the onset of structural disorder enables a dramatic increase in the proton conductivity (by two to three orders of magnitude). Because most solid acids do not decompose until temperatures >250∘C, they can be used as excellent fuel cell electrolytes in the temperature window between the onset of the superprotonic phase transition and the onset of decomposition. Thus, SAFCs enable operation of high-performance PEM-like fuel cells at temperatures greater than 100∘C. Haile et al. at the California Institute of Technology have largely been responsible for the development of SAFC technology over the last 15 years. Some addi- tional information on solid-acid electrolytematerials is provided in Chapter 9, Section 9.1.5.

Because SAFCs can operate at intermediate temperatures (100–200∘C), they combine many of the advantages of PEMFCs and PAFCs. Like PEMFCs, they are based on a solid electrolyte that can be made thin and is (relatively) mechanically strong. Like PAFCs, the higher operating temperature of the SAFC enables somewhat greater tolerance for CO and other fuel-stream impurities. Indeed, the company SAFCell, which is currently working to

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296 OVERVIEW OF FUEL CELL TYPES

commercialize SAFC technology, has demonstrated SAFCs running on propane as well as reformed diesel fuel. The main issues associated with the SAFC include preventing degra- dation of the solid-acid electrolyte during long-term operation and decreasing the amount of precious metal catalyst needed in the electrodes.

8.7.10 Redox Flow Batteries

A redox (reduction–oxidation) flow battery is a rechargeable battery that uses liquid fuel and liquid oxidant. A redox flow battery is very similar to a fuel cell since it stores the fuel and oxidant in separate tanks outside of reaction cells. Liquid fuel from the fuel tank is pumped to the anode and undergoes oxidation (the fuel is stripped of electrons). On the other side of the device, the liquid oxidant undergoes reduction (gains electrons) at the cathode. Depleted fuel and oxidant are sent back to the same storage tanks after the reaction. Therefore, each tank stores a mixture of fresh fuel (or oxidant) and used fuel (or oxidant).

A key feature of the redox flow battery is the reversibility of the reaction, which enables these systems to be rechargeable. While fuel and oxidant keep flowing through the anode and cathode, the direction of the electron flow can be reversed (from discharge to charge) by applying a charging voltage to the cell. This reverses the half-cell reactions in the anode and cathode. During charging, depleted fuel (or oxidant) can therefore be reconverted to fresh fuel (or oxidant). Ensuring reversibility in redox flow batteries requires a clever selection of fuel and oxidant chemistries. One famous example is the all-vanadium redox flow battery system. Vanadium is a transition metal that can exist in many different oxidation states (e.g., V5+, V4+, V3+, V2+). When vanadium oxide (V2O5) is dissolved in sulfuric acid (H2SO4), all four vanadium oxidation states can exist in the aqueous electrolyte in the form of VO2

+, VO2+, V3+ and V2+. By using this electrolyte as both the liquid fuel and liquid oxidant, the following half-cell reactions can be exploited at the anode and the cathode, respectively:

Anode: V2+ → V3+ + e− (8.11)

Cathode: VO2 + + 2H+ + e− → VO2+ + H2O (8.12)

Then, the overall reaction becomes

V2+ + VO2+ + 2H+ → V3+ + VO2+ + H2O (8.13)

Here, water and protons are required to maintain the charge balance (they are provided by the sulfuric acid electrolyte). The reactions are reversed in charge mode. You can eas- ily see that the reaction requires the exchange of protons between the anode and cathode. Therefore, a proton exchange membrane is placed between the anode and the cathode. This makes the redox flow battery very similar to a PEMFC in principle even though the cell structure and materials are different. Challenges associated with redox flow batteries include system complexity as well as low energy density and power density. Nevertheless, commercialization is now underway for several large-scale energy storage systems and back-up power supply systems (with sizes up to 1MW in power and several MWh in energy storage) due to the relatively cheap price of redox flow batteries compared to common solid-state secondary batteries such as lithium-ion batteries.

