Fuel cell fundamentals [3 ed.] 9780471741480, 0471741485

Table of contents :
Content: I. Fuel cell principles --
1. Introduction --
2. Fuel cell thermodynamics --
3. Fuel cell reaction kinetics --
4. Fuel cell charge transport --
5. Fuel cell mass transport --
6. Fuel cell modeling --
7. Fuel cell characterization --
II. Fuel cell technology --
8. Overview of fuel cell types --
9. Overview of fuel cell systems --
10. Fuel cell system integration and subsystem design --
11. Environmental impact of fuel cells.

Citation preview

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

This book is printed on acid-free paper. ♾ Copyright © 2016 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at www.wiley.com/go/permissions. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with the respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor the author shall be liable for damages arising herefrom. For general information about our other products and services, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley publishes in a variety of print and electronic formats and by print-on-demand. Some material included with standard print versions of this book may not be included in e-books or in print-on-demand. If this book refers to media such as a CD or DVD that is not included in the version you purchased, you may download this material at http://booksupport.wiley.com. For more information about Wiley products, visit www.wiley.com. Library of Congress Cataloging-in-Publication Data is available: ISBN 9781119113805 (Cloth) ISBN 9781119114208 (ePDF) ISBN 9781119114154 (ePub) Cover Design: Wiley Cover Illustrations: Ryan O’Hayre Cover Image: Glacial abstract shapes © ppart/iStockphoto Printed in the United States of America 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 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11

3

What Is a Fuel Cell? / 3 A Simple Fuel Cell / 6 Fuel Cell Advantages / 8 Fuel Cell Disadvantages / 11 Fuel Cell Types / 12 Basic Fuel Cell Operation / 14 Fuel Cell Performance / 18 Characterization and Modeling / 20 Fuel Cell Technology / 21 Fuel Cells and the Environment / 21 Chapter Summary / 22 Chapter Exercises / 23 v

vi

CONTENTS

2

Fuel Cell Thermodynamics 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

3

3.9 3.10 3.11 3.12 3.13 3.14 3.15

4

Thermodynamics Review / 25 Heat Potential of a Fuel: Enthalpy of Reaction / 34 Work Potential of a Fuel: Gibbs Free Energy / 37 Predicting Reversible Voltage of a Fuel Cell under Non-Standard-State Conditions / 47 Fuel Cell Efficiency / 60 Thermal and Mass Balances in Fuel Cells / 65 Thermodynamics of Reversible Fuel Cells / 67 Chapter Summary / 71 Chapter Exercises / 72

Fuel Cell Reaction Kinetics 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

77

Introduction to Electrode Kinetics / 77 Why Charge Transfer Reactions Have an Activation Energy / 82 Activation Energy Determines Reaction Rate / 84 Calculating Net Rate of a Reaction / 85 Rate of Reaction at Equilibrium: Exchange Current Density / 86 Potential of a Reaction at Equilibrium: Galvani Potential / 87 Potential and Rate: Butler–Volmer Equation / 89 Exchange Currents and Electrocatalysis: How to Improve Kinetic Performance / 94 Simplified Activation Kinetics: Tafel Equation / 97 Different Fuel Cell Reactions Produce Different Kinetics / 100 Catalyst–Electrode Design / 103 Quantum Mechanics: Framework for Understanding Catalysis in Fuel Cells / 104 The Sabatier Principle for Catalyst Selection / 107 Connecting the Butler–Volmer and Nernst Equations (Optional) / 108 Chapter Summary / 112 Chapter Exercises / 113

Fuel Cell Charge Transport 4.1 4.2 4.3 4.4 4.5

25

Charges Move in Response to Forces / 117 Charge Transport Results in a Voltage Loss / 121 Characteristics of Fuel Cell Charge Transport Resistance / 124 Physical Meaning of Conductivity / 128 Review of Fuel Cell Electrolyte Classes / 132

117

CONTENTS

4.6 4.7 4.8 4.9

5

Fuel Cell Mass Transport 5.1 5.2 5.3 5.4

6

237

What Do We Want to Characterize? / 238 Overview of Characterization Techniques / 239 In Situ Electrochemical Characterization Techniques / 240 Ex Situ Characterization Techniques / 265 Chapter Summary / 268 Chapter Exercises / 269

II

FUEL CELL TECHNOLOGY

8

Overview of Fuel Cell Types 8.1 8.2 8.3 8.4 8.5

203

Putting It All Together: A Basic Fuel Cell Model / 203 A 1D Fuel Cell Model / 206 Fuel Cell Models Based on Computational Fluid Dynamics (Optional) / 227 Chapter Summary / 230 Chapter Exercises / 231

Fuel Cell Characterization 7.1 7.2 7.3 7.4 7.5

167

Transport in Electrode versus Flow Structure / 168 Transport in Electrode: Diffusive Transport / 170 Transport in Flow Structures: Convective Transport / 183 Chapter Summary / 199 Chapter Exercises / 200

Fuel Cell Modeling 6.1 6.2 6.3 6.4

7

More on Diffusivity and Conductivity (Optional) / 153 Why Electrical Driving Forces Dominate Charge Transport (Optional) / 160 Quantum Mechanics–Based Simulation of Ion Conduction in Oxide Electrolytes (Optional) / 161 Chapter Summary / 163 Chapter Exercises / 164

Introduction / 273 Phosphoric Acid Fuel Cell / 274 Polymer Electrolyte Membrane Fuel Cell / 275 Alkaline Fuel Cell / 278 Molten Carbonate Fuel Cell / 280

273

vii

viii

CONTENTS

8.6 8.7 8.8 8.9

9

Solid-Oxide Fuel Cell / 282 Other Fuel Cells / 284 Summary Comparison / 298 Chapter Summary / 299 Chapter Exercises / 301

PEMFC and SOFC Materials 9.1 9.2 9.3 9.4 9.5 9.6

303

PEMFC Electrolyte Materials / 304 PEMFC Electrode/Catalyst Materials / 308 SOFC Electrolyte Materials / 317 SOFC Electrode/Catalyst Materials / 326 Material Stability, Durability, and Lifetime / 336 Chapter Summary / 340 Chapter Exercises / 342

10 Overview of Fuel Cell Systems

347

10.1 10.2 10.3 10.4 10.5

Fuel Cell Subsystem / 348 Thermal Management Subsystem / 353 Fuel Delivery/Processing Subsystem / 357 Power Electronics Subsystem / 364 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 11.1 11.2 11.3 11.4 11.5 11.6

Fuel Reforming Overview / 394 Water Gas Shift Reactors / 409 Carbon Monoxide Clean-Up / 411 Reformer and Processor Efficiency Losses / 414 Reactor Design for Fuel Reformers and Processors / 416 Chapter Summary / 417 Chapter Exercises / 419

393

CONTENTS

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 14.1 14.2 14.3 14.4 14.5 14.6

481

Life Cycle Assessment / 481 Important Emissions for LCA / 490 Emissions Related to Global Warming / 490 Emissions Related to Air Pollution / 502 Analyzing Entire Scenarios with LCA / 507 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 D.2 D.3 D.4 D.5 D.6 D.7

Atomic Orbitals / 533 Postulates of Quantum Mechanics / 534 One-Dimensional Electron Gas / 536 Analogy to Column Buckling / 537 Hydrogen Atom / 538 Multielectron Systems / 540 Density Functional Theory / 540

ix

x

CONTENTS

E Periodic Table of the Elements

543

F

545

Suggested Further Reading

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.

xi

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 concepts 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 Education, for the support that made this book possible. We would also like to acknowledge Honda 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 recognized 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. Professor 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 significant 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. Special 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 laboratories [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) Stationary 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

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. …

xv

NOMENCLATURE

Symbol

Meaning

Common Units

A Ac a ASR C Cdl c∗ c c

Area Catalyst area coefficient Activity Area specific resistance Capacitance Double-layer capacitance Concentration at reaction surface Concentration Constant describing how mass transport affects concentration losses Heat capacity Diffusivity Electric field Thermodynamic ideal voltage Thermodynamic ideal voltage Temperature-dependent thermodynamic voltage at reference concentration Helmholtz free energy Faraday constant Generalized force Reaction rate constant Friction factor

cm2 Dimensionless Dimensionless Ω ⋅ cm2 F F mol∕cm2 mol∕m3 V

cp D E E Ethermo ET F F Fk f f

J∕mol ⋅ K cm2 ∕s V∕cm V V V J, J∕mol 96, 485 C∕mol N Hz, s−1 Dimensionless

xvii

xviii

NOMENCLATURE

Symbol

Meaning

Common Units

G, g g ΔG‡ ΔGact H H, h HC HE h ℏ hm i J Jˆ JC j j0 j 00

Gibbs free energy Acceleration due to gravity Activation energy barrier Activation energy barrier Heat Enthalpy Gas channel thickness Diffusion layer thickness Planck’s constant Reduced Planck constant, h∕2𝜋 Mass transfer convection coefficient Current Molar flux, molar reaction rate Mass flux Convective mass flux Current density Exchange current density Exchange current density at reference concentration Limiting current density Fuel leakage current Boltzmann’s constant Length Molar mass Mass flow rate Generalized coupling coefficient between force and flux Mass Heat capacity flow rate Number of moles Avogadro’s number Number of electrons transferred in the reaction Number of moles of gas Power or power density Pressure Heat Charge Adsorption charge Adsorption charge for smooth catalyst surface Fundamental charge Ideal gas constant Resistance Faradaic resistance

J, J∕mol m∕s2 J∕mol, J J∕mol, J J J, J∕mol cm cm 6.63 × 10−34 J ⋅ s 1.05 × 10−34 J ⋅ s m∕s A mol∕cm2 ⋅ s g∕cm2 ⋅ s, kg∕m2 ⋅ s kg∕m2 ⋅ s A∕cm2 A∕cm2 A∕cm2

jL jleak k L M M Mik m mcp N NA n ng P P Q Q Qh Qm q R R Rf

A∕cm2 A∕cm2 1.38 × 10−23 J∕K m g∕mol, kg∕mol kg∕s Varies kg kW∕kg ⋅ ∘ C Dimensionless 6.02 × 1023 mol−1 Dimensionless Dimensionless W or W∕cm2 bar, atm, Pa J, J∕mol C C∕cm2 C∕cm2 1.60 × 10−19 C 8.314 J∕mol ⋅ K Ω Ω

NOMENCLATURE

Symbol

Meaning

Common Units

Re S, s S∕C Sh T t U u u¯ V V V 𝑣 𝑣 𝑣 W X x x𝑣 yx Z z

Reynolds number Entropy Steam-to-carbon ratio Sherwood number Temperature Thickness Internal energy Mobility Mean flow velocity Voltage Volume Reaction rate per unit area Velocity Hopping rate Molar flow rate Work Parasitic power load Mole fraction Vacancy fraction Yield of element X Impedance Height

Dimensionless J∕K, J∕mol ⋅ K Dimensionless Dimensionless K, ∘ C cm J, J∕mol cm2 ∕V ⋅ s cm∕s, m∕s V L, cm3 mol∕cm2 ⋅ s cm∕s s−1 , Hz mol∕s, mol∕min J, J∕mol W Dimensionless mol vacancies∕mol sites Dimensionless Ω cm

Greek Symbols Symbol

Meaning

Common Units

𝛼 𝛼 𝛼∗ 𝛽 𝛾 Δ 𝛿 𝜀 𝜀FP 𝜀FR 𝜀H 𝜀O 𝜀R 𝜀 𝜀̇

Charge transfer coefficient Coefficient for CO2 equivalent Channel aspect ratio Coefficient for CO2 equivalent Activity coefficient Denotes change in quantity Diffusion layer thickness Efficiency Efficiency of fuel processor Efficiency of fuel reformer Efficiency of heat recovery Efficiency overall Efficiency, electrical Porosity Strain rate

Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless m, cm Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless s−1

xix

xx

NOMENCLATURE

Symbol

Meaning

Common Units

𝜂 𝜂act 𝜂conc 𝜂ohmic λ λ 𝜇 𝜇 𝜇̃ 𝜌 𝜌 𝜎 𝜎 𝜏 𝜏 𝜑 𝜑 𝜔

Overvoltage Activation overvoltage Concentration overvoltage Ohmic overvoltage Stoichiometric coefficient Water content Viscosity Chemical potential Electrochemical potential Resistivity Density Conductivity Warburg coefficient Mean free time Shear stress Electrical potential Phase factor Angular frequency (𝜔 = 2𝜋f )

V V V V Dimensionless Dimensionless kg ⋅ m/s J, J/mol J, J/mol Ω cm kg∕cm3 , kg∕m3 S∕cm, (Ω ⋅ cm)−1 Ω∕s0.5 s Pa V Dimensionless rad/s

Superscripts Symbol

Meaning

0 eff

Denotes standard or reference state Effective property

Subscripts Symbol

Meaning

diff E, e, elec f (HHV) (LHV) i P P R rxn SK SYS

Diffusion Electrical (e.g., Pe , Welec ) Quantity of formation (e.g., ΔHf ) Higher heating value Lower heating value Species i Product Parasitic Reactant Change in a reaction (e.g., ΔHrxn ) Stack 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 simplest example, the combustion of hydrogen: H2 + 12 O2 ⇌ H2 O

(1.1)

3

INTRODUCTION

H2O(1/g)

O2(g)

Fuel cell H2(g)

Electricity

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 a matter 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 the molecules. 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 reaction by somehow harnessing the electrons as they move from high-energy reactant bonds H2

H2

H2O

O2

1

H2O

3

2

4

2 Potential energy

4

1 Reactants (H2/O2)

3 4 Products (H2O)

Reaction progress

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 bonds must first be broken, requiring energy input (2) before hydrogen–oxygen bonds are formed, leading to energy output (3, 4).

WHAT IS A FUEL CELL?

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 hydrogen 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.

Potential energy (KJ/mol)

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.

–100

1

3

–200 –300

2

–400 –436 –500 74 100

200

Internuclear distance (pm)

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 internuclear repulsion.

5

6

INTRODUCTION

1.2

A SIMPLE FUEL CELL

In a fuel cell, the hydrogen combustion reaction is split into two electrochemical half reactions: (1.2) H2 ⇌ 2H+ + 2e− 1 O 2 2

+ 2H+ + 2e− ⇌ H2 O

(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 material 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 recombine with protons and bubbling oxygen gas to produce water 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

e–

H+

H2

O2

Figure 1.4. A simple fuel cell.

A SIMPLE FUEL CELL

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

(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 supplied by a device per unit mass. Typical units are W/g or kW/kg.

7

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

Fuel cell, battery

(a)

Chemical energy 1

Electrical energy 4

Heat energy 2

Mechanical energy 3

Combustion engine

(b) Fuel tank Battery Fuel cell or combustion engine Work out

Work out

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

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 refueling, 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, batteries, 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 operate 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.”

9

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

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 current 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 disadvantages. 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 specialized 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, further improvements are required if fuel cells are to compete in portable and automotive applications. Combustion engines and batteries generally outperform fuel cells on a volumetric 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

Gravimetric power density (W/kg)

10000

Fuel cell (portable)

1000

IC engines (automotive)

Fuel cell (automotive)

IC engine (portable)

Li-ion battery

100 Lead-acid battery

10 0.01

0.1

1

10

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

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

11

INTRODUCTION

50 Gasoline

45 Gravimetric energy density (MJ/kg)

12

40 35 30 Ethanol

25 Methanol

20 15

Hydrogen, 7500PSI (including system)

10 5

Hydrogen, metal hydride (low)

Hydrogen, 3500PSI (including system)

0 0

5

Hydrogen, metal hydride (high)

Hydrogen, liquid (including system)

10

15

20

25

30

35

Volumetric Energy Density (MJ/L)

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 difficult to use directly and usually require reforming. These problems can reduce fuel cell performance and increase the requirements for ancillary equipment. Thus, although gasoline 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 significant 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. 2. 3. 4. 5.

Phosphoric acid fuel cell (PAFC) Polymer electrolyte membrane fuel cell (PEMFC) Alkaline fuel cell (AFC) Molten carbonate fuel cell (MCFC) Solid-oxide fuel cell (SOFC)

FUEL CELL TYPES

TABLE 1.1. Description of Major Fuel Cell Types PEMFC

PAFC

AFC

MCFC

SOFC

Electrolyte

Polymer membrane

Liquid H3 PO4 Liquid KOH Molten (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 Ceramic based 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 principles, 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 polymer membrane 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 PEMFC membrane. As we have already seen, the electrochemical half reactions in an H2 –O2 PEMFC are H2 → 2H+ + 2e− 1 O 2 2

+ 2H+ + 2e− → H2 O

(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− → H2 O + 2e− 1 O 2 2

+ 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.

13

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 the materials 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

Hydrog

en Oxygen

Anode

Cathode

Electrolyte

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

BASIC FUEL CELL OPERATION

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. Electrons 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: (1.8) H2 → 2H+ + 2e− 1 O 2 2

+ 2H+ + 2e− → H2 O

(1.9)

The hydrogen reaction is an oxidation reaction because electrons are being liberated 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.

15

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 electrode. 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 producing 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 external circuit 4. Product removal from the fuel cell 3

1 2

4

3

2

4

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

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 combination with porous electrode structures. Flow field plates contain many 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 electrode geometries influence fuel cell performance is an exercise in mass transport, diffusion, and fluid mechanics. The materials aspects of flow structures and electrodes 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 electrochemical 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 electrochemical 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 maintain 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 conductive 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 electrons. 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

17

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 overlooked. 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)

Fuel cell voltage (V)

18

Activation region

Ohmic region

Mass transport region

(Chapter 3)

(Chapter 4)

(Chapter 5)

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 unavoidable losses. Three major losses influence the shape of this i–V curve; they will be described in Chapters 3–5.

FUEL CELL PERFORMANCE

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 thermodynamically 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.

1.2 Power density curve

Fuel cell voltage (V)

1.0

0.6 0.5

0.8

0.4

0.6 i-V curve

0.3

0.4 0.2 0.2

Fuel cell power density (W/cm2)

0.7

0.1

0

0 0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Current density (A/cm2)

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 corresponding 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.

19

20

INTRODUCTION

The current supplied by a fuel cell is directly proportional to the amount of fuel consumed (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 thermodynamically 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 concentration 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

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 underlying 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 environmental 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 system! In the final chapter of this book, you will 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 conversion 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 completely eliminated. Currently, it is unclear when, if ever, the hydrogen economy will become a reality. Various 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

21

22

INTRODUCTION

Solar power

O2

O2 H2 storage

Sun Electrolyzer

Fuel cell

Water Wind power

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) separated 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 reaction 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

• 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− → 12 O2 + 2e− (d) CH4 + 4O2− → CO2 + 2H2 O + 8e− (e) O2− + CO → CO2 + 2e− (f) 12 O2 + H2 O + 2e− → 2(OH)− (g) H2 + 2(OH)− → 2H2 O + 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 compressed 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. Calculate how much energy (in kilojoules per mole of product) is released by the reaction H2 + 12 O2 ⇌ H2 O

(1.12)

23

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 temperature 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 corresponding 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. The metal 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 (c) 1.0 Wh/ 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 thermodynamics 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 theories 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 quantities. For example, Nobel Prize–winning physicist Richard Feynman wrote in his Lectures 25

26

FUEL CELL THERMODYNAMICS

on Physics: “It is important to realize that in modern physics today, we have no knowledge 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 experience. 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 thermodynamics 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

H2 tank

Microscopic view

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

THERMODYNAMICS REVIEW

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 surroundings: via heat (Q) or work (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 = p dV

(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, however, we ignore electrical work. Considering only mechanical work, we can rewrite the expression for the internal energy change of a system as dU = dQ − p dV 2.1.4

(2.5)

Second Law

The second law of thermodynamics introduces the concept of entropy. Entropy is determined 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

27

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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 log Ω

(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 particular 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 the most “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 unfortunate 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) Z! = N! (Z − N)!(N!)

(2.7)

THERMODYNAMICS REVIEW

(b)

(a)

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! == 1.62 × 105 97!3!

(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 separate 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 thermodynamic potentials. You are already familiar with one thermodynamic potential: the internal

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FUEL CELL THERMODYNAMICS

energy of a system. We can combine results from the first and the second laws of thermodynamics (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 − p dV

(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 variables T and p are related to variations in the independent variables (S and V): ) ( dU =T (2.13) dS V (

dU dV

) = −p

(2.14)

S

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 mathematical way to accomplish this conversion using a Legendre transform. A step-by-step transformation of U begins by defining a new thermodynamic potential G(T, p) as follows: ) ) ( ( dU dU S− V (2.15) G=U− dS V dV S 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 − S dT + p dV + V dp

(2.17)

Since we know that dU = T dS – p dV, we can see that dG = −S dT + 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

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 ) ( dU V (2.19) H=U− dV S 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 + p dV + 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 = −S dT − p dV

(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 environmental temperature T from a negligible initial volume after subtracting what the environment automatically supplied.

31

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FUEL CELL THERMODYNAMICS

–TS

U

Internal energy

U = energy needed to create a system

+pV

H

Enthalpy

H = U + pV

F

Helmholtz free energy

F = U –TS

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

G

Gibbs free energy

G = U + pV –TS

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

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 diagram to help remember the relationships. Copyright © 2000 by Addison Wesley Longman. Reprinted by permission of Pearson Education, Inc. (Figure 5.2, p. 151, from An 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: Δ̂grxn , Δ̂srxn , Δ𝑣̂ 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

a process. For example, the overall reaction in a H2 –O2 fuel cell, H2 + 12 O2 → H2 O

(2.25)

has Δ̂grxn = −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 H2 O 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 ( ) ( ) 1 mol H2 −237 kJ 5 mol O2 × × = −2370 kJ (2.26) mol H2 (1∕2) mol O2 Of course the intrinsic (per-mole) Gibbs free energy of this reaction is still Δ̂grxn = −237 kJ∕mol H2 . 2.1.7

Standard State

Because most thermodynamic quantities depend on temperature and pressure, it is convenient 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, Δĥ 0 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 Δĥ 0 and Δ̂s0 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 Δ̂g0 changes much more strongly with temperature (as shown in Equation 2.39) and therefore Δ̂g0 values should always be adjusted by temperature 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.)

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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 voltage equations no longer apply. To distinguish between reversible and nonreversible fuel cell voltages in this book, we will use the symbols E and V, where E represents a reversible (thermodynamically 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 originate? 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

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 combustion. 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 Δĥ rxn ) 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 approximations. Therefore, enthalpy-of-reaction values are normally calculated by computing the formation enthalpy differences between reactants and products. A standard-state formation enthalpy Δĥ 0f (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; Δĥ 0rxn may be calculated as [ ] [ ] Δĥ 0rxn = m Δĥ 0f (M) + n Δĥ 0f (N) − a Δĥ 0f (A) + b Δĥ 0f (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, Δ̂s0rxn , using standard entropy values ŝ 0 for the species taking part in the reaction. See Example 2.1 for details.

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FUEL CELL THERMODYNAMICS

Example 2.1 A direct methanol fuel cell uses methanol as fuel instead of hydrogen. Calculate the Δĥ 0rxn and Δ̂s0rxn for the methanol combustion reaction: CH3 OH(liq) + 32 O2 → CO2 + 2H2 O(liq)

(2.32)

Solution: From Appendix B, the Δĥ 0f and ŝ 0 values for CH3 OH, O2 , CO2 , and H2 O are given in the following table.

Chemical Species

Δĥ 0f (kJ/mol)

CH3 OH(liq) O2 CO2 H2 O(liq)

–238.5 0 –393.51 –285.83

ŝ 0 [J/(mol⋅K)] 127.19 205.00 213.79 69.95

Following Equation 2.31, the Δĥ 0rxn for methanol combustion is calculated as [ ] [ ] Δĥ 0rxn = 2 Δĥ 0f (H2 O(liq) ) + Δĥ 0f (CO2 ) − 32 Δĥ 0f (O2 ) + Δĥ 0f (CH3 OH(liq) ) ] [ = [2(−285.83) + (−393.51)] − 32 (0) + (−238.5) = −726.67 kJ∕mol

(2.33)

Similarly, Δ̂s0rxn is calculated as ] [ ] [ Δ̂s0rxn = 2̂s0 (H2 O(liq) ) + ŝ 0 (CO2 ) − 32 ŝ 0 (O2 ) + ŝ 0 (CH3 OH(liq) ) [ ] = [2(69.95) + (213.79)] − 32 (205.00) + (127.19) = −81.00 J∕(mol ⋅ K) 2.2.2

(2.34)

Temperature Dependence of Enthalpy

The amount of heat energy that a substance can absorb changes with temperature. It follows that a substance’s formation enthalpy also changes with temperature. The variation of enthalpy with temperature is described by a substance’s heat capacity: Δĥ f = Δĥ 0f +

T

∫T0

cp (T) dT

(2.35)

where Δĥ f is the formation enthalpy of the substance at an arbitrary temperature T, Δĥ 0f 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

constant-pressure heat capacity of the substance (which itself may be a function of temperature). If phase changes occur along the path between T0 and T, extra caution must 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: T

ŝ = ŝ 0 +

∫T0

cp (T) T

dT

(2.36)

From Equations 2.31, 2.35, and 2.36, Δĥ rxn and Δ̂srxn for any reaction at any temperature can be calculated as long as the basic thermodynamic data (Δĥ 0f , 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, Δĥ 0f 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 environment 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.) If G 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 Δ̂grxn values as we calculated Δĥ rxn and Δ̂srxn values. In fact, we can calculate Δ̂grxn values directly from Δĥ rxn and Δ̂srxn 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)

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FUEL CELL THERMODYNAMICS

Differentiating this expression gives dG = dH − T dS − S dT

(2.38)

Holding temperature constant (isothermal process, dT = 0) and writing this relationship in terms of molar quantities give Δ̂g = Δĥ − T Δ̂s (2.39) Thus, for an isothermal reaction, we can compute Δ̂g in terms of Δĥ and Δ̂s. 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 Δ̂g values at different reaction temperatures.

Example 2.2 Determine the approximate temperature at which the following reaction is no longer spontaneous: CO + H2 O(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: Δ̂grxn (T) = Δĥ rxn (T) − T Δ̂srxn (T) = 0

(2.41)

To get an approximate answer, we can assume that Δĥ rxn and Δ̂srxn are independent of temperature (heat capacity effects are ignored). In this case, the temperature dependence of Δ̂grxn is approximated as Δ̂grxn (T) = Δĥ 0rxn − T Δ̂s0rxn

(2.42)

From Appendix B, the Δĥ 0f and ŝ 0 values for CO, CO2 , H2 , and H2 O are given in the table below.

Chemical Species CO CO2 H2 H2 O(g)

Δĥ 0f ( kJ/mol) –110.53 –393.51 0 –241.83

ŝ 0 [J/(mol⋅K)] 197.66 213.79 130.68 188.84

WORK POTENTIAL OF A FUEL: GIBBS FREE ENERGY

Following Equation 2.31, Δĥ 0rxn is calculated as [ ] [ ] Δĥ 0rxn = Δĥ 0f (CO2 ) + Δĥ 0f (H2 ) − Δĥ 0f (CO) + Δĥ 0f (H2 O) = [(−393.51) + (0)] − [(−110.53) + (−241.83)] = −41.15 kJ∕mol

(2.43)

Similarly, Δ̂s0rxn is calculated as [ ] [ ] Δ̂s0rxn = ŝ 0 (CO2 ) + ŝ 0 (H2 ) − ŝ 0 (CO) + ŝ 0 (H2 O) = [(213.79) + (130.68)] − [(197.66) + (188.84)] = −42.03 J∕(mol ⋅ K)

(2.44)

This gives Δ̂grxn (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. However, 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 ≈ 979 K ≈ 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 unfavorable. 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.

39

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FUEL CELL THERMODYNAMICS

From Equation 2.17, remember that we define a change in Gibbs free energy as dG = dU − T dS − S dT + p dV + 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 − (p dV + dWelec )

(2.48)

which yields dG = −S dT + 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 pressure 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 operating 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 thermodynamic (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

External work, W

A thermodynamic device (engine) with internal chemical reaction

Products at T0 , p0

Reactants at T0 , p0 Heat flux, Qrev Isothermal and isobaric environment at T0 , p0

Figure 2.4. Diagram of a reversible thermodynamic engine (or heat engine) operating under constant 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 thermodynamics: (2.52) HReactants (T0 , p0 ) = HProducts (T0 , p0 ) − Qrev + W 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)

41

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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 Q = rev T0 T0

(2.55)

Qrev = T0 ΔS(T0 , p0 ) Plugging Equation 2.55 into 2.53 and solving for W, we have W = −ΔH(T0 , p0 ) + Qrev = −ΔH(T0 , p0 ) + T0 ΔS(T0 , p0 )

(2.56)

= −ΔG(T0 , p0 ) 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 thermodynamic 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 thermodynamic 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, if ΔG 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 reaction 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 limited by several kinetic phenomena. These phenomena are covered in Chapters 3–5. Before

WORK POTENTIAL OF A FUEL: GIBBS FREE ENERGY

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 Δ̂g = −nFE

(2.59)

Thus, the Gibbs free energy sets the magnitude of the reversible voltage for an electrochemical reaction. For example, in a hydrogen–oxygen fuel cell, the reaction H2 + 12 O2 ⇌ H2 O

(2.60)

has a Gibbs free-energy change of –237 kJ/mol under standard-state conditions for liquid water product. The reversible voltage generated by a hydrogen–oxygen fuel cell under standard-state conditions is thus E0 = − =−

Δ̂g0rxn nF −237, 000 J∕mol (2 mol e− ∕mol reactant)(96, 485 C∕mol)

(2.61)

= +1.23 V where E0 is the standard-state reversible voltage and Δ̂g0rxn 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.

43

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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 ⇌ CH3 OH + H2 O

+0.03

O2 + 4H+ + 4e− ⇌ 2H2 O

+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 potentials 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: ∑ 0 0 = Ehalf (2.62) Ecell reactions 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 electrochemistry (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 electrons, 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 = nNA q = nF

(2.63)

WORK POTENTIAL OF A FUEL: GIBBS FREE ENERGY

What we call Faraday’s constant is really the quantity NA ×q: F = NA × q = (6.022 × 1023 electrons∕mol) × (1.60 × 10–19 C∕electron) = 96, 485 C∕mol Interestingly, the fact that Faraday’s constant is a large number has important technological 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 electrons 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 (2.64) A + 2B → C + 2e− ΔGrxn 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 A + B → 12 C + e− 21 ΔGrxn (2.65) 2 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 ΔG of the original reaction. As long as n and ΔG are kept consistent with the reaction 2 stoichiometry, you should not suffer any confusion. For example, the standard-state potential of the hydrogen–oxygen fuel cell is determined by E0 = −0.000 H2 → 2H+ + 2e− + 12 (O2 + 4H+ + 4e− → 2H2 O)

E0 = +1.229

= H2 + 12 O2 → H2 O

0 Ecell = +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 direction 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

45

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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 reaction will 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 (CH3 OH) as fuel instead of hydrogen: (2.67) CH3 OH + 32 O2 → CO2 + 2H2 O Calculate the standard-state reversible potential for a direct methanol fuel cell. Solution: We break this overall reaction into two electrochemical half reactions: CH3 OH + H2 O ⇌ CO2 + 6H+ + 6e− 3 (O2 + 4H+ + 4e− ⇌ 2H2 O) 2

E0 = −0.03 E0 = +1.229

CH3 OH + 32 O2 → CO2 + 2H2 O

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 reaction by 32 to get a balanced reaction, we did not multiply the E0 value by 32 . The E0 values are independent of reaction amounts.

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

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, contributions 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 = −S dT + V dp from which we can write

(

dG dT

(2.68)

) = −S

(2.69)

p

For molar reaction quantities, this becomes ( ) d(Δ̂g) = −Δ̂s dT p

(2.70)

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

) = p

Δ̂s nF

(2.72)

47

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 +

Δ̂s (T − T0 ) nF

(2.73)

Generally, we assume Δ̂s 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 Δ̂s. As Equation 2.73 indicates, if Δ̂s for a chemical reaction is positive, then ET will increase with temperature. If Δ̂s is negative, then ET will decrease with temperature. For most fuel cell reactions Δ̂s 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, Δ̂srxn = −44.34 J/(mol⋅K) (for H2 O(g) as product). The variation of cell voltage with temperature is approximated as ET = E0 +

−44.34 J∕(mol ⋅ K) (T − T0 ) (2)(96, 485)

0

= E − (2.298 × 10

−4

(2.74)

V∕K)(T − T0 )

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 voltage 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 = −S dT + V dp This time, we note

(

dG dp

(2.75)

) =V

(2.76)

T

Written for molar reaction quantities, this becomes ( ) d(Δ̂g) = Δ𝑣̂ dp T

(2.77)

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

Temperature (K) 300 400 500 600 700 800 900 1000 1100 1200 1.30

CO CH3OH

Standard potential (V)

1.25 1.20 1.15

C2H4

1.10 1.05 1.00 0.95

H2 H2O(g)

CH4 C

CO2 H2 H2O(l) C

CO

0.9 100 200 300 400 500 600 700 800 900 1000

Temperature (°C)

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 Δ̂g = −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: ( ) Δ𝑣̂ dE =− (2.79) dp T nF 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 voltage 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 ( ) Δng RT dE =− (2.80) dp T nFp 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 .

49

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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 ( ) 𝜕G (2.81) 𝜇i𝛼 = 𝜕ni T, p,nj≠i where 𝜇i𝛼 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 chemical 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 = 𝜇i0 + RT ln ai

(2.82)

where 𝜇i0 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

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 =

∑

𝜇i dni =

i

∑ (𝜇i0 + RT ln ai ) dni

(2.83)

i

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 concentration (number per volume) or the activity (normalized concentration) of species i in a system: 𝜇i = 𝜇i0 + 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 activity 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.

51

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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 a molar basis for species A, Δ̂g for this reaction may be calculated from the chemical potentials of the various species participating in the reaction (assuming a single phase): 0 + n𝜇N0 ) − (𝜇A0 + b𝜇B0 ) + RT ln Δ̂g = (m𝜇M

an am M N a1A abB

(2.85)

Recognizing that the lumped standard-state chemical potential terms represent the standard-state molar free-energy change for the reaction, Δ̂g0 , the equation can be simplified to a final form: am anN Δ̂g = Δ̂g∘ + RT ln M (2.86) a1A abB 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: Δ̂g = −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: m n RT aM aN ln 1 b E=E − nF aA aB 0

(2.88)

For a system with an arbitrary number of product and reactant species, this equation takes the general form ∏ 𝑣i aproducts RT 0 E=E − (2.89) ln ∏ 𝑣i nF areactants Always take care to raise the activity of each species by its corresponding stoichiometric coefficient (𝑣i ). For example, if a reaction involves 2Na+ , the activity of Na+ must be raised to the power of 2 (e.g., a2Na+ ). 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 H2 O 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

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. Remember it forever. As an example of the utility of this equation, we will apply it to the familiar hydrogen– oxygen fuel cell reaction: (2.90) H2 + 12 O2 ⇌ H2 O We write the Nernst equation for this reaction as E = E0 −

aH2 O RT ln 2F a a1∕2 H2 O

(2.91)

2

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 (aH2 O = 1). This yields RT 1 E = E0 − (2.92) ln 2F p p1∕2 H2 O 2

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) 1 ln (2)(96, 485) (3)(5 × 0.21)1∕2

(2.93)

= 1.254 V

PRESSURE, TEMPERATURE, AND NERNST EQUATION The Nernst equation accounts for the same pressure effects that were previously discussed 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.