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OTHER FUEL CELLS 297

8.7.11 Electrolysis and Reversible Fuel Cell–Electrolyzers

The electrolysis reaction is typically a fuel cell reaction run in the reverse direction. In water electrolysis, an electric current is applied to split water molecules into oxygen and hydro- gen. This overall reaction is the reverse of the hydrogen–oxygen fuel cell reaction. In an electrolysis cell, the names for the positive and negative terminals are reversed as compared with that for a fuel cell, such that the positive terminal is the anode and the negative termi- nal is the cathode. This apparent “reversal” of the electrode nomenclature should not cause confusion if one recalls that electrons always flow into the cathode and out of the anode. This applies for fuel cells, batteries, electrolysis cells, etc. When the source of electricity is renewable power, water electrolysis can be one method for producing renewable hydro- gen. Electrolyzers based on water electrolysis are in small-scale commercial use today to provide oxygen onboard submarines and hydrogen for specific segments of the merchant hydrogen market that require high-purity hydrogen.

In PEM water electrolysis, the PEM fuel cell reaction is run in reverse. As shown in Figure 8.17, the anode and cathode reactions are

Anode: H2O → 1 2 O2 + 2H+ + 2e−

Cathode: 2H+ + 2 e− → H2 (8.14)

At the positive terminal (anode), water reacts to form oxygen molecules, protons, and electrons. The PEM electrolyte conducts protons across it. An external power source is applied to drive electrons to flow through an external circuit from the positive terminal (anode) to the negative terminal (cathode). At the negative terminal (cathode), electrons that have traversed the external circuit combine with the protons that have been conducted through the electrolyte to produce hydrogen. The overall reaction is

Overall: H2O → 1 2 O2 + H2 (8.15)

Chapter 4, Section 4.5.2, discussed how, in PEM fuel cells, protons drag water with them across the electrolyte. Similarly, in a PEM electrolyzer, water may be transported across the

--O2 1 2

Figure 8.17. Schematic diagram of a single cell of a PEM water electrolyzer that electrochemically converts water to hydrogen and oxygen.

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298 OVERVIEW OF FUEL CELL TYPES

electrolyte as well. As a result, the hydrogen exhaust stream at the cathode may need to be dehumidified or dried prior to storage at high pressures.

Similar to the design of PEM fuel cells, several individual PEM electrolysis cells are typically connected together into an electrolyzer stack, so as to create hydrogen and oxy- gen in large enough quantities to be useful. Also, similar to the system needs discussed in Section 10.1 for PEM fuel cell stacks, a PEM electrolyzer stack requires several subsys- tems to manage mass and energy flows into and out of the stack. Within a PEM electrolyzer system, these subsystems may include

1. a subsystem for storing, purifying, and delivering purified water to the anode,

2. a subsystem for managing power electronics, including controllers and sensors for the stack and conversion of AC power from the grid to DC power for use in the stack (i.e., a rectifier),

3. a subsystem for storing oxygen gas produced at the anode, and

4. a subsystem for dehumidifying and storing hydrogen produced at the cathode.

Based on surveys of PEM electrolyzer manufacturers, studies have estimated sys- temwide electrical efficiencies to be about 54 kWh of electricity per kilogram of hydrogen (kg H2) in the near term and 50 kWh/kg H2 for future systems [48a 48b]. These estimates are for PEM electrolyzers that produce and store hydrogen only and vent oxygen to the atmosphere.

Some system designs for electrolyzers also allow the same electrolyzer device to oper- ate as a fuel cell. These devices are referred to as reversible fuel cell–electrolyzers. The reversible fuel cell–electrolyzer can be used as a fuel and oxidant storage device when oper- ated as an electrolyzer and as a power system when operated as a fuel cell. The same hard- ware is used for the electrochemical stack, but the direction of current flow changes between electrolyzer and fuel cell operation. Reversible fuel cell–electrolyzers can be benchmarked against systems that combine a separate electrolyzer as one piece of hardware and a separate fuel cell device as another piece of hardware. Compared with these systems, the reversible fuel cell–electrolyzer may exhibit a lower electrical efficiency and lifetime but is expected to have a lower mass and volume. In other words, the reversible fuel cell–electrolyzer is expected to have a higher gravimetric and volumetric energy density, concepts discussed in greater detail in Chapter 10. These design features may be especially important to space flight and aeronautical applications.