53

54

FUEL CELL THERMODYNAMICS

At an arbitrary temperature T ≠ T0 , the Nernst equation must be modified as ∏ 𝑣i aproducts RT ln ∏ 𝑣i E = ET − nF areactants

(2.94)

where ET is given from Equation 2.73 as ET = E0 +

Δ̂s (T − T0 ) nF

(2.95)

Thus, the full expression describing how the reversible cell voltage varies with temperature, pressure, and activity can be written as ∏ 𝑣i aproducts Δ̂s RT E=E + (T − T0 ) − ln ∏ 𝑣i nF nF areactants 0

(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? Perhaps 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) 1 ln (2)(96, 485) (1)(0.21)1∕2

(2.97)

= 1.219 V 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

_

+

H2 2H+ + 2e– H2 (100 atm)

_

H+

2H+ + 2e– H2

H2 (10–8 atm)

+

Porous Pt Electrolyte Porous Pt electrode membrane electrode

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-lowpressure vacuum compartment separated by a composite platinum–electrolyte–platinum membrane structure. This “hydrogen fuel cell” contains no oxygen to react with the hydrogen, 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) 10−8 ln (2)(96, 485) 100

(2.98)

= 0.296 V 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 chemical 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

55

56

FUEL CELL THERMODYNAMICS

electrolyte to the vacuum compartment, where they react with electrons in the second platinum 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 electrons 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 compartment 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 chemical 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 electrochemical potential: 𝜇̃ = 𝜇i + zi F𝜙i = 𝜇0i +RT ln ai + zi F𝜙i (2.99) where 𝜇̃ 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 𝜇̃ i

i

∑ i

( −

)

products

𝑣i 𝜇i

( −

products

∑

) 𝑣i 𝜇̃ i

i

∑ i

=0 )

reactants

𝑣i 𝜇i

(at equilibrium)

(2.100)

= −zi F Δ𝜙i reactants

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: 𝜇̃ i = 𝜇i0 + RT ln ai + zi F𝜙i = 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 n moles of electrons move from the anode to the cathode per mole of chemical reaction, then

Δ𝜙e−

∏ 𝑣i aproducts Δ̂g0 RT =E=− − ln ∏ 𝑣i nF nF areactants

(2.102)

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

which gives

∏ 𝑣i aproducts RT E=E − ln ∏ 𝑣i nF areactants 0

(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 equilibria. As an example, we will derive the Nernst equation for the Cu2+ concentration cell illustrated in Figure 2.7.

V

e–

e–

SO42–

SO42– Cu2+

Cu2+

[Cu12+ ] = 10M

1

Cu12+ + 2e1–

Cu1

[Cu22+] = 10 –5M

Cu2

Cu22+ + 2e2–

2

Figure 2.7. Copper concentration cell.

In this concentration cell, we have two electrolyte baths containing different concentrations of Cu2+ ions (with counterbalancing SO4 2– ions for ionic charge balance),

57

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+ concentration difference between the two baths. Because of the high concentration of Cu2+ ions in bath 1, we have the reaction + 2e−1 → Cu1 Cu2+ 1

(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: + 2e−2 (2.105) Cu2 → Cu2+ 2 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 + 2e−1 → Cu2+ + 2e−2 + Cu1 Cu2 + Cu2+ 1 2

(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 0 1 1 𝜇̃ Cu 2+ = 𝜇Cu2+ + RT ln aCu2+ + 2F𝜑Cu2+ 2 0 2 2 𝜇̃ Cu 2+ = 𝜇Cu2+ + RT ln aCu2+ + 2F𝜑Cu2+

𝜇̃ e1− = 𝜇e0− + RT ln a1e− − 1F𝜑1e− = 𝜇e0− − 1F𝜑1e− 𝜇̃ e2− = 𝜇e0− + RT ln a2e− − 1F𝜑2e− = 𝜇e0− − 1F𝜑2e− 1 0 0 𝜇̃ Cu = 𝜇Cu + RT ln a1Cu = 𝜇Cu 2 0 0 𝜇̃ Cu = 𝜇Cu + RT ln a2Cu = 𝜇Cu

(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 0 0 2 2 0 2 + 𝜇Cu (𝜇Cu 2+ + RT ln aCu2+ + 2F𝜑Cu2+ + 2𝜇e− − 2F𝜑e− ) 0 0 1 1 0 1 − (𝜇Cu + 𝜇Cu 2+ + RT ln aCu2+ + 2F𝜑Cu2+ + 2𝜇e− − 2F𝜑e− ) = 0

(2.108)

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

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 a1Cu2+ + 2F𝜑1Cu2+ − RT ln a2Cu2+ − 2F𝜑2Cu2+ = 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 2+ = 𝜑2 2+ . Applying this final simplification yields Cu

Cu

a1 RT ln

Cu2+

a2

= 2F(𝜑1e− − 𝜑2e− ) = 2FE

(2.110)

Cu2+

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 concentration cell.

2.4.5

Summary

Let us briefly summarize the effects of non-standard-state conditions on reversible electrochemical cell voltages. In the past few pages, we have used classical thermodynamics 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

Δ̂s nF

(2.111)

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

dE dp

) =− T

Δng RT nFp

=−

Δ𝑣̂ nF

(2.112)

• The variation of the reversible cell voltage with chemical activity (chemical composition, concentration, etc.) is given by the Nernst equation: ∏ 𝑣i aproducts RT E=E − ln ∏ 𝑣i nF areactants 0

(2.113)

59

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 H2 O 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 =

Δ̂g Δĥ

(2.116)

At room temperature and pressure, the H2 –O2 fuel cell has Δ̂g0 = −237.17 kJ/mol and Δĥ 0HHV = −285.83 kJ/mol. This yields a 83% reversible HHV efficiency for the H2 –O2 fuel cell at STP: −237.17 𝜀thermo, fc = = 0.83 (2.117) −285.83

FUEL CELL EFFICIENCY

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: T − TL (2.118) 𝜀Carnot = H TH 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 vaporization, the Δĥ rxn for a hydrogen–oxygen fuel cell is significantly different, depending on whether vapor or liquid water product is assumed. When liquid water is produced, Δĥ 0rxn = −286 kJ/mol; when water vapor is produced, Δĥ 0rxn = −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 Δĥ 0rxn involving liquid water is called the higher heating value (HHV), while the Δĥ 0rxn 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 book will make use of the HHV. Thus, we should rewrite Equation 2.116 to explicitly reflect this fact: 𝜀thermo, fc =

Δ̂g Δĥ HHV

(2.119)

In these efficiency calculations, it is important to note that Δ̂g should still be calculated by properly accounting for phase transitions. Thus, for a hydrogen–oxygen fuel cell operating above 100∘ C, the calculation of Δ̂g should use formation enthalpies and entropies for water vapor. Below 100∘ C, the calculation of Δĥ HHV should use the formation enthalpies and entropies for liquid water. You should recognize that calculating Δ̂g based on water vapor above 100∘ C, while simultaneously using Δĥ HHV (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%.

61

FUEL CELL THERMODYNAMICS

Temperature (K) 1.00

300 400 500 600 700 800 900 1000 1100 1200

0.90 Reversible efficiency (HHV)

62

H2/O2 fuel cell

0.80 0.70 0.60 0.50 0.40

Carnot cycle

0.30 0.20 100 200 300 400 500 600 700 800 900 1000 Temperature (°C)

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 H2 O(l) vs. H2 O(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

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 voltage 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 irreversible 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 thermodynamically 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 are most 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 provided 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

63

FUEL CELL THERMODYNAMICS

cell supplied with 1.5 times more 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 =

Δ̂g Δĥ HHV

)

( ) ( i∕nF ) V E 𝑣fuel

(2.124)

For a fuel cell operating under a constant-stoichiometry condition, this equation simplifies to ) ( ( )( ) Δ̂g 1 V (2.125) 𝜀real = E 𝜆 Δĥ HHV 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 1 1

0.9 0.8 j–V curve

0.7 0.6

0.6 0.5 0.4

ε, constant stoichiometry (λ=1.1)

0.4

Efficiency

0.8 Cell voltage (V)

64

0.3 0.2

0.2

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

0.1 0

0 0

0.5

1

1.5

2

Current density (A/cm2)

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

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 creation 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, consumes hydrogen and oxygen to generate water, heat, and electricity. Although hydrogen and oxygen are consumed during operation, water, heat, and electricity are produced in correspondingly 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. From Equation 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 ̂ = Ph + Pe = Ph + V × i Pin = |Δh|𝑣

(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 ̂ −V ×i Ph = Pin − Pe = |Δh|𝑣 ⎛ 𝜆 ||Δĥ || ⎞ = ⎜ | | − V ⎟ × i = (𝜆EH − V) × i ⎜ nF ⎟ ⎝ ⎠

(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

65

FUEL CELL THERMODYNAMICS

̂

voltage, V, and current, i. The term EH = |ΔnFh| in Equation 2 is known as the “thermoneutral voltage.” EH represents an “ideal” voltage calculated from the enthalpy of reaction, g| ) is calculated from the similarly to how the ideal reversible voltage of a fuel cell (E0 = |Δ̂ nF 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 =

̂ 286, 000 J∕mol |Δh| = = 1.48 V nF 2 × 96, 485 C∕mol

If this fuel cell is operating at 0.7 V and 10 A under STP conditions with 100% fuel utilization (λ = 1), it generates 7 W of electrical power (Pe = 0.7 V × 10 A = 7 W) and 7.8 W 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.

EH E0 Voltage (V)

66

EH-V

V iout i

Current (A)

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 = |ΔnFh| ) 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

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: i 𝑣out = 𝑣in ± (2.129) nF 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: i i = 𝑣Air,in × 𝑤O2 − nF nF 1, 000, 000 A = 20 mol∕s × 0.21 − = 1.6 mol∕s 4 × 96, 485 C∕mol

𝑣O2 ,out = 𝑣O2 ,in −

(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 ) ( i 1000 kA = = 5.18 mol∕s 𝑣H2 O = 𝑣H2 = nF 2 × 96485 C∕mol 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 = nF𝑣air 𝑤O2 = 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 + 12 O2 → H2 O + Electricity

Electrolyzer mode:

H2 O + Electricity → H2 + 12 O2

(2.131)

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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 Δ̂g

(2.132)

For water electrolysis at room temperature and pressure, we have Δ̂g0 = 237.17 kJ/mol and Δĥ 0HHV = 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! 286 𝜀thermo,electrolyzer = = 1.2 (2.133) 237 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 Δĥ = Δ̂g 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 associated 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 thermodynamic 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

hydrogen is consumed in the fuel cell mode is inevitably less than the ideal STP thermodynamic 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 differences. 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 = |ΔnFh| . 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 consumption 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 operating voltage V and current i) versus the chemical “power” produced by the electrolyzer (as given by EH and i): ̂ i Ph,electrolysis = Pe, in − Pchem, out = V × i − |Δh| nF ( ) ̂ |Δh| = V− × i = (V − EH ) × i nF

(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 hydrogen 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 produced 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 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. 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 (T Δs losses). As shown in Figure 2.12, the situation is reversed for an electrolyzer. Thus, the ideal thermodynamic efficiency of an electrolyzer increases with 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.

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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 = |ΔnFh| ) represents the heat loss in the fuel cell. Under the electrolyzer 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 consumption 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

Figure 2.12. Reversible HHV efficiency of H2 O electrolysis compared to an H2 –O2 fuel cell. The thermodynamic efficiency of electrolysis increases with increasing temperature, while thermodynamic 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 performance. • 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.

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• 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 Δ̂g = −nFE. • ΔG scales with reaction amount whereas Δ̂g and E do not scale with reaction amount. • E varies with temperature as dE∕dT = Δ̂s∕nF. For fuel cells, Δ̂s is generally negative; therefore, reversible fuel cell voltages tend to decrease with increasing temperature. E varies with pressure as dE∕dp = −Δng RT∕(nFp) = −Δ𝑣∕nF ̂ • The Nernst equation describes how E varies with reactant/product activities: ∏ 𝑣i aproducts RT E = E0 − ln ∏ 𝑣i nF areactants • The Nernst equation intrinsically includes the pressure effects on reversible cell voltage but does not fully account for the temperature effects. • Ideal HHV fuel cell efficiency 𝜀thermo = Δ̂g∕Δĥ HHV . • 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 efficiency is given by the product of individual efficiencies. • A fuel cell satisfies the laws of energy and mass conservation. Accordingly, the thermal 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 2.2

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

2.3

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

2.4

Why does ΔG 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

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 reaction of the form zeA + 1A + bB ⇌ mM + nN + zeC (2.137)

2.7

̂ ever be Can the thermodynamic efficiency of a fuel cell, as defined by 𝜀 = Δ̂g∕Δh, 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 voltage V (volts). (a) Write expressions for the output mass flux (mol/s) of methanol (𝑣MeOH, out ), air (𝑣air, out ), water (𝑣H2 O, 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 Δĥ rxn and Δ̂srxn were independent of temperature. 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 CO2 H2 H2 O(g)

29.2 37.2 28.8 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 expression 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

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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 measured 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 (𝑣H2 O, 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 Δĥ rxn = –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 (c) 444.4cm2 (d) 666.6 cm2 2.15 For the fuel cell described above in problem 2.14, assuming operation on pure hydrogen 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 (c) 32.2 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. (c) E is constant. (d) This cannot be determined.

CHAPTER EXERCISES

2.17 Given the following half-cell reactions: 1. O2− + CO(g) → CO2 (g) + 2e− 2. 2O2− → 4e− + O2 (g) 3. 8e− + 2H2 O(g) + CO2 (g) → 4O2− + CH4 (g) 4. 12 O2 (g) + H2 O(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 electrolysis 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, how much 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 utilization, 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.)

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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 thermodynamically 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 transfer 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.

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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 reactions. 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. Electrochemical reactions, like the HOR, (3.1) H2 ⇌ 2H+ + 2e− 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 current i evolved by an electrochemical reaction is a direct measure of the rate of the electrochemical reaction. The unit of current is the ampere; an ampere is a coulomb per second (C∕s). From Faraday’s law, dQ (3.2) i= dt 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 = nF

dN = nF𝑣 dt

(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.) – – – –

+ + + +

– – – –

+ + + +

H2

2e–

Electrode

2H + 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

Example 3.1 Assuming 100% fuel utilization, how much current can a fuel cell produce if provisioned with 5 sccm H2 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=

p(dV∕dt) dN = dt RT

(3.4)

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

(1 atm)(0.005 L∕min) dN = = 2.05 × 10−4 mol H2 ∕min (3.5) dt [0.082 L ⋅ atm∕(mol ⋅ K)](298.15 K)

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 dN = (2)(96,485 C∕mol)(2.05 × 10−4 mol H2 ∕min)(1 min∕60 s) = 0.659 A dt (3.6) Thus, a flow rate of 5 sccm H2 is sufficient to sustain 0.659 A of current, assuming 100% fuel utilization. i = nF

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 electrochemical 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.

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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: (3.8) Qtot = i1 t1 + i2 t2 = (2 A)(3600 s) + (5 A)(7200 s) = 43,200 C 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 43200 C = = 0.224 mol H2 nF (2)(96,485 C∕mol)

(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 ): i j= (3.10) A 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=

3.1.6

j i 1 dN = = A dt nFA nF

(3.11)

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 influencing 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 chemical species: (3.12) Ox + e− ⇌ Re

INTRODUCTION TO ELECTRODE KINETICS

Increasing electron energy

Electrode Electrolyte

Electrode Electrolyte

Electrode Electrolyte

e– Fermi level

e– Fermi level Fermi level

Increasing electrode potential (Voltage)

Negative (relative) electrode potential

Equilibrium electrode potential

Positive (relative) electrode potential

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

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Reactants (H2 + O2)

Free energy

82

∆G‡

∆Grxn

Products (H2O) Reaction progress

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, releasing 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

…H 1

e– Electrode (M)

H+

2

Electrolyte

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 reaction, 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.

83

FUEL CELL REACTION KINETICS

2

Free energy

84

1

a ∆G1 ‡ ∆G2 ‡

(M…H)

∆Grxn

(M + e–) + H+

Distance from interface

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: ‡ (3.14) Pact = e−ΔG1 ∕(RT) 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 f1 e−ΔG1 ∕(RT)

(3.15)

where J1 is the reaction rate in the forward direction (reactants → products), c∗R is the reactant surface concentration (mol∕cm2 ), and f1 is the decay rate to products. The decay rate

CALCULATING NET RATE OF A REACTION

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 THE DECAY RATE (OPTIONAL) As was mentioned above, the decay rate 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: Pa→p f1 = (3.16) 𝜏a Here, Pa→p is the probability that the activated state will decay to the product state and 𝜏a is the lifetime of the activated state. Both decay rates to products (f1 ) and decay rates to reactants (f2 ) can be computed. In general, the decay rates are determined by the curvature of the free-energy surface in the vicinity of the activated state. For simplicity, it is often assumed that there is an equal likelihood of conversion to the reactant (r) or product (p) states (Pa→p = Pa→r = 12 ). In addition, 𝜏a can often be approximated as h∕2kT, where k is Boltzmann’s constant and h is Planck’s constant. In these cases, the decay rate to products and reactants are equal, reducing to f1 = f 2 =

kT h

(3.17)

Combining this simplified decay rate expression with our reaction rate equation 3.15 yields the following reduced expression for reaction rate: J1 = c∗R

3.4

kT −ΔG‡1 ∕(RT) e h

(3.18)

CALCULATING NET RATE OF A REACTION

When evaluating the overall rate of a reaction, we must consider the rates for both the forward and reverse directions of the reaction. The net rate is given by the difference in rates between the forward and reverse reactions. For example, the chemisorbed hydrogen reaction (Equation 3.13) can be split into forward and reverse reactions: Forward reaction: M · · · H → (M + e− ) + H+

(3.19)

Reverse reaction: (M + e− ) + H+ → M · · · H

(3.20)

with corresponding reaction rates given by J1 for the forward reaction and J2 for the reverse reaction. The net reaction rate J is defined as J = J1 − J2

(3.21)

85

86

FUEL CELL REACTION KINETICS

In general, the rates for the forward and reverse reactions may not be equal. In our example of the chemisorbed hydrogen reaction, the free-energy diagram in Figure 3.5 shows that the activation barrier for the forward reaction is much smaller than the activation barrier for the reverse reaction (ΔG‡1 < ΔG‡2 ). In this situation, it stands to reason that the forward reaction rate should be much greater than the reverse reaction rate. Using our reaction rate formula (Equation 3.15), the net reaction rate J may be written as ‡ ‡ (3.22) J = c∗R f1 e−ΔG1 ∕(RT) − c∗P f2 e−ΔG2 ∕(RT) where c∗R is the reactant surface concentration, c∗p is the product surface concentration, ΔG‡1 is the activation barrier for the forward reaction, and ΔG‡2 is the activation barrier for the reverse reaction. From the figure, it is obvious that ΔG‡2 is related to ΔG‡1 and ΔGrxn . In calculating the relationship between these activation energies, it is imperative to be careful with signs: ΔG quantities are always calculated as final state – initial state. For both ΔG‡1 and ΔG‡2 , the final state is the activated state; thus, activation barriers are always positive. If signs are properly accounted for, then ΔGrxn = ΔG‡1 − ΔG‡2

(3.23)

Equation 3.22 can then be expressed in terms of only the forward activation barrier ΔG‡1 : ‡

‡

J = c∗R f1 e−ΔG1 ∕(RT) − c∗P f2 e−(ΔG1 −ΔGrxn )∕(RT)

(3.24)

Thus, Equation 3.24 states that the net rate of a reaction is given by the difference between the forward and reverse reaction rates, both of which are exponentially dependent on an activation barrier, ΔG‡1 . 3.5 RATE OF REACTION AT EQUILIBRIUM: EXCHANGE CURRENT DENSITY For fuel cells, we are interested in the current produced by an electrochemical reaction. Therefore, we want to recast these reaction rate expressions in terms of current density. Recall from Section 3.1.3 that current density j and reaction rate J are related by j = nFJ. Therefore, the forward current density can be expressed as ‡

j1 = nFc∗R f1 e−ΔG1 ∕(RT)

(3.25)

and the reverse current density is given by ‡

j2 = nFc∗P f2 e−(ΔG1 −ΔGrxn )∕(RT)

(3.26)

POTENTIAL OF A REACTION AT EQUILIBRIUM: GALVANI POTENTIAL

At thermodynamic equilibrium, we recognize that the forward and reverse current densities must balance so that there is no net current density (j = 0). In other words, j1 = j 2 = j 0

(at equilibrium)

(3.27)

We call j0 the exchange current density for the reaction. Although at equilibrium the net reaction rate is zero, both forward and reverse reactions are taking place at a rate which is characterized by j0 ; this is called dynamic equilibrium. 3.6

POTENTIAL OF A REACTION AT EQUILIBRIUM: GALVANI POTENTIAL

Another way to understand the equilibrium state of a reaction is presented in Figure 3.6, which revisits our chemisorbed hydrogen system. Figure 3.6a is a simplified version of

Chemical free energy

(a)

+

∆Grxn

(M + e–) + H+

Distance from interface Electrical energy

(b)

(M…H)

–nF∆ϕ Distance from interface

= Chemical + electrical energy

(c)

∆G‡ j0 Distance from interface

Figure 3.6. At equilibrium, the chemical free-energy difference (a) across a reaction interface is balanced by an electrical potential difference (b), resulting in a zero net reaction rate (c).

87

FUEL CELL REACTION KINETICS

Figure 3.5, showing the chemical free-energy path for the chemisorbed hydrogen reaction. The lower free energy of the product state ([M + e− ] + H+ ) compared to the reactant state (M · · · H) leads to unequal activation barriers for the forward- versus reverse-reaction directions. Therefore, as we have previously discussed, we expect the forward reaction rate to proceed faster than the reverse reaction rate. However, these unequal rates quickly result in a buildup of charge, with e− accumulating in the metal electrode and H+ accumulating in the electrolyte. The charge accumulation continues until the resultant potential difference Δ𝜙 across the reaction interface (as shown in Figure 3.6b) exactly counterbalances the chemical free-energy difference between the reactant and product states. This balance expresses the thermodynamic statement of electrochemical equilibrium that we developed in Equation 2.100. The combined effect of the chemical and electrical potentials is shown in Figure 3.6c, where the net force balance leads to equal rates for the forward and reverse reactions. As you have previously seen, the speed of this equilibrium reaction rate is captured in the exchange current density j0 . Recall that before the buildup of the interfacial potential (Δ𝜙), the forward rate was much faster than the reverse rate. The buildup of an interfacial potential effectively equalizes the situation by increasing the forward activation barrier from ΔG‡1 to ΔG‡ , while decreasing the reverse activation barrier from ΔG‡2 to ΔG‡ . We can write the forward and reverse current densities at equilibrium as ‡

j1 = nFc∗R f1 e−(ΔG

‡

j2 = nFc∗P f2 e−(ΔG

)∕(RT)

(3.28)

−ΔGrxn −nFΔ𝜙)∕(RT)

‡

= nFc∗P f2 e−(ΔG

)∕(RT)

(3.29)

While we have discussed Figure 3.6 in terms of the hydrogen reaction, it could just as easily represent the situation for the oxygen reaction at a fuel cell cathode. As in the hydrogen reaction, a difference in chemical free energy between the reactant and product states at the cathode will lead to an electrical potential difference. At equilibrium, the two force contributions balance, leading to a dynamic equilibrium with zero net reaction. In optional Section 3.14 of this chapter, a more detailed view incorporating both the anode and the cathode interfaces is presented. As shown in Figure 3.7, the sum of the interfacial electrical potential differences at the anode and cathode yields the overall thermodynamic equilibrium voltage for the fuel cell.

Voltage (V)

88

∆ϕcathode ∆ϕanode Anode

Eo Electrolyte Distance (x)

Cathode

Figure 3.7. One hypothetical possibility for the shape of the fuel cell voltage profile, since scientists can determine E0 but not Δ𝜙anode or Δ𝜙cathode . The Galvani potentials at the anode and cathode of a fuel cell must sum to give the overall thermodynamic cell voltage E0 .

POTENTIAL AND RATE: BUTLER–VOLMER EQUATION

The anode (Δ𝜙anode ) and cathode (Δ𝜙cathode ) interfacial potentials shown in Figure 3.7 are called Galvani potentials. For reasons we will not discuss, the exact magnitude of these Galvani potentials are as-yet unknowable. While scientists know that the anode and cathode Galvani potentials must sum to give the net thermodynamic voltage of the fuel cell as a whole (E0 = Δ𝜙anode + Δ𝜙cathode ), they are unable to determine how much of this potential may be attributed to the anode interface versus the cathode interface. Thus, Figure 3.7 illustrates only one possible view of the fuel cell voltage profile. As a homework problem, you will sketch other possible voltage profiles.

3.7

POTENTIAL AND RATE: BUTLER–VOLMER EQUATION

A distinguishing feature of electrochemical reactions is the ability to manipulate the size of the activation barrier by varying the cell potential. Charged species are involved as either reactants or products in all electrochemical reactions. The free energy of a charged species is sensitive to voltage. Therefore, changing the cell voltage changes the free energy of the charged species taking part in a reaction, thus affecting the size of the activation barrier. Figure 3.8 illustrates this idea. If we neglect to benefit from the full Galvani potential across a reaction interface, we can bias the system energetics such that the forward reaction rate is favored. By sacrificing part of the thermodynamically available cell voltage, we can produce a net current from our fuel cell. The Galvani potentials at the anode and the cathode must both be reduced (though not necessarily in equal amounts) to extract a net current from a fuel cell. It is important to understand the scale of Figure 3.8, which focuses on a nanometer-sized dimension right at the interface between the anode and the electrolyte. Thus, the Galvani potential, which is shown to increase linearly across the 1–2 nm thickness of the anode–electrolyte interface in Figure 3.8b, is in actuality an almost perfectly abrupt voltage “step,” when shown at a larger scale in Figure 3.9. As shown in Figure 3.9, reductions to both the anode and cathode Galvani potentials (which are necessary to favorably “bias” the anode and cathode reactions in the forward direction) combine to yield a smaller net fuel cell voltage. Figure 3.8 is a detailed view of what is happening only at the anode–electrolyte interface. An analogous detailed view for the cathode–electrolyte interface is not shown but would be similar to Figure 3.8, although the size of the voltage step would not necessarily be identical. A full detailed picture including both the anode and cathode processes is provided by Figure 3.19 in an optional section at the end of this chapter. As shown in Figure 3.8c, decreasing the Galvani potential by 𝜂 reduces the forward activation barrier (ΔG‡1 < ΔG‡ ) and increases the reverse activation barrier (ΔG‡2 > ΔG‡ ). A careful inspection of the figure shows that the forward activation barrier is decreased by 𝛼nF𝜂, while the reverse activation barrier is increased by (1 − 𝛼)nF𝜂. The value of 𝛼 depends on the symmetry of the activation barrier. Called the transfer coefficient, 𝛼 expresses how the change in the electrical potential across the reaction interface changes the sizes of the forward versus reverse activation barriers. The value of 𝛼 is always between 0 and 1. For “symmetric” reactions, 𝛼 = 0.5. For most electrochemical reactions, 𝛼 ranges from about 0.2 to 0.5.

89

FUEL CELL REACTION KINETICS

Chemical free energy

(a) (M…H)

∆Grxn

+ Electrical energy

(b)

(M + e–) + H+ Distance from interface

–nFη

–nF∆ϕ

Distance from interface

=

–αnFη

(c) Chemical + electrical energy

90

∆G‡ –nFη

∆G1‡

∆G2‡

Distance from interface

Figure 3.8. If the Galvani potential across a reaction interface is reduced, the free energy of the forward reaction will be favored over the reverse reaction. While the chemical energy (a) of the reaction system is the same as before, changing the electrical potential (b) upsets the balance between the forward and reverse activation barriers (c). In this diagram, reducing the Galvani potential by 𝜂 reduces the forward activation barrier ((ΔG‡1 < ΔG‡ ) and increases the reverse activation barrier (ΔG†2 > ΔG† ).

At equilibrium, the current densities for the forward and reverse reactions are both given by j0 . Away from equilibrium, we can write the new forward and reverse current densities by starting from j0 and taking into account the changes in the forward and reverse activation barriers: (3.30) j1 = j0 e(𝛼nF𝜂∕(RT)) j2 = j0 e−(1−𝛼)nF𝜂∕(RT)

(3.31)

j = j0 (e𝛼nF𝜂∕(RT) − e−(1−𝛼)nF𝜂∕(RT) )

(3.32)

The net current (j1 – j2 )is then

k

k

k

Figure 3.9. Extracting a net current from a fuel cell requires sacrificing a portion of both the anode and cathode Galvani potentials. In this figure, the anode Galvani potential is lowered by 𝜂act, A , while the cathode Galvani potential is lowered by 𝜂act, C . As the figure indicates, 𝜂act, A and 𝜂act, C are not necessarily equal. For a typical H2 –O2 fuel cell, 𝜂act, C is generally much larger than 𝜂act, A . Compare the detail view in this figure with Figure 3.8b. You should realize that these figures are showing the same thing, although Figure 3.8 is plotted with units of energy (ΔG = nFV), while Figure 3.9 is plotted with units of voltage (V).

91

k

92

FUEL CELL REACTION KINETICS

Although it may not be obvious, this equation assumes that the concentrations of reactant and product species at the electrode are unaffected by the presence of a net reaction rate. (Remember that j0 depends on c∗R and c∗P ; see Equations 3.25 and 3.26.) In reality, however, a net reaction rate will likely affect the surface concentrations of the reactant and product species. For example, if the forward reaction rate increases dramatically while the reverse reaction rate decreases dramatically, the reactant species surface concentration will tend to become depleted. In this case, we can explicitly reflect the concentration dependence of the exchange current density in our equation as follows: ( ∗ ) cR 𝛼nF𝜂∕(RT) c∗P −(1−𝛼)nF𝜂∕(RT) 0 (3.33) e − 0∗ e j = j0 c0∗ cP R where 𝜂 is the voltage loss, n is the number of electrons transferred in the electrochemical reaction, c∗R and c∗P are the actual surface concentrations of the rate-limiting species in the reaction, and j00 is measured at the reference reactant and product concentration values c0∗ and c0∗ . Effectively, j00 represents the exchange current density at a “standard R P concentration.” Equation 3.32 (or 3.33), known as the Butler–Volmer equation, is considered the cornerstone of electrochemical kinetics. It is used as the primary departure point for most attempts to describe how current and voltage are related in electrochemical systems. Remember it forever. The Butler–Volmer equation basically states that the current produced by an electrochemical reaction increases exponentially with activation overvoltage. Activation overvoltage is the label given to 𝜂, recognizing that 𝜂 represents voltage which is sacrificed (lost) to overcome the activation barrier associated with the electrochemical reaction. Thus, the Butler–Volmer equation tells us that if we want more electricity (current) from our fuel cell, we must pay a price in terms of lost voltage. Figure 3.10 shows the functional form of the Butler–Volmer equation. Two distinct regions are indicated where simplifications of Equation 3.32 lead to easier kinetic treatment. These simplifications will be discussed in Section 3.9. THE ACTIVATION OVERVOLTAGE, 𝜼act To clarify that 𝜂 represents a voltage loss due to activation, it is typically given the subscript act, as in 𝜂act . This distinguishes it from other voltage losses that you will read about in the upcoming chapters (which are also given the symbol 𝜂). From now on, we refer to the activation loss appearing in the Butler–Volmer equation as 𝜂act , the activation overvoltage. While we derived the Butler–Volmer equation using a specific reaction example, in reality the Butler–Volmer equation is fundamentally applicable only for single-electron transfer events. Nevertheless, the Butler–Volmer equation generally serves as an excellent approximation for most single-step electrochemical reactions, and even for multistep electrochemical reactions where the rate-determining step is intrinsically much slower than the other steps. However, for more complex multistep reactions where several steps have

POTENTIAL AND RATE: BUTLER–VOLMER EQUATION

Figure 3.10. Relationship between 𝜂 and j as given by the Butler–Volmer equation. The fine solid lines show the individual contributions from the forward (j1 ) and reverse (j2 ) current density terms while the dark solid line shows the net current density (j) given by the complete Butler–Volmer equation. Note that the Butler–Volmer curve is distinctly linear at low current density and distinctly exponential at high current density. In these regions, simplifications of the Butler–Volmer equation (as developed in Section 3.9) may be used. Note that the direction (sign) on the 𝜂 axis is switched in this figure to enable direct comparison with Figure 3.11.

approximately the same intrinsic rate, modifications to the Butler–Volmer equation are required. While important, such treatments are beyond the scope of this book. Even for these complex multistep reactions, however, Butler–Volmer kinetics often proves to be an excellent first approximation. For simple electrochemical systems, variations between reactions can be treated in terms of variations in kinetic parameters such as 𝛼 and j0 using the Butler–Volmer equation. As far as fuel cell performance is concerned, reaction kinetics induces a characteristic, exponentially shaped loss on a fuel cell’s j–V curve, as shown in Figure 3.11. This curve was

1.2

Theoretical EMF or ideal voltage

Cell voltage(V)

ηact j0 = 10–2

j0 = 10–5

0.5

j0 = 10–8 Current density (mA/cm2)

1000

Figure 3.11. Effect of activation overvoltage on fuel cell performance. Reaction kinetics typically inflicts an exponential loss on a fuel cell’s j–V curve as determined by the Butler–Volmer equation. The magnitude of this loss is influenced by the size of j0 . (Curves calculated for various j0 values with 𝛼 = 0.5, n = 2, and T = 298.15 K.)