8.8 SUMMARY COMPARISON

Currently, none of the fuel cell types is ready for widespread mass-market commercial application. Until significant cost, power density, reliability, and durability improvements are made, fuel cells will remain a niche technology. Of the five primary fuel cell types we have discussed, PEMFCs and SOFCs offer the best prospects for continued improvement and eventual application. While PAFCs and AFCs benefited from early historical develop- ment, the other fuel cell types have caught up and offer further advantages that will likely make them more attractive in the long run. Due to their high energy/power density and low

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CHAPTER SUMMARY 299

TABLE 8.1. Comparison Summary of the Five Major Fuel Cell Types

Electrical Power Fuel Efficiency Density Power Internal Balance Cell Type (%) (mW/cm2) Range (kW) Reforming CO Tolerance of Plant

PAFC 40 150–300 50–1000 No Poison (<1%) Moderate

PEMFC 40–50 500–2500 0.001–1000 No Poison (<50 ppm) Low-moderate

AFC 50 150–400 1–100 No Poison (<50 ppm) Moderate

MCFC 45–55 100–300 100–100,000 Yes Fuel Complex

SOFC 50–60 250–500 10–100,000 Yes Fuel Moderate

operating temperature, thePEMFCand theDMFCappear uniquely suited for portable power applications. Both the PEMFC and the SOFC can be applied to residential power and other small-scale stationarypower applications.High-power applications (above250kWor so) are best served by SOFC and combined-cycle (SOFC–turbine) technology. High-temperature fuel cells offer attractive efficiency and fuel flexibility advantages. They also generate higher quality waste heat, which can be used in combined applications. While all fuel cells operate best on hydrogen, those operating at higher temperatures offer improved impurity tolerance and the possibility of internal reforming of hydrocarbon fuels to yield hydrogen. Table 8.1 summarizes the major benefits and characteristics of the five fuel cell classes discussed in this chapter, while Figure 8.18 provides a convenient graphical summary.

8.9 CHAPTER SUMMARY

This chapter briefly covered the five major fuel cell types. Different electrolytes lead to dif- ferences in reaction chemistry, operating temperature, cell materials, and cell design. These differences lead to important distinctions between the relative advantages, disadvantages, and characteristics of the five fuel cell types.

• The five major fuel cell types are phosphoric acid fuel cell (PAFC), polymer elec- trolyte membrane fuel cell (PEMFC), alkaline fuel cell (AFC), molten carbonate fuel cell (MCFC), and solid-oxide fuel cell (SOFC). They differ from one another on the basis of their electrolyte.

• You should be able to identify and discuss the important differences in reaction chem- istry, operating temperature, cell design, catalyst, and electrode material for each of the five major fuel cell types.

• You should be able to write the H2–O2 anode and cathode half reactions for each of the five fuel cell classes.

• PAFC advantages include technological maturity, reliability, and low electrolyte cost. Disadvantages include the requirement for expensive platinum catalyst, poisoning susceptibility, and corrosive liquid electrolyte.

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300 OVERVIEW OF FUEL CELL TYPES

(a)

(b)

H2O

1 2 O2

1 2 O2CO

CO2

1 2 H2O

1 4 CO2+

1 4 CH4

SOFC

1 2 O2

1 2O2 O2-2e

1 2 O2 O2-2e

O2-

Anode CatalystCatalyst Electrolyte

Cathode

Temperature Reducing agent Oxidising agent

H2 O 2- H2O 2e

-++

1 2 H2O

1 4 CO2+ + 2e

-+14 CH4 O 2-

MCFC

CO3 2-

H2O CO2+

2CO2

1 2 O2 CO2+

CO3 2-H2 + 2e

-++H2O CO2 1 2 O2 CO3

2-2e-++ CO2

1 2 O2 CO3

2-2e-++ CO2

CO3 2-CO + 2CO2 2e

O2- CO2CO + 2e -+

Temperature Reducing agent Oxidising agente

Anode CatalystCatalyst Electrolyte

Cathode

1 3 H2O

H2O

2H2O

H2O

+13 CH3OH

1 3 CO2

PAFC

AFC

DMFC

1 2 O2

H2O 1 2 O2 +

2H+

H2 2e -+2H+

2H+

H2O 1 2 O2 2e

-++ 2H+ H2 2e

-+2H+

2OH-

H2O 1 2 O2 2e-++ 2OH-

H2O2H+ 1 2 O2 2e-++

+ H2 2H2O 2e -+2OH-

SPFC/ PEM

1 3 H2O+

1 3 CH3OH

1 3 CO2 2e

-++ 2H+

Figure 8.18. Graphical comparison of the main fuel cell classes. (see color insert)

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CHAPTER EXERCISES 301

• PEMFC advantages include high power density, low operating temperature, and good start–stop cycling durability. Disadvantages include the requirement for expensive platinum catalyst, high-cost membrane and cell components, poor poison tolerance, and water management issues.

• AFC advantages include improved cathode performance, non-precious-metal catalyst potential, and inexpensive electrolyte/cell materials. Disadvantages include system complexities introduced by the requirement for water removal at the cathode, occa- sional replenishment of the KOH electrolyte, and the requirement for pure H2 and O2 gas. (The AFC cannot tolerate even atmospheric levels of CO2.)

• MCFC advantages include fuel flexibility, non-precious-metal catalyst, and the production of high-quality waste heat for cogeneration applications. Disadvantages include system complexity introduced by the requirement for CO2 recycling, a corrosive molten electrolyte, and relatively expensive cell materials.

• SOFC advantages include fuel flexibility, non-precious-metal catalyst, completely solid-state electrolyte, and the production of high-quality waste heat for cogeneration applications. Disadvantages include system complexity introduced by the high oper- ating temperature, high-temperature cell-sealing difficulties (especially under thermal cycling), and relatively expensive cell components/fabrication.

• While all fuel cells run best on H2 gas, the high-temperature fuel cells can also run on simple hydrocarbon fuels or CO via direct electro-oxidation or internal reforming.

• Historically, the PAFC and the AFC benefited from extensive research and develop- ment. Today, the PEMFC and the SOFC appear poised to best meet potential appli- cations. PEMFCs are especially suited for portable and small stationary applications while SOFCs appear suited for distributed-power and utility-scale power applications.

CHAPTER EXERCISES

Review Questions

8.1 (a) Why is nickel used in many high-temperature fuel cells? (b) In SOFC anodes, why is YSZ mixed with the nickel? (c) In MCFC anodes, why is chromium added to the nickel?

8.2 What do you think is the single most significant advantage of high-temperature fuel cells compared to low-temperature fuel cells? Defend your answer.

8.3 Draw a diagram similar to the one in Figure 8.9 for an SOFC operating on CO fuel. Show both the anode and cathode half reactions clearly, as well as the reactants, prod- ucts, and ionic species.

Calculations

8.4 Given the information in the caption of Figure 8.8, calculate the volumetric power density (W/m3) of the MCFC system assuming the average height of the system is 3 m.

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302 OVERVIEW OF FUEL CELL TYPES

8.5 The fuel cell car shown in Figure 8.4 can travel 500 km at 100 km/h with a full tank of hydrogen. Given the information in the caption of Figure 8.4, estimate the average power output from the fuel cell during the travel. Assume that hydrogen is an ideal gas and the fuel cell efficiency is 55%.