93

94

FUEL CELL REACTION KINETICS

calculated by starting with Ethermo and then subtracting 𝜂act . The functional dependence of 𝜂act on j was given by the Butler–Volmer equation 3.32. The magnitude of the activation loss (in other words, the size of 𝜂act ) depends on the reaction kinetic parameters. The loss especially depends on the size of j0 , as shown in Figure 3.11. Having a high j0 is absolutely critical to good fuel cell performance. As we will now discuss, there are several effective ways to increase j0 . Example 3.3 If a fuel cell reaction exhibits 𝛼 = 0.5 and n = 2 at room temperature, what activation overvoltage is required to increase the forward current density by one order of magnitude and decrease the reverse current density by one order of magnitude? Solution: Since 𝛼 = 0.5, the reaction is symmetric. We can look at either the forward or reverse term in the Butler–Volmer equation to calculate the overvoltage necessary to cause an order-of-magnitude change in current density. Using the forward term, 10j1 j0 (e𝛼nF𝜂act2 ∕(RT) ) = j1 j0 (e𝛼nF𝜂act1 ∕(RT) ) 10 = e𝛼nFΔ𝜂act ∕(RT)

(3.34)

where we have defined Δ𝜂act as the change in activation overvoltage (𝜂act2 − 𝜂act1 ) necessary to increase the forward current density 10-fold. Solving for Δ𝜂act gives Δ𝜂act =

(8.314)(298.15) RT ln 10 = ln 10 = 0.059 V 𝛼nF (0.5)(2)(96,485)

(3.35)

Thus, an activation overvoltage of approximately 60 mV is required to increase the forward current density by one order of magnitude and decrease the reverse current density by one order of magnitude for this reaction. If the exchange current density for this reaction was 10−6 A∕cm2 , increasing the net current density by six orders of magnitude to 1 A∕cm2 (a typical fuel cell operating current density) would require an activation overvoltage of 6 × 60 mV = 0.36 V. Activation overvoltage penalties of 0.3–0.4 V are therefore quite typical for operating fuel cells. 3.8 EXCHANGE CURRENTS AND ELECTROCATALYSIS: HOW TO IMPROVE KINETIC PERFORMANCE Improving kinetic performance focuses on increasing j0 . To understand how we can increase j0 , recall how j0 is defined. Remember that j0 represents the “rate of exchange” between the reactant and product states at equilibrium. We can define j0 from either the forward- or reverse-reaction direction. Taking the forward reaction for simplicity (see Equation 3.25) and including the concentration effects, ‡

j0 = nFc∗R f1 e−ΔG1 ∕(RT)

(3.36)

EXCHANGE CURRENTS AND ELECTROCATALYSIS: HOW TO IMPROVE KINETIC PERFORMANCE

By including reactant concentration effects in j0 , we must then use Equation 3.32 for the Butler–Volmer equation. Examining Equation 3.36, it is clear that we cannot change n, F, f1 (not significantly), or R. Therefore, we have only three ways to increase j0 . In fact, there are four major ways to increase j0 , although the fourth method is not apparent from our equation: 1. Increase the reactant concentration c∗R . 2. Decrease the activation barrier ΔG‡1 . 3. Increase the temperature T. 4. Increase the number of possible reaction sites (i.e., increase the reaction interface roughness). Each of these is discussed below. 3.8.1

Increase Reactant Concentration

In the last chapter, we noted that the thermodynamic benefit to increasing reactant concentration is minor, due to the logarithmic form of the Nernst equation. In contrast, the kinetic benefit to increasing reactant concentration is significant, with a linear rather than logarithmic impact. By operating fuel cells at higher pressure, we can increase the concentrations of the reactant gas species, improving the kinetics commensurately. Unfortunately, the kinetic penalty due to decreasing reactant concentration is likewise significant. In real fuel cells, kinetic reactant concentration effects generally work against us for several reasons. First, most fuel cells use air instead of pure oxygen at the cathode. This leads to an approximate 5× reduction in the oxygen kinetics compared to pure oxygen operation. Second, as will be discussed in Chapter 5, reactant concentrations tend to decrease at fuel cell electrodes during high-current-density operation (due to mass transport limitations). Essentially, the reactants are being consumed at the electrodes faster than they can be replenished, causing the local reactant concentrations to diminish. This depletion effect leads to further kinetic penalties. This interaction between kinetics and mass transport is the heart of the concentration loss effect described in Chapter 5. 3.8.2

Decrease Activation Barrier

As is apparent from Equation 3.36, decreasing the size of the activation barrier ΔG‡1 will increase j0 . A decrease in ΔG‡1 represents the catalytic influence of the surface of the electrode: A catalytic electrode is one which significantly lowers the activation barrier for the reaction. Because ΔG‡1 appears as an exponent, even small decreases in the activation barrier can cause large effects. Using a highly catalytic electrode therefore provides a way to dramatically increase j0 . How does a catalytic electrode lower the activation barrier? By changing the free-energy surface of the reaction. If you recall Figure 3.5, the size of the activation barrier for the hydrogen charge transfer reaction is related to the shape of the [M · · · H] and [(M + e− ) + H+ ] free-energy curves. Thus, the free-energy curves shown in Figure 3.5 will

95

96

FUEL CELL REACTION KINETICS

depend on the nature of the electrode metal, M. Different free-energy curves and therefore different activation barriers arise, depending on the chemical nature of the M · · · H bond. For the case of the hydrogen charge transfer reaction, an intermediate-strength bond provides the greatest catalytic effect. Why is an intermediate-strength bond most effective? If the [M · · · H] bond is too weak, then it is difficult for hydrogen to bond to the electrode surface in the first place, and it is furthermore difficult to transfer charge from the hydrogen to the electrode. On the other hand, if the [M · · · H] is too strong, the hydrogen bonds too well to the electrode surface. We then find it difficult to liberate free protons (H+ ), and the electrode surface becomes clogged with unreactive [M · · · H] pairs. The optimal compromise between bonding and reactivity occurs for intermediate-strength [M · · · H] bonds. This peak in catalytic activity coincides with platinum group metals and their neighbors, such as Pt, Pd, Ir, and Rh. See Section 3.13 on the Sabatier principle for more information on what makes for the best catalysts. CHOICE OF CATALYST ALSO AFFECTS 𝜶 Note that the value of 𝛼 will also be affected by the choice of catalyst. Recall that 𝛼 is based on the symmetry of the free-energy curve in the vicinity of the activated state. Therefore, changes in the electrode free-energy curve can also be expected to change 𝛼. The Butler–Volmer equation predicts that increasing 𝛼 will result in a higher net current density. Therefore, catalysts with a high 𝛼 should be desired over catalysts with a low 𝛼. Generally, 𝛼 is difficult to quantify and changes only slightly with choice of catalyst, so it is often overlooked compared to other catalytic effects. 3.8.3

Increase Temperature

Equation 3.36 shows that increasing the temperature of reaction will also increase j0 . By increasing the reaction temperature, we are increasing the thermal energy available in the system; all particles in the system now move about and vibrate with increased intensity. This higher level of thermal activity increases the likelihood that a given reactant will possess sufficient energy to reach the activated state, thus increasing the rate of reaction. Like changing the activation barrier, changing the temperature has an exponential effect on j0 . In reality, the complete story about temperature is a little more complicated than described here. At high overvoltage levels, increasing the temperature can actually decrease the current density. This effect is explained for the interested reader in a future dialogue box. 3.8.4

Increase Reaction Sites

Although not evident from Equation 3.36, the fourth method for increasing j0 is to increase the number of available reaction sites per unit area. It is helpful to remember that j0 represents a current density, or a reaction current per unit area. Current densities are generally based on the plane, or projected geometric area of an electrode. If an electrode surface is extremely rough, the true electrode surface area can be orders of magnitude larger than the

SIMPLIFIED ACTIVATION KINETICS: TAFEL EQUATION

geometric electrode area. As far as the kinetics are concerned, a highly rough electrode surface provides many more sites for reaction than a smooth electrode surface. Therefore, the effective j0 of a rough electrode surface will be greater than the j0 of a smooth electrode surface simply because of the greater surface area. This relationship can be summarized by the equation A (3.37) j0 = j′0 ′ A where j′0 represents the intrinsic exchange current density of a perfectly smooth electrode surface. The ratio A∕A′ expresses the surface area enhancement of a real electrode (area A) compared to an ideally smooth electrode (area A′ ) . This definition has the benefit that j′0 can be considered an intrinsic property of an electrode for a specific electrochemical reaction. For example, the standard state j′0 for the HOR on platinum in sulfuric acid is widely considered to be around 10−3 A∕cm2 . A platinum catalyst electrode with an effective surface area 1000 times greater than smooth platinum would therefore show an effective j0 for the HOR of approximately 1 A∕cm2 . 3.9

SIMPLIFIED ACTIVATION KINETICS: TAFEL EQUATION

When dealing with fuel cell reaction kinetics, the Butler–Volmer equation often proves unwieldy. In this section, we simplify the Butler–Volmer expression via two useful approximations. These approximations apply when the activation overvoltage (𝜂act ) in the Butler–Volmer equation is either very small or very large: • When 𝜂act Is Very Small. For small 𝜂act (less than about 15 mV at room temperature or, more fundamentally, when j > j0 ), the second exponential term in the Butler–Volmer equation becomes negligible. In other words, the forward-reaction direction dominates, corresponding to a completely irreversible reaction process. The Butler–Volmer equation simplifies to j = j0 e𝛼nF𝜂act ∕(RT)

(3.39)

97

98

FUEL CELL REACTION KINETICS

solving this equation for 𝜂act yields nact = −

RT RT ln j0 + ln j 𝛼nF 𝛼nF

(3.40)

a plot of 𝜂act versus lnj should be a straight line. Determination of j0 and 𝛼 is possible by fitting the line of 𝜂act versus ln j or log j. For good results, the fit should persist for at least one order of magnitude in current, preferably more. If this equation is generalized in the form (3.41) 𝜂act = a + b log j it is known as the Tafel equation, and b is called the Tafel slope. Like its relative, the Butler–Volmer equation, this equation is also quite important to electrochemical kinetics. Actually, the Tafel equation predates the Butler–Volmer equation. It was first developed as an empirical law based on electrochemical observations. It was only much later that the Butler–Volmer kinetic theory provided an explanation for the Tafel equation from basic principles! For fuel cells, we are primarily interested in situations where large amounts of net current are produced. This situation corresponds to the case of an irreversible reaction process in which the forward-reaction direction dominates. Therefore, the second simplification of the Butler–Volmer equation (the Tafel equation) proves more useful in most discussions. An example of a Tafel plot showing the linear 𝜂 vs. ln j behavior of a typical electrochemical reaction is shown in Figure 3.12. At high overvoltages, the linear Tafel equation applies very well to the curve. However, at low overvoltages, the Tafel approximation deviates from Butler–Volmer kinetics. From the slope and intercept of a linear fit to this plot, you should be able to calculate j0 and 𝛼. (Note that most Tafel plots give 𝜂act vs. log j. Be aware of the conversion necessary to switch from log j to ln j.)

η (V)

0.25 0.20 0.15

Slope = RT/αnF

Butler–Volmer (forward current)

0.10 Fit to Tafel equation

0.05 0 –14 –13 –12 –11 ln|j0| –9

–8

–7

–6

–5

–4

ln |j | ( j in A/cm2)

Figure 3.12. The j−𝜂 representation of a hypothetical electrochemical reaction. At high overvoltages, a linear fit of the kinetics to the Tafel approximation allows determination of j0 and 𝛼. The Tafel approximation deviates from Butler–Volmer kinetics at low overvoltages.

SIMPLIFIED ACTIVATION KINETICS: TAFEL EQUATION

Example 3.4 Calculate j0 and 𝛼 for the hypothetical reaction in Figure 3.12. Assume that the kinetic response depicted in the figure is for an electrochemical reaction at room temperature with n = 2. Solution: Using the linear Tafel fit of the data in Figure 3.12, we can extract both j0 and 𝛼. From the figure, the j-axis intercept of the Tafel line gives ln j0 = −10. Therefore, (3.42) j0 = e−10 = 4.54 × 10−5 A∕cm2 Approximating the Tafel slope of this figure gives Slope ≈

0.25 − 0.10 = 0.05 −5 − (−8)

(3.43)

From the Tafel equation, this slope is equal to RT∕𝛼nF. Solving for 𝛼 gives 𝛼=

(8.314)(298.15) RT = = 0.257 slope × nF (0.05)(2)(96,400)

(3.44)

Thus, 𝛼 for this reaction is fairly small at 0.257, and j0 is moderate at 4.54 × 10−5 A∕cm2 . These kinetic parameters signify a moderate-to-slow electrochemical reaction.

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 opposite effect on the activation overvoltage. At high enough overvoltage levels, this “bad” temperature effect actually outweighs the “good” temperature effect. Since this reversal only occurs at high overvoltage levels, we can use the Tafel approximation of the Butler–Volmer equation to further discuss the situation: j = j0 e𝛼nF𝜂act ∕(RT)

(3.45)

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

j = Ae−ΔG1 ∕(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.

99

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 transport) usually outweigh this reaction kinetics effect. Nonetheless, the phenomenon provides 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 variations 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 cathode, 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 relatively 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),

Theoretical EMF or ideal voltage 1.2

Cell voltage (V)

100

Anode activation loss Cathode activation loss

0.5

Current density (mA/cm2)

1000

Figure 3.13. Relative contributions to activation loss from H2 –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

TABLE 3.1. Standard-State (T ≈ 300 K, 1 atm) Exchange Current Densities for Hydrogen Oxidation Reaction on Various Metal Surfaces j′0 (A/cm2 )

Surface

Electrolyte

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 intermediates 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 oxygen kinetics, dramatically reducing the oxygen activation losses. The reactivity of hydrocarbon 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.

101

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/cm2 )

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 temperature 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: (3.48) H2 + 2OH− → 2H2 O + 2e− −

Compare this, yet again, to the hydrogen oxidation reaction in a SOFC, where O2 is the charge carrier: (3.49) H2 + O2− → H2 O + 2e− 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

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 fabricated 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 important 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. The major 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

TPB’s

Catalytic electrode particles

Electrolyte

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.

103

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. As with 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 electrolyte 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, platinum 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 equivalent 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 engineers. 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 electron bond. The three-dimensional (3D) surface drawn around the hydrogen molecule in

QUANTUM MECHANICS: FRAMEWORK FOR UNDERSTANDING CATALYSIS IN FUEL CELLS

(a)

(c)

(b)

(d)

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 overlapping 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 Gaussian1 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 quantum 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 emerging bonds between platinum and hydrogen lead to weakening of the hydrogen–hydrogen bond and ultimately to complete separation. Thus, the platinum catalyst facilitates the separation of the hydrogen molecule 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. In most PEMFC environments, it is believed that the hydrogen ions are removed from the platinum surface by binding to water molecules, forming hydronium ions (H3 O+ ). Figure 3.16 illustrates this reaction sequence. 1 Gaussian is a computational tool predicting energies, molecular structures, and vibrational frequencies of molecular systems by Gaussian Inc.

105

106

FUEL CELL REACTION KINETICS

(a)

(b)

(c)

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 formation 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 surface 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 platinum 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 completely separated into individual oxygen atoms.

THE SABATIER PRINCIPLE FOR CATALYST SELECTION

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 simplified. Various details, including the influence of voltage, platinum surface structure, and the involvement of additional water molecules, are ignored. For example, more sophisticated 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 factors 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 exponentially, 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 catalyst materials are plotted together in this fashion, a characteristic “volcano” type curve 2 Also, the spin states of the electrons in platinum influence the energy required to break the oxygen bonds. See Appendix D for further explanations.

107

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 information 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 equilibrium 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 behavior 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 probably guess that the Butler–Volmer equation must collapse to the Nernst equation under

CONNECTING THE BUTLER–VOLMER AND NERNST EQUATIONS (OPTIONAL)

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

CR∗ CR0∗

exp

𝛼nF𝜂∕(RT)

−

CP∗ CP0∗

) 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

∏

(

CR∗ ,i CR0∗,i

)𝑣i exp

𝛼nF𝜂∕(RT)

−

∏

(

CP∗ ,i

)𝑣i

CP0∗,i

) 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: (3.52) H2 ↔ 2H+ + 2e− Cathode: 2H+ + 2e− + 12 O2 ↔ H2 O

(3.53)

Using Equation 3.51, we can then write the reaction kinetics associated with each electrode’s reaction as follows: Anode: ⎛ C ∗,A A A A A ⎜ H2 j = j0 exp2𝛼 F𝜂 ∕(RT) − ⎜ C0∗,A ⎝ H2

(

∗,A CH + 0∗,A C H+

)2 (

Ce∗−,A C0∗,A e−

)2 exp−2(1−𝛼

A )F𝜂 A ∕(RT)

⎞ ⎟ ⎟ ⎠

(3.54)

Cathode: 1 ⎛( ∗,C )2 ( ∗,C )2 ⎛ ∗,C ⎞ 2 ⎞ ∗,C CH CO CH+ C ⎜ − e 2 ⎟ 2O C C 2𝛼 C F𝜂 C ∕(RT) −2(1−𝛼 C )F𝜂 C ∕(RT) ⎟ ⎜ j = j0 ⎜ exp − 0∗,C exp ⎟ 0∗,C 0∗,C ⎟ Ce0∗− ,C ⎜⎝ CO CH O ⎜ CH+ ⎟ ⎠ 2 2 ⎝ ⎠ (3.55) Here the superscripts A and C in the equations stand for the anode and the cathode, respectively.

109

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 independently and that they are typically not equal to one another, 𝜂 A ≠ 𝜂 C . 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∗,A ∗,C ment of a few important parameters such as concentrations of protons (CH + , CH+ ), elec∗,A ∗,C ∗,A ∗,C ∗,C trons (Ce− , Ce− ), hydrogen (CH ), oxygen (CO ), and water (CH O ), or the overvoltages 2 2 2 (𝜂 A , 𝜂 C ). Some of these parameters, such as C∗,A , C∗,C , and C∗,C , may be specified by the H2

O2

H2 O

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 condition, Equation 3.54 becomes

0=

(

⎛ C ∗,A H

jA0 ⎜ ⎜

2

exp 0∗,A

2𝛼F𝜂 A ∕(RT)

∗,A CH + 0∗,A C

−

⎝ CH

)2 (

)2 exp−2(1−𝛼

A )F𝜂 A ∕(RT)

exp−2(1−𝛼

A )F𝜂 A ∕(RT)

e−

H+

2

Ce∗−,A C0∗,A

⎞ ⎟ ⎟ ⎠

(3.56)

After rearranging this, we obtain ∗, A CH 2

0∗,A CH 2

( exp

2𝛼F𝜂 A ∕(RT)

=

∗,A CH + 0∗,A C

)2 (

Ce∗−,A C0∗,A

)2 (3.57)

e−

H+

Applying the natural logarithm function to both sides of the equation yields ⎛ C ∗, A ⎞ 2𝛼 A F𝜂 A H ln ⎜ 0∗2,A ⎟ + = ln ⎜C ⎟ RT ⎝ H2 ⎠

(

∗,A CH + 0∗,A C

)2

( + ln

Ce∗−,A C0∗,A

)2 −

e−

H+

2(1 − 𝛼 A )F𝜂 A RT

(3.58)

After rearranging, we obtain ⎛ C ∗,A ⎞ 2F𝜂 A H = − ln ⎜ 0∗2,A ⎟ + ln ⎟ ⎜C RT ⎝ H2 ⎠

(

∗, A CH + 0∗,A C H+

)2

( + ln

Ce∗−,A C0∗,A

)2 (3.59)

e−

Or, upon using the definition of activity, 𝜂A =

RT (− ln(a∗H,A ) + ln (a∗H,+A )2 + ln (ae∗−,A )2 ) 2 2F

(3.60)

In a similar fashion, we obtain the following starting from Equation 3.55 for the cathode: ) ( )2 ( RT ∗,C 1 ∗,C ) 2 + ln(aH ) (3.61) − ln a∗H,+C − ln (ae∗−,C )2 − ln (aO 𝜂C = 2 2O 2F

Chemical free energy

CONNECTING THE BUTLER–VOLMER AND NERNST EQUATIONS (OPTIONAL)

H2

Transport through conductors

∆Grxn,anode

2H++2e–

2H++2e–+ 1 O2 2 ∆Grxn,cathode

Cathode

Electrolyte

Free energy

Anode

H2O

–nFηanode

–nF∆φcathode

–nFηcathode –nFηanode

–nF∆φanode

Anode

Electrolyte

Cathode

–nFη anode –αnFη cathode

–αnFηanode

ΔG‡cathode

Free energy

ΔG‡anode

–nFη anode ∆G‡1,anode

∆G‡1,cathode

∆G ‡2,anode

Anode

–nFη anode –nFη cathode

∆G‡2,cathode

Electrolyte

Cathode

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.

111

112

FUEL CELL REACTION KINETICS

Now we will combine Equations 3.60 and 3.61 by adding them: ( ∗,C )2 ( ∗,C ) 2 ⎞ ⎛ a∗H,CO aH+ ae− ⎟ ⎜ RT 2 𝜂A + 𝜂C = − ln − ln ln ⎟ ∗,A ∗,A ( )1 2F ⎜⎜ aH+ ae− ⎟ ∗,A ∗,C 2 ⎠ ⎝ aH2 aO2

(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 (𝜂 A + 𝜂 C = E0 − E). Now we have ( ∗,C ) 2 ( ∗,C )2 ⎞ ⎛ a∗H,CO aH+ ae− ⎟ RT ⎜ 2 E=E − − ln − ln ln ⎟ ⎜ 1 ∗ , A ∗ , A ( ) 2F ⎜ aH+ ae− ⎟ ∗,A ∗,A 2 ⎠ ⎝ aH2 aO2 0

(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. The Nernst 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 current 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 performance 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

• 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 density 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 temperature, (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 structured 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 = j0 e𝛼nF𝜂act ∕(RT) . In a generalized 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 chemistry, 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) conductivity, 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 O + 2H+ + 2e− ⇌ H2 O 2 2

113

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 –O2 fuel cell, if we increase the overall rate of the fuel cell reaction, H2 + 12 O2 ⇌ H2 O which is made up of the half reactions H2 ⇌ 2H+ + 2e− 1 O 2 2

+ 2H+ + 2e− ⇌ H2 O

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 interface, 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 H 2 2

Cathode:

H+ + e− + 14 O2 ↔ 12 H2 O

↔ H+ + e−

CHAPTER EXERCISES

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 using multistep processes such as Anode: H2 ↔ 2H+ + 2e− Cathode: 2H+ + 2e− + O2 ↔ H2 O2 ,ad H2 O2 ,ad ↔ H2 O + 12 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 ( ∗ ) CR CP∗ 𝛼nF𝜂∕(RT) −(1−𝛼)nF𝜂∕(RT) exp − 0∗ exp j = j0 CR0∗ CP (a) If the concentrations of the reactant (CR∗∗ ) and product (CP∗∗ ) species at zero current density (or equilibrium) are not equal to the reference concentrations (CR0∗ and CP0∗ ), 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 (CR∗∗ and CP∗∗ ) as reference concentrations. 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?

115

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 (CH3 OH) fuel instead of H2 , what would be the minimum amount (in grams) of methanol required for 100 h of life again assuming 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 = 10−8 A∕cm2 at 300 K and j0 = 10−4 A∕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 question: 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 electrochemical generation of electricity: the production and consumption of charge via electrochemical 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 flowing 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

A

JA

A 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 zi F 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 = zi FJ

(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

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 ∑ Mik Fk (4.2) Ji = k

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

e–

–

+

e– H+

–+

e– H+

–+ +

H2 e– H+

Anode

H

– + O2

e– H+

–+

e– H+

–+

Electrolyte

Cathode

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 electrons 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 (relatively) 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.

119

120

FUEL CELL CHARGE TRANSPORT

cell, protons and electrons accumulate at the anode, while protons and electrons are consumed 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. However, 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 dominates 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 dV j=𝜎 (4.3) dx 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 driving 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 pressure 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=

Diffusion

Concentration gradient, dc∕dx

Diffusivity D

Convection

Pressure gradient, dp∕dx

Viscosity 𝜇

𝜎 dV |zi |F dx

dc dx Gc dp J= 𝜇 dx J = −D

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

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 V L

(4.4)

( ) L 𝜎

(4.5)

j=𝜎 Solving for V yields V=j

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 ( ) L = iR (4.6) V=i A𝜎 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. Length = L

Area = A

j

j

R = L/Aσ

V V 0

V = jL/σ = iR 0

x

L

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 conductor. 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.

121

FUEL CELL CHARGE TRANSPORT

Voltage (V)

Because this voltage loss obey’s Ohm’s law, we call it an “ohmic” loss. Like the activation 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

Eo Anode

Electrolyte

Cathode

Distance (x)

Voltage (V)

(a)

η act,C η act,A Anode

V Electrolyte

Eo

Cathode

Distance (x) (b)

Voltage (V)

122

η ohmic

Anode

Electrolyte

o V E Cathode

Distance (x) (c)

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 by E0 . (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

(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 conductivity 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 cm2 , 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: L 0.01 cm = 0.01 Ω = −1 𝜎A (0.10 Ω ⋅ cm−1 )(10 cm2 ) 0.005 cm = = 0.005 Ω (0.10 Ω−1 ⋅ cm−1 )(10 cm2 )

Case (a): Rionic = Case (b): Rionic

(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.

123

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 normalize 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; resistance 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 discussing 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 (4.11) 𝜂ohmic = j(ASRohmic )

1.2

Cell voltage (V)

124

Theoretical EMF or ideal voltage Ohmic loss: ηohmic = iRohmic Rohmic = 0.50 Ω Rohmic = 0.75 Ω

0.5

Rohmic = 1.0 Ω Current (A)

1.0

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

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 A 1 = 1 cm 2 R1 = 0.1 Ω

Fuel cell 2 A 2 = 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 = R1 A1 = (0.1 Ω)(1 cm2 ) = 0.1 Ω ⋅ cm2 ASR2 = R2 A2 = (0.02 Ω)(10 cm2 ) = 0.2 Ω ⋅ cm2

(4.14)

125

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 (4.17) 𝜂2,ohmic = i2 (R2 ) = (10 A)(0.02 Ω) = 0.2 V 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 metals, 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

• 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, especially 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 contact 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 resistance 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.

127

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FUEL CELL CHARGE TRANSPORT

Rinterconnect

Anode

Ranode

Relectrolyte Rcathode

Electrolyte

Rinterconnect

Cathode

Figure 4.7. The total ohmic resistance presented by a fuel cell is actually a combination of resistances, 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 electrolyte resistance usually accounts for most of the total. If electrolyte thickness cannot be reduced, decreasing ohmic loss depends on finding high-𝜎 ionic conductors. Unfortunately, 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 electrolyte 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 accommodates 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)ci ui

(4.19)

PHYSICAL MEANING OF CONDUCTIVITY

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 material. 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 conduction 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. Pretend 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, however, 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 missing 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 differences 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.

129

130

FUEL CELL CHARGE TRANSPORT

e–

M+

e–

e–

M+

M+

e– M+

M+

M+

e–

e–

M+

e– M+

e–

M+

e–

e–

M+

e– e– M+

M+

e–

e–

e–

M+

e– M+

e–

M+

M+

e–

M+

e–

M+

e–

e–

(a)

A–

C+

A–

C+

A–

C+

A–

C+

A–

C+

A–

C+

C+

A–

A–

C+

A– C+

A–

C+

A–

C+

A–

C+

A–

C+

A–

C+

A–

A– C+

C+

A–

C+

Vacancy

A–

C+

Interstitial (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 position 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 imperfections, impurities, etc.): q𝜏 u= (4.20) m 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 conductivity (Equation 4.19) gives |z F|c q𝜏 𝜎= e e (4.21) m 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

densities are generally on the order of 1028 atoms/m3 , which yields molar carrier concentrations 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). Typical 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 = Do e−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 conductivity (Equation 4.19) gives c(zi F)2 D 𝜎= (4.24) RT 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 electrolytes 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 conductivity 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.

131

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 materials 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 temperature) is an example of an ionic liquid. Pure H3 PO4 at 50∘ C is another example of an ionic liquid. At room temperature, H3 PO4 is a somewhat waxy, white crystalline solid. However, when heated above 42∘ C it becomes a viscous ionic liquid consisting of H+ ions, PO4 3– ions, and H3 PO4 molecules. Almost all aqueous/liquid electrolyte fuel cells use a matrix material to support or immobilize 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 electrolyte, 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 H3 PO4 electrolytes or pure H3 PO4 (an ionic liquid). Molten carbonate fuel cells use molten (K/Li)2 CO3 immobilized in a supporting

REVIEW OF FUEL CELL ELECTROLYTE CLASSES

matrix. The (K/Li)2 CO3 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 driving 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 = zi q

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 (4.26) FD = 6𝜋𝜇rv 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): | 𝑣 | | = |zi |q ui = || | 6𝜋𝜇r dV∕dx | |

(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 mobilities, 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 solutions the H+ ion tends to be hydrated by one or more water molecules. This ionic species is therefore better thought of as H3 O+ or H ⋅ (H2 O)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)ci ui

(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 Mobility, u (cm2 /V ⋅ s)

Anion

H+ (H3 O+ )

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

Cation

Source: From Ref. [6a].

Mobility, u (cm2 /V ⋅ s)

133

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 H3 PO4 is experimentally 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 H3 PO4 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 approximate 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 (4.29) cOH− = (0.1 mol∕L)(1 L∕1000 cm3 ) = 1 × 10−4 mol∕cm3 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

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) (4.32) 𝜎T = APEM e−Ea ∕kT 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 conductors 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, ensuring 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 polymer 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. Increasing 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.

135

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. Persulfonated 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 polytetrafluoroethylene (Teflon). However, unlike Teflon, Nafion includes sulfonic acid (SO3 – H+ ) functional 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 structure 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 (H3 O+ ) 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 content 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): p a𝑤 = 𝑤 (4.33) pSAT 1 The

Nafion model reviewed here was suggested by Springer et al. [8]

REVIEW OF FUEL CELL ELECTROLYTE CLASSES

Nafion

Polytetraflouroethylene (PTFE) F

F

F

F

F

F

F

F

C

C

C

C

C

C

C

C

F

F

F

F

F

O

F

F

F

C

F

F

C

CF3

n

n

O F

C

F

C

=

O= S

n

m

F F O–

H+

O

(a)

H2O

+

H 3O

–

SO3

1nm

(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 conduction 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 represents the saturation water vapor pressure for the system at the temperature of operation. The data in Figure 4.11 can be represented mathematically as { 0.043 + 17.18a𝑤 − 39.85a2𝑤 + 36.0a3𝑤 ) ( λ = 14 + 4 a𝑤 − 1

for 0 < a𝑤 ≤ 1 for 1 < a𝑤 ≤ 3

(4.34)

137

FUEL CELL CHARGE TRANSPORT

14 12 10 − λ = H2O/SO3

138

8 6 4 2 0 0

0.2

0.4

0.6

0.8

1

Water vapor activity ( pw /pSAT)

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 temperature 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 × 10−5 T 2 + 1.4454 × 10−7 T 3

(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] −5 ×802 +1.4454×10−7 ×803

pSAT = 10−2.1794+0.02953×80−9.1837×10

= 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

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 =

pH2 O𝑤 pSAT

=

xH2 O × ptotal xH2 O,SAT × ptotal

=

0.1 = 0.65 0.154

(4.37)

Nafion Conductivity Is Highly Dependent on Water Content. As previously mentioned, 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 experimentally determined relationships may be summarized as )] [ ( 1 1 − 𝜎(T, λ) = 𝜎303K (λ) exp 1268 303 T

(4.38)

𝜎303K (λ) = 0.005193λ − 0.00326

(4.39)

where

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

0.12 0.1

σ (S/cm)

0.08 0.06 0.04 0.02 0 0

5

10

15

20

25

λ = H O/SO 2

3

Figure 4.12. Ionic conductivity of Nafion versus water content λ according to Equations 4.38 and 4.39 at 303 K.

139

FUEL CELL CHARGE TRANSPORT

–0.6

100˚C

50˚C

0˚C

–0.7

log(σ) [log(S/cm)]

140

–0.8 –0.9 –1 –1.1 –1.2 –1.3 2.6

2.8

3

3.2

3.4

3.6

3.8

3

1/T (x10 K) Figure 4.13. Ionic conductivity of Nafion versus temperature according to Equation 4.38 when λ = 22.

membrane thickness (tm ) as tm

ASRm =

∫0

𝜌(z)dz =

tm

∫0

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 = nH2 O ∕H+ ). Obviously, 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. Commonly, it is assumed that ndrag changes linearly with λ as ndrag = nSAT drag

λ 22

for 0 ≤ λ ≤ 22

(4.41)

2 Actually, 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

where nSAT ≈ 2.5. Knowledge of the electro-osmotic drag coefficient allows us to estimate drag the water drag flux from anode to cathode when a net current j flows through the PEMFC: JH2 O,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 diffusion 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 JH2 O,back

diffusion

=−

𝜌dry Mm

Dλ

dλ dz

(4.43)

where 𝜌dry is the dry density (kg/m3 ) 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 JH2 O = 2nSAT drag

𝜌dry j λ dλ D (λ) − 2F 22 Mm λ 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.

141

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, JH2 O and λ. For convenience, we will set JH2 O = 𝛼NH2 = 𝛼(j∕2F), where 𝛼 is an unknown that denotes the ratio of water flux to hydrogen flux. After rearrangement, Equation 4.44 becomes ( ) jM λ dλ m = 2nSAT − 𝛼 drag 22 dz 2F𝜌dry Dλ

(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− (mol∕m3 ) = 3

𝜌dry (kg∕m3 ) Mm (kg∕mol)

(4.48)

where 𝜌dry is the dry density of Nafion (kg/m3 ) and Mm is the Nafion equivalent weight (kg/mol).

REVIEW OF FUEL CELL ELECTROLYTE CLASSES

In a similar fashion, water content, λ (H2 O∕SO3 – ), can be converted to water concentration in Nafion as CH2 O (mol∕m3 ) = λ

𝜌dry (kg∕m3 )

(4.49)

M m (kg∕mol)

Typically, Nafion has an equivalent weight of around ∼ 1–1.1 kg∕mol and a dry density of ∼ 1970 kg∕m3 . Thus, the estimated charge density for Nafion would be CSO− (mol∕m3 ) = 3

1970 kg∕m3 = 1970 mol∕m3 1 kg∕mol

(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 measured as )] [ ( 1 1 Dλ = exp 2416 − 303 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.

Water diffusivity, Dλ (cm2/s)

4

x 10 −6

3.5 3 2.5 2 1.5 1 0.5 0

0

5

10

λ (H2O/SO3-)

15

Figure 4.14. Water diffusivity Dλ in Nafion versus water content λ at 303 K.

143

144

FUEL CELL CHARGE TRANSPORT

Even though this is an ordinary differential equation on λ, we may not solve it analytically 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 )] [ ( 1 1 Dλ = 10−6 exp 2416 − 303 353 × (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 [ ] jMm nSAT drag 11𝛼 11𝛼 z = λ(z) SAT + C exp 22 F 𝜌dry Dλ 2.5 ndrag [ ] ) ( 0.7 A∕cm2 × (1.0 kg∕mol) × 2.5 + C exp z (22 × 96, 485 C∕mol) × (0.00197 kg∕cm3 ) × (3.81 cm2 ∕s) = 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} )] [ ( 1 1 − × exp 1268 303 353 = 0.0404 + 0.0216 exp(109.8z)

(4.55)

REVIEW OF FUEL CELL ELECTROLYTE CLASSES

Figure 4.15b shows the result. Finally, we can determine the area-specific resistance of the membrane using Equation 4.40: tm

∫0

dz = 𝜎[λ(z)] ∫0

0.0125

dz = 0.15 Ω ⋅ cm2 0.0404 + 0.0216 exp(109.8z) (4.56) Thus, the ohmic overvoltage due to the membrane resistance in this PEMFC is approximately ASRm =

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 conduction 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 Y2 O3 , 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 Zr4+ 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

145

FUEL CELL CHARGE TRANSPORT

15 14

−) Water content λ (HO/SO 2 3

13 12 11 10 9 8 7

0

0.002

Anode

0.004

0.006

0.008

Membrane thickness(cm)

0.01

0.012

Cathode

(a) 0.13 0.12 Local conductivity (S/cm)

146

0.11 0.1 0.09 0.08 0.07 0.06

0

0.002 Anode

0.004 0.006 0.008 Membrane thickness(cm)

0.01 0.012 Cathode

(b) 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

Vacancy Zr 4+

Zr 4+

Zr 4+

O 2–

O2–

O 2–

Zr 4+

O2–

O2–

O2–

Zr 4+

O2–

Zr 4+

Zr 4+

O2–

Zr 4+

Zr 4+

Zr 4+ O2–

O2– Y 3+

O2–

Zr 4+ O2–

O2– O2–

O2–

˚ Zr 4+

Zr 4+ O2–

O2–

Zr 4+

O2–

Zr 4+

Zr 4+

Zr 4+

Zr 4+

O 2–

Y 3+ O2–

Zr 4+

(a)

O2–

Zr 4+

O2–

Zr 4+

O2–

Zr 4+

(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.)

log(σT ) (Ω–1 · cm–1 K)

2.4 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6

6

7

8

9

10

11

12

13

14

15

%Y2O3 Figure 4.17. YSZ conductivity versus %Y2 O3 (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.

147

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)2 D 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 explanation behind diffusivity will be detailed in forthcoming sections. For now, however, it is sufficient to know that carrier diffusivity in SOFC electrolytes is exponentially temperature dependent: (4.60) D = D0 e−ΔGact ∕(RT) where D0 is a constant (cm2 /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)2 D0 e−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

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 extrinsic 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)2 D0 e−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 = ASOFC e−ΔGact ∕RT

(4.64)

Similarly to Equation 4.32, the term ΔGact ∕RT can instead be written as Ea ∕kT, yielding 𝜎T = ASOFC e−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

149

FUEL CELL CHARGE TRANSPORT

4 3

log(σT ) (Ω–1 · cm–1 K)

150

2 1

ΔGact=0.60eV

0 –1 –2 Gd-doped ceria Y-stabilized zirconia

–3 –4 0.6

0.8

1.0

1.2

ΔGact=0.89eV

1.4

1000/T K

1.6

1.8

2.0

–1

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 vacancies become the source of ionic conduction in this material. To create 8YSZ, 8 mol % Y2 O3 is combined with 92 mol % ZrO2 . The chemical formula of 8YSZ can therefore be represented as 0.92(ZrO2 ) + 0.08(Y2 O3 ) = Zr0.92 Y0.16 O2.08 . 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 Zr0.92 Y0.16 O2.08 V0.08 . One

REVIEW OF FUEL CELL ELECTROLYTE CLASSES

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/cm3 ) by applying knowledge 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: no

(4.66) V where co is the concentration of oxygen sites in the material (mol/cm3 ), 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 density (𝜌, g∕cm3 ) as M V= (4.67) 𝜌 c𝑣 = x𝑣 co = x𝑣

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 = 21.4 cm3 6.15 g∕cm3 (

c𝑣 = 0.037 vacancies∕O site

2.16 mol O sites∕mol YSZ ( ) 21.4 cm3 ∕mol YSZ

(4.68) ) (4.69)

3

= 0.0037 mol vacancies∕cm

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 fashion. 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 ) = 20.5 cm3 4

(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.