8.6 Consider an SOFC system with an electrical efficiency of 55%. Suppose the SOFC rejects heat at 800∘C. (a) If a heat engine takes this input heat from the fuel cell and rejects it at 100∘C, what

is the Carnot efficiency of this heat engine?

(b) Assume that the practical efficiency of the heat engine is 60% of the Carnot effi- ciency. In this case, if the heat engine and fuel cell are combined, what would be the net electrical efficiency of the combined system?

8.7 The overall reaction in the DMFC is CH3OH + (3∕2)O2 → CO2 + 2H2O. The number of electrons transferred per mole of water produced in a direct methanol fuel cell is:

(a) 2

(b) 3

(d) 6

CHAPTER 9

PEMFC AND SOFC MATERIALS

The purpose of this chapter is to provide a general introduction to the various materials options associated with PEMFC and SOFC technologies. The development and design of optimal materials for specific fuel cell applications is an area of intense research activity—indeed, you will soon find that there are a bewildering array of possible materials and design options. Although our discussions of PEMFCs have so far mostly focused on Nafion–electrolyte-based membranes in combination with traditional Pt/C catalysts, you will learn in this chapter that there are literally dozens of other candidate polymer electrolyte and catalyst material combinations, each offering potential advantages (but also often disadvantages). In a similar fashion, our discussions of SOFCs have so far focused on YSZ-based electrolytes in combination with Ni–YSZ anodes and LSM cathodes. Again, you will learn in this chapter that there are many interesting and sometimes highly compelling alternative SOFC electrolyte, anode, and cathode materials.

This chapter is organized to provide an overview of the most common candidate PEMFC electrolyte, electrode, and catalyst materials and a comparable overview of the most com- mon SOFC electrolyte, electrode, and catalyst materials. A common theme you will see throughout the chapter is the dual requirement for high performance and high stability. Any candidate fuel cell material (whether it is an electrolyte, electrode, or catalyst mate- rial) must not only deliver high performance but also be stable and durable in the fuel cell environment. Because fuel cell environments are often quite harsh, meeting these dual requirements is challenging.

303

304 PEMFC AND SOFC MATERIALS

9.1 PEMFC ELECTROLYTE MATERIALS

As youwill recall fromChapter 4, electrolytematerials must conduct ions, but not electrons. They must also be gas impermeable (to prevent the anode and cathode gases from mixing) and yet as thin as possible (to minimize resistance).

Most PEMFC electrolytes are based on thin polymeric membranes that conduct H+ ions. Many of these polymer materials rely on water-based vehicle mechanisms for ionic trans- port (refer back to Section 4.5.2 for a detailed discussion of these transport mechanisms). Because water is often intimately involved in the ionic transport chain in these electrolyte materials, ionic conductivity tends to be extremely sensitive to the level of hydration. Oper- ation under dry conditions or at temperatures greater than 100∘C is therefore severely limited, if not impossible. Because of these hydration and temperature issues, sophisticated water and temperature management schemes are crucially required for most PEMFC-based systems.

Designing a polymer electrolyte material capable of operating above 100∘C is highly desirable, as this dramatically simplifies water management, while simultaneously improv- ing electrochemical performance and impurity tolerance (recall from Chapter 3 how elec- trochemical reaction rates increase exponentially with temperature). In addition to reducing hydration dependence and increasing operating temperature, candidate PEMFC electrolyte materials must also possess high ionic conductivity and goodmechanical properties (so they can be processed into thin, durable membranes) and must be highly stable/durable in the PEMFC environment and reasonably inexpensive. In the following subsections, several of the potential polymer-based electrolyte materials will be briefly discussed.