151

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 traditional 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 materials are known as “mixed ionic–electronic conductors” (MIECs) or, more simply, “mixed conductors.” Many doped metal oxide ceramic materials exhibit both electronic and ionic conductivity. 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 electronic 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 geometry 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 electrons, 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: Electron hole formation:

′ + Vo− 2Oxo → 2SrLa

′ null → 2SrLa + 2h⋅

MORE ON DIFFUSIVITY AND CONDUCTIVITY (OPTIONAL)

Standard Electrode: Only TPBs are active for reaction

MIEC Electrode: Entire surface is active for reaction

e–

e–

O2

O2

O2

O2

O2–

O2–

Electrolyte

O2– Electrolyte

(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 (Vo⋅⋅ ) is formed for every two Sr2+ dopant substitutions. 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 purposes 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

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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

Concentration of gray atoms

154

Jnet

Distance (x)

(a) ΔX

A

JA+ JA– (c1) (c2) A (b) 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)

(concentration) of gray atoms available to hop times the hopping rate: JA+ = 12 𝑣c1 Δx

(4.71)

where JA+ is the forward flux through plane A (mol/cm2 ⋅ s), v is the hopping rate (s–1 ), c1 is the volume concentration (mol/cm3 ) 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 (4.72) JA− = 12 𝑣c2 Δx 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 = 12 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 (c1 − c2 ) Δx 1 2 Δc = − 2 𝑣(Δx) Δx 1 2 dc = − 2 𝑣(Δx) (for small x) dx

Jnet = − 12 𝑣(Δx)2

(4.74)

Comparison with the traditional diffusion equation J = −D(dc∕dx) allows us to identify what we call the diffusivity as (4.75) D = 12 𝑣(Δx)2 We therefore recognize that the diffusivity embodies information about the intrinsic hopping 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.)

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FUEL CELL CHARGE TRANSPORT

(a) C+

C+

C+

C+

C+

–

A +

C

(b) Free energy

156

∆Gact

Distance

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 = 𝑣0 e−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 (4.77) D = 12 (Δx)2 𝑣0 e−ΔGact ∕(RT) or, lumping all the preexponential constants into a D0 term. D = D0 e−ΔGact ∕(RT) 4.6.2

(4.78)

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)

Free energy

dV 1 zF ∆x dx 2

Voltage gradient dV zF dx

∆G’act

zF ∆x

1 ∆x 2

dV dx

∆x Distance

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 12 zF Δx(dV∕dx). [In other words, ΔG′act =

ΔGact − 12 zF Δx(dV∕dx).]

reduced by 12 zF Δx(dV∕dx) while the activation barrier for the “reverse” hop is increased by 12 zF Δx(dV∕dx). (We are assuming that the activated state occurs exactly halfway between the two lattice positions, or in other words that 𝛼 = 12 .) The forward-(𝑣1 ) and reverse-(𝑣2 ) hopping-rate expressions are therefore 𝑣1 = 𝑣0 exp

] [ − ΔGact − 12 zF Δx (dV∕dx) [

𝑣2 = 𝑣0 exp

− ΔGact +

RT 1 zF Δx (dV∕dx) 2

]

(4.79)

RT

This voltage gradient modification to the activation barrier turns out to be small. In fact, 1 2

dV zF Δx ≪1 RT dx

157

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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 ) ( dV zF Δx 𝑣1 ≈ 𝑣0 e−ΔGact ∕(RT) 1 + 12 RT dx ) ( dV 1 zF −ΔGact ∕(RT) 𝑣2 ≈ 𝑣0 e 1− 2 Δx RT 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− = 12 Δx(c1 𝑣1 − c2 𝑣2 ) (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 ) czF dV Δx RT dx ) ( dV czF 1 = 2 (Δx)2 𝑣0 e−ΔGact ∕(RT) RT dx

Jnet = 12 Δx 𝑣0 e−ΔGact ∕(RT)

(

(4.82)

Recognizing the first group of terms as our diffusion coefficient D, we thus have Jnet =

czFD dV RT dx

(4.83)

Comparing this to the conduction equation J=

𝜎 dV zF dx

we see that 𝜎 and D are related by 𝜎=

c(zF)2 D 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 underpinnings 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)

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): 𝜇̃i = 𝜇i0 + RT ln ai + zi F𝜙i If we assume that activity is purely related to concentration (ai = ci ∕c0 ), then the electrochemical potential can be written as 𝜇̃i = 𝜇i0 + RT ln

ci + zi F𝜙i c0

(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: ) ( [ ( )] d ln ci ∕c0 𝜕 𝜇̃ dV (4.86) Ji = −Mi𝜇 = −Mi𝜇 RT + zi F 𝜕x dx dx The concentration term in the natural logarithm can be processed by remembering the chain rule of differentiation: d[ln(ci ∕c0 )] c0 d(ci ∕c0 ) 1 dci = = dx ci dx ci 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: Mi𝜇 RT dci dV Ji = − − Mi𝜇 zi F (4.88) ci dx dx Comparing the concentration gradient term in this equation to our previous expression for diffusion allows us to identify Mi𝜇 in terms of diffusivity: Mi𝜇 RT ci

=D

Dci Mi𝜇 = 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: Mi𝜇 zF =

c (zF)2 D 𝜎 , where 𝜎 = i |z|F RT

(4.90)

159

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FUEL CELL CHARGE TRANSPORT

By using the electrochemical potential, we arrive at the same result as before. Interestingly, 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 conductors 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 concentration 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)2 D RT

(4.93)

The maximum possible sustainable charge flux due to a concentration gradient across a material is c (4.94) jc = zFD 0 L

QUANTUM MECHANICS–BASED SIMULATION OF ION CONDUCTION IN OXIDE ELECTROLYTES (OPTIONAL)

where L is the thickness of the material and c0 is the bulk concentration of charge carriers. The voltage, V, that would be required to produce an equivalent charge flux can be calculated from j𝑣 = jc c0

(zF)2 D RT

c V = zFD 0 L L

Solving for V gives V=

RT zF

(4.95)

(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 conductivities. 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 techniques 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 calculated 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 (unoccupied) “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

161

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 different 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 determining 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 calculation, 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 configuration, 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 lowest 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.

Relative energy

162

ΔE

m

Migration path

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

2.4

–4.2 Experiment

–4.3

log(σT ) (Ω –1 · cm–1 K)

KMC 2.2

–4.4

2.1

–4.5

2.0

–4.6

1.9

–4.7

1.8

–4.8

1.7

–4.9

1.6

6

8

10 12 mole % Y2O3

14

log D/D0

2.3

16

Figure 4.24. Logarithmic plot of conductivity times T versus mol% Y2 O3 in YSZ comparing experiment (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 conductivity 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 resistance. • 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.

163

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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)ci ui . • Metals and ion conductors show vastly different structures and conduction mechanisms, leading to vastly different conductivities. • Ion conductivity even in good electrolytes is generally four to eight orders of magnitude 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 balance 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 crystal lattice. Natural (intrinsic) defect concentrations are generally low, so higher (extrinsic) 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 reaction 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)2 D∕(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

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 conductivities 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 electrolyte. Is there any change in the figure?

4.7

Draw a fuel cell voltage profile similar to those shown in Figure 4.4 that simultaneously 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 absolute 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 thickness (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 counterproductive! In other words, at a given current density, an optimal electrolyte thickness 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 electrolyte 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/cm2 . 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.

165

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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 = 12 a. (b) Calculate the intrinsic carrier concentration in the electrolyte given Δh𝑣 = 1 eV. (Assume vacancies are the dominant carrier.) (c) From your answers in (a) and (b), calculate the intrinsic conductivity of this electrolyte at 1000∘ C. 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 Y2 O3 . 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 backdiffusion moves water from the anode to the cathode (d) Electro-osmotic drag moves water from the anode to the cathode while backdiffusion 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 (ΔG ) for act

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 continuously 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 determined 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 electrodes 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, however, 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 insulates 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. Convection 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 “effective” at transporting species than diffusion. For example, at STP, the maximum likely diffusive O2 flux across a 500-μm-thick porous electrode is ≈ 4 × 10–5 mol∕(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

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 diffusivedominated 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 Flow channel

Anode electrode

H2

H2

H+

Concentration

H2

O2

Diffusion layer

c0H2

Anode

Electrolyte

Cathode

c*H2 Flow channel

Electrode

Distance 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 (c0H ) at the flow channel to a much 2 lower value (c∗H ) at the catalyst layer. The magnitude of the H2 gas velocity in the flow channel 2 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.

169

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 demarcation 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 concentration gradients, leading to diffusive transport across the electrode. From this figure, you can see that the electrochemical reaction leads to reactant depletion (and product accumulation) at the catalyst layer. In other words, c∗R < c0R and c∗P > c0P , where c∗R , c∗P represent the catalyst layer reactant and product concentrations, respectively, and c0R , c0P 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

Flow structure

Flow channel

Anode Catalyst electrode layer Electrolyte

Reactants (R) In

JR

j rxn

JP

Concentration

Products (P) Out

c0R

JR

c*P

JP

c0P

Reaction in catalyst layer consumes R, generates P

c*R δ

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 c0P ). The consumption/generation of species (at a rate given by jrxn ) within the catalyst layer leads to reactant depletion and product accumulation (c∗R < c0R and c∗P > c0P ). 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 concentration, 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 c0P ). As soon as the fuel cell begins producing current, however, the electrochemical reaction leads to depletion of reactants (and accumulation

171

FUEL CELL MASS TRANSPORT

Flow channel

Catalyst layer

Anode electrode

c0R

t2

Concentration

172

t1

t =0

t3

t ∞ t ∞ c0P

c*P

t3

c*R

t2

t1

t =0

δ 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, (5.1) j = nF Jdiff 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

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]: )b

( T p ⋅ Dij = a √ Tci Tcj

( (pci pcj )1∕3 (Tci Tcj )5∕12

1 1 + Mi Mj

)1∕2 (5.2)

where p is the total pressure (atm), Dij is the binary diffusion coefficient (cm2 /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 nonpolar gases, such as H2 , O2 , and N2 . For pairs involving H2 O (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)

2.016 28.964 28.013 31.999 28.010 44.010 18.015

33.3 132.4 126.2 154.4 132.9 304.2 647.3

12.80 37.0 33.5 49.7 34.5 72.8 217.5

H2 Air N2 O2 CO CO2 H2 O 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] 1.5 Deff (5.3) ij = 𝜀 Dij

173

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, = Dij . Often, Equation 5.3 is modified to include tortuosity 𝜏 as porosity is 1 and Deff ij 𝜏 Deff ij = 𝜀 Dij

(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 tortuosity 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 effective diffusivity proves more accurate [14]: Deff ij = 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 dc (5.6) dx For the steady-state situation shown in Figure 5.4, this equation becomes (written for the flux of a diffusing reactant) Jdiff = −D

Jdiff = −Deff

c∗R − c0R 𝛿

(5.7)

where c∗R is the catalyst layer reactant concentration, c0R is the bulk (flow channel) reactant concentration, δ is the electrode (diffusion layer) thickness, and Deff is the effective reactant 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 = nF Deff c∗R = c0R −

c∗R − c0R 𝛿

j𝛿 nF Deff

(5.8) (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 Deff . 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

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 represents 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 activation (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 correctly 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 oxygen 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 .

175

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): 𝑣

i RT Πaproducts E=E − ln i nF Πa𝑣reactants

0

(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, c0R = 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 c0R values: 0 ∗ − ENernst 𝜂conc, Nernst = ENernst ) ( ( ) RT RT 1 1 0 = E − ln 0 − E0 − ln ∗ nF c nF cR R

=

(5.12)

0 RT cR ln ∗ nF cR

0 ∗ is the Nernst voltage using c0 values and ENernst is the Nernst voltage using where ENernst 0 ∗ c values. Recall that cR can be described in terms of the limiting current density (from Equation 5.10), j 𝛿 c0R = L eff (5.13) nFD

and that c∗R can be described in terms of the diffusion Equation 5.9, c∗R = c0R − =

j𝛿 nFDeff

jL 𝛿 j𝛿 − eff nFD nFDeff

(5.14)

TRANSPORT IN ELECTRODE: DIFFUSIVE TRANSPORT

Thus, the ratio c0R ∕c∗R can be written as c0R c∗R

=

jL 𝛿∕nFDeff jL 𝛿∕nFDeff − j𝛿∕nFDeff

=

jL jL − j

(5.15)

Substituting this result into our expression for 𝜂conc provides the final result: 𝜂conc,Nernst =

j RT ln L nF jL − j

(5.16)

Note that this expression is valid only for j < jL (j should never be greater than jL anyway). For 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.

k

k

k

PUTTING IT ALL TOGETHER: A BASIC FUEL CELL MODEL

Activation loss (Chapter 3)

Ohmic loss (Chapter 4)

Current density (A/cm2)

Cell voltage (V)

Cell voltage (V)

Cell voltage (V)

Reversible voltage (Chapter 2)

205

Current density (A/cm2)

Net fuel cell performance Cell voltage (V)

Cell voltage (V)

Concentration loss (Chapter 5)

Current density (A/cm2)

Current density (A/cm2)

Current density (A/cm2)

Figure 6.1. Pictorial summary of major factors that contribute to fuel cell performance. The overall 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 No leakage loss

Cell voltage (V)

k

FC with leakage loss

j leak 0.0

1.0

Measured current density (A/cm2)

Figure 6.2. Pictorial illustration of the effect of a leakage current loss on overall fuel cell performance. 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.

k

k

k

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 )] ) ( jL −(j ASRohmic ) − c ln ) ( jL − j + jleak

k

(6.4)

The most noticeable effect of leakage current is to reduce a fuel cell’s open-circuit voltage 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 unlimited 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].

k

k

k

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 2

10 A/cm2

j0 (H2 )

0.10 A∕cm

j0 (O2 )

10−4 A∕cm2

0.10 A/cm2

α(H2 )

0.50

0.50

α(O2 )

0.30

0.30

ASRohmic

2

0.01Ω ⋅ cm

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

Typical PEMFC

Typical SOFC

Theoretical EMF or ideal voltage 1.2

Cell voltage (V)

k

Cell voltage (V)

1.2 1 0.8 0.6 0.4

k

0.8 0.6 0.4 0.2

0.2 0

Theoretical EMF or ideal voltage

1

0

0.5

1

1.5

2

0

0

0.5

Current density (A/cm2)

1

1.5

2

Current density (A/cm2)

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 PEMFCs 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.

k

k

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 H2 O and H2 flowing into/out of the anode, the H2 O, N2 , and O2 flowing into/out of the cathode, and the H2 O 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 flux 14 = flux 5 = flux 1 − flux 4 = flux 8 − flux 13

(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, J + j C = H = JHA = 2JOC = SH 2 2 2O 2F 2

k

(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; JHA stands for the net flux of H2 in the anode (in 2 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, JOC stands for the 2 C (mol/s ⋅ cm2 ) net flux of oxygen at the cathode. Also, note that the water generation rate SH 2O at the cathode is equal to the net hydrogen flux. (For each mole of hydrogen that is consumed, 1 mol of water will be produced.) In an analogous manner, the following water flux balance must also be satisfied: flux 2 − flux 3 = flux 6 − flux 7 = flux 12 − flux 9 − flux 5 anode

membrane

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, JHA

2O

= JHM O = JHC

k

2

2O

−

j 2F

(6.8)

k

k

A 1D FUEL CELL MODEL

209

y z

14

a

b

H2 H2 O

c

Convection O2 H2O N2

8

1

Flow structure Porous electrode

d

Diffusion Electro-osmotic drag Electronic conduction Ionic conduction

9 2

X

5

+

10

6 7

11

3

12 4

H2 O H2

Anode

X

N2 H2O + O2

13

Electrolyte

H2

2H+ + 2e-

1 2H+ + 2e- + --O 2 2 H2O

Cathode (a)

y z

k

a

b

c

d

Convection O2

H2 H2 O Flow structure Porous electrode

10

2

X

5

+

X

8

+ 4

9

k

Electronic conduction Ionic conduction

7

3

H 2O H2

Diffusion

N2

6

1

H2 + O21 --O + 2e2 2

H2O + 2eO2-

N2 O2

Anode

Electrolyte

Cathode (b)

Figure 6.4. Flux details for (a) 1D PEMFC model and (b) 1D SOFC model. (a) In a PEMFC, water (H2 O) and protons (H+ ) transport through the electrolyte. (b) In a SOFC, oxygen ions (O2– ) transport through the electrolyte.

where JHA O , JHM O , and JHC O represent the net flux into the anode catalyst layer, across the 2 2 2 electrolyte, and out of the cathode catalyst layer, respectively, and j∕2 F represents the water generation rate at the cathode due to electrochemical reaction.

k

k

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=

JHM O 2

j∕2 F

(6.9)

Using Equation 6.9, we can write Equation 6.8 in terms of j and 𝛼: JHC

2O

=

j (1 + 𝛼) 2F

(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: JHA O JHM O JHC O JHM+ j 2 2 2 A C = = JH = 2JO = = = 2 2 2F 2 𝛼 𝛼 1+𝛼

k

(6.11)

This is the master flux balance equation for our PEMFC model. The flux balance principle 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: j (6.12) = JOM2− = JHA = 2JOC = −JHA O 2 2 2 2F 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 concentration 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

k

k

k

H2 + O2− → H2 O + 2e− — —

(6) 2H+ + 1 O2 + 2e− → H2 O(l) O2 + 2e− → O2−

−

(3,5) e , O (3,5) e− , H+ (3,5) e− , O2− (6) H+ , H2 O(l) a 2−

O

(3,5) e− , H+ −

(3,5) e , O (3) e−

(3,5) e− , O2− (3) e− (3) e−

H2 , H2 O(g) (5) H2 , H2 O(g) , H2 O(l) (5) H2 , H2 O(g) (6) H2 O(l) —

(5) N2 , O2 , H2 O(g) , H2 O(l) (5) N2 , O2 (6) N2 , O2 , H2 O(g) , H2 O(l) N 2 , O2 (2) N2 , O2 , H2 O(g) , H2 O(l) (2) N2 , O2

(1) H2 , H2 O(g) (1) H2 , H2 O(g) , H2 O(l) (1) H2 , H2 O(g) — —

(1) N2 , O2 , H2 O(g) , H2 O(l) (1) N2 , O2 (1) N2 , O2 , H2 O(g) , H2 O(l) (1) N2 , O2 (1) N2 , O2 , H2 O(g) (1) N2 , O2

2−

2−

→ H2 O + 2e−

—

—

1

2

— 2

—

—

(5) O2 + 2e− → O2−

2

1

(4) H2 → 2H+ + 2e−

(5) H2 + O

(3) e−

(6) H2 , H2 O(g) , H2 O(l)

(1) H2 , H2 O(g) , H2 O(l) 2−

(3) e−

(2) H2 , H2 O(g)

(1) H2 , H2 O(g)

—

(3) e−

(2) H2 , H2 O(g) , H2 O(l)

(1) H2 , H2 O(g) , H2 O(l)

Note: Six key assumptions, numbered 1–6 in parentheses, lead to the simplified model shown in Table 6.3. a To be precise, this water transport phenomenon is due to electro-osmotic drag (see Chapter 4). For convenience, it has been categorized as conduction due to its close relationship with proton conduction.

Flow channels

Electrode

Catalyst

Cathode

Electrolyte

Catalyst

Electrode

Flow channels

Anode

Electrochemical Reaction

Conduction

Diffusion

Convection

k

Domains

TABLE 6.2. Description of Full PEMFC (or SOFC, in italics) Model

k

211

k

k

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:

k

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, diffusion 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 conduction 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 combined 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 PEMFCs 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 departure 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.

k

k

k

A 1D FUEL CELL MODEL

213

TABLE 6.3. Description of Simplified PEMFC (or SOFC, in italics) Model Domains

Convection

Diffusion

Conduction

Electrochemical Reaction

Flow channels

—

—

—

—

Electrode

—

H2 , H2 O(g)

—

—

—

H2 , H2 O(g)

—

—

Anode

Catalyst

Electrolyte

—

—

—

—

—

—

—

H2 + O2− → H2 O(g) + 2e−

—

H2 O(g)

H+ , H2 O(g)

—

2−

O Cathode Catalyst

Electrode

Flow channels

k

—

—

—

2H+ + 12 O2 + 2e− → H2 O(g)

—

—

—

1 O 2 2

—

N2 , O2 , H2 O(g)

—

—

—

N 2 , O2

—

—

—

—

—

—

+ 2e− → O2−

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 requirements, 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. Actually, we have already learned all the required governing equations in previous chapters.

k

k

k

214

FUEL CELL MODELING

By solving these governing equations, we can determine how the concentrations of H2 , O2 , H2 O, 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 , H2 O, and N2 . We start with a modified form of the basic diffusion model that was described by Equation 5.7: Ji =

k

−pDeff ij dx

i

RT

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 = ci RT). Recall how the effective diffusivity is obtained using Equations 5.2–5.5 based on the measured/assumed porosity of the Deff ij 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 H2 O). 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 consumption 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): dc (6.14) Ji = −Dij i dz 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

k

dcj dz

(6.15)

k

k

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 diffusion process. At low density, multicomponent gas diffusion can be approximated by the Maxwell–Stefan equation [38]: ∑ xi Jj − xj Ji dxi = RT dz pDeff j≠i

(6.16)

ij

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 is the effective binary diffusivity (m2 ∕s). (K), p is the total gas pressure (Pa), and Deff ij 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]. k

Electrolyte. Having used the diffusion equations to describe gas transport in the electrodes, 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: JOM2− =

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: A e−ΔGact ∕(RT) 𝜎 = SOFC (6.19) T 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

k

k

k

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: JHM O = 2nSAT drag 2

𝜌dry d𝜆 j 𝜆 D − 2 F 22 Mm 𝜆 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 =

jc0O RT 2 ln 4𝛼F j0 cO2

(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 k

𝜂cathode =

j RT ln C 4𝛼F j0 p xO2

(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 H2 O transport in the anode as JHA 2 JHA

2

=

O=

k

−pA Deff H ,H 2

2O

dxH2

RT

dz

−pA Deff H2 ,H2 O

dxH2 O

RT

dz

(6.23)

k

k

A 1D FUEL CELL MODEL

217

Using Equation 6.12, we can relate JHA and JHA O to the fuel cell current density j. When 2 2 we integrate Equation 6.23, however, we need to provide boundary conditions. Fortunately, we know (or can impose) the values of xH2 and xH2 O 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

jRT 2 FpA Deff H2 ,H2 O

xH2 O (z) = xH2 O |a + z

jRT 2 FpA Deff H2 ,H2 O

(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

jRT 2 FpA Deff H2 ,H2 O

xH2 O |b = xH2 O |a + tA

k

jRT 2 FpA Deff H2 ,H2 O

(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: jRT (6.26) xO2 |c = xO2 |d − tC C 4Fp Deff O ,N 2

2

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

⎤ ⎡ ⎥ ⎢ j RT = ln ⎢ { )} ⎥ ( 4𝛼F ⎢ j pC x | − tC jRT∕ 4FpC Deff ⎥ O2 d O2 ,N2 ⎦ ⎣0

(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 tM T =j 𝜎 ASOFC e−ΔGact ∕(RT)

k

(6.28)

k

k

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

C

Cathode pressure, p (atm)

1

Anode pressure, pA (atm)

1

Effective hydrogen (or water) diffusivity,

Deff (m2 /s) H2 ,H2 O

Effective oxygen diffusivity, Deff (m2 /s) O ,N

2 × 10−5

Transfer coefficient, α

0.5

2

2

2

Exchange current density, j0 (A∕cm )

0.1

Electrolyte constant, ASOFC (K∕Ω ⋅ m)

9 × 107

Electrolyte activation energy, ΔGact (kJ∕mol)

100

Electrolyte thickness, tM (μm)

20

A

Anode thickness, t (μm)

k

1 × 10−4

50

C

Cathode thickness t (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 ⎤ ⎡ ⎥ ⎢ j tM T RT = Ethermo − j − ln ⎢ { [ ( )]} ⎥ −ΔG ∕(RT) act 4𝛼F ⎢ j pC x | − tC jRT∕ 4FpC Deff ASOFC e ⎥ O2 d O2 ,N2 ⎦ ⎣0 (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 = 0.5 A∕cm

2

104 cm2 m2

) (9 × 10 K ⋅ 7

2

Ω-1

= (0.5 A∕cm )(0.176 Ω cm2 ) = 0.088 V

k

(0.00002 m)(1073 K) ⋅ m-1 )e−(100,000 J∕mol)∕(8.314 J∕mol⋅K×1073 K) (6.30)

k

k

A 1D FUEL CELL MODEL

219

⎡ 0.5 A∕cm2 (8.314 J∕mol ⋅ K) (1073 K) ⎢ 𝜂cathode = ln ⎢ 2 4 × 0.5 × 96485 C∕mol ⎢ 0.1 A∕cm ⋅ 1 atm × 101300 Pa∕atm ⎣ ⎤ ⎥ 1 ⎥ × ⎥ 5000 A∕m2 × 8.314 J∕ (mol ⋅ K) × 1073 K 0.210 − 0.0008 m ⎥ 2 (4 × 96,485 C∕mol) × 101,325 Pa × 0.00002 m ∕s ⎦ = 0.158 V

(6.31) V = 1.0 V − 0.088 V − 0.158 V = 0.754 V

(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. PEMFC Model 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: k

JHA = 2

JHA

2

O=

−pA Deff H ,H 2

2O

RT RT

k

dz

−pA Deff H ,H 2

dxH2

2O

dxH2 O 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.

k

k

220

FUEL CELL MODELING

These equations look exactly like the SOFC anode Equations 6.23. One significant and important difference, however, is that JHA O is unknown in our PEMFC model since we do 2 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

jRT 2 FpA Deff H2 ,H2 O

xH2 O (z) = xH2 O |a − z

𝛼 ∗ jRT

(6.34)

(6.35)

2 FpA Deff H ,H

2O

2

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): jRT xH2 |b = xH2 |a − tA (6.36) A 2 Fp Deff H ,H O 2

xH2 O |b = xH2 O |a − tA

2

𝛼 ∗ jRT 2

k

(6.37)

2 FpA Deff H ,H

2O

In a similar manner, we can obtain the oxygen and water concentrations at the cathode– membrane interface “c”: xO2 |c = xO2 |d − tC

jRT 2 FpC Deff O2 ,H2 O

xH2 O |c = xH2 O |d + tC

(1 + 𝛼 ∗ )jRT 2 FpC Deff O ,H 2

(6.38)

(6.39)

2O

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): 11𝛼 ∗ 𝜆(z) = SAT + C exp ndrag ( +C exp

(

jMm nSAT drag

) z

=

11𝛼 ∗ 2.5

22 F𝜌dry D𝜆 ( ) j A∕cm2 × 1.0 kg∕mol × 2.5

)

22 × 96,485 C∕mol × 0.00197 kg∕cm3 × D𝜆 (cm2 ∕s) ( ) ( ) 0.000598 ⋅ j A∕cm2 ⋅ z(cm) ∗ = 4.4𝛼 + C exp D𝜆 (cm2 ∕s)

z(cm)

(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

(

𝜆|c = 𝜆(tM ) = 4.4𝛼 ∗ + C exp

k

(

0.000598 ⋅ j A∕cm

) 2

D𝜆 (cm2 ∕s)

⋅ tM (cm)

)

(6.41) (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 surrounding 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 𝜆 = 10 + 4aW

for

0 < aW ≤ 1 for

1 < aW ≤ 3

(6.43) (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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FUEL CELL MODELING

TABLE 6.5. Physical Properties of PEMFC Used in Example Physical Properties

Values

Thermodynamic voltage, Ethermo (V)

1.0

2

Operating current density, j (A∕cm )

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, xH2 O

0.1

C

Cathode pressure, p (atm)

3

A

Anode pressure, p (atm) Effective Effective

3

hydrogen (or water) diffusivity, Deff (cm2 /s) H2 ,H2 O oxygen (or water) diffusivity, Deff (cm2 /s) O2 ,H2 O

0.149 0.0295

Water diffusivity in Nafion, Dλ (cm2 ∕s)

3.81 × 10−6

Transfer coefficient, α

0.5 2

Exchange current density, j0 (A∕cm )

k

0.0001

M

125

Anode thickness, t (μm)

350

Cathode thickness tC (μm)

350

Gas constant, R (J∕mol ⋅ K)

8.314

Faraday constant, F (C∕mol)

96,485

Electrolyte thickness, t (μm) A

Since a𝑤 |b = pA xH2 O |b ∕pSAT , combining Equations 6.43 and 6.37 gives ( ) pA 𝛼 ∗ jRT A 𝜆|b = 14a𝑤 |b = 14 xH2 O |a − t pSAT 2 FpA Deff

k

(6.45)

H2 ,H2 O

Similarly, combining Equations 6.39 and 6.44 for the cathode side yields ( ) ∗ ) jRT pC + 𝛼 (1 𝜆|c = 10 + 4a𝑤 |c = 10 + 4 xH2 O |d + tC pSAT 2 FpC Deff

(6.46)

O2 ,H2 O

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 generated. 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 3 atm 𝜆|b = 14 0.307 atm ( × 0.1 − 0.00035 m

𝛼 ∗ × 0.5 A∕0.0001 m2 ⋅ 8.314 J∕mol K × 343 K (2 × 96,485 C∕mol) (3 × 101,325 Pa)(0.149 × 0.0001 m2 ∕s)

= 13.68 − 0.781𝛼 ∗

)

(6.47)

3 atm 𝜆|c = 10 + 4 0.307 atm ( × 0.1 + 0.00035 m

(1 + 𝛼 ∗ ) × 0.5 A∕0.0001 m2 ⋅ 8.314 J∕mol ⋅ K × 343 K (2 × 96,485 C∕mol)(3 × 101,325 Pa)(0.0295 × 0.0001 m2 ∕s)

= 15.04 − 1.127𝛼 ∗

)

(6.48)

and Equations 6.41 and 6.42 become

k

𝜆|b = 𝜆(0) = 4.4𝛼 ∗ + C ( ) 0.000598 × 0.5 A∕cm2 × 0.0125 cm ∗ 𝜆|c = 4.4𝛼 + C exp 3.81 × 10−6 = 4.4𝛼 ∗ + 2.667C

(6.49)

(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: { [ ( )] } 0.000598 × 0.5 𝜎(z) = 0.005193 4.4𝛼 + C exp z − 0.00326 3.81 × 10−6 )] [ ( 1 1 × exp 1268 − 303 343 = 0.0704 + 0.0266 exp(78.48z) (6.51) Finally, we can determine the resistance of the membrane using Equation 4.40: tm

ASRm =

∫0

dz = 𝜎(z) ∫0

0.0125

dz 0.0704 + 0.0266 exp(78.48z)

= 0.109 Ω ⋅ cm2

(6.52)

Thus, the ohmic overvoltage due to the membrane resistance in this PEMFC is approximately 𝜂ohmic = j × ASRm = 0.5 A∕cm2 × 0.109 Ω ⋅ cm2 = 0.0505 V

k

(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 (typical 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 concentration 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).

k

k

and we can compute the cathodic overvoltage using Equation 6.27 as ⎡ 0.5 A∕cm2 (8.314 J∕mol ⋅ K)(343 K) ⎢ ln ⎢ 𝜂cathode = 2 4 × 0.5 × 96485 C∕mol ⎢ 0.0001 A∕cm × 3 atm × 101300 Pa∕atm ⎣

⎤ ⎥ 1 ⎥ ×( ) ⎥ 5000 A∕m2 × 8.314 J∕mol ⋅ K × 343 K 0.19 − 0.00035m ⎥ (4 × 96,485 C∕mol) (3 × 101,325 Pa)(0.0295 × 0.0001 m2 ∕s) ⎦ = 0.135 V

(6.54)

Finally, we find the fuel cell voltage as V = 1.0 V − 0.0505 V − 0.135 V = 0.810 V

(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 boundaries “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 oxygen 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 =

JOC

2 ,outlet

JOC

(6.56)

+ JNC

2 ,outlet

2 ,outlet

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 JOC ,outlet and JNC ,outlet with known values. From the SOFC flux 2 2 balance Equation 6.12, we know that JOC

2 ,outlet

k

= JOC

2 ,inlet

− JOC = JOC

2 ,inlet

2

−

j 4F

(6.57)

Commonly, in fuel cell operation, the oxygen inlet flux JOC ,inlet (and the hydrogen inlet 2 flux) are regulated according to the stoichiometric number. The concept of a stoichiometric number is briefly introduced in the text box that follows. From the definition of the stoichiometric number, (6.58) JOC ,inlet = 𝜆O2 JOC 2

2

Plugging the above equation into Equation 6.57 allows us to solve for JOC ,outlet in terms 2 the stoichiometric number: JOC

2 ,outlet

Finding JNC

2 ,outlet

= (𝜆O2 − 1)JOC = (𝜆O2 − 1) 2

j 4F

(6.59)

is easier. Since there is no nitrogen consumption, JNC

2 ,outlet

= JNC

2 ,inlet

= 𝜔JOC

2 ,inlet

= 𝜔𝜆O2 JOC = 𝜔𝜆O2 2

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

(6.61)

(1 + 𝜔)𝜆O2 − 1

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 stoichiometric 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 wasteful, 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 hydrogen 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 JHA

𝜆O2 =

2

JO2 ,inlet JOC

(6.62)

2

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:

k

V = Ethermo − 𝜂ohmic − 𝜂cathode = Ethermo − j

k

tM T ASOFC e−ΔGact ∕(RT)

⎤ ⎡ ⎥ ⎢ ⎥ ⎢ j RT − ln ⎢ ( )⎥ 4𝛼F ⎢ 𝜆O2 − 1 jRT ⎥ C − tC ⎥ ⎢ j0 p C Deff − 1 + 𝜔) 𝜆 (1 4Fp ⎦ ⎣ O2 O2 ,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 ⎡ 0.5 A∕cm2 (8.314 J∕mol ⋅ K)(1073 K) ⎢ ln ⎢ ncathode = 2 4 ⋅ 0.5 ⋅ 96485 C∕mol ⎢ 0.1 A∕cm ⋅ 1 atm × 101300 Pa∕atm ⎣ ⎤ ⎥ 1 ×( )⎥ ⎥ 5000 A∕m2 × 8.314 J∕mol ⋅ K × 1073 K 1.5 − 1 − 0.0008 m ⎥ 2 (1 + 3.76) 1.5 − 1 (4 × 96,485 C∕mol)(101,325 Pa)(0.00002 m ∕s) ⎦ = 0.228 V

(6.64) V = 1.0 V − 0.088 V − 0.228 V = 0.684 V

k

(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 represent 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.

k

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 difficult. 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 programs 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

k

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228

FUEL CELL MODELING

Air outlet

Air inlet Hydrogen inlet

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.

k

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 addition 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 produced 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 Temperature, T

100% 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.

k

k 1.2 1.1 1

Voltage (V)

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2

0

100

200

300

400

Current

500

600

700

800

900

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) Air Outlet O2 0214B 0.2

k

k

0.1

0

0

Air Inlet

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 visualization 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 experimental 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

k

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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.

k

• 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 thermodynamically 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. Computational 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

k

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FUEL CELL MODELING

Current density (A/cm2)

Cell voltage (V)

Cell voltage (V)

Cell voltage (V)

232

Current density (A/cm2)

(b)

(c )

Cell voltage (V)

Cell voltage (V)

(a)

Current density (A/cm2)

Current density (A/cm2)

(d )

Current density (A/cm2)

(e)

Figure 6.12. Curves for problem 6.1.

(c) A PEMFC severely limited by poor reaction kinetics (d) A PEMFC with an extremely low ohmic resistance (e) A SOFC suffering from reactant starvation

k

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 typical 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 atmospheric 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.

k

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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.

k

6.8

(a) Plot the complete j–V curve for the 1D SOFC model example in the text assuming 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 problem, we consider it. (a) Using a linear approximation of Butler–Volmer equations for the anode as j = j0

p 2𝛼F 𝜂 p0 RT act

k

(6.66)

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234

FUEL CELL MODELING

show that the anodic overvoltage can be modeled as 𝜂anode =

RT 2𝛼F

j

( j0

pA

xH2 ||a −

tA

)

jRT

(6.67)

2 FpA Deff H ,H 2

2O

(b) Based on the information from Table 6.4, plot the anodic and cathodic overvoltages 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 overvoltages 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. k

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 stoichiometric 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? (c) Derive an equation that describes the power density (P) as a function of current density (j) 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?

k

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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: at the cathode of this fuel cell given DO2 ,air = 0.2 cm2 ∕s, (a) Calculate Deff O2 ,air 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 for this “flooded” fuel cell cathode. to 0.26 and increases τ to 3, calculate Deff O2 ,air (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? =P = 1 atm: 6.16 Consider a pure H –O fuel cell at T = 80∘ C and P 2

k

2

cathode

anode

(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?

k

k

k

k

k

k

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 performance 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.

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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, electrochemically 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 performance 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 characterization 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 dramatically depending on factors like the operating conditions and testing procedures. To ensure that j–V performance comparisons are fair, identical operating conditions, testing procedures 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 performance, the scientific community will reserve their applause.

OVERVIEW OF CHARACTERIZATION TECHNIQUES

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 performance in terms of the various major loss categories: activation loss, ohmic loss, concentration 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 Deff .

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

239

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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 electrochemical 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, characterizing 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 characterization 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 temperature, 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

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 misnomer. 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 galvanostatic 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 measurement. 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.

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• 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 properties 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 electrochemical 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 perturbation (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 conducted 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 measurements. 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 spectroscopy 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

Exhaust Fu e

Pressure gauges

lc

el

l

Electric measurement

0.00 sccm 0.00 sccm

50.0 C

Humidifier

Mass flow controller 50.0 C

Heater

(a)

Power source/temperature controller with feedback loop Power supply

Exhaust

Tube furnace

Temperature sensors Pressure gauges

l

el

lc

e Fu

Electric measurement

Gas line heaters

0.00 sccm 0.00 sccm Mass flow controller

50.0 C

50.0 C

Optional humidifiers (necessary for proton conducting ceramic electrolytes)

(b) 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.

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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, pressure, 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 controlling these carbon residues become essential in fuel cell test stations operating at high temperatures with hydrocarbon fuels. SOFC testing brings unique experimental requirements 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

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 current 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. Therefore, care must be taken to fully document measurement operating conditions, testing procedures, 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 outlets. 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 compressors, 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

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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 stoichiometrically with the current so that the ratio between reactant supply and current consumption is always fixed. This method is known as the fixed-stoichiometry condition. 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 compression 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 concentration 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 activation 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 coefficient 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 sophisticated 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

1

Voltage (V)

Voltage (V)

1 0.8 0.6 0.4

IV curve Activation loss

0.2 0 10–2

0.8 0.6

IV curve Tafel fitting

0.4 0.2

10–1 Current density (A/cm2)

0 10–2

100

10–1

Current density (A/cm2)

(a)

(b)

Activation loss

Voltage (V)

1

Ohmic and concentration loss

0.8 0.6

IV curve Activation loss

0.4 0.2 0

0

0.2

0.4

0.6

0.8

1

Current density (A/cm2) (c)

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 perturbation, 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.

247

Voltage (V)

FUEL CELL CHARACTERIZATION

V0

t Current (A)

248

i0 Phase shift (ϕ) t

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 frequency 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) cos(𝑤t) = Z0 i0 cos(𝑤t − 𝜙) 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=

V0 ej𝑤t = Z0 ej𝜙 = Z0 (cos 𝜙 + j sin 𝜙) i0 e(j𝑤t−j𝜙)

(7.5)

The impedance of a system can therefore be expressed in terms of an impedance magnitude Z0 and a phase shift 𝜙, or in terms of a real component (Zreal = Z0 cos 𝜙) and an imaginary component (Zimag = Z0 sin ϕj). 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

Small-signal voltage perturbation Cell voltage (V)

Probe pseudolinear portion of i--V curve

Current (A) Yields smallsignal 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 frequencies, 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. (Consider 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 hypothetical 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 semicircular 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

249

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FUEL CELL CHARACTERIZATION

Anode Ohmic activation losses losses

0

0

ZΩ

Cathode activation losses

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 relative 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

–Z imag R

0

0

R

Z real

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 diagram. Recall from the complex definition of impedance that the impedance of a system can be represented in terms of its real component Z0 cos 𝜙 and its imaginary component ( jZ0 sin 𝜙): Z = Z0 cos 𝜙 + jZ0 sin 𝜙 (7.7) A Nyquist diagram plots the real component of impedance versus the imaginary component 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 interface. 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 reaction, 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.

251

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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 separation 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 dV (7.8) i=C dt For a sinusoidal voltage perturbation (V = V0 ej𝑤t ), this gives d(V0 ej𝑤t ) = C(j𝑤)V0 ej𝑤t dt

(7.9)

V0 ej𝑤t V(t) 1 = = j𝑤t i(t) j𝑤C C(j𝑤)V0 e

(7.10)

i(t) = C which yields an impedance of Z=

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: (7.11) Zseries = Z1 + Z2 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

–Z imag C

Decreasing ω

R

0

Z real

R

0

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 resistor. 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 simplification of reaction kinetics (Equation 3.40): 𝜂act = −

RT RT ln i0 + ln i 𝛼nF 𝛼nF

(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 RT 1 = di 𝛼nF i

(7.14)

Substituting i = i0 e𝛼nF𝜂act ∕(RT) into this expression yields ( Zf = Rf =

RT 𝛼nF

)

1 i0

e𝛼nF𝜂act ∕(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

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FUEL CELL CHARACTERIZATION

the resistive Faradaic impedance. Just like combining parallel resistances, the parallel combination of two impedance elements is given by

For our case, this becomes

1 1 1 = + Zparallel Z1 Z2

(7.16)

1 1 + j𝑤Cdl = Z Rf

(7.17)

1 1∕Rf + j𝑤Cdl

(7.18)

Thus Z=

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; frequency 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. Cdl

–Zimag

Rf

ω = 1/Rf C dl Decreasing ω

0 0

Rf

Z real

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

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 𝜎 Z = √ i (1 − j) 𝑤

(7.19)

where σi in this equation is the Warburg coefficient for a species i (not the conductivity) and is defined as ) ( RT 1 𝜎i = (7.20) √ √ 0 (ni F)2 A 2 ci Di 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 ( √ ) 𝜎i j𝑤 Z = √ (1 − j) tanh 𝛿 (7.21) Di 𝑤

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FUEL CELL CHARACTERIZATION

–Z imag

ec re as in g

ω

Zw

D

256

0

Slope = 1.0

Z real

0

Figure 7.10. Circuit diagram and Nyquist plot for a Warburg element used to model diffusion processes. The impedance response is a diagonal line with a slope of 1. Impedance increases from left to right with decreasing frequency.

–Z imag Follows infinite Warburg for ω > 10Di/δ

Zw Decreasing ω

Z = δσi

2 Di

Slope = 1.0 0 0

Z real

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 equivalent 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

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

Finite (porous bounded) Warburg

√ (𝜎i ∕ 𝑤)(1 − j) √ √ (𝜎i ∕ 𝑤)(1 − j) tanh(δ j𝑤∕Di )

Series impedance elements

Zseries = Z1 + Z2

Parallel impedance elements

1∕Zparallel = 1∕Z1 + 1∕Z2

Infinite Warburg

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. 2. 3. 4.

Anode activation Cathode activation Cathode mass transfer 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 compared 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 activation 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 significantly 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.

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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

–Z imag

0

–Z imag

Z real

0 (a)

0 Z real

0 (b)

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 dominate 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 overvoltage 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 pitfalls. 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).

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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 = (0.25 A)(0.10 Ω) = 0.025 V (7.22) 𝜂act = 𝜂tot − 𝜂ohmic = 0.43 V − 0.025 V = 0.405 V 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 = (1.0 A)(0.10 Ω) = 0.10 V (7.23) 𝜂act = 𝜂tot − 𝜂ohmic = 0.58 V − 0.1 V = 0.48 V 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: ) ) ( ( RT RT ln i0 + ln i 𝜂act = − 𝛼nF 𝛼nF (7.24) ( ) RT 0.48 V = − ln i0 𝛼nF Substitution into a similar equation for point a allows us to solve for α: For point a: ) ) ( ( RT RT ln i0 + ln i 𝜂act = − 𝛼nF 𝛼nF ) ( RT ln 0.25 0.405 V = 0.48 V + 𝛼nF 𝛼 = 0.239 for T = 300 K, n = 2

(7.25)

Substituting α back into the equation for point b yields i0 : ( 0.48 V = −

(8.314) (300) (0.239)(2)(96400)

) ln i0

(7.26)

−4

i0 = 1.4 × 10 A 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

Cell voltage (V)

1.2 1.0

a

0.8

b c

0.6 0.4 0.2 0.0 0.0

0.5

1.0

1.5

2.0

2.5

Current (A)

–Z imag

0

–Z imag

–Z imag

0

0.10Ω

1.1Ω

Zreal

0 0.10Ω

(a)

0.40Ω

(b)

Zreal

0.10Ω 0.30Ω

(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.

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FUEL CELL CHARACTERIZATION

Cdl

RΩ

Rf

ZW

Current (A)

(a)

0.5

t

0.0 (b)

Voltage (V)

262

1.0

0.7 0.6 t (c)

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

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 0.10 V = = 0.2 Ω i 0.50 A

(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

263

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 generally 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 contribution 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 adsorption 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 Current

Voltage V2

Time V1

V2

Voltage

V1 (a)

(b)

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

Current (μA)

200 150 100 50

Qh

0

Qh

–50

–100 –150 –200 0

300 600 900 1200 1500 1800 Potential (mV vs. hydrogen anode)

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 hydrogen 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 =

Qh measured active catalyst surface area = geometric surface area Qm Ageometric

(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 surface 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.

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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 infiltration 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 pressures. 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 cos 𝜃 r

(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 characterization 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

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=

2p2 I ΔV − Δp Δt (p1 + p2 )Δp

(7.32)

where I is a constant.

7.4.4

Structure Determinations

Significant information about microstructure, porosity, pore size distribution, and interconnectedness 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 information 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 catalyst 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, bonding, 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

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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 electrochemical 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 measurement, (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 immediate 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 implement 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

• 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 analysis, 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 measurement.

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 (represented 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

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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 element 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

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8.1

INTRODUCTION

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As described in the first chapter of this book, there are five major types of fuel cells, differentiated from one another on the basis of their electrolyte: 1. 2. 3. 4. 5.

Phosphoric acid fuel cell (PAFC) Polymer electrolyte membrane fuel cell (PEMFC) Alkaline fuel cell (AFC) Molten carbonate fuel cell (MCFC) 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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8.2

PHOSPHORIC ACID FUEL CELL

In the PAFC, liquid H3 PO4 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: Cathode:

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H2 → 2H+ + 2e− 1 O + 2H+ + 2e− → H2 O 2 2

(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, H3 PO4 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, H3 PO4 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 H2 S, is around 50 ppm (parts per million).

H

Porous graphite anode

e-

H2

H3PO4 in SiC matrix Porous graphite cathode

2H+ + 2ePt/C catalyst

H+ 1 + 2e- + 2H+ --O 2 2

H2 O

O2

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.

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PAFC Advantages • Mature technology • Excellent reliability/long-term performance • Electrolyte is relatively low cost

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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, usually 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: Cathode:

H2 → 2H+ + 2e− 1 O + 2H+ + 2e− → H2 O 2 2

(8.2)

A schematic diagram of a PEMFC is provided in Figure 8.3. Figure 8.4 gives a photograph 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 transparent. It is coated on either side with a thin layer of platinum-based catalyst and porous

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OVERVIEW OF FUEL CELL TYPES

H2 Porous carbon anode

e-

H2

Polymer electrolyte Porous carbon cathode

2H+ + 2ePt/C catalyst

H+

--12 O2 + 2e- + 2H+

H2 O

O2

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.

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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 liquid 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-based materials 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 oxidizes methanol (CH3 OH) to provide electricity. 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 characteristics. 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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(a)

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Figure 8.4. (a) Rendering of the 2015 Hyundai ix35 fuel cell car power train. The PEMFC stack generates 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 system will 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 manufacturers 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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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: Cathode:

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H2 + 2OH− → 2H2 O + 2e− 1 O + 2e− + H2 O → 2OH− 2 2

(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 electrolyte, 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 temperatures 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 → CO3 2− + H2 O

(8.4)

H2 Porous carbon anode e-

Aqueous KOH electrolyte Porous carbon cathode

2H2O + 2e-

H2 + 2OH-

Pt/C or Ni catalyst

OH- H2O

--12 O2 + 2e- + H2O

2OH-

O2

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

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Figure 8.6. Photograph of United Technologies Corporation(UTC) AFC. These fuel cell units supplied 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, K2 CO3 can begin to precipitate out of the electrolyte (due to its lower solubility), leading to significant 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 electrolyte membrane materials have been developed that partially mitigate the CO2 instability issue associated with AFC operation. 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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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, Li2 CO3 and K2 CO3 , 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: Cathode:

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H2 + CO3 2− → CO2 + H2 O + 2e− 1 O + CO2 + 2e− → CO2− 3 2 2

(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 mixture 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 maintaining the operating temperature of the MCFC. A schematic diagram of a MCFC is provided in Figure 8.7. Figure 8.8 gives a photograph of a 2.5-MW pressurized MCFC system. CO2 H2 Porous nickel/ chrome H2 + CO23 e-

Molten carbonate in ceramic matrix Porous nickel oxide

CO2 + H2O + 2eCO 32-

--12 O2 + CO2 + 2e-

CO23

O2 CO2 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 catalytic 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)

Distribution Panel

Electricity Supply

EBOP

End User

District Heating Medium Temperature Water Return

Heat Exchanger 60˚C

Heat Supply

MBOP

2.5 ~ 2.8 MW

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Communication

Ex-PLC

120˚C

Hot Water Supply Pump

Medium Temperature Water Supply

Water Supply

Fuel Supply

STACK

Fuel Supply Device

Water Supply Pump

(b)

Figure 8.8. Photograph of a 2.5-MW MCFC system. (a) The system can power roughly 3500 individual 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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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 electrodes, 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 the MCFC provides fuel flexibility. The MCFC 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/ shutdown 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 applications, efficiencies could reach close to 90%. MCFC Advantages • Fuel flexibility • Nonprecious metal catalyst • High-quality waste heat for cogeneration applications

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MCFC Disadvantages • Must implement CO2 recycling • Corrosive, molten electrolyte • Degradation/lifetime issues • Relatively expensive materials

8.6

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SOLID-OXIDE FUEL CELL

The SOFC employs a solid ceramic electrolyte. The most popular SOFC electrolyte material 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: Cathode:

H2 + O2− → H2 O + 2e− 1 O + 2e− → O2− 2 2

(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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283

H2 Porous nickel/YSZ cermet e-

Solid ceramic electrolyte Porous mixed conducting oxide

H2 + O2-

H2O + 2eO2 -

--12 O2 + 2e-

O2-

O2

Figure 8.9. Schematic of H2 –O2 SOFC. The ceramic electrolyte is solid state. A nickel–YSZ cermet 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.

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Figure 8.10. Photograph of Siemens-Westinghouse 220-kW hybrid SOFC/micro gas-turbine system. 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 surface area of the anode structure. The cathode electrode is usually a mixed ion-conducting

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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 materials 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 electrical 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 disadvantages associated with high-temperature operation while maintaining the most significant 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. k

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 classification. 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 nonstandard fuel cell technologies.

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8.7.1

285

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 H2 O 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 CH3 OH + H2 O → CO2 + 6H+ + 6e−

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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 rechargeable 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, resulting 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 poison. Current DMFCs exhibit power densities of about 30–100 mW/cm2 . (Compare this to H2 PEMFC power densities of 500–2500 mW/cm2 .) In addition to the considerable activation 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 CH3 OH + 6OH− → CO2 + 5H2 O + 6e− While at the cathode, the oxygen reduction proceeds as 3 O + 3H2 O + 6e− → 6OH− 2 2

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OVERVIEW OF FUEL CELL TYPES

(a)

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(b)

Figure 8.11. Recent example prototype portable DMFC systems. (a) A 20-W DMFC notebook computer 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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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: Cathode: Overall:

HCOOH → CO2 + 2H+ + 2e− 1 O + 2H+ + 2e− → H2 O 2 2 HCOOH + 12 O2 → CO2 + H2 O

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 borohydride fuel cell uses a solution of sodium borohydride (NaBH4 ) or, alternatively, ammonium borohydride (NH4 BH4 ) 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 k

Cathode: Anode: Overall:

2O2 + 4H2 O + 8e− → 8OH− NaBH4 + 8OH− → NaBO2 + 6H2 O + 8e− NaBH4 + 2O2 → NaBO2 + 2H2 O

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 (powdered) 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 H2 O. 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 + 2H2 O → NaBO2 + 4H2

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OVERVIEW OF FUEL CELL TYPES

The hydrogen can then be used in a conventional H2 /O2 PEMFC fuel cell according to Anode: Cathode:

H2 → 2H+ + 2e− 1∕2O2 + 2H+ + 2e− → H2 O

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, microorganisms, 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 electrode 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) conditions, microorganisms at the anode can oxidize sugars to produce carbon dioxide, protons, and electrons, as shown in the following equation: k

C12 H22 O11 + 13H2 O → 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 circuit. 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

8.7.3

289

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

O2/electrolyte

Fuel/electrolyte solution

solution

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k

Anode

Cathode

Depletion Diffusion Depletion zone (mixing) zone zone

Figure 8.12. A membraneless fuel cell design based on a Y-shaped microfluidic channel configuration 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 oxygen 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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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, membraneless 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

k

Metal–Air Cells

The metal–air cell is essentially halfway between a fuel cell and a battery. The anode “fuel” in a metal–air cell is a solid (powdered), highly reactive metal. 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 achieve much higher energy density than most 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 schematically 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

+ O2

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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291

At the anode, zinc is oxidized by OH– ions in the electrolyte to create ZnO and electrons are liberated: Anode Reaction:

Zn + 2OH− → ZnO + H2 O + 2e−

(8.7)

The electrons pass through an external circuit, providing useful work, before recombining at the cathode with O2 and water to produce fresh OH– ions: Cathode Reaction:

1 O 2 2

+ H2 O + 2e− → 2OH−

(8.8)

The OH– ions transport back through the electrolyte, thus completing the circuit: Overall Reaction:

1 O 2 2

+ 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.

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8.7.5

Single-Chamber SOFC

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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 platinum 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 cathode 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 developed, 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 Sm0.5 Sr0.5 CoO3-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), significantly 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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OVERVIEW OF FUEL CELL TYPES

O2

Fuel Selective anode

Fuel + Air

Electrolyte Selective cathode

O2

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.

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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 performance. 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 sealing 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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293

Air

Cathode Electrolyte Anode Partially combusted products (e.g., H2, CO) Flame: combustion chemistry and heat release

Burner 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.

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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 producing 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, including S (reduced to SO2 ), C and CO (reduced to CO2 ), hydrogen (reduced to H2 O), and

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OVERVIEW OF FUEL CELL TYPES

Porous anode separator

YSZ electrolyte

Liquid tin anode

Porous cathode

O2-

H2 or hydrocarbon fuel SnO

SnO + Fuel Sn + H2O, CO 2, SO2, etc.

O2 (from air)

O2-

Sn + O22e-

SnO +

--12 O2 + 2e-

O2-

O2H 2O, CO2, SO2

k

Sn e-

O2-

e-

Figure 8.16. Operating principle of the liquid-tin anode SOFC (LTA-SOFC). Based on a conventional 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 + H2 O). While the LTA-SOFC is still under preliminary development, it appears to be an extremely attractive technology for fuel-flexible power generation 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 characteristics 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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295

The most common proton-conducting ceramic electrolytes include acceptor-doped perovskite compositions based on BaZrO3 and BaCeO3 . Like SOFC electrolytes, PCFC electrolytes require elevated temperatures to facilitate ion conduction, since the hopping process associated with ionic motion in these materials has a relatively high activation energy. However, 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 designing 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 materials (especially new cathode materials) that work at lower temperatures and are compatible with PCFC electrolytes. Early PCFCs used the same electrode materials developed 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 k

Solid-Acid Fuel Cells

Solid-acid fuel cells (SAFCs) use a solid proton-conducting electrolyte based on an inorganic 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 (H2 SO4 ) is reacted with cesium sulfate (Cs2 SO4 ) salt, the solid acid CsHSO4 is produced: 1 H SO4 2 2

+ 12 Cs2 SO4 → 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 (typically 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 additional information on solid-acid electrolyte materials 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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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 degradation of the solid-acid electrolyte during long-term operation and decreasing the amount of precious metal catalyst needed in the electrodes. 8.7.10

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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 (V2 O5 ) is dissolved in sulfuric acid (H2 SO4 ), 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: Cathode:

V2+ → V3+ + e− +

+

(8.11) −

VO2 + 2H + e

→ VO

2+

+ H2 O

(8.12)

Then, the overall reaction becomes V2+ + VO2 + + 2H+ → V3+ + VO2+ + H2 O

(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 easily 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 1 MW 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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8.7.11

297

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 hydrogen. 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 terminal 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 hydrogen. 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: Cathode:

k

H2 O → 12 O2 + 2H+ + 2e− 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:

H2 O →

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

--12 O2

Figure 8.17. Schematic diagram of a single cell of a PEM water electrolyzer that electrochemically converts water to hydrogen and oxygen.

k

k

k

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 oxygen 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 subsystems 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.

k

Based on surveys of PEM electrolyzer manufacturers, studies have estimated systemwide 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 operate 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 operated as an electrolyzer and as a power system when operated as a fuel cell. The same hardware 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 development, 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

k

k

k

CHAPTER SUMMARY

299

TABLE 8.1. Comparison Summary of the Five Major Fuel Cell Types

k

Electrical Power Fuel Efficiency Density Power Internal Cell Type (%) (mW/cm2 ) Range (kW) Reforming CO Tolerance

Balance of Plant

PAFC

40

150–300

Moderate

PEMFC

40–50

AFC

50–1000

No

Poison (100∘ C) membranes, various inorganic materials 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. Nanoparticles/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 conductivity 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 composite membranes at high temperatures, however, does not match that of Nafion at normal operating temperatures, for example, 80∘ C. Furthermore, like Nafion, these composite membranes still have to be hydrated to maintain high proton conductivity, and their mechanical 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 H2 SO4 or H3 PO4 , and normal salts, such as K2 SO4 . 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 CsHSO4 and CsH2 PO4 [84–86]. These materials are solid (although typically disordered rather than fully crystalline) at room temperature and can be formed into membrane structures. Conduction in solid-acid membranes relies on a rotational diffusion transfer mechanism, where protons are passed along between rotationally mobile tetrahedral oxy-anion groups (such as SO4 2– or PO4 3– ) [87]. The proton conductivity of solid acids increases by several orders of magnitude (>10–2 S ⋅ cm1 ) upon undergoing a phase transition between 100 and 200∘ C [87]. Solid-acid membranes are generally thermally and electrochemically stable under 200∘ C. Accordingly, solid-acid membranes have been proposed as electrolytes for intermediate-temperature PEMFCs. However, a reducing environment (as experienced at a fuel cell anode) may accelerate solid-acid membrane decomposition, especially in the presence of typical fuel cell catalysts. This decomposition can also lead to the formation of species such as H2 S that cause irreversible catalytic poisoning, rendering the fuel cell inoperable [88]. The fabrication of thin solid-acid membranes is difficult due to the poor mechanical properties and poor ductility of solid-acid materials, although composite-based

307

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PEMFC AND SOFC MATERIALS

membranes can be fabricated [89]. Other significant issues related to water solubility and thermal expansion have also not yet been overcome. 9.2

PEMFC ELECTRODE/CATALYST MATERIALS

Fuel cell electrodes serve a dual purpose: They must efficiently deliver/collect electrons from the fuel cell and also deliver/collect reactant/product species from the fuel cell. Because of these dual requirements, fuel cell electrodes must simultaneously provide high electrical conductivity and high porosity. Additionally, high catalytic activity is desired, especially in the vicinity of the electrode–electrolyte interface. Unfortunately, PEMFC catalysts are often based on expensive noble metal materials (like platinum), and it is

Pt/C catalyst Ethylene glycol Nafion solution

~130 ˚C and 300psi

Ink formation

Ink application

Hot press bonding

1

2

3

Carbon cloth electrode

Pt/C catalyst layer

Nafion electrolyte

Figure 9.3. Typical PEMFC MEA fabrication process. (1) Pt /C catalysts are mixed with water, 5% Nafion solution, and ethylene glycol to form a catalyst ink. (2) The catalyst ink is applied to the electrolyte membrane using one of several techniques. (3) Carbon cloth or carbon paper electrodes are hot-press bonded onto either side of the catalyst-coated membrane. Detail drawing shows the desired final MEA microstructure.

PEMFC ELECTRODE/CATALYST MATERIALS

therefore desirable to use as little of these catalyst materials as possible. For this reason, PEMFC electrode structures are typically fabricated using a dual-layer approach: A thin (typically 10–30 μm thick) but highly active catalyst layer usually consisting of a mixture of expensive porous Pt/C catalyst and electrolyte material is deposited directly on the surface of the electrolyte. A much thicker (typically 100–500 mm thick) inexpensive, porous, and electrically conductive electrode layer (without any catalyst) is then bonded on top of the expensive catalyst layer to provide protection and facilitate current collection. This results in a dual-layer catalyst/electrode structure, as shown in the Figure 9.3 detail. (See the dialogue box on PEMFC MEA fabrication later in this chapter for further information on how these structures are made.) This dual-layer structure maximizes catalytic activity, gas access, product removal, and electrical conductivity, while minimizing costs. Regardless of the exact materials chosen for the catalyst and electrode layers, this dual-layer approach is followed in most PEMFC MEA designs. The dual-layer PEMFC design approach is discussed in more detail in the next section. PEMFC MEA FABRICATION 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 pore, 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 geometry with significant intermixing, or blending, of the electrode and electrolyte phases (along with gas porosity). Fabricating these nanostructured electrode–electrolyte interfaces is a delicate process; it is perhaps more art than science. The prototypical PEMFC MEA fabrication approach is illustrated in Figure 9.3. The basic idea is to maximize the effectiveness of the expensive Pt catalyst by deploying ultra-small (2–3 nm) platinum particles on a high-surface-area carbon powder. This powder is then mixed with extra polymer electrolyte material to create a blended composite material that maximizes the opportunities for all three phases (gas pores, catalytically active electrode, and electrolyte) to intimately mix. This approach was pioneered by Wilson and Gottesfeld at Los Alamos National Laboratory in the early 1990s [90]. Variations on the theme have since developed, but the basic approach is as follows: 1. Ink Formulation. A catalyst “ink” is formulated containing the catalyst-loaded carbon material mixed with a 5% Nafion solution, water, and glycerol to control viscosity. 2. Ink Deposition. The ink is deposited onto both sides of an electrolyte membrane via one of several methods including spray deposition, painting, and screen printing. Screen printing generally provides the greatest thickness control and reproducibility. 3. Electrode Attachment. Porous carbon cloth or carbon paper electrodes are bonded to both sides of the membrane via hot-press embossing (at ∼120–140∘ C, 70–90 atm). These porous electrodes serve to protect the catalyst layer and hold it in place.

309

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9.2.1

Dual-Layer (Gas Diffusion Layer/Catalyst Layer) Approach

As discussed previously, almost all PEMFC designs utilize a dual-layer catalyst/electrode structure, as illustrated in Figure 9.3. In this section, we discuss in more detail the design requirements for both the catalyst layer and the electrode layer in this dual-layer approach. Catalyst Layer Requirements. Since PEMFC catalysts are often extremely expensive, they are generally not used by themselves in the catalyst layer. Instead, nanoscale (2–3 nm) particles of catalyst are typically decorated onto a high-surface-area carbon powder (such as Vulcan XC-72). By immobilizing the catalyst particles on a high-surface-area carbon support, a very small amount of catalyst material can be used to create an extremely large amount of effective catalyst surface area. Development of better (or cheaper) catalyst materials and better high-surface-area carbon support materials is an area of active research. The catalyst particles themselves must provide high activity (and durability), as their task is to facilitate the electrochemical fuel cell reactions. Meanwhile, the supporting carbon material must provide an inexpensive, stable, corrosion-resistant, and electrically connected porous support structure. The carbon support essentially “wires” the catalyst particles to the overlying fuel cell electrode. Electrolyte material is also typically added to the catalyst layer in order to wire the catalyst particles to the fuel cell electrolyte as well. The ions liberated (or consumed) by the electrochemical reactions on the catalyst particles are percolated through electrolyte pathways to the bulk electrolyte membrane. Meanwhile, the electrons liberated (or consumed) by electrochemical reactions on the catalyst particles are percolated through the high-surface-area carbon particle network to the protective, porous electrode overlayer. Key requirements for the catalyst layer therefore include: • • • • • • •

High catalytic activity High surface area/high density of triple-phase boundaries (TPBs) Percolating electrical and ionic conductivity High stability/corrosion resistance Excellent poison/impurity tolerance Minimal degradation Low cost (if possible!)

Gas Diffusion Layer Requirements. The thicker, protective second layer of the fuel cell electrode structure is often referred to as the “gas diffusion layer,” or GDL, reflecting its role in permitting gas to diffuse into the catalyst layer, while simultaneously providing protection and electrical connectivity. The GDL also plays a significant role in determining the removal of liquid water from the fuel cell. For this reason, many researchers feel it should be called a “porous transport layer” rather than a gas diffusion layer, as this more accurately reflects its role in managing liquid water transport and electrical transport in addition to gas transport. The exact material chosen for the GDL, its porosity, its thickness, and its relative hydrophilic or hydrophobic nature can all have a significant impact on PEMFC performance.

PEMFC ELECTRODE/CATALYST MATERIALS

Key requirements for the GDL include: • • • • • •

High electrical conductivity High gas permeability High stability/corrosion resistance Facilitation of water removal Good mechanical properties Low cost

In the next section, some of these materials issues associated with the choice of GDL are briefly discussed. Afterwards, further sections will discuss PEMFC catalyst materials options that extend beyond the established Pt/C standard.

9.2.2

GDL Electrode Materials

Most PEMFCs employ carbon-fiber-based GDL materials. The two most common GDL materials are carbon fiber cloths (woven) and carbon fiber papers (nonwoven). Carbon fiber materials are chosen due to their good electrical conductivity and high porosity (typically >70% porous). Furthermore, carbon fiber materials exhibit excellent stability and corrosion resistance along with good mechanical properties. Both carbon-cloth and carbon-paper materials exhibit significant anisotropy in electrical conductivity. In-plane electrical conductivity tends to be significantly higher than through-plane conductivity (typically by a factor of 10–50 times). As shown in Figure 9.4, in-plane conductivity is more important, since the average in-plane conduction path length for an electron transported through the GDL is 10 times higher than the through-plane conduction path length. Therefore, in-plane conductivity becomes an important figure of merit for GDL materials, and fiber-sheet assemblies (with consequently high in-plane conductivity) provide a sensible solution.

Flow channels

2-5 mm

r H2

O2 o -

e

GDL (~300 μm) Catalyst layer (~30 μm) Electrolyte

Figure 9.4. Gas and electron transport within the fuel cell GDL. In the GDL, lateral (in-plane) transport is more important than vertical (out-of-plane) transport. For example, electrons generated under the middle of a fuel cell flow channel must be transported laterally 1–2 mm, but must only transport ∼300 μm vertically to reach the current collecting rib structures. Similarly, gas from the flow channel must transport ∼1–2 nm laterally, but only ∼300 μm vertically to reach reaction zones under the channel ribs.

311

312

PEMFC AND SOFC MATERIALS

Carbon Cloth. Carbon fiber cloths are produced using a textile process that weaves carbon fiber filaments (“yarn”) into a thin, flexible, fabric-like material. Carbon cloths tend to possess mechanical resiliency (they are highly pliable), low density (∼0.3 g∕cm3 ), and high permeability (∼50 Darcys) [91]. The carbon cloths deployed in PEMFC GDL applications are typically 350–500 μm thick but can compress significantly (30–50%) when clamped into a fuel cell assembly. Importantly, this compression can significantly change their electrical and gas permeation properties. Carbon Paper. Carbon fiber paper materials are produced by bonding a random, “haystack” like arrangement of carbon fibers into a thin, stiff, lightweight paperlike sheet. Because carbon paper is not woven, a binder material (typically a carbonized resin) is needed to maintain mechanical integrity. This binder material, or “webbing,” fills some of the pores between individual fibers, and thus carbon paper materials tend to be denser (∼0.45 g∕cm3 ) and less permeable (∼10 Darcys) than their carbon cloth counterparts [91]. Additionally, carbon fiber paper tends to be stiff and somewhat brittle, rather than soft and compliant. The carbon paper GDLs deployed in PEMFC applications are typically 150–250 mm thick and, because of their stiffness, experience significantly less compression (10–20%) than carbon cloth when clamped into a fuel cell assembly. Hydrophobic Treatment. GDL materials must enable the removal of liquid water from the fuel cell. If liquid water accumulates in the fuel cell catalyst or GDL layers, it will eventually block reactant supply and cause a fuel cell’s performance to deteriorate. This phenomenon is known as “flooding.” In order to prevent flooding, most GDL materials are treated with polytetrafluoroethelyne (PTFE, or Teflon) in order to increase hydrophobicity. PTFE loadings between 5 and 30% are typically added to most carbon-fiber fuel cell GDLs. PTFE is most commonly applied by dipping the GDL into an aqueous PTFE suspension and then baking the treated GDL in an oven at 350–400∘ C to remove the residual solvent and sinter/fix the PTFE particles to the GDL fibers. PTFE loading is controlled by adjusting the concentration of the aqueous PTFE suspension. Microporous Layers. An increasingly common practice is to improve the interface between the GDL and the catalyst layer by applying an intermediate “microporous layer” in between them. This microporous layer provides a transition between the large-scale porosity (10–30 μm pores) of the GDL and the fine-scale porosity (10–100 nm pores) of the catalyst layer. The microporous layer can also improve the wicking of liquid water from the catalyst layer and decrease the electrical contact resistance between the GDL and the catalyst layer. The microporous layer is typically formed by mixing submicrometer-sized particles of graphite with a polymeric binder, usually PTFE. A thin layer of this mixture is applied to one side of the GDL and heat-treated, resulting in a thin, uniform, microporous graphitic layer ∼20–50 μm thick. The microporous treated face of the GDL is then bonded to the catalyst-coated electrolyte membrane, resulting in a so-called seven-layer MEA or “Electrode Los-Alamos Type” (ELAT). (The seven layers are anode GDL, anode microporous layer, anode catalyst layer, electrolyte, cathode catalyst layer, cathode microporous layer, cathode GDL.)

PEMFC ELECTRODE/CATALYST MATERIALS

Other GDL Materials. While carbon cloth and carbon paper are the dominant GDL materials used in most PEMFC designs, researchers have occasionally examined other options. One option is to completely eliminate the GDL altogether. This option only appears to work if an extremely fine current collector layer (e.g., a finely patterned metal mesh) is used to collect electrons from the catalyst layer, as the in-plane resistance of the catalyst layer is too high to enable significant lateral electrical transport. A second option is to use an expanded metal mesh or porous metal foam material in place of the typical carbon fiber GDL. However, metal-based GDL materials present significant challenges: They tend to corrode, they are too hydrophilic, and the available range of porosity is typically too coarse.