9.1.1 Perfluorinated Polymers (e.g., Nafion)

Currently, perfluorinated polymers like Nafion are the most popular and important elec- trolytes for PEMFC and direct methanol fuel cell applications. In addition to Nafion, other perfluorinated polymer materials include Neosepta-F™ (Tokuyama), Gore-Select™ (W. L. Gore and Associates, Inc.), Flemion™ (Asahi Glass Company), Asiplex™ (Asahi Chem- ical Industry), and Dow. Nafion, the prototypical material in this class, was described in detail in Section 4.5.2 of this textbook; therefore, we will not spend significant time dis- cussing it here. However, as a brief review, recall that Nafion has a backbone structure similar to polytetrafluoroethylene (Teflon™). However, unlike Teflon, Nafion includes sul- fonic acid (SO3

−H+) functional groups. The Teflon backbone providesmechanical strength, while the sulfonic acid (SO3

–H+) groups provide charge sites for proton transport. Because Nafion is based on a Teflon-like polymer, it is extraordinarily stable and durable. In addi- tion, Nafion exhibits extremely high ionic conductivity. To maintain this extraordinary conductivity, however, Nafion must be fully hydrated with liquid water [49, 50]. Usually, hydration is achieved by humidifying the fuel and oxidant gases provisioned to the fuel cell. Because of this hydration requirement, Nafion membranes are typically restricted to operating temperatures below 100∘C [51, 52]. Furthermore, because Nafion conductivity decreases markedly on dehydration, Nafion-based fuel cell systems must typically imple- ment careful water management schemes to ensure full hydration during operation. These

PEMFC ELECTROLYTE MATERIALS 305

water management schemes can add considerable bulk and cost to Nafion-based fuel cell systems. Additionally, Nafion itself is extremely costly (currently>$400∕m2) [53]. Finally, the low-operating-temperature requirement leads to significant electrocatalytic issues. In general, Nafion-based fuel cells require noble metal catalysts [54] and demonstrate poor CO and S tolerance [55].

9.1.2 Sulfonated Hydrocarbon Polymers (e.g., Polyetheretherketone = PEEK)

Chemically and thermally stable aromatic hydrocarbon polymers can be employed as the polymer backbone for proton-conducting polymer electrolytes. There are several advan- tages to using hydrocarbon polymers instead of perfluorinated polymers:

• Hydrocarbon polymers are more diverse and less expensive than perfluorinated polymers.

• Hydrocarbon polymers containing polar groups have high water uptakes over a wide temperature range, thus improving high-temperature hydration.

• Hydrocarbon polymers are more easily recycled by conventional methods.

Disadvantages of hydrocarbon-based polymers include:

• Hydrocarbon polymers are generally less stable (both chemically and thermally) than perfluorinated polymers.

• Hydrocarbon polymers generally show lower ionic conductivity than perfluorinated polymers under traditional operating conditions (high humidity, 60–80∘C).

One of the most common aromatic polymer approaches involves polyetheretherketone (PEEK)materials, which are based on a family of nonfluorinated polymers made up of ether and ketone units. Depending on the relative ratios of ether to ketone, membrane chemistries are described with abbreviations such as PEEK, PEEKK, PEK, and PEKKEK. However, we will refer to the entire family of membranes with the generic shorthand PEEK.

The structure of sulfonated PEEK is provided in Figure 9.1 as a representative example of this class of materials. Like Nafion, these aromatic hydrocarbon membranes are typically sulfonated to provide proton conductivity. Because hydrocarbon membranes are nonfluo- rinated, however, they are significantly cheaper to produce than their Nafion counterparts. Unfortunately, in spite of the economic benefits, hydrocarbon membranes have been unable to match the ionic conductivity performance of their perfluorinated polymer competitors.

O O C n

SO3H

Figure 9.1. General chemical structure of sulfonated PEEK.

306 PEMFC AND SOFC MATERIALS

At temperatures below 100∘C, for example, the ionic conductivity of PEEK membranes is typically 10–100 times lower than that of Nafion [56]. At high temperatures (>150∘C), the performance of PEEK approaches that of Nafion, indicating that PEEK and other aromatic hydrocarbon membranes may show promise in higher temperature applications. Additional details on PEEK and related hydrocarbon membrane candidates are provided by several recent reviews [57, 58].