9.2.3

PEMFC Anode Catalysts

Platinum (for H2 Fuel Cells). In a standard H2 fuel cell, PEMFC anode catalysts must facilitate the hydrogen oxidation reaction (HOR): H2 → 2H+ + 2e− Currently, the best electrocatalyst for the HOR is platinum (Pt). The extremely high activity of Pt is believed to be due to a nearly optimal bonding affinity between Pt and hydrogen. The bonding is strong enough to promote facile absorption of H2 from the gas phase onto a Pt surface and subsequent electron transfer, but the bonding is weak enough to allow desorption of the resultant H+ ion into the electrolyte. In contrast, metals like W, Mo, Nb, and Ta form too strong a bond with H2 , resulting in a stable hydride phase. Metals like Pb, Sn, Zn, Ag, Cu, and Au, on the other hand, form too weak a bond with H2 , resulting in little or no absorption. Although Pt is expensive, it proves to be an exceptionally effective catalyst for the HOR. Using the well-developed Pt/C catalyst approach, whereby ultrasmall (2–3 nm) Pt particles are supported on a high-surface-area carbon powder, only extremely small amounts of Pt catalyst are required. Typical Pt loadings in PEMFC anodes have thus been successfully reduced to around 0.05 mg Pt/cm2 . At these levels, the anode Pt catalyst expense is relatively modest compared to the expense associated with other components in the fuel cell. For example, a 50 kW automotive fuel cell stack operating at a power density of 1.0 W/cm2 would require about 2.5 g of Pt for the anode catalyst. At a price of $1200/ounce, this represents a Pt materials cost of ∼$100. Novel methods for depositing the platinum catalyst (for example, employing ultrathin sputter-deposited Pt layers) may be able to reduce anode Pt loading levels even further. Thus, using Pt-based anode catalysts may be perfectly feasible for H2 -fueled PEMFCs, although issues associated with catalyst durability and degradation bear careful scrutiny (as will be discussed in Section 9.5 of this chapter). Platinum Alloys (for Direct Alcohol Fuel Cells). While Pt is a perfectly acceptable (and indeed entirely viable) catalyst for H2 -fueled PEMFC anodes, pure platinum catalysts are not acceptable for direct methanol or direct ethanol fuel cell anodes. Direct alcohol fuel cell reactions like the methanol oxidation reaction are complex and proceed by a series

313

314

PEMFC AND SOFC MATERIALS

of individual steps. Some of these reaction steps can lead to the formation of undesirable intermediates, such as CO, which act as poisons. CO poisons pure Pt catalysts by strongly and irreversibly absorbing on the Pt surface. As absorbed CO builds up on the Pt surface, further electrochemical reaction is prevented. CO tolerance is provided by alloying the catalyst with a secondary component, such as Ru, Sn, W, or Re. Ruthenium (Ru) is considered to be the most effective at providing tolerance. Addition of Ru to the Pt surface creates new absorption sites capable of forming OHads species. These OHads species react with the bound CO species to produce CO2 and H+ , thereby removing the CO poison. Researchers have also identified nanoscale RuOx Hy phases as perhaps playing a role in the improved methanol oxidation characteristics of PtRu catalysts [92]. More recently, ternary catalysts consisting of Pt, Ru, and a third element (such as W or Mo) have been identified that may prove even more effective than PtRu [93]. Although PtRu alloys work remarkably well for the methanol oxidation reaction, they prove ineffective for the ethanol oxidation reaction. Ethanol poses an additional problem, due to the need to catalyze the cleavage of a carbon–carbon bond. The most effective ethanol oxidation catalysts tend to be based on Pt–Sn alloys, or even non-noble metal Sn-based alloys [93, 94], although effective ethanol oxidation is usually only achieved in alkaline fuel cell environments. PEMFC-based direct alcohol fuel cells are technologically attractive because of the higher energy densities and improved logistics of liquid fuels compared to hydrogen. However, the insufficiencies of current alcohol oxidation electrocatalysts mean that achievable efficiencies and power densities remain unacceptably low. Improved alcohol oxidation catalysts are therefore an area of vigorous research. 9.2.4

PEMFC Cathode Catalysts

Regardless whether a PEMFC is fueled by hydrogen, a liquid alcohol, or another fuel source, the reaction proceeding at the cathode will be the oxygen reduction reaction (ORR): 1∕2 O2 + 2H+ + 2e− → H2 O Like the anode HOR reaction, the dominant catalyst of choice for the cathode ORR is currently Pt. Unfortunately, Pt is considerably less active for the ORR than for the HOR. This means that significantly higher Pt loading levels are required in PEMFC cathodes. While Pt loading levels at the anode have been successfully reduced to ∼0.05 mg Pt/cm2 , cathode loading levels are currently 8–10 times higher, at about 0.4–0.5 mg Pt/cm2 . At these loading levels, pure Pt cathode catalysts are too expensive for large-scale PEMFC applications. Significant effort is therefore underway to reduce catalyst costs in PEMFC cathodes. To meet projected cost targets for automotive PEMFC commercialization, cathode Pt loadings should be reduced from about 0.40 to 0.10 mg Pt/cm2 without a loss in cell voltage or durability, while maintaining maximum power density and cell efficiency [95]. Approaches to reduce catalyst costs in PEMFC cathodes have generally followed three basic strategies: (1) optimize current Pt/C catalysts (by decreasing Pt particle size and improving Pt distribution/dispersion), (2) develop new Pt alloy catalysts that are even more

PEMFC ELECTRODE/CATALYST MATERIALS

active for the ORR than pure Pt, or (3) develop inexpensive, Pt-free catalysts, even if they are less active than Pt catalysts. Platinum. The performance of platinum cathode catalysts are typically quantified using two related metrics: mass activity and specific activity. Mass activity, i∗m(0.9V) , describes the amount of current produced in a fuel cell at a voltage of 0.9V per unit mass of cathode catalyst (measured under standard automotive PEMFC fuel cell conditions, typically 100 kPa O2 , 80∘ C, full hydration). Typical units for mass activity are A/mg Pt. Specific activity i∗s(0.9V) describes the amount of current produced in a fuel cell at a voltage of 0.9V per unit surface area of cathode catalyst (again measured under standard automotive PEMFC fuel cell conditions, typically 100 kPa O2 , 80∘ C, full hydration). Typical units for specific activity are μA∕cm2 Pt. State-of-the-art Pt cathode catalysts can attain mass activity values around 0.16 A/mg Pt and specific activity values around 200 μA∕cm2 Pt [96]. Mass activity and specific activity are related via the specific surface area (s∗ ) of the catalyst: i∗ m(0.9V) = i∗ s(0.9 V) × s∗

(9.1)

where s∗ is the specific surface area of the catalyst (catalyst surface area per unit mass). Efforts to optimize current Pt/C catalysts focus on further decreasing Pt catalyst particle size and further improving Pt catalyst distribution/dispersion. The basic idea is to increase s∗ , the amount of active surface area per unit mass of Pt, by deploying smaller and better dispersed Pt particles. Unfortunately, there appear to be limits to this particle size refinement approach. Current Pt/C catalysts employ Pt particle sizes as small as 2–3 nm, yielding specific surface area values of around 80–90 m2 /g Pt. However, further decreases in Pt particle size do not appear to lead to further improvements in mass activity. This is because even though the specific surface area (s∗ ) continues to increase with decreasing particle size, the specific activity (i∗s(0.9 V) ) is actually observed to decrease with decreasing Pt particle size. In other words, Pt particles below 2–3 nm in size appear to become less active catalysts. This unfortunate Pt particle-size “deactivation” effect is hypothesized to be caused by size-dependent changes in the adsorption of oxygen-containing species, OHads , which are frequently believed to decrease the O2 reduction reaction activity [96]. Because of these particle size effects, it appears that further decreases in Pt particle dimensions below ∼2–3 nm are counterproductive, and therefore further decreases in cathode Pt loading below 0.4 mg Pt/cm2 may prove infeasible using pure Pt catalysts. Further decreases in Pt particle dimensions also lead to accelerated instability and degradation issues, because there are strong energetic driving forces for ultra-small Pt particles to coarsen or corrode, leading to substantial decreases in catalytic performance over time. These issues will be dealt with in more detail in Section 9.5 of this chapter. Platinum Alloys. Because of the likely insufficiency of pure Pt catalysts for PEMFC cathodes, substantial research has been directed toward the development of Pt alloy catalysts that are even more active for the ORR than pure Pt. A number of Pt alloy catalysts have been investigated for PEMFC cathode applications, including Pt–Ni, Pt–Cr, Pt–Co, Pt–Mn, Pt–Fe, and Pt–Ti, usually in a 75–25% ratio (75% Pt, 25% second metal). Although catalytic activity comparisons have proved notoriously

315

316

PEMFC AND SOFC MATERIALS

difficult and even contentious, there is a general consensus that certain Pt alloys, like Pt3 Cr and Pt3 Co, do indeed show enhanced ORR specific activity compared to pure Pt, perhaps by as much as a factor of 2–4. Pt–Co catalysts appear especially attractive, and various studies are examining different compositions in the Pt–Co alloy system. Although Pt alloy catalysts appear to offer a potentially feasible route to enhance activity and lower PEMFC cathode catalyst costs, they also create several new complications: • Compared to pure Pt catalysts, Pt alloy catalysts have proven harder to deploy as extremely high surface area (small particle size) dispersions on carbon supports. • Pt alloy catalysts contain transition metals (such as Co, Cr, Fe, Ni, Ti), which can poison the PEMFC if they leach from the catalyst. • The mixed composition of Pt alloy catalysts may make them more susceptible to accelerated degradation, corrosion, and deactivation. Of the above-listed concerns, leaching is probably the single greatest issue. In order to obviate leaching, researchers have introduced a preleaching process designed to remove base metal deposited on the carbon surface or poorly alloyed to the Pt prior to MEA preparation [97]. Pre-leached Pt alloy catalysts have been shown to yield dramatically lower poisoning rates than their unleached counterparts, while still retaining a significant activity advantage compared to pure Pt catalysts. Impressively, Pt–Co alloys have also shown particle coarsening/sintering/degradation rates that are actually lower than pure Pt catalysts, indicating that degradation issues may also be ameliorated by moving to these alloy blends. Non-platinum ORR Catalysts. Yet another approach to PEMFC cathode catalyst design is to develop inexpensive, Pt-free catalyst materials. The basic idea is to trade decreased catalytic activity for decreased cost. However, any candidate Pt-free catalyst, no matter how inexpensive, must still be reasonably active. Preliminary estimates indicate that even a “zero-cost” cathode catalyst must have a volumetric catalytic activity no worse than 1/10 that of Pt. The reason is that there are limits to how much catalyst we can load into a fuel cell. If a catalyst is 10 times less active than Pt, then we need to load 10 times more of it into the cathode. This can only be accomplished by increasing the thickness of the catalyst layer. However, as the catalyst layer thickness increases, electrical and gas transport resistances also increase, so there is a trade-off. At most, catalyst layer thickness can only be increased by a factor of 10 or so compared to the state of the art before ionic transport, mass transport, and electrical resistance losses become unacceptably large. Due to the relatively harsh, acidic environment of the PEMFC, finding stable non-noble metal candidate cathode materials is a real challenge. In fact, the acid stability criterion alone rules out all non–noble metals and most, if not all, oxides. Only a few potential non-noble metal catalyst materials have so far emerged, and none have yet been able to obtain even 1/10 the activity of Pt. Candidates thus far investigated include metal-macrocycles, heteropoly acid catalysts, and high-surface-area doped carbons, each briefly described here: Metal Macrocycles. Metal macrocycles are materials in which a transition metal ion, typically Fe or Co, is stabilized by several nitrogen atoms bound into an aromatic

SOFC ELECTROLYTE MATERIALS

or graphite-like carbon structure. These man-made structures emulate, or are often compared to, the active center of hemoglobin. Examples of such macrocycle catalysts are polymerized iron phthalocyanine and cobalt methoxytetraphenylporphyrin [98]. Heterpolyacid Catalysts. Heteropoly acids (HPAs) are a large and diverse class of oxidatively stable inorganic oxides that have attracted a great deal of interest as potential PEMFC electrocatalysts. Currently, vanadium and iron substituted HPA catalysts have shown the most potential [100], although ORR activity values are still too low for practical use. In addition to activity concerns, HPA materials are water soluble, so permanent absorption and immobilization of these catalysts within the PEMFC catalyst layer have proven challenging. Doped Carbon. High-surface-area carbon materials, doped with Fe, N, B, or a variety of other elements, have exhibited some of the best ORR activities of any of the non-Pt catalyst alternatives. In these materials, pyridine-type bond formation and 𝜋-electron delocalization (caused by the heterovalent atomic doping) is hypothesized to lead to ORR catalytic activity. Doped carbon catalysts have two important positives: They are relatively inexpensive, and they can be produced with extremely high surface area. Even the most successful of these doped carbon catalysts, however, still appear to be at least 50 times less active than Pt [95]. None of the above catalysts have come close to achieving 1/10 the activity of Pt in the acidic PEMFC environment, and all have also exhibited considerable stability/degradation concerns. Especially concerning is the tendency for non-Pt-based ORR catalysts to produce a significant amount of peroxide intermediate. These peroxide intermediates are known to cause significant degradation of most PEMFC electrolyte materials, and thus their formation must be avoided. While the outlook for platinum-free catalysts for PEMFC cathodes remains rather dim, it is worthwhile to note that the situation is considerably different for alkaline-based fuel cells. In an alkaline environment, the number of potential ORR catalysts increases significantly. This is for two reasons: (1) many more metals and oxides are stable in alkaline media and (2) the kinetics of the ORR are significantly improved in alkaline media. For these reasons, a number of non-platinum catalysts such as nickel, silver, transition metal oxides, and various chevrel-phase chalcogenides (which contain molybdenum, usually with selenium) have proven to be interesting alternatives. However, as was discussed previously in Chapter 8, the switch from an acidic fuel cell to an alkaline fuel cell also brings new concerns and challenges, including issues with CO2 degradation of the electrolyte. The reader is referred to a number of excellent reviews for further discussion of alkaline-based fuel cell catalysts and materials [100, 101].

9.3

SOFC ELECTROLYTE MATERIALS

In this section, we switch focus from PEMFC materials to SOFC materials. SOFCs are based on crystalline oxide ceramic electrolyte materials that conduct ions via defect hopping mechanisms at high temperatures. Unlike PEMFC electrolytes, then, SOFC electrolytes are not sensitive to membrane hydration and do not necessarily require sophisticated water management systems. In absolute terms, however, ion conductivity in ceramic oxide electrolytes is well below that of most polymeric proton conductors. To obtain sufficiently high

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ion conduction through oxide membranes, it is typically necessary to operate SOFC devices at temperatures in excess of 700 or 800∘ C. There are a number of candidate SOFC electrolyte materials, most notably yttria-stabilized zirconia (YSZ) and gadolinia-doped ceria (GDC). We briefly discussed both of these materials in Section 4.5.3. YSZ is the best known SOFC electrolyte material, and it possesses a number of compelling advantages, including excellent chemical stability and chemical inertness. YSZ also possesses one of the highest fracture toughness values of all the metal oxides. Most importantly for fuel cells, YSZ shows reasonably good ionic conductivity (at sufficiently high temperatures) and little or no electronic conductivity. GDC, in contrast, shows significantly higher ionic conductivity than YSZ but also shows significant electronic conductivity under reducing conditions. Thus, its suitability for fuel cell environments is still being debated. In response to this stability challenge, however, GDC/YSZ “multilayer” electrolytes have been explored that utilize GDC on the cathode side and YSZ on the anode side. Thin-film GDC/YSZ electrolyte assemblies have been shown to deliver power densities as high as 400 mW/cm2 at temperatures as low as 400∘ C [102]. Both YSZ and GDC will be discussed in more detail in the subsections that follow. In addition to the fluorite crystal-structure-based materials, such as YSZ and GDC, there are many potential SOFC materials from the doped perovskite family. These doped perovskites follow a general formula ABO3 , where A and B are metal atoms such as barium, zirconium, or cerium. Intriguingly, some doped perovskites provide O2– conductivity, while others provide H+ conductivity. As these materials are also oxide-based ceramic electrolytes, they will also be discussed in more detail later in this chapter. Figure 9.5 provides a comparison of the major ion-conducting electrolyte materials for fuel cells, showing representative examples of four key materials groups: polymeric proton conductors, solid acids, oxide ion conductors, and proton-conducting oxides. As discussed previously for characterizing ion conductivity [recall Figure 4.18], log(σT) is plotted versus 1/T. A more exhaustive discussion of a broader range of O2– - and H+ -conducting ceramic electrolyte materials is provided in the subsections that follow.

9.3.1

Yttria-Stabilized Zirconia

YSZ is arguably the most important electrolyte material for solid-oxide fuel cells. YSZ is created by doping ZrO2 with a certain percentage (typically around 8 mol %) Y2 O3 . The fluorite crystal structure of the zirconia host (the same as calcium fluorite CaF2 with the general formula AO2 ) is retained, as shown in Figure 9.6. In this figure, the light-colored spheres are oxygen anions while the darker spheres are the cations. In YSZ, each time two zirconium cations (Zr4+ ) are replaced by two yttrium cations (Y3+ ), one oxygen site (O2– ) will be left vacant to maintain charge balance. As you learned in Section 4.5.3, increasing the yttria content increases the number of these vacant oxygen sites and thereby leads to significant O2– conductivity. Replacing one atom with another one of different valence is referred to as aliovalent doping. If more vacancies are available, then more oxide ions can be transported per time unit, and hence the conductivity will increase. However, there is an upper limit to the

SOFC ELECTROLYTE MATERIALS

3 2

log( σT) (KΩ–1· cm–1)

1 0 –1 –2 –3 Nafion117: n=16 CsHSO4 BYZ YSZ

–4 –5 –6 0.7

1.2

1.7

2.2

2.7

3.2

3.7

1000/T (K–1)

Figure 9.5. Conductivity of a proton-conducting polymer (Nafion), a solid acid (CsH2 O4 ), an oxide ion conductor (YSZ), and a proton-conducting oxide (BZY) as a function of 1/T.

Figure 9.6. The fluorite crystal structure exhibited by stabilized zirconia and by doped ceria.

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amount of doping, beyond which conductivity begins to decrease rather than continue to increase. With increasing defect concentration the electrostatic interaction between dopants and vacancies increases, ultimately impeding oxide vacancy formation and oxide vacancy mobility. In fact, vacancies and dopants may form low-energy associations. The closer the spacing between vacancies and dopants, the more associations will be formed. Closer vacancy–dopant distances are linked to bigger barriers for oxide ion mobility, or stronger associations between vacancies and dopants. The balance between increased vacancy concentration for improved conductivity and the simultaneous formation of impeding associations results in a conductivity peak at a concentration of 6–8% Y2 O3 , on a molar basis. 9.3.2

Doped Ceria

Doped ceria is another common oxygen-ion-conducting ceramic material with characteristics compatible with SOFC applications. Doped ceria materials are obtained by doping ceria (CeO2 ) with a second aliovalent lanthanide metal, yielding a general form denoted by Ce1-δ (Ln)𝛿 O2-1∕2𝛿 . The primary advantage of doped ceria is that it generally shows higher ionic conductivity than YSZ. This relative conductivity advantage is particularly important at lower temperatures. Ionic conductivity is highly dependent on the type and concentration of the dopant ions, and in the case of ceria, doping with Sm or Gd gives the highest values of conductivity. Samaria- and gadolinia-doped ceria materials are often abbreviated SDC and GDC, respectively. The optimal dopant concentrations for SDC and GDC are typically in the range of 10–20%. For example, a typical electrolyte formulation for SOFC applications is Ce0.9 Gd0.1 O1.95 , which is commonly abbreviated as GDC10 or CGO10. GDC10 has an ionic conductivity of 0.01 S ⋅ cm−1 at 500 ∘ C [103]. Like stabilized zirconia, doped ceria exhibits the fluorite structure. Figure 9.7 shows the ionic conductivity of GDC20 (Ce0.8 Gd0.2 O1.9 ) as well as that of several other electrolyte materials discussed in this chapter. It is instructive to understand the factors that give rise to GDC’s higher conductivity relative to YSZ. This is primarily due to the relative sizes of the dopant ions as compared to the sizes of the primary ions they replace. Recall that aliovalent doping results in oxygen ion conductivity by creating vacancies, and that conductivity increases with doping concentration up to a certain peak point, after which it starts to decrease. This decline in conductivity occurs because of the increased interaction between the dopant ions and the oxygen vacancies. Originally, it was thought that this interaction was primarily a Coulombic effect, as both the dopant ion and the vacancy act as if they are oppositely charged species within a neutral lattice [106]. However, if the effect is purely Coulombic, then all dopants with the same relative charge (e.g., Y3+ , Sc3+ , La3+ ) should give rise to exactly the same level of conductivity, which is clearly not the case. Instead, it turns out that size, in addition to charge, is of primary importance. It has been shown that the major interaction between these point defects is through the elastic strain introduced into the crystal lattice by a mismatch between the size of the dopant ion and the ion that it replaces. To make a good oxygen ion conductor, it further appears that leaving the crystal lattice as undisturbed as possible is highly desirable. Thus, the best dopants are ones that closely match the host

SOFC ELECTROLYTE MATERIALS

3

YSZ (Zr0.92Y0.08O2-δ)

2.5

ScSZ (Zr0.907Sc0.093O2-δ) GDC(Ce0.8Gd0.2O2-δ)

log( σT) (KΩ–1· cm–1)

2

LSGM(La0.8Sr0.2Ga0.76Mg0.19Co0.05O3-δ)

1.5

BIMEVOX(Bi2V0.9Cu0.1O5.5-δ)

1

LAMOX(La1.8Dy0.2Mo2O9) BYZ(BaZr0.8Y0.2O3-δ)

0.5 0 –0.5 –1 –1.5 0.75

0.95

1.15

1.35

1.55

1.75

1.95

1000/T (K–1)

Figure 9.7. Ionic conductivity of representative examples from the various electrolyte materials groups discussed in this chapter. Conductivity is oxygen ionic, with the exception of BZY, where it is protonic [103–105].

ion in size. In the case of GDC, the host and dopant ions are very close in size (much more so than in the case of YSZ), leading to higher maximum effective dopant levels and higher ionic conductivity [106]. Unfortunately, doped ceria materials do have several significant disadvantages in SOFC electrolyte applications. The primary disadvantage of doped ceria arises from the fact that, under reducing conditions (i.e., at the anode), Ce4+ is partially reduced to Ce3+ . This induces n-type electronic conductivity, which can lead to partial internal electronic short circuits, and this problem increases with increasing temperatures. A second disadvantage is that ceria chemically expands under reducing conditions (due to nonstoichiometry with respect to its normal valency in air), and this lattice expansion can lead to mechanical failure [103]. Experiments have shown that GDC10 is more resistant to reduction than GDC20. The electronic and ionic conductivities of GDC10 as functions of temperature are shown in Figure 9.8. This figure shows that the electronic conductivity at the anode side will be greater than the ionic conductivity for temperatures greater than about 550∘ C [103]. The disadvantages of doped ceria can be partially solved by adopting a multilayer approach where, for example, a GDC layer facing the cathode is combined with another solid electrolyte (e.g., YSZ) facing the anode. However, multilayer cells also have performance problems due to formation of reaction products with low conductivity at the interface between the electrolyte layers, as well as the mismatch in thermal expansion between the electrolyte layers, which can result in microcracks. To summarize the preceding discussion, the advantages of GDC over YSZ are best realized at lower temperatures, where the higher conductivity of GDC is most pronounced, and where the disadvantages associated with electrical conductivity and mechanical instability are suppressed.

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3 2 log( σT) (KΩ–1· cm–1)

322

1 0 –1 –2

Electronic conductivity

–3

Ionic conductivity

–4 –5

0.5

1

1.5 1000/T

2

2.5

(K–1)

Figure 9.8. Ionic and electronic conductivities of CGO10 (GDC10) in reducing atmosphere (10% H2 , 2.3% H2 O) [103].

9.3.3

Bismuth Oxides

Bismuth oxide (Bi2 O3 ) exhibits polymorphism, meaning that it has the ability to exist in more than one crystal structure. In fact, Bi2 O3 has four crystallographic polymorphs, including a monoclinic crystal structure, designated 𝛼-Bi2 O3 , at room temperature. This monoclinic structure transforms to the cubic-fluorite-type crystal structure, 𝛿-Bi2 O3 , when heated above 727∘ C, where it remains until the melting point of 824∘ C is reached (two other metastable intermediate phases exist and are referred to as 𝛽 and 𝛾). The high-temperature 𝛿 phase is the primary reason why Bi2 O3 is considered to be a promising SOFC electrolyte material, since its ionic conductivity is among the highest ever measured in an oxygen ion conductor [107]. At 750 ∘ C, the conductivity of 𝛿-Bi2 O3 is typically about 1 S ⋅ cm−1 , which is far higher than YSZ or even GDC! The 𝛿-phase conductivity is predominantly ionic with O2– being the main charge carrier. The exceptionally high conductivity arises from the fact that 𝛿-Bi2 O3 has an intrinsically “defective” fluorite-type crystal structure in which two of the eight oxygen sites in the unit cell are naturally vacant. This results in a very high (25%) oxygen vacancy content. The exceptionally high conductivity of 𝛿-Bi2 O3 has triggered efforts to stabilize the high-temperature 𝛿 phase at low temperatures. Stabilization is achieved by substituting some of the bismuth atoms with rare-earth dopants (such as Y, Dy, or Er) and/or with higher-valency cations such as W or Nb. The resulting doped Bi2 O3 materials retain high ionic conductivity at lower temperatures. The maximum conductivity in the binary systems is observed for Er- and Y-containing materials, namely, Bi1-x Erx O1.5 with Er concentrations of approximately 20% and Bi1-x Yx O1.5 with Y concentrations in the 23–25% range [103]. In addition to stabilized 𝛿-Bi2 O3 , high ionic conductivity is also characteristic of stabilized 𝛾-bismuth vanadate (𝛾-Bi4 V2 O11 ), giving rise to what is known as the BIMEVOX class of materials. Compared with 𝛿-Bi2 O3 , the BIMEVOX family possesses better phase stability at moderate temperatures. The stabilization of 𝛾-bismuth vanadate is accomplished by partially substituting the vanadium with transition metal cations such as Cu, Ni, or Co. Examples of highly conductive BIMEVOX ceramics include Bi2 V1-x Cux O5.5-𝛿 (the

SOFC ELECTROLYTE MATERIALS

conductivity of which is shown in Figure 9.7) and Bi2 V1-x Nix O5.5-𝛿 , with the best conductivities achieved for doping concentrations in the 7–12% range. Unfortunately, a fair amount of progress still needs to be made before 𝛿-Bi2 O3 - and BIMEVOX-based materials can be practically used in SOFC systems. While doped 𝛿-Bi2 O3 materials show significantly improved stability compared to pure Bi2 O3 , these materials are still metastable at temperatures below 500–600∘ C, and thus they undergo a slow phase transformation and lose their conductivity with time. Other disadvantages of 𝛿-Bi2 O3 -based materials include high electronic conductivity, volatilization of bismuth oxide at moderate temperatures, high corrosion activity, and low mechanical strength. As for BIMEVOX materials, the disadvantages include high chemical reactivity and low mechanical strength. 9.3.4

Materials Based on La2 Mo2 O9 (LAMOX Family)

The parent compound of what is known as the LAMOX series is La2 Mo2 O9 . Like bismuth oxide, it exhibits polymorphism, resulting in a phase transition at high temperatures, which is accompanied by a dramatic increase in conductivity. At around 600∘ C La2 Mo2 O9 transitions from an 𝛼 to a 𝛽 phase, at which point the ionic conductivity increases by approximately two orders of magnitude, reaching about 0.03 S ⋅ cm−1 at ∼720∘ C [106]. Like other novel materials being investigated for use in SOFCs, LAMOX materials still require a fair amount of development before they will be ready for practical use. However, they are interesting materials because of the unique mechanism that leads to their high conductivity, a mechanism known as lone-pair substitution (LPS). The LPS concept is interesting because it potentially provides a new approach to develop alternative oxygen ion conductors. A lone pair is a valence electron pair that is not bonded or shared with other atoms. Researchers Lacorre et al. have proposed that the high conductivity of 𝛽-La2 Mo2 O9 can be explained in the context of the cubic lattice structure of 𝛽-SnWO4 , where electron lone pairs act as structural elements within the crystal. It is believed that La2 Mo2 O9 is structurally similar to 𝛽-SnWO4 (Sn2 W2 O8 ), except that Sn is replaced by La and W is replaced by Mo. While 𝛽-SnWO4 has lone pairs associated with the Sn2+ cations, when La3+ cations are substituted, one oxygen ion and one oxygen vacancy are instead created, giving rise to the high oxygen mobility in 𝛽-La2 Mo2 O9 . Like the Bi2 O3 materials discussed earlier, La2 Mo2 O9 is a good ion conductor only in its high-temperature 𝛽 phase. As with Bi2 O3 , however, the high-temperature 𝛽 phase can be stabilized to lower temperatures by doping. Examples of stabilized, high-conductivity compositions include La1.7 Bi0.3 Mo2 O9-𝛿 , La2 Mo1.7 W0.3 O9-𝛿 , and La2 Mo1.95 V0.05 O9-𝛿 . Like doped ceria, however, LAMOX materials are susceptible to reduction, and their electronic conductivity increases with temperature. Thus, their potential as SOFC electrolyte materials is best suited to oxidizing conditions and intermediate temperatures. Some La2 Mo2 O9 -based materials also exhibit degradation at moderate oxygen pressures. Alternative doping and other strategies are currently being investigated to help address these issues. 9.3.5

Oxygen-Ion-Conducting Perovskite Oxides

Perovskite oxide materials follow the general formula ABO3 , where A and B are metal atoms and O is oxygen. The perovskite crystal structure is shown in Figure 9.9.

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La

Figure 9.9. Perovskite structure, proton-conducting BaZrO3 .

O

Ga

exhibited

by

oxygen-ion-conducting

LaGaO3

and

by

The perovskite structure leads to a wide range of possible ion-conducting materials because there are two different metal cation sites available for dopant substitutions. Perovskite oxides can exhibit oxygen ion conductivity and/or proton conductivity. In this section, we will discuss oxygen-ion-conducting perovskites. Of the major oxygen-ion-conducting perovskites investigated to date, lanthanum gallate (LaGaO3 ) has so far emerged as the most promising candidate for SOFC electrolyte applications. High oxygen ionic conductivity in LaGaO3 is achieved by substituting some of the lanthanum with alkaline earth elements such as strontium, calcium, or barium. As we discussed in an earlier section, minimum lattice distortion yields the highest oxygen ion mobility. Because of this, strontium is the best choice out of the three candidate dopants listed above. The oxygen vacancy concentration (and hence conductivity) can be further increased by substituting some of the gallium with divalent metal cations, such as Mg2+ . These dual substitutions gives rise to complex oxide stoichiometries like La1-x Srx Ga1-y Mgy O3-𝛿 , which is known as the LSGM series. Ionic conductivity in LSGM is maximized for Sr dopant concentration in the 10–20% range and Mg dopant concentration in the 15–20%, for example, La0.9 Sr0.1 Ga0.8 Mg0.2 O3-𝛿 . Moreover, it has been shown that the conductivity of LSGM can be further enhanced by introducing small amounts (below 3–7% concentration) of an additional transition metal dopant cation that has variable valence, such as cobalt, onto the gallium sites [103–108]. This additional doping further increases the ionic conduction in LSGM, while producing little to no increase in the electronic conductivity. The conductivity of LSGM is entirely ionic and higher than that of YSZ over a very wide range of oxygen partial pressure and over a wide range of temperatures up to about 1000∘ C. LSGM-based SOFCs can therefore operate at somewhat lower temperatures than their YSZ-based counterparts. At much lower temperatures (100 μm), and it acts as the structural support for the entire MEA. This dense, thick electrolyte material is created first and fired to provide strength. Subsequently, thin, porous anode and cathode electrode layers are deposited on either side of the electrolyte via spray coating, dip coating, or tape casting,

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and the MEA is fired again. Electrolyte-supported SOFCs tend to possess good mechanical properties—they are typically mechanically strong, resistant to delamination, and resistant to thermal shock. Unfortunately, the thick electrolyte leads to high ohmic resistance (or necessitates higher temperature operation). Cathode-Supported MEA. In the cathode-supported MEA design, the cathode is the thickest part of the fuel cell (tcathode >1 mm), and it acts as the structural support for the entire MEA. This thick, porous cathode structure is created first by mixing the cathode electrode powder (typically LSM) with binders and a pore former (typically carbon black or starch) and then by extruding, die pressing, or tape casting the desired cathode shape. Subsequently, a thin (10–30 μm), finely textured, mixed interfacial layer of LSM + YSZ is applied to one side of the cathode. The purpose of this interfacial layer is to create a large number of triple-phase boundary sites by intimately mixing the ion-conducting (YSZ) and electrically conducting (LSM) phases. This layer also has finer porosity than the thick structural part of the cathode, making the deposition of a dense electrolyte (the next step) easier. In the next step, electrolyte slurry is made by mixing the electrolyte powder (typically YSZ) with dispersants and a solvent (typically an alcohol). A thin, dense coating of this slurry is then applied on top of the cathode via spray coating, dip coating, or screen printing, and the cathode + electrolyte structure is fired. Extreme care must be taken to optimize the deposition and firing of this thin ( 90∘ C) the fuel cell components. (This last requirement ensures that the sealant will not infiltrate into the porous electrodes or flow out from the fuel cell stack during fabrication or operation of the cell.) Additionally, the sealant must be an electrical insulator, since it contacts both the anode and the cathode. The choice of sealing material depends on the stack design and on the materials choices that have been made for the other cell components. For planar stacks with ceramic interconnects, sealants are typically made of ceramic glasses. In particular, ceramic glasses are desired which can bond to the facing cell materials and have essentially the same thermal expansion behavior as the rest of the cell components. This approach is attractive as it leads to the fabrication of monolithic stacks made entirely of ceramic. For planar stacks with metal interconnects, on the other hand, alkali-based soft glasses with low glass transition temperature are often employed. These glasses contain a large amount of alkali or alkaline earth oxides in addition to silica. Unfortunately, they can occasionally migrate inside the fuel cell stack and react with other cell components.