9.1.3 Phosphoric Acid Doped Polybenzimidazole (PBI)

Phosphoric acid doped polybenzimidazole (PBI) exhibits a proton conduction mechanism that does not require the presence of liquid water and thus operates effectively above 100∘C. The structure of PBI is shown in Figure 9.2. PBI by itself is an aromatic hydrocarbon mate- rial, not too different from PEEK. However, rather than directly sulfonating the polymer chain, ionic conductivity is created by doping the material with a strong acid (typically H3PO4). This conduction strategy makes use of acid–base complexation between the rel- atively basic polymer and the strong doping acid. Proton conduction is believed to occur primarily via a free-acid vehicle mechanism involving H3PO4 [59, 60], although proton transfer mediated by segmental motion may also contribute to the observed conductivity [61, 62]. PBI membranes show conductivity values similar to Nafion[∼0.02 S ⋅ cm−1 at 80∘C, 40% relative humidty (RH)], although with considerably reduced humidity depen- dence [63]. PBI was first considered for fuel cell use in 1995 [64]. Compared to Nafion, PBI has several important advantages. It possesses remarkable thermal stability with a glass transition temperature over 430∘C [65, 66]. Compared to Nafion, PBI membranes are reported to have higher mechanical strength [67] and are approximately 100 times less expensive to produce [68]. Finally, PBI membranes may be operated in fuel cells up to approximately 200∘C, leading to considerably improved impurity tolerance (espe- cially to CO) [69, 70], single-phase (steam) water production, and the production of higher quality waste heat [71]. In spite of these considerable advantages, PBI presents several con- cerns, including durability issues related to acid leaching [72–74], oxidative degeneration of the membrane [72], and the slow kinetics of the oxygen reduction reaction in phosphoric acid environments [72]. Additionally, optimizing electrocatalyst inks with PBI blends has proved challenging compared to Nafion/Pt/C catalyst ink counterparts [72, 75]. Additional details on PBI and related membrane chemistries are provided by several recent reviews [72, 75–77].

H H NN

N N

Figure 9.2. Chemical structure of poly-2,2′-m-(phenylene)-5,5′-bibenzimidazole, commonly called polybenzimidazole, or PBI.

PEMFC ELECTROLYTE MATERIALS 307

9.1.4 Polymer–Inorganic Composite Membranes

In an effort to synthesize high-temperature (>100∘C) membranes, various inorganic mate- rials have been incorporated into existing polymer membranes such as Nafion. Inclusion of a hydroscopic oxide (e.g., SiO2 or TiO2) increases water retention at high temperatures [78, 79]. Consequently, such composite membranes exhibit appreciable conductivity up to 140∘C, at temperatures where pure Nafion is unusable because of water loss. Nanopar- ticles/microparticles of proton-conducting materials such as phosphates (e.g., zirconium phosphates) and heteropolyacids (e.g., phosphotungstic acid and silicotungstic acid) have also been infiltrated into polymer membranes to increase water retention and proton con- ductivity at high temperatures [80]. Fuel cells incorporating these composite membranes have demonstrated promising power densities of over 600 mW/cm2 at high temperature (>100∘C) using humidified hydrogen and oxygen. The proton conductivity of these com- posite membranes at high temperatures, however, does not match that of Nafion at nor- mal operating temperatures, for example, 80∘C. Furthermore, like Nafion, these composite membranes still have to be hydrated tomaintain high proton conductivity, and their mechan- ical integrity needs improvement. Further information on composite membranes can be obtained from several recent review articles [81–83].

9.1.5 Solid-Acid Membranes

Solid acids are not polymeric materials. Nevertheless, they are included in this section because they represent a potentially interesting class of low- and intermediate-temperature proton conductors that could be employed in fuel cell designs closely resembling traditional PEMFCs.

Solid acids are compounds partway between normal acids, such as H2SO4 or H3PO4, and normal salts, such as K2SO4. When some of a normal acid’s hydrogen atoms are replaced by alternative cations in the solid-acid form, the material acts as a proton donor. The most widely investigated solid acids for fuel cell use include Cs