9.5

MATERIAL STABILITY, DURABILITY, AND LIFETIME

As with almost any other device, durability and lifetime are critical issues for determining the eventual success of fuel cell technology. Commercial targets for vehicular-based fuel cell power require ∼5000 h of stable operation, while for stationary power applications the target is greater than 50,000 h. Significantly, these long-term operation and performance targets have already been demonstrated, indicating that there is no fundamental limitation to the long-term stability and durability of fuel cell technology. However, most long-term

MATERIAL STABILITY, DURABILITY, AND LIFETIME

durability demonstrations have been conducted under near-ideal operating conditions or with impractical amounts of expensive materials such as noble metals. As a result, the durability and lifetime issues of fuel cells under practical or commercial constraints continue to be critical areas of research and improvement. The key durability considerations for PEMFC and SOFC technologies are described in further detail in the following sections. 9.5.1

PEMFC Materials Durability and Lifetime Issues

PEMFC durability depends strongly on the operating conditions. Conditions that maximize durability include constant-load operation at relative humidity close to 100% and at temperatures around 75∘ C. Under these conditions, well-optimized PEMFC stacks can operate for over 40,000 h with less than 10% cumulative efficiency and power loss. Under these optimized conditions, durability is primarily governed by the slow degradation of the GDL’s water removal capacity. Other durability issues include membrane degradation and Pt particle growth. In real-world applications, PEMFC systems will be exposed to less than ideal operating conditions, including load variability, start–stop cycling, imperfect humidification, temperature fluctuations, and occasional fuel or oxidant starvation. Under these conditions, degradation is greatly accelerated, and a large number of durability problems can become critical. Following is a description of the primary degradation mechanisms that can occur in these situations. Membrane Degradation. Chemical degradation due to chemical attack of the electrolyte membrane by free radicals is among the leading causes of membrane failure. Hydroxy (.OH) and hydroperoxy (.OOH) radicals are the most likely drivers of membrane chemical degradation since they are among the most reactive chemical species known. Radical-induced chemical degradation leads to reduced mechanical strength and reduced proton conductivity of the membrane. Hydroxy (.OH) and hydroperoxy (.OOH) radicals are believed to arise from the decomposition of H2 O2 , which itself is created from incomplete reduction of oxygen in the PEMFC cathode. Radicals can also be generated by reactant crossover through the membrane, which leads to molecular H2 and O2 reacting on the surface of the Pt catalyst [116]. Other leading causes of membrane degradation include mechanical failure and ionic contamination. Mechanical failure can arise from pinholes or foreign materials introduced during MEA manufacturing as well as from stresses developed during temperature and humidity cycling. As for ionic contamination, sources of contaminant ions include metal bipolar plates, humidifiers, and air itself. The membrane easily absorbs ionic contaminants because the sulfonate sites have a stronger affinity for almost all metal ions (except for Li+ ) than for protons. Since protons are therefore displaced by these metal ion contaminants, this process leads to a net loss in proton conductivity [116]. Electrode/Catalyst Degradation. Pt dissolution and particle growth result in a reduction in electrochemically active surface area and therefore lead to catalyst performance loss during extended operation. Pt dissolution is a significant problem at intermediate potentials but is negligible at low and high potentials. At lower potentials (i.e., under the conditions of normal H2 /air fuel cell operation), the solubility of platinum in acid is quite low. At higher potentials, upon exposure to air, PtO is formed and the resulting oxide layer insulates the

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platinum particles from dissolution. At intermediate potentials, however, the uncovered Pt catalyst surface is prone to high rates of platinum dissolution. A second major degradation issue arises from corrosion of the carbon-based catalyst support. At elevated temperatures, particularly at the cathode, carbon atoms are able to react with oxygen atoms and/or water to generate gaseous products such as CO and CO2 , which then leave the cell. During PEMFC start-up and shutdown local cathode potential can reach as high as 1.5 V, and this high oxidative potential accelerates carbon corrosion. Carbon oxidation permanently removes carbon from the cell, leading to a reduction in the catalyst support surface area and a consequent increase in electrical resistance and a loss or agglomeration of “electrically connected” Pt particles. In the extreme, a complete structural collapse of the electrode is even possible. Other electrode durability issues include possible oxygen evolution and GDL degradation. If oxygen evolution (from the electrochemical oxidation of water) occurs at the anode, it can react with residual hydrogen, resulting in significant damage. Meanwhile, chemical surface oxidation of the GDL by water and other radicals can trigger a decrease in hydrophobic character, leading to a substantial decrease in water removal capacity and hence higher mass transport losses. Bipolar Plate and Seal Degradation. The release of undesirable contaminants from the bipolar plates can cause serious poisoning of the membrane and catalyst. For graphite and graphite composite plates, the corrosion and release of contaminants is generally not observed under normal operating conditions, but it is conceivable under start–stop or fuel starvation conditions. However, corrosion is much more likely in the case of metal-based bipolar plates. The degree of corrosion/contaminant release depends on the specifics of the metal alloy, the operating voltage, and the relative humidity. PEMFC seal degradation is mostly avoidable, and if it occurs, it is often because of inappropriate seal material selection. For example, silicone seals in direct contact with perfluorosulfonic acid membranes (e.g., Nafion) suffer from acidic decomposition. 9.5.2

SOFC Materials Durability and Lifetime Issues

The commercial requirements for SOFC systems (particularly for stationary SOFC applications) often require cell and stack lifetimes as high as 50,000–100,000 h with very small degradation rates. In order to achieve this level of durability, degradation mechanisms associated with both individual cell components and the total stack must be carefully understood and addressed. Total stack durability requires excellent compatibility between all the stack materials during processing, fabrication, and operation. During SOFC operation stack materials are often exposed to temperatures as high as 1000∘ C, while during SOFC fabrication stack materials are often exposed to temperatures as high as 1400∘ C [117]. At these high temperatures, degradation is driven primarily by changes in material morphology, microstructure, and phase. In particular, sintering and agglomeration processes as well as chemical reactions and interdiffusion across interfaces or through grain boundaries can contribute significantly to SOFC aging and degradation during operation. Component-specific durability issues are detailed next. Electrolyte Degradation. During prolonged operation, YSZ-based electrolytes exhibit a nontrivial decrease in ionic conductivity as a function of operating time. This is partly

MATERIAL STABILITY, DURABILITY, AND LIFETIME

because optimal YSZ electrolyte compositions (around 8 mol% Y2 O3 ) exist in a two-phase field at typical SOFC operating temperatures, and thus they tend to undergo a slow phase separation that results in conductivity degradation. Other causes of long-term conductivity degradation include the growth of precipitates and the formation of resistive layers at grain boundaries due to grain boundary segregation. In ceria-based electrolytes, significant degradation problems can arise because of the GDC/YSZ multilayer architecture that is typically required for these cells (recall Section 9.4.2). At the interface between the GDC and YSZ layers, solid solutions with poor ionic conductivity can be formed. In LSGM electrolytes, a major durability concern arises from instability in reducing atmospheres. This instability results in Ga depletion through the volatilization of gallium oxide and a subsequent permanent decrease in ionic conductivity. Anode Degradation. For Ni/YSZ cermet materials (the most common SOFC anode), the cermet microstructure plays a critical role in determining long-term stability. In particular, the particle size and distribution of the Ni and YSZ phases, porosity, surface area, connectivity of the Ni particles, and abundance of triple-phase boundary sites are all major determinants of anode performance. The major mechanism behind Ni/YSZ anode degradation is agglomeration, coarsening, and/or oxidation of the Ni particles, which lead to a reduction in electrical conductivity and the number of triple-phase boundaries. Additionally, for hydrocarbon-fuel operation, sulfur poisoning and carbon deposition lead to a substantial reduction in the rate of electrochemical reaction, excessively high anode losses, and deterioration in cell performance. Cathode Degradation. For LSM–YSZ composite materials (the most common SOFC cathode), oxidative degradation and secondary reaction are the primary degradation mechanisms. In particular, chemical reaction at the cathode–YSZ electrolyte interface can occur, resulting in the formation of undesirable La2 Zr2 O7 and SrZrO3 phases. Both these phases have low conductivity, and their existence at the interface between the cathode and electrolyte leads to an increase in cell resistance and activation losses and thereby degradation of the cell performance. Formation of these reaction products can be minimized by carefully limiting the operating temperature—the lower the operating temperature, the lower the rate of degradation. Interconnect and Sealant Degradation. For LaCrO3 -based ceramic interconnects, one of the main degradation mechanisms stems from the very low thermal conductivity of the interconnect material. This poor thermal conductivity can potentially lead to severe thermal gradients in the fuel cell stack. To make matters worse, ceramic interconnects expand differently upon heating in oxidizing versus reducing environments, and they are exposed to both environments as they connect the anode to the cathode. This differential expansion behavior can create fairly severe stresses across the interconnect material, eventually leading to the catastrophic failure of the stack. For chromium-based metallic interconnects, degradation is primarily caused by chromium volatilization. As discussed in Section 9.4.7, chromium-based metallic interconnects form a protective layer of chromia scale during operation at high temperatures in oxidizing (cathode) environments. This chromia scale is actually good, because it protects against further corrosion, but at high temperatures, chromia species such as CrO3 or CrO2 (OH)2 can be volatilized (evaporated) from the scale. These volatilized

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chromia species can then be subsequently reduced and deposited at the cathode–electrolyte interface (often in the form of Cr2 O3 ), thereby blocking the three-phase boundaries at the LSM–YSZ–gas interface and severely degrading cathode activity [115]. For sealants, both ceramic glass and alkali-glass sealants primarily degrade because of reaction with other fuel cell components and the stresses arising from TEC mismatches with the rest of the fuel cell. Again, the trend toward lower operating temperature SOFCs will reduce reaction rates and lower the rate of sealant degradation, leading to more durable SOFC structures. Fully understanding and extrapolating damage accumulation over time are very important, as solid-oxide fuel cells in stationary applications will be expected to last decades or longer. 9.6

CHAPTER SUMMARY

This chapter provided an overview of the various PEMFC and SOFC materials and materials issues. A review of the most commonly used materials today was provided in addition to a description of newer materials and current research efforts. Advantages, disadvantages, and current state of development as well as lifetime and durability issues were discussed for each class of materials. • For PEMFC electrolytes, Nafion and other perfluorinated polymers are the primary materials in use today. Alternative materials include sulfonated hydrocarbon polymers, phosphoric acid doped polybenzimidazole (PBI), polymer–inorganic composite membranes, and solid-acid membranes. • Advantages of the alternative PEMFC electrolyte materials typically include lower costs and the ability to extend the operational temperature range beyond 100∘ C due to improved water retention and/or reduced humidity dependence. Disadvantages include typically lower ionic conductivity under “traditional” PEMFC operating conditions (in the case of hydrocarbon polymers and polymer–inorganic composites), oxidative degeneration (in the case of PBI), or membrane decomposition under reducing conditions (in the case of solid acid membranes). • PEMFC electrodes and catalysts are typically fabricated using a dual-layer approach. The catalyst layer is very thin (typically 10–30 μm in thickness) in order to minimize the amount of expensive catalyst used (typically platinum). The gas diffusion layer (GDL) is the much thicker electrode layer (typically 100–500 mm thick) and is inexpensive, porous, and electrically conductive. • The GDL layer in most PEMFCs is made of carbon fiber cloth or carbon fiber paper. Carbon fiber materials exhibit good electrical conductivity, high porosity, and good mechanical properties. GDL materials are typically treated with polytetrafluoroethelyne (PTFE, or Teflon) in order to increase hydrophobicity and in turn enhance the removal of liquid water from the fuel cell. It is also common to include a microporous layer between the GDL and the catalyst layer in order to provide a transition between the large-scale porosity of the former and the fine-scale porosity of the latter. The microporous layer is typically formed by mixing submicrometer-sized particles of graphite with PTFE.

CHAPTER SUMMARY

• Viable alternative GDL materials remain limited. Metal-based GDL materials have been investigated but continue to face significant challenges as they tend to corrode and to be too hydrophilic. • The standard PEMFC anode catalyst in H2 fuel cells is platinum; ultra-small Pt particles are typically supported on a high-surface-area carbon powder in order to minimize the amount of Pt needed. In direct alcohol fuel cells, however, intermediates such as CO can be formed, which irreversibly absorb on pure Pt. In this case, CO tolerance is achieved by alloying Pt with a secondary component like Ru (in the case of methanol fuel cells) or Sn (in the case of ethanol fuel cells). • PEMFC cathodes require a much higher (∼8–10X more) Pt catalyst loading levels compared to PEMFC anodes, leading to high catalyst expense. Alternatives include Pt alloys (e.g., Pt–Co catalysts) as well as Pt-free catalysts such as metal-macrocycles, heteropoly acids (HPAs), and doped carbons. Unfortunately, Pt-free cathode catalysts have generally not yet been shown to be viable alternatives. • SOFC electrolytes are based on oxide ceramics, and YSZ remains the most common electrolyte material today. The conductivity of YSZ is relatively high and is entirely ionic. Maximum conductivity is obtained with compositions containing ∼8% yttria on a molar basis. Other advantages of YSZ include its chemical and mechanical stability and its relatively low coefficient of thermal expansion. • Many other classes of SOFC electrolyte materials are being actively investigated. These include doped ceria materials such as gadolinia-doped ceria (GDC), which is a promising alternative (with higher conductivity than YSZ) at low SOFC operating temperatures. Perovskite oxides such as LSGM are promising for intermediate-temperature operation (in the 700–1000∘ C range). Materials based on bismuth oxides and bismuth vanadate (the latter are known as BIMEVOX materials) also draw significant interest because of the exceptionally high conductivity of a specific crystallographic polymorph of each of the two materials. These bismuth-oxide-based materials, however, remain saddled with many disadvantages, including chemical instability and low mechanical strength. Similarly, the LAMOX family of materials has drawn interest because of the unusual mechanism that leads to their high conductivity, but they remain in the early stages of development. • Some doped perovskites act as proton conductors, as opposed to oxygen ion conductors. Yttrium-doped barium zirconate (BZY) is currently the leading ceramic proton-conducting electrolyte material. • Like PEMFC electrodes, SOFC electrodes also typically employ a dual-layer approach. The first layer is very fine and thin (typically 10–30 μm in thickness) and catalytically active, in order to maximize triple-phase boundary sites. The second layer is much thicker (100 μm–2 mm thick) and provides mechanical support, electrical conductivity, and high porosity for gas access. • Ni/YSZ is currently the primary anode material in SOFCs. It meets the anode material requirements and is particularly well suited for SOFCs based on YSZ electrolytes due to the close match in the thermal expansion coefficient. Disadvantages include susceptibility to sulfur and carbon poisoning.

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• The desire to directly use hydrocarbon fuels in SOFCs has led to active efforts to evaluate anode materials than can suppress carbon deposition. Materials under active development include doped ceria (e.g., CG40) and doped perovskites (e.g., LSCV, which is typically used with YSZ in LSCV–YSZ composite anodes). Other materials under consideration include mixed ionic–electronic conductors (MIECs) such as pyrochlore-type oxides and the tungsten bronze family. • SOFC cathodes are responsible for the electrochemical reduction of oxygen and their conductivity needs to be both ionic and electronic. Strontium-doped LaMnO3 perovskite (LSM) is the most common material in use today. It is typically used in composite cathodes with another material than can provide good ionic conductivity (e.g., LSM–YSZ composite cathodes). Other cathode materials include MIEC (mixed ionic–electronic conductivity) materials, such as La1-x Srx Co1-y Fey O3-𝛿 (LSCF). • In SOFC stacks, interconnects provide electrical connection from the anode of one cell to the cathode of the next cell. Matching the interconnect material’s thermal expansion coefficient with that of the electrolyte is critical to the mechanical stability of the stack. Interconnects are typically either ceramic (usually based on LaCrO3 ) or metallic (usually chromium based, such as the Cr–5Fe–1Y2 O3 alloy). An appropriate sealant material is also needed in planar SOFC stacks. Sealants are typically made from ceramic glasses or soft glasses. • The durability and lifetime considerations of fuel cells continue to present challenges under practical operating conditions and typical commercial constraints. This chapter examined both total stack durability and the degradation mechanisms associated with each primary component in PEMFC and SOFC devices.

CHAPTER EXERCISES Review Questions 9.1

Discuss the advantages and disadvantages of hydrocarbon-based polymer electrolyte membranes compared to perfluorosulfonated membranes.

9.2

Define/describe the following terms or acronyms: (a) ELAT, (b) PTFE, (c) GDL, (d) GDC10, (e) MEA, (f) Ni–YSZ, (g) TPB, (h) aliovalent.

9.3

Why might a “Pt-free” catalyst not work for a PEMFC cathode, even if it was zero-cost?

9.4

Discuss the trade-offs between ceramic oxide ion and polymeric proton-conducting electrolyte membranes in fuel cells. Focus your discussion on performance, stability, and fuel variety.

9.5

As discussed in Chapter 4, to increase conduction of the fuel cell electrolyte, one can increase the temperature of operation, decrease the electrolyte thickness, or choose a material with higher ionic conductivity. Discuss the trade-offs between these approaches. Illustrate your discussion with examples of specific materials choices.

CHAPTER EXERCISES

9.6

For automotive applications, a 5000-h fuel cell durability target has been established. Based on the amount of time the average American drives a car each day, how many years would a fuel cell car “engine” last given this 5000-h durability target?

9.7

For stationary applications, a 50,000-h fuel cell durability target has been established. If a stationary fuel cell power plant operates 24 h a day, 7 days a week, year-round, how many years would a stationary fuel cell power plant last, given this 50,000-h durability target?

9.8

You are asked to design the highest-performing fuel cell in terms of power density, regardless of cost. Suggest the best materials for electrode, electrolyte, catalyst, and interconnect (if necessary) for a high-power-density PEM, a high-power-density SOFC, and a high-power-density proton-conducting oxide electrolyte fuel cell, respectively.

Calculations 9.9

In a PEM fuel cell, there are catalyst particles dispersed in the catalyst layer (CL) to increase the electrochemically active area. You model this as j0 = j00 𝛿 where 𝛿 is the catalyst layer thickness. As you increase the CL thickness, the activation losses will therefore decrease, but the concentration losses will increase. Using this simple model and considering only activation and concentration losses, find an expression for the optimum CL thickness to minimize voltage losses. Your answer should depend on current and other relevant parameters. Hint: Concentration losses become significant at high current, where activation losses are approximately given by the Tafel equation: 𝜂act =

j RT log 𝛼nF j0

9.10 As was discussed in Chapters 4, for reasonable performance, fuel cells should achieve an area-specific resistance of no more than 0.15 Ω ⋅ cm2 . In SOFCs, this target ASR value can be achieved either by operating at higher temperatures or by reducing the thickness of the electrolyte. Calculate and plot the thickness required for YSZ and GDC membranes to achieve a specific resistance ASR of 0.15 Ω ⋅ cm2 as a function of temperature. In this plot add curves for a polymeric electrolyte membrane as well as other oxide ion- and proton-conducting membranes. Discuss the results. 9.11 Low-temperature SOFCs may be built by fabricating arrays of thin-film windows consisting of a thin layer of a mixed electronically and ionically conducting cathode on top of a pure ionically conducting electrolyte, on top of a porous layer of platinum serving as the anode. Assume that the cathode is the bottleneck (i.e., rate controlling). There is a simultaneous flow of electrons in the cathode film plane, accompanied by a

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O2–

e–

d

d Figure 9.12. Mixed conductive cathode, oxide ions flow perpendicular to cathode, electrons flow parallel to the plane.

flow of oxide ions normal to the mixed conducting cathode film. The best SOFC performance can be achieved if the combined resistance (simple additive sum of ionic and electronic resistances) is a minimum. Show that there is a minimum resistance in a thin cathode film conductive to electrons and ions. Express the combined resistance of this plate as a function of thickness t, and determine t at the point of minimum combined resistance. Assume the following material and geometric parameters. Assume the following geometry (see Figure 9.12): Ions travel perpendicular to the cross section with dimension d × d for a distance t; electrons travel parallel to the plane with cross section d × t for a distance d. For the electronic conductivity assume σe = 100 S∕cm. From the figures provided in this chapter choose appropriate values for 𝜎 of several mixed electronic and oxide ion-conducting materials. Provide a numerical value for t assuming 𝜎ion = 1 × 10−2 S∕cm and d = 1cm. 9.12 The catalytic activity of a fuel cell catalyst sometimes depends strongly on the interface between the catalyst and the electrolyte. Researchers have found that an SOFC with a gadolinia-doped ceria (GDC) electrolyte promotes superior catalytic performance compared to YSZ. However, GDC cannot be used for the entire thickness of

Cathode

GDC YSZ

Anode Figure 9.13. Composite electrolyte consisting of a layer of GDC and one layer of YSZ with catalyst particles decorating the electrolyte surface.

CHAPTER EXERCISES

the electrolyte because of stability issues. Therefore, you want to evaluate the merits of using an “interlayer” of GDC coated on top of a YSZ electrolyte to improve catalytic activity, as shown in Figure 9.13. While this GDC interlayer will reduce the activation losses, it will cause an increase in cell resistance because it represents an additional layer of material. Assume j = 0.5 A∕cm2 and T = 400∘ C. Under these conditions, you have determined the additional data in the following table. GDC j0 A Layer thickness

4 × 10 0.29 50 nm

YSZ −3

A∕cm2

2 × 10−3 A∕cm2 0.22 20 nm

From the figures provided in this chapter, you will also need to estimate appropriate values of σ (for both YSZ and GDC) at T = 400∘ C. (a) Calculate the ohmic and cathodic activation losses for an SOFC without the GDC interlayer. (b) Calculate the ohmic and cathodic activation losses of an SOFC with the GDC interlayer. (c) Sketch the voltage profile as a function of distance through the electrolyte in each case. 9.13 Nafion 117 has a conductivity of approximately 0.1 S/cm and a thickness of approximately 200μm. You have developed a solid-oxide electrolyte (O2– conducting) with the following properties: • Carrier concentration c = 10–3 mol∕cm3 • Eact = 0.6 eV • D0 = 10–5 cm2 ∕s You want to operate your SOFC electrolyte at 400∘ C and to have a comparable ASR to the Nafion 117 membrane. What thickness must your solid membrane be to have the same ASR as Nafion 117?

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CHAPTER 10

OVERVIEW OF FUEL CELL SYSTEMS

In this chapter, we move beyond the single-fuel-cell unit to the complete fuel cell system. The ultimate goal of any fuel cell system is to deliver the right amount of power to the right place at the right time. To meet that goal, a fuel cell system generally includes a set of fuel cells in combination with a suite of additional components. Multiple cells are required since a single fuel cell provides only about 0.6–0.7 V at operational current levels. Other components besides the fuel cells themselves are needed to keep the cells running. These components include devices that provide the fuel supply, cooling, power regulation, and system monitoring, to name a few. Often, these devices can take up more room (and cost) than the fuel cell unit itself. Those that draw electrical power from the fuel cell are called ancillaries, or parasitic power devices. The target application strongly dictates fuel cell system design. In utility-scale stationary power generation, where reliability and energy efficiency are at a premium, there is a strong incentive to include beneficial system components. In portable fuel cell systems, where mobility and energy density are at a premium, there is a strong incentive to minimize system components. The two example fuel cell systems shown in Figure 10.1 compare these two different design approaches. This chapter covers the major subsystems included in a typical fuel cell system design. These subsystems, some of which are illustrated in Figure 10.1, include the following: • • • •

The fuel cell subsystem The thermal management subsystem The fuel delivery/processing subsystem The power electronics subsystem

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Air in Control

Passive air from ambient

Air supply

Power Conditioned er regulation power out w o Fuel cell P /inversion stack Exhau st System Hot water cooling/ for building Reformer heat recovery

Air

H2

Fuel tank

Exhaust

H2in

Planar fuel cell stack Power

Metal hydride tank

Control valve

Control/power regulation

Conditioned power out Fuel in

(a)

(b)

Figure 10.1. Schematic of two fuel cell systems: (a) stationary residential-scale fuel cell system, (b) portable fuel cell system.

In addition to detailing these subsystems, this chapter also discusses other relevant system design issues such as system pressurization, humidification, and portable fuel cell sizing.

10.1

FUEL CELL SUBSYSTEM

As you have learned, the voltage of a single fuel cell is limited to about 1 V. Furthermore, we recognize that, under load, the output voltage of a single hydrogen fuel cell is typically 0.6–0.7 V. This range generally corresponds to an operational “sweet spot” where the electrical efficiency of the fuel cell is reasonable (around 45%) and the power density of the fuel cell is near its maximum. However, most real-world applications require electricity at several, tens, or even hundreds of volts. How do we get 0.6-V fuel cells to supply the high-voltage requirements of real-world applications? One option is to interconnect multiple fuel cells in series. Connected in series, fuel cell voltages sum. This technique, known as fuel cell “stacking,” permits fuel cell systems to meet any voltage requirement. In addition to building voltage, fuel cell stacks are often designed with these goals in mind: • • • • •

Simple and inexpensive to fabricate Low-loss electrical interconnects between cells Efficient manifolding scheme (for reactant gas distribution) Efficient cooling scheme (especially for high-power stacks) Reliable sealing arrangements between cells

FUEL CELL SUBSYSTEM

Membrane Electrode Flow structure Fuel Oxidant Ion flux

Figure 10.2. Vertical stack interconnection. Fuel cells are serially interconnected via bipolar plates. A bipolar plate simultaneously acts as the anode of one cell and cathode of the neighboring cell. In this diagram, the flow structures, which must be conductive, act as bipolar plates.

Figure 10.2 illustrates the most common form of fuel cell interconnection, referred to as vertical or bipolar plate stacking. In this configuration, a single conductive flow structure or plate is in contact with both the fuel electrode of one cell and the oxidant electrode of the next, connecting the two fuel cells in series. The plate serves as the anode in one cell and the cathode in the next cell, hence the name bipolar plate. Bipolar stacking is similar to how batteries are stacked on top of one another in a flashlight. Bipolar stacks have the advantage of straightforward electrical connection between cells and exhibit extremely low ohmic loss due to the relatively large electrical contact area between cells. The bipolar plate design leads to fuel cell stacks that are robust. Most conventional PEMFC stacks adapt this configuration. Bipolar configurations can be hard to seal. Consider the fuel cell assembly shown in Figure 10.3. It should be apparent from this 3D view that gas will leak out the edges of the porous and gas-permeable electrodes unless edge seals are provided around every cell in the stack. A common way to provide edge seals is to make the electrolyte slightly larger in the planar direction than the porous electrodes and then fit sealing gaskets around both sides. This technique is illustrated in Figure 10.4. Under compression, the edge gaskets create a gas-tight seal around each cell. Planar interconnection designs also have been explored as alternatives to vertical stacking. In planar configurations, cells are connected laterally rather than vertically. While planar designs are less amenable to large-scale power systems because of their increased electrical resistance losses, the format yields form factor advantages for certain portable applications such as laptop computers or cell phones. Planar designs are also used with ceramic fuel cells because it can be easier to fabricate a few smaller cells linked together in a planar design rather than making a single large cell that may be more susceptible to lower manufacturing yields, cracking, and/or other material failure. This approach is sometimes referred to as a “window pane” design, whereby, for example, a few smaller cells are linked laterally, emulating the appearance of a window with a few panes. Figure 10.5 illustrates two

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Hydrogen channels Negative connection

–

+ Positive connection Oxygen channels Figure 10.3. A 3D view of a fuel cell bipolar stack. Unless edge seals are provided around each cell, it is clear that this stack will leak.

Edge-sealing gasket

Anode

Assembly

Electrolyte Cathode

Edge-sealing gasket Figure 10.4. An example of a sealing method that incorporates gaskets around the edges of each cell.

possible planar interconnection configurations. The upper diagram presents the so-called banded electrolyte design, in which the cathode of one cell is electrically connected to the anode of another cell across (or around) the electrolyte. Such construction can yield better volumetric packaging compared to conventional vertical stacks in low-power applications. However, the most critical disadvantage of this configuration is that interconnections must ultimately cross from one side of the electrolyte to the other. These cross-electrolyte interconnections are made at the outer perimeter of a cell array by “edge tabs” or by routing breaches through the central area of the electrolyte. Interconnection at the perimeter

FUEL CELL SUBSYSTEM

Membrane Electrode Banded

Flow structure Fuel Oxidant Ion flux

“Flip-flop” Figure 10.5. Planar series interconnection. Two planar interconnection schemes are shown, the banded and flip-flop designs. In contrast to the banded configuration, the flip-flop scheme has single-level interconnects that never cross the electrolyte plane.

limits design flexibility and may require longer conductor lengths and thereby may increase resistive losses. Breach interconnection through the electrolyte presents an extremely difficult challenge with respect to local sealing, and the problem is particularly severe for polymer electrolytes that may deform grossly as a function of humidity level. To overcome the challenges associated with the banded electrolyte design, the planar flip-flop configuration has been proposed. The lower diagram in Figure 10.5 illustrates such a configuration. The most prominent feature of the flip-flop design is the interconnection of electrodes from two different cells on the same side of the electrolyte. For SOFCs, sealing issues, as well as materials and manufacturing constraints, can make the planar and vertical stacking arrangements shown in Figures 10.2 and 10.5 less desirable. Although these designs have been successfully implemented for SOFCs, a stacking arrangement that minimizes the number of seals may be preferred due to historical challenges with matching the thermal expansion coefficient of the seals with that of the cells. One highly successful method to minimize seals is to employ a tubular geometry, as shown in Figure 10.6. Tubular geometries can be especially useful for high-temperature fuel cells, which encounter large temperature gradients. Over larger temperature gradients, the impact of any difference in thermal expansion coefficients of materials is magnified, mechanical stresses on materials are greater, and the risk of material cracking is higher. As a result, sealing can be more challenging for high-temperature fuel cells. In part to reduce the surface area required for sealing, the SOFC systems from Siemens-Westinghouse Inc. use a tubular design, whereby the sealing surface is just at the tips of the tubes. A photograph of a Siemens-Westinghouse tubular fuel cell stack is shown in Figure 10.7. While the fuel cell stack is the primary component of the fuel cell subsystem, additional equipment often is needed external to the stack to ensure its proper operation. This equipment is still considered part of the fuel cell subsystem. One example of such additional equipment is an external humidifier, which may be needed to help supply PEMFCs with humidified inlet gases. As discussed in Chapter 4, Section 4.5.2, PEM membrane conductivity is a function of water content. To control the level of humidity in the membrane and therefore its conductivity, some PEMFC stack subsystems employ an external humidifier. For example, to control the humidity level of inlet air to the cathode, automotive PEMFC stack subsystems have employed tubular humidifiers and plate-frame membrane humidifiers upstream of the cathode [118].

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Cell interconnect

Array continues Fuel

Anode Electrolyte Air

Cathode

Fuel

Fuel

Air

Air

End view

Air

Side view

Figure 10.6. End and side views of tubular SOFC design employed by Siemens-Westinghouse. Air is fed through the inside of the tubes, while the fuel stream is fed along the outside of the tubes. Series stacking is accomplished by the continuation of more cells in the same plane as the electrode and electrolyte, while parallel stacking can be accomplished by the addition of cells in the plane perpendicular to the electrode and electrolyte.

Nickel felt attachment

Figure 10.7. Photograph and end-on detail of a small (24-cell) stack of Siemens-Westinghouse tubular SOFCs. Each tube is 150 cm long with a diameter of 2.2 cm.

THERMAL MANAGEMENT SUBSYSTEM

10.2

THERMAL MANAGEMENT SUBSYSTEM

As we know, fuel cells are usually only about 30–60% electrically efficient at typical operating power densities. Energy not converted into electrical power is available as heat from the fuel cell stack and its exhaust gases. This heat is sometimes referred to as heat dissipated by electrochemical processes, or electrochemical waste heat. If the rate of heat generation is too high, the fuel cell stack can overheat. If stack cooling is not sufficient, the stack may exceed its recommended operating temperature range, or thermal gradients may arise within the stack. Thermal gradients within the stack can have a negative effect on cell performance by causing cells to operate at different voltages and by enhancing degradation mechanisms. Cooling the stack can help the stack to operate within its optimal temperature range and to avoid thermal gradients. Cooling the stack is also important from the perspective of heat recovery for both internal fuel cell system heating and heating demand sources external to the fuel cell system. For example, heat can be recovered from the stack (and other parts of the fuel cell system) for preheating cold inlet streams and for heating upstream endothermic fuel reforming processes (discussed in the following sections). Internal reuse of heat within the fuel cell system can be one of the most important factors influencing overall fuel cell system efficiency. Fuel cell system heat also can be recovered for heating processes external to the fuel cell system, such as heating buildings and industrial processes, and can thereby displace heat generation and consequent fuel consumption by other devices. Heat recovery for both internal and external heating is discussed in greater detail in Chapter 12. For all of these reasons, the design of a fuel cell system’s thermal management subsystem is crucial. EXOTHERMIC AND ENDOTHERMIC REACTORS Some of the chemical reactors in the fuel cell system produce heat; their reactions are exothermic. Other reactors are endothermic; their reactions require heat to be added. Endothermic reactors are heat sinks and require heat to be conveyed to them from exothermic reactors or other heat sources. Thermal management subsystem design can involve either “passive” or “active” cooling of the fuel cell stack. The choice between these two approaches can strongly depend on fuel cell type, size, and operating strategy. Small, low-temperature fuel cells (such as PEMFCs) frequently can rely on passive cooling, which typically includes 1. cooling via natural convection of air against the external surface area of the fuel cell stack and 2. cooling via the free or forced convection of reactant and/or product gases through the fuel cell stack at air-to-fuel ratios determined by electrochemical limitations (not thermal limitations). Small, high-temperature fuel cell stacks also can cool passively using approaches 1 and 2 as well as radiative cooling. In contrast to this, other types of fuel cell systems are more

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likely to require active cooling. A few examples of systems that are more likely to require active cooling are (A) medium- to larger-size high-temperature fuel cells (such as SOFCs and MCFCs); (B) larger, low-temperature fuel cells (such as PEMFCs and PAFCs); and (C) fuel cell systems that are rapidly ramped up and/or down in electrical power output. Active cooling typically involves 1. the addition of at least one other cooling stream (on top of the reactant and product streams) that absorbs heat via forced convection of a fluid against or through the fuel cell stack and/or 2. running existing reactant and/or product streams at higher flow rates than that needed for electrochemical reaction alone so as to enhance forced convection. One example of an actively cooled stack is a high-power-density, ∼80-kilowatt-electric (kWe) automotive PEMFC stack, which tends to be operated to electrically ramp quickly, and to employ active liquid cooling. As mentioned, low-power portable PEMFC systems ( 100 W) generally require active cooling. Figure 10.8 shows an example of a bipolar plate design that includes additional channels for active air cooling. As another design approach, automotive PEMFC stacks may be designed with electrochemical cells interspersed in a rough ratio of one-to-one with “cooling cells” that use bipolar plate channels to flow liquid coolant (rather than flowing reactant or product gases) [118a]. An actively cooled stack also will need ancillary devices such as fans, blowers, or pumps to circulate the added fluid cooling stream. Unfortunately, this ancillary device will Additional internal channels for cooling

Flow channels for gas routing Figure 10.8. Examples of fuel cell bipolar plates with additional internal channels provided for integrated cooling of fuel cell stack.

THERMAL MANAGEMENT SUBSYSTEM

consume some of the electric power generated by the fuel cell stack, which is referred to as parasitic power. The choice of fan, blower, or pump depends on the required cooling rate, overcoming any pressure drop in the coolant channels, and meeting overall system electrical efficiency, weight, and volume requirements. Generally, fans and blowers are used for circulating gases; pumps are used for circulating liquids. The effectiveness of a particular cooling device can be evaluated by considering the amount of heat removal it accomplishes compared to the electrical power it consumes: Effectiveness =

heat removal rate electrical power consumed by fan, blower, or pump

(10.1)

Effectiveness ratios of 20–40 are generally attainable for well-designed cooling systems. High-power-density PEMFC stacks often employ active liquid cooling (such as with water) instead of active cooling with a gas (such as air). The volumetric heat capacities of liquids are much greater than the volumetric heat capacities of gases. For example, the volumetric heat capacity of water (∼4.2 J∕(cm3 K)) is about 3000 times higher than that for air (∼0.0013 J∕(cm3 K)). As a result, water can carry away a much higher quantity of heat for the same volumetric flow rate, assuming other variables are held constant. Thus, active liquid cooling is frequently used when the volume of the fuel cell stack is constrained (for example, in vehicular applications). In a liquid-based cooling system, the fluid is typically part of a closed loop, i.e., the fluid is continuously circulated and only periodically replenished if some of it escapes or evaporates. If the cooling liquid is water, it must be deionized so that it cannot carry an electric current. Most automotive fuel cell stacks (in the range of 50–90 kWe) are liquid cooled using either deionized water or a water–glycol mixture. By contrast, high-temperature fuel cells, such as MCFCs and SOFCs, tend to operate at much higher temperatures and therefore employ different cooling designs. In fact, the heat dissipated by electrochemical processes is often recovered within the fuel cell system for internal heating of different endothermic processes. The high-temperature heat dissipated by the fuel cell may be used internally within the fuel cell system 1. to provide heat for the reactions at the cells themselves, 2. to preheat inlet gases, and/or 3. to provide heat for upstream endothermic processes. Depending on the application, MCFCs and SOFCs most commonly are actively cooled via (1) the addition of a separate cooling stream and/or (2) running reactant and/or product streams at higher flow rates. Heat released by one part of the fuel cell system often can be recovered for a useful purpose. Heat released by the stack can be recovered for (1) internal fuel cell system heating and (2) external heating. Examples of internal heating include preheating the inlet gases to the fuel cell stack and vaporizing water to humidify inlet gases for the stack. Examples of external heating include using an automotive fuel cell system to provide space heating for the passengers in a vehicle or using a stationary fuel cell system to provide space heating and hot water for a building. Heat recovery for both internal system heating and external

355

356

OVERVIEW OF FUEL CELL SYSTEMS

heating is discussed in detail in Chapter 12. Heat can be recovered not only from the fuel cell stack but also from other system components, as discussed in Chapter 12. Example 10.1 The fuel cell system shown on the left of Figure 10.1 is an MCFC that produces 200 kW of electric power with an electrical efficiency of 52% based on the higher heating value (HHV) of natural gas fuel it consumes. (1) Calculate the quantity of heat released by the fuel cell. Assume that any energy not produced as electric power from the fuel cell stack is released as heat. (2) You would like to use the heat released by the fuel cell to heat a building. Assume that you can recover 70% of the available heat for this purpose, with 30% of the available heat lost to the surroundings. Calculate the amount of heat recovered and the amount lost to the environment. Solution: 1. As discussed in Chapter 2, the real electrical efficiency of the fuel cell stack is described by Pe (10.2) 𝜀R = ̇ ΔH(HHV),fuel where Pe is the electrical power output of the fuel cell stack. We assume any energy that is not produced as electric power from the stack is produced as heat. This assumes that the parasitic power draw from pumps, compressors, and other components is negligible. The amount of heat released by the fuel cell is the maximum quantity of recoverable heat (dḢ MAX ). The maximum heat recovery efficiency (𝜀H,MAX ) is 𝜀H,MAX = 1 − 𝜀R = 1 − 0.52 = 0.48 = 48%

(10.3)

The amount of heat released by the fuel cell is dḢ MAX =

(1 − 𝜀R )Pe (1 − 0.52)200 kW = = 185 kW 𝜀R 0.52

(10.4)

2. The amount of heat recovered is 0.70 × 185 kW = 130 kW and the amount of heat lost to the environment is 0.30 × 185 kW = 55 kW. Example 10.2 Different thermal management subsystem design options are being explored for the fuel cell system described in Example 10.1. Exhaust gases from the fuel cell stack are used to heat upstream fuel reforming processes and to preheat inlet streams. After internally exchanging heat with these processes, the anode exhaust gas is cooled down to 300∘ C. At this juncture, the anode exhaust gas stream carries one-third of the systemwide recoverable heat. One option being considered for cooling this stream is forced air convection using a 2-kWe compressor to blow air against the anode exhaust gas stream in a gas-to-gas heat exchanger. (1) Please calculate the amount of recoverable heat in the anode exhaust stream. (2) Assuming 100% efficient heat transfer between the hot anode exhaust gas and the cold, coolant air stream in

FUEL DELIVERY/PROCESSING SUBSYSTEM

the heat exchanger, please calculate the effectiveness of the compressor. (3) Please comment on the appropriateness of actively cooling this anode exhaust stream with liquid water compared with air and with liquid vs. gaseous forced convection. Solution: 1. The amount of recoverable heat in the anode off-gas stream is 1∕3 × 130 kW = ∼43 kW. 2. The effectiveness of the compressor is 43 kW∕2 kW = ∼22. 3. Because a portion of the anode exhaust gas heat is available at temperatures as high as 300∘ C, cooling this stream with liquid water could be challenging because water changes phase at standard pressure at 100∘ C. A liquid water coolant stream available at an inlet 25∘ C ambient temperature, with a reasonable volumetric flow rate, may undergo a phase change to steam and significantly expand in volume. Other liquids with higher boiling points may be more appropriate as cooling fluids. The volumetric heat capacity of liquids is generally much higher than that for gases (about ∼3000 times higher for water as for air), and therefore more heat can be extracted in the same volume with liquids. Heat exchanger volume is therefore expected to be less for a gas-to-liquid heat exchanger compared with a gas-to-gas heat exchanger. At the same time, attention also must be paid to liquid vaporization temperatures, especially in high-temperature applications. 10.3

FUEL DELIVERY/PROCESSING SUBSYSTEM

Providing fuel for a fuel cell is often the most difficult task that a system designer faces. Almost all practical fuel cells today use hydrogen or compounds containing hydrogen as fuel. As a result, there are effectively two main options for fueling a fuel cell: 1. Use hydrogen directly. 2. Use a hydrogen carrier. A hydrogen carrier is a convenient chemical species that is used to convey hydrogen to a fuel cell. For example, methane, CH4 , is a convenient hydrogen carrier because it is far more readily available than hydrogen. If hydrogen is used directly, it must be created first, via one of several processes that we will learn more about in Chapter 11, and stored before use. For stationary fuel cell systems, availability is one of the most important criteria affecting the choice of fuel. By contrast, for portable fuel cells, the storage effectiveness of the fuel is critical. Storage effectiveness can be measured using (1) gravimetric energy density and (2) volumetric energy density: stored enthalpy of fuel total system mass stored enthalpy of fuel Volumetric energy density = total system volume

Gravimetric energy density =

(10.5) (10.6)

357

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OVERVIEW OF FUEL CELL SYSTEMS

These metrics express the energy content stored by a fuel system relative to the fuel system size. These metrics can be used regardless of whether a direct H2 storage system or a H2 carrier system is employed. Some of the major options for fueling are now discussed in more detail. 10.3.1

H2 Storage

In a H2 storage system, the fuel cell is supplied directly with H2 gas. There are several major advantages to direct hydrogen supply: • • • •

Most fuel cell types run best on pure H2 . Impurity/contaminant concerns are greatly reduced. The fuel cell system is simplified. Hydrogen has a long storage “shelf life” (except for liquid H2 ).

Unfortunately, H2 is not a widely available fuel. Furthermore, H2 storage systems are still not as energy dense as petroleum fuel storage. The three most common ways to store hydrogen are: 1. As a compressed gas 2. As a liquid 3. In a metal hydride Each of these storage options is briefly discussed below. Table 10.1 summarizes typical characteristics of each of the three direct H2 storage methods as well as a hybrid cryo-compressed gas storage option [119–121, 121a] developed for vehicles by Lawrence Livermore National Laboratories (LLNL). The LLNL approach provides improved energy densities compared to standard gas compression, but with less stringent cooling requirements compared to standard cryogenic liquid hydrogen storage. TABLE 10.1. Comparison of Various Direct H2 Storage Systems

Storage System

Mass Storage Volumetric Gravimetric Volumetric Efficiency Storage Density Storage Energy Storage Energy (% kg H2 /kg (kg H2 /L Density Density storage) storage) (kWh/kg) (kWh/L)

Compressed H2 , 300 bars

3.1

0.014

1.2

0.55

Compressed H2 , 700 bars

4.8

0.033

1.9

1.30

Cryogenic liquid H2

14.2

0.043

5.57

1.68

Cryo-compression tank (LLNL)

7.38

0.045

2.46

1.51

Metal hydride (conservative)

0.65

0.028

0.26

1.12

Metal hydride (optimistic)

2.0

0.085

0.80

3.40

Note: The mass and volume of the entire storage system (tank, valves, tubing, and regulators) are taken into account in these data.

FUEL DELIVERY/PROCESSING SUBSYSTEM

HYDROGEN STORAGE EFFICIENCY The effectiveness of a direct hydrogen storage system can also be measured by (1) hydrogen mass storage efficiency and (2) hydrogen volume storage density. These two parameters describe the amount of hydrogen that can be stored in a direct storage system relative to the storage system size: mass of H2 stored × 100% total system mass mass of H2 stored Volume storage density = total system volume

Mass storage efficiency =

(10.7) (10.8)

Examples of these values are shown for different H2 storage technologies in Table 10.1. • Compressed H2 . This is the most straightforward way to store hydrogen. The H2 is compressed to very high pressures inside specially designed gas cylinders. Storage efficiencies are rather modest but generally improve with cylinder size and increased pressurization. Current cylinder technology permits storage pressures as high as 700 bars. However, high-pressure storage can introduce significant safety problems. Additionally, the act of pressurizing the H2 is energy intensive. Approximately 10% of the energy content of H2 gas must be expended to pressurize it to 300 bars. Fortunately, as the storage pressure increases still further, the losses do not increase at the same rate. The additional energy expended to further compress the H2 is balanced by the fact that more H2 is stored. • Liquid H2 . If hydrogen gas is cooled to 22 K, it will condense into a liquid. Liquefaction permits H2 storage at low pressure. Liquid hydrogen has the highest mass storage density of the direct H2 storage options, about 0.071 g∕cm3 . The storage container must be a thick, double-walled reinforced vacuum insulator to maintain the cryogenic temperatures. Therefore, volumetric storage efficiencies are modest, although mass storage efficiencies can be impressive. (For this reason, liquid H2 is frequently used as a fuel for rocket propulsion in space flight, where gravimetric energy density is especially important.) Perhaps most problematically, H2 liquefaction is extremely energy intensive; the energy required to liquefy H2 is approximately 30% of the energy content of the H2 fuel itself. • Metal Hydride. Common metal hydride materials include iron, titanium, manganese, nickel, and chromium alloys. Ground into extremely fine powders and placed into a container, these metal alloys work like “sponges” and can absorb large quantities of H2 gas usually by dissociating the H2 molecules into H atoms, which are then absorbed within the alloy. Upon heating, the hydrides will release their stored H2 gas. Metal hydrides can absorb incredibly large quantities of H2 . In fact, H gas atoms can be packed inside some metal hydrides in a manner that achieves a higher volumetric energy density than liquid hydrogen! Unfortunately, the hydride materials

359

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OVERVIEW OF FUEL CELL SYSTEMS

themselves are quite heavy, so gravimetric energy density is modest. Furthermore, the materials are expensive. Metal hydride storage may be most attractive for certain portable applications.

10.3.2

Using a H2 Carrier

Using an H2 carrier instead of hydrogen gas can permit significantly higher gravimetric and volumetric energy storage densities. These H2 carriers are especially attractive for portable and mobile applications. H2 carriers may include methane (CH4 ), methanol (CH3 OH), sodium borohydride (NaBH4 ), formic acid (HCO2 H), and gasoline (Cn H1.87n ). Hydrogen carriers are also attractive for stationary applications. Because H2 gas does not occur naturally on its own, it must be derived from another hydrogen-containing compound. Unlike natural gas or oil, we cannot “drill” for hydrogen. Thus, most stationary fuel cells operate on more widely available fuels like natural gas (which is primarily composed of methane) or biogas. Using these carrier fuels, fuel cells can still offer high electrical efficiency, modularity, and low emissions compared to existing power plant options. Unfortunately, most H2 carriers are not directly usable in a fuel cell, i.e., the H2 carrier species does not directly react at the fuel cell’s anode via electrochemical oxidation. Instead, most H2 carriers must be chemically processed to produce H2 gas, which is then fed to the fuel cell. A few H2 carriers are directly usable. One example is methanol, which is used in direct methanol fuel cells (DMFCs). (Chapter 8, Section 8.7.1, introduces the reader to the operation of DMFCs.) To compare the “effectiveness” of H2 carriers in providing fuel for a fuel cell, it is important to consider how much of the energy stored in the original carrier is actually usable by the fuel cell. For example, the energy density of methanol is considerably greater than that of compressed hydrogen, but a fuel cell may only be able to convert 20% of methanol’s energy into electricity, whereas it could convert 50% of compressed hydrogen’s energy into electricity. In this case, the effectiveness of the methanol fuel compared to hydrogen is only 0.40. An H2 carrier system’s effectiveness is defined as the percentage of a carrier’s energy that can be converted into electricity in a fuel cell compared to the percentage of the energy in hydrogen gas that can be converted into electricity: Carrier system effectiveness =

% conversion of carrier to electricity % conversion of H2 to electricity

(10.9)

Adjustment by this effectiveness value permits a fair comparison between the storage energy density of a direct H2 system and an H2 carrier system for portable fuel cells. Returning to our methanol example, methanol reforming requires a 50% molar mixture of methanol and water, according to the reaction CH3 OH + H2 O → CO2 + 3H2

(10.10)

If a hypothetical methanol fueling system consists of a 1-L 50% methanol–50% water (by moles) fuel reservoir plus an additional 1-L reformer, the net volumetric energy density for the fueling system would be 1.72 kWh∕L (3.4 kWh for 1L of a 50–50 methanol–water mixture as shown in Table 10.2 divided by 2 L for the volume of fuel reservoir plus

FUEL DELIVERY/PROCESSING SUBSYSTEM

TABLE 10.2. Comparison of Various Carrier H2 Storage Systems

Storage System

Gravimetric Storage Energy Density (kWh/kg)

Volumetric Storage Energy Density (kWh/L)

Carrier Effectiveness

Direct methanol (50% molar mix with H2 O)

4

3.4

0.40

Reformed methanol (50% molar mix with H2 O)

2

1.7

0.70

Reformed NaBH4 (30% molar mix with H2 O)

1.5

1.5

0.90

Note: The mass and volume of the entire storage system (tank, valves, reformer, etc.) are taken into account in these data.

reformer = 1.72 kWh/L). If we assume that the effectiveness ratio for utilizing the energy content carried in this fuel–water mixture is 0.7, then this methanol fuel system would be equivalent to a direct hydrogen system that has a volume storage energy density of 1.2 kWh∕L. On a gravimetric basis, this methanol fuel system might be equivalent to a direct hydrogen system with a gravimetric energy density of 1.4 kWh∕kg. The storage metrics and effectiveness of several carrier fuel storage systems are detailed in Table 10.2. As alluded to earlier in this section, there are two major ways to utilize hydrogen carriers. They can be electro-oxidized directly in a fuel cell to generate electricity (but only if they are relatively simple, easily reacted species) or they can be reformed (chemically processed) into hydrogen gas, which is then used by the fuel cell to produce electricity. Reforming can be further subdivided according to whether (1) it occurs in a chemical reactor outside the fuel cell (external reforming) or (2) it occurs at the catalyst’s surface inside the fuel cell itself (internal reforming). These three options are now briefly discussed. • Direct Electro-Oxidation. Direct electro-oxidation is attractive primarily because it is simple. No additional external chemical reactors or other components are required compared to a normal H2 –O2 fuel cell, although different catalysts, electrolytes, and electrode materials may need to be used. Examples of fuels that can be directly electro-oxidized in a fuel cell include methanol, ethanol, and formic acid. Chapter 8, Section 8.7.1, introduces the reader to the operation of these types of fuel cells. In direct electro-oxidation, electrons are directly stripped from a fuel molecule. The extra steps required to first reform the fuel into hydrogen are thus avoided. As an example, the reaction chemistry of the direct methanol fuel cell was presented in Chapter 8, Section 8.7.1. Unfortunately, fuel cells operating directly on non-hydrogen fuels suffer significant power density and electrical energy efficiency reductions due to kinetic complications. Because of these complications, a fuel cell operating directly on a non-hydrogen fuel needs to be much larger than a fuel cell operating on hydrogen to provide the same power. In some designs, the size is larger by a factor of 10, in which case the energy density gains produced by switching to a carrier fuel are offset. A careful examination of the balance between fuel reservoir size, fuel cell size, and fuel efficiency is required to determine whether direct electro-oxidation of a carrier fuel makes sense.

361

362

OVERVIEW OF FUEL CELL SYSTEMS

• External Reforming. Fuel processors use heat, often in combination with catalysts and steam, to break down H2 carrier fuels to H2 . During a fuel reforming process, additional species such as CO and CO2 may also be produced. At best, these side-products dilute the H2 gas fed to the fuel cell, slightly lowering performance. At worst, they can act as poisons to the fuel cell, severely reducing performance. In such cases, additional processing steps are required to increase the H2 content of the gas and remove the poisons before the reformate (i.e., the reformed H2 gas mixture) is fed to the fuel cell. Some of these chemical processes release heat (exothermic), while others require heat to be supplied (endothermic). For high-temperature fuel cells, the required heat may be supplied by the fuel cell stack itself. For low-temperature fuel cells, some of the incoming fuel may be burned to provide high enough temperature heat. The size and complexity of an external fuel processor depend on the type of fuel reformed, whether impurities or poisons need to be removed, and how much reformate needs to be produced. Figure 10.9 shows a few examples of external fuel processors. Chapter 11 discusses in detail the design of fuel processor subsystems. • Internal Reforming. In internal reforming, the reforming process occurs inside the fuel cell stack itself, at the surface of the anode’s catalysts. Internal reforming typically is implemented in high-temperature fuel cells using certain fuels. In these cells, the high-operating-temperature catalysts work not only to facilitate electro-oxidation at the anode but also to facilitate fuel reforming reactions. In a typical internal reforming scheme, the H2 carrier gas is mixed with steam before being fed to the fuel cell anode. The gas and steam react over the anode catalyst surface to produce H2 , CO, and CO2 . The CO typically reacts with more steam via the water gas shift reaction to produce further H2 . The water gas shift reaction is discussed in greater detail in Chapter 11 and is shown in Equation 11.4. Compared to external reforming, internal reforming presents several potential advantages. These advantages may include reduced system complexity (the need for an external chemical reactor is eliminated), reduced system capital cost because the external reactor is not needed, and direct heat transfer between endothermic reforming reactions and exothermic electrochemical reactions. In some designs, internal reforming may also lead to higher system efficiency and higher conversion efficiency. Direct electro-oxidation is very suitable for portable applications, where simple systems, minimal ancillaries, low power, and a long run time are needed. Fuel reforming is most frequently applied in stationary applications, where fuel flexibility is important and the excess heat also can be used either by the system or by sources of heat demand outside the system. Currently, on-board fuel reforming technology appears less attractive for automotive applications. In 2004, the U.S. Department of Energy decided to discontinue on-board fuel processor R&D for fuel cell vehicles.

10.3.3

Fuel Delivery/Processing Subsystem Summary

Fuel cell type and application ultimately determine the best fuel delivery subsystem for a given situation. For stationary applications such as distributed generation, fuel processing

FUEL DELIVERY/PROCESSING SUBSYSTEM

(a)

(b) Figure 10.9. Two examples of external reformers. (a) A Honda Home Energy Station that generates hydrogen from natural gas for use in fuel cell vehicles, while supplying electricity and hot water to the home through fuel cell cogeneration functions. This unit, located in New York, is a second-generation model (developed in collaboration with Plug Power Inc.), which unifies a natural gas reformer and pressurizing units into one compact component to reduce the volume. The unit can produce up to 2 standard cubic meters of hydrogen per hour. (b) A Pacific Northwest National Laboratory microfuel processor that converts methanol into hydrogen and carbon dioxide. The system includes a catalytic combustor, a steam reformer, two vaporizers, and a recuperative heat exchanger embedded in a device no larger than a dime! When first built, it was the smallest integrated catalytic fuel processor in the world.

363

364

OVERVIEW OF FUEL CELL SYSTEMS

subsystems may operate on locally available fuels such as natural gas, which is composed primarily of methane, or biogas. For transportation systems, compressed gas H2 storage is currently a leading candidate. For small portable fuel cells, metal hydride storage, which exhibits relatively high volumetric storage energy densities, and direct electro-oxidation of fuels (especially direct methanol) are leading candidates. While direct H2 fuel delivery subsystems are relatively simple, carrier-gas-based fuel processing subsystems can be quite complex. Because of their complexity, fuel processing subsystems will be discussed in greater detail in Chapter 11. Table 10.3 summarizes the relative storage energy densities, advantages, disadvantages, and applications of the major fuel delivery/processing subsystems. Note that these tendencies were extrapolated from real-world subsystems. Storage densities vary considerably, depending on the details of the system design, size, and intended application.

10.4

POWER ELECTRONICS SUBSYSTEM

The power electronics subsystem consists of (1) power regulation, (2) power inversion, (3) monitoring and control, and (4) power supply management. These four tasks of the power electronics subsystem will be discussed in detail in the following four sections. Fuel cell power conditioning generally involves two tasks: (1) power regulation and (2) power inversion. Regulation means providing power at an exact voltage and maintaining that voltage constant over time, even as the current load changes. Inversion means converting the DC power provided by a fuel cell to AC power, which most electronic devices consume. For almost all fuel cell applications, power regulation is essential. For most stationary and automotive fuel cell systems, inversion is also essential. Stationary systems supply electricity to the surrounding AC electric grid and/or to building AC power grids. Automotive systems often need to invert DC power to AC power for an AC electric motor, which tends to be more efficient, lower in capital cost, and more widely available than a DC motor. Inversion is unnecessary for some portable fuel cell applications: For example, a fuel cell laptop uses DC power directly. Unfortunately, power conditioning comes at a price, in terms of both economics and efficiency. Power conditioning will typically add about 10–15% to the capital cost of a fuel cell system. Also, power conditioning reduces the electrical efficiency of a fuel cell system by about 5–20%. Careful selection of the optimal power conditioning solution for a given application is essential. Power regulation and power inversion are discussed next.

10.4.1

Power Regulation

Most applications require electric power that is delivered at a specific voltage level and that is stable over time. Unfortunately, the electric power provided by a fuel cell is not perfectly stable; a fuel cell’s voltage is highly dependent on temperature, pressure, humidity, and flow rate of reactant gases. Cell voltage changes dramatically, depending on the current load. For example, looking at the polarization curve of a single cell, as shown in Figure 1.10, you can see that voltage can experience roughly a 2-to-1 decline with current draw. Also, even

365

Moderate Low

Biogas

Low

Neat hydrogen

Methane

Low

High

Direct methanol

Reformed gasoline

Low

Metal hydride

Moderate–high

Moderate–high

Cryogenic H2

Reformed methanol

Moderate

Compressed H2

Fuel System

Gravimetric Storage Energy Density Fuel Availability

High

Moderate

Moderate

Low

Low

Low

Low

Moderate

Low–moderate

High

High

Low

Moderate

Low

Low

High

Low

Moderate

Moderate

High

Fuels for Stationary Generation Applications

Low

Moderate–high

High

High

Moderate

Moderate

High

Fuel Suitability for Fuel Cell

Fuel Systems for Mobile Applications

Volumetric Storage Energy Density

Best for high-temperature fuel cells

Best for high-temperature fuel cells

Must have H2 source!

Expensive, hard to reform

For transportation applications

For portable applications

Expensive, heavy

Liquefaction is energy intensive

For transportation

Comments

TABLE 10.3. Qualitative Summary of Various Fuel/Fuel System Choices for Mobile and Stationary Fuel Cell Applications

366

OVERVIEW OF FUEL CELL SYSTEMS

if multiple fuel cells are carefully stacked together in series, the voltage of the system will often not be exactly what is desired for a given application. For these reasons, fuel cell power is generally regulated using DC–DC converters. A DC–DC converter takes a fluctuating DC fuel cell voltage as input and converts it to fixed, stable, specified DC voltage output. There are two major types of DC–DC converters: step-up converters and step-down converters. In a step-up converter, the input voltage from a fuel cell is stepped up to a higher fixed output voltage. In a step-down converter, the input voltage from a fuel cell is stepped down to a lower fixed output voltage. In either case, regardless of the value of the input voltage (and even if it changes in time), it will be stepped to the converter’s specified output voltage, within certain limits. While a step-down converter sounds reasonable, a step-up converter seems impossible. Are we getting something for nothing? The answer is no! In either case, total power must be conserved, minus some losses. For example, a typical step-up converter might step a fuel cell stack’s input from 10 V and 20 A to an output of 20 V and 9 A. Although the voltage has been boosted by a factor of 2, the current has been cut by slightly more than one-half. You can calculate the efficiency of this converter by comparing the output power to the input power: Efficiency =

output power 20 V × 9 A = = 0.90 input power 10 V × 20 A

(10.11)

This step-up converter is 90% efficient. DC–DC converters are generally 85–98% efficient. Step-down converters are typically more efficient than step-up converters, and converter efficiency improves as the input voltage increases. For this reason, fuel cell stacking is important. While theoretically possible, it would be extremely inefficient to take a single fuel cell at 0.5 V and step it up to 120 V. Figure 10.10 illustrates several examples of the voltage and current relationships for step-up and step-down converters. In a fuel cell, a step-up converter can be used to maintain a constant voltage, regardless of the load. This idea is shown schematically in Figure 10.11. Keep in mind that, as we just discussed, stepping up the voltage lowers the current output commensurately. Thus, as shown by the arrows, point X on the fuel cell j–V curve corresponds to point X ′ on the step-up converter curve, while point Y on the fuel cell j–V curve corresponds to point Y ′ on the step-up converter curve.

10.4.2

Power Inversion

In most stationary applications, such as utility or residential power, the fuel cell will be connected to the surrounding electricity grid or must meet the needs of common household appliances. In these cases, AC rather than DC power is required. Depending on the exact application, either one-phase or three-phase AC power will be required. Utilities and large industrial customers require three-phase power, whereas most residences and businesses need only single-phase AC power. Fortunately, both single-phase and three-phase power inversion technologies are well developed and highly efficient. Similar to DC–DC converters, DC–AC inverters are typically 85–97% efficient. Figure 10.12 introduces a typical single-phase inverter solution, known as pulse-width modulation. In pulse-width modulation, a series of switches trigger periodic DC voltage

POWER ELECTRONICS SUBSYSTEM

8A

8W 7W

3.5 A 2V 1V Current

Voltage

Power

Current

Before conversion

Voltage

Power

After conversion (a)

6W

5.5 W

3V

2.75 A

2A

Current

2V

Voltage

Power

Current

Before conversion

Voltage

Power

After conversion (b)

Figure 10.10. Example current–voltage–power relationships for (a) a step-up converter and (b) a step-down converter. 1.2

Step-up converter output = 1V X'

1 Cell voltage (V)

Y'

0.8

X

0.6

Y

0.4 0.2 0 0

0.5

1

1.5

2

Current density (A/cm2)

Figure 10.11. A DC–DC converter may be used to transform a fuel cell’s variable j–V curve behavior into a constant-voltage output. Up conversion to the higher fixed-voltage output of the converter is accompanied by a commensurate reduction in current, as shown by points X vs. X ′ and Y vs. Y ′ .

367

368

OVERVIEW OF FUEL CELL SYSTEMS

Voltage Time

Current Time Figure 10.12. Pulse-width voltage modulation allows DC to be transformed into an approximately sinusoidal current waveform.

pulses through a regulator circuit. By varying the width of these pulses (starting with a few short pulses and then increasing the pulse widths before decreasing them again), a reasonable approximation to a sine wave can be created in the resulting current response. 10.4.3

Monitoring and Control System

A large fuel cell system is essentially a complex electrochemical processing plant. During operation, many variables such as stack temperature, gas flow rates, power output, cooling, and reforming need monitoring and control. A fuel cell control system generally consists of three separate aspects: a system-monitoring aspect (gauges, sensors, etc., that monitor the conditions of the fuel cell), a system actuation aspect (valves, pumps, switches, etc., that can be regulated to impose changes on the system), and a central control unit, which mediates the interaction between the monitoring sensors and the control actuators. The objective of the central control unit is usually to keep the fuel cell operating at a stable, specified condition. The central control unit can be regarded as the “brains” of the fuel cell system. Most control systems use feedback algorithms to maintain the fuel cell at a stable operating point. For example, a feedback loop might be implemented between a fuel cell stack temperature sensor and the thermal management subsystem. In such a feedback loop, if the control unit senses that the temperature of the fuel cell stack is increasing, it might increase the flow rate of cooling air through the stack. On the other hand, if the fuel cell stack temperature decreases, the control system might reduce the cooling airflow rate. A schematic diagram of a simple fuel cell system with a control system is shown in Figure 10.13. 10.4.4

Power Supply Management

Power supply management is the part of the power electronics subsystem used to match the fuel cell system’s electrical output with that demanded by the load. Fuel cells can have

CASE STUDY OF FUEL CELL SYSTEM DESIGN: STATIONARY COMBINED HEAT AND POWER SYSTEMS

Out Heat recovery In Exhaust Fuel in

Reformer Temp.

Fuel cell stack Temp.

Conditioner/inverter

AC power

Load

DC power V

Air in Control system

Figure 10.13. Schematic diagram of a simple fuel cell system with a control system.

a slower dynamic response than other electronic devices, such as batteries and capacitors, because of lag times in system components such as pumps, compressors, and fuel reformers or limitations in thermal and mechanical stresses on the fuel cell stack, especially at high temperatures. Fuel cell systems can operate with or without energy buffers such as batteries or capacitors. Without any energy buffers, the response of fuel cell systems may be anywhere between seconds to hours. With energy buffers, the system’s response time can be reduced to milliseconds. Power supply management also incorporates a strategy for serving a changing electric load. A midsized car consumes 25 kW of electrical power on average but up to 120 kW at its peak. A fuel cell system’s power supply must be designed and controlled to supply power even under large fluctuations in load. In distributed generation applications, power supply management also may incorporate a strategy for the fuel cell system to interact with the local grid and to respond to changes in electrical demand from the buildings it serves. 10.5 CASE STUDY OF FUEL CELL SYSTEM DESIGN: STATIONARY COMBINED HEAT AND POWER SYSTEMS Taking what we have just learned about the four major fuel cell subsystems, we will now review this knowledge in the context of a stationary combined heat and power (CHP) fuel cell system design. Many stationary fuel cell systems are designed to convert the chemical energy in a fuel into both electrical power and useful heat—a scenario known as combined heat and power. Figure 10.14 shows a diagram of a stationary CHP fuel cell system, showing the primary chemical reactors, mass flows, and heat flows (a process diagram) associated with this system. This particular fuel cell system uses a hydrogen fuel cell stack and consumes natural gas fuel. This fuel cell system provides both electricity and heat for a building.

369

k

Stream splitter Natural gas stream Anode exhaust Cathode exhaust Heat stream Air stream Electricity line Water line

Electricity storage

DC/AC inverter AC electric grid

6

Water heating system

Space heating system

Boost regulator

DC electricity

Cathode exhaust

System exhaust N 2 CO2 H 2O condenser

k

Liquid H 2 O

H 2O Catalytic after-burner

5

N2 O2

Anode exhaust

H2 N 2 CO2 H 2O

Natural gas compressor

Fuel cell anode

1

Fuel cell cathode

2

H2 N 2

Water pump

Steam generator

Preheater

Catalytic fuel reformer

Water gas H N CO 2 2 4 shift CO2 H 2O reactor

H2 O CO clean-up CO 3

2

Air

compressor

1

2

3

4

5

6 Reference Figure 12.6 and Table 12.1

Figure 10.14. Process diagram of CHP fuel cell system.

370

k

k

CASE STUDY OF FUEL CELL SYSTEM DESIGN: STATIONARY COMBINED HEAT AND POWER SYSTEMS

The fuel cell system illustrated in Figure 10.14 contains all four primary subsystems previously introduced in this chapter: (1) the fuel processing subsystem, (2) the fuel cell subsystem, (3) the power electronics subsystem, and (4) the thermal management subsystem. The fuel processing subsystem consists of the streams of flowing gases (illustrated by arrows) and the series of chemical reactors (illustrated by cylinders). The fuel cell subsystem is shown by the fuel cell stack, the pump and compressors, and the stack’s coolant loop. The power electronics subsystem incorporates the thin, dark-shaded electricity lines and connecting boxes in the upper-right corner. The thermal management subsystem is represented by dashed heat stream lines with arrows and physically includes a network of heat exchangers, flowing fluids, and pumps. COMBINED HEAT AND POWER Combined heat and power, or cogeneration, is the simultaneous production of electricity and heat from the same energy source. A CHP power plant produces both electric power and heat. This heat can be recovered for a useful purpose, such as warming a building space or water, or for an industrial process. For CHP plants, it is useful to define the term overall efficiency (𝜀O ). The overall efficiency is the sum of the electrical efficiency (𝜀R ) of the power plant and its heat recovery efficiency (𝜀H ): 𝜀O = 𝜀R + 𝜀H < 100%

(10.12)

where 𝜀O cannot exceed 100%. Combined heat and power fuel cell systems have achieved 𝜀R = 50% and 𝜀H = 20% for 𝜀O = 70% [122]. Another important term for CHP power plants is the heat-to-power ratio (H∕P). The H∕P is the ratio of retrievable ̇ to net system electrical power (Pe,SYS ): heat (dH) H dḢ = P Pe,SYS

(10.13)

For the CHP fuel cell system above, H∕P = 𝜀H ∕𝜀R = 0.20∕0.50 = 0.40. The H∕P varies for different types of power plant designs, usually between 0.25 and 2. As another example, your educational institution or company may use a CHP natural gas power plant to provide electricity and heat to your campus. For such a plant, typical values are 𝜀R = 40% and 𝜀H = 20%, 𝜀O = 60%, and H∕P = 0.50.

NATURAL GAS FUEL Natural gas is one of the most common fuels for heating buildings and for fueling power plants. Natural gas is primarily composed of methane (CH4 ). A sample composition of dry, desulfurized natural gas fuel is shown in Table 10.4. The gas constituent is listed on the left and the molar percent composition is listed on the right. Actual natural gas composition varies by region according to the source of the gas (the gas field from which it is

371

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OVERVIEW OF FUEL CELL SYSTEMS

extracted) and regulations regarding its purity. Actual natural gas will also at a minimum contain trace quantities of sulfur compounds. Sulfur compounds occur naturally in gas fields but are also added by gas supplier companies as an odorant. TABLE 10.4. Sample Composition of Dry, Desulfurized Natural Gas Fuel Constituent Methane (CH4 ) Ethane (C2 H6 ) Carbon dioxide (CO2 ) Nitrogen (N2 ) Propane (C3 H8 ) Butane (C4 H10 ) Pentane (C5 H12 ) Carbon monoxide (CO) Oxygen (O2 ) Hydrogen (H2 ) Water (H2 O)

Percent Molar Composition (%) 96.74% 1.64% 0.91% 0.45% 0.19% 0.05% 0.02% 0.00% 0.00% 0.00% 0.00%

Note: Natural gas is typically greater than 90% methane (CH4 ), but compositions vary by region. It typically contains a small percentage of more complex hydrocarbons (HC), including ethane (C2 H6 ), propane (C3 H8 ), butane (C4 H10 ), and pentane (C5 H12 ). Actual natural gas will also contain trace sulfur compounds.

The four subsystems shown in Figure 10.14 perform several functions: 1. The fuel processing subsystem chemically converts a hydrocarbon (HC) fuel such as natural gas into a hydrogen- (H2 -) rich gas. This subsystem also purifies the gas to remove or reduce any poisons such as carbon monoxide (CO) or sulfur compounds. For example, in Figure 10.14, the reactor labeled 3, “CO clean-up,” purifies the stream of CO. This purified gas can then be tolerated by sensitive catalysts (such as platinum) at the fuel cell’s electrode and within the fuel processor’s downstream chemical reactors. Finally, this subsystem takes any excess fuel and oxidant not consumed by the fuel cell and recycles them within the system. Figure 10.14 shows the anode and cathode off-gas being combusted in a catalytic afterburner to recover heat internally within the fuel cell system. 2. The fuel cell subsystem consists primarily of a fuel cell stack (labeled 1 in Figure 10.14) that converts a H2 -rich gas and oxidant into DC electricity and heat, along with pumps and compressors that convey reactants and products, and the heating and cooling loops required for the stack and these streams. 3. The power electronics subsystem, shown in Figure 10.14 by the electricity lines, converts the fuel cell’s DC electrical power to AC power used in the building. The power electronics subsystem also balances the building’s electrical demand with the electricity supplied by the fuel cell system by using an energy storage device, such as a battery or capacitor, or by relying on the surrounding AC electrical grid.

CASE STUDY OF FUEL CELL SYSTEM DESIGN: STATIONARY COMBINED HEAT AND POWER SYSTEMS

4. The thermal management subsystem, shown in Figure 10.14 by the dashed heat streams, captures heat released by the fuel cell stack and by the fuel processing subsystem. This heat is used either to warm other system components (such as a steam generator) or to heat the building. Excess heat is rejected to the environment. The following sections briefly discuss this stationary fuel cell system’s four primary subsystems to give a better understanding of their design.

10.5.1

Fuel Processor Subsystem

The details of the fuel processor subsystem are shown in Figure 10.15. The main purpose of this fuel processor is to convert a HC fuel (such as CH4 ) into a H2 -rich gas. The system consists of a series of catalytic chemical reactors, heat management devices, reactant and product delivery streams, and extraction equipment. First, liquid water is heated and converted to steam in a steam generator (labeled 1). Steam could be needed for several downstream processes, including humidifying the fuel cell’s inlet gases and providing a reactant for the fuel processor. Second, compressed natural gas fuel is combined with compressed air and/or steam and warmed in a preheater (labeled 2). Third, the fuel mixture enters a fuel reformer (labeled 3), where it reacts at high temperature (>600∘ C), often in the presence of a catalyst, producing a H2 -rich stream (referred to as the reformate stream). Fourth, the reformate stream enters a water gas shift reactor (labeled 4), which increases the quantity of H2 in the stream and decreases the CO content. Fifth, in the CO clean-up reactor (labeled 5), the reformate is stripped of CO via either chemical reaction or physical separation, so that the CO will not poison the fuel cell. Sixth, in the afterburner section (labeled 6), exhaust exiting from the fuel cell anode and cathode is combusted catalytically to recover heat for other fuel processing stages and/or to provide heat to a source of thermal demand inside or outside the fuel cell system. Depending on the H2 utilization of the fuel cell, a large quantity of H2 may be available at the fuel cell’s exhaust outlet, between 5 and 45% of incoming fuel energy. Also, combustion of H2 in the afterburner produces water in the form of steam, which can be reused in other parts of the system. Finally, as shown in Figure 10.15, after the catalytic afterburner, a condenser converts steam back to liquid water by cooling this stream. The condenser can be used to capture the latent heat of condensation. In a fuel cell system, a condenser is important both for recapturing heat and for recovering liquid water to achieve neutral system water balance. Regardless of the source of fuel, almost all fuel processor subsystem designs will incorporate (1) an afterburner, (2) a steam generator, and (3) a condenser to achieve a higher overall, systemwide efficiency. Within the fuel processor industry, the fuel reformer’s efficiency (𝜀FR ) is often described in terms of the higher heating value (HHV) of H2 in the reformate exiting the fuel reformer (ΔH(HHV),H2 ) compared with the HHV of fuel entering the fuel reformer (ΔH(HHV),fuel ), including any fuel that must be combusted to provide energy for the reformer itself: 𝜀FR =

ΔH(HHV),H2 ΔH(HHV),fuel

(10.14)

373

k

Cathode exhaust

System exhaust N 2 CO2 H 2O condenser

H 2O Catalytic afterburner 6

Liquid H 2 O

N2 O2

Anode exhaust

H2 N 2 CO2 H 2O

Fuel cell Fuel cell anode cathode

Natural gas compressor

k

Water pump

Steam generator 1

Preheater

CO2 H2 N 2

Catalytic fuel reformer 3

2

Water gas H 2 N 2 CO shift reactor CO2 H 2O 4

CO clean-up

Air

compressor Cathode exhaust Air stream Water line

Figure 10.15. Fuel processing subsystem.

374

k

Stream splitter Natural gas stream Anode exhaust

5

H2 O

k

CASE STUDY OF FUEL CELL SYSTEM DESIGN: STATIONARY COMBINED HEAT AND POWER SYSTEMS

(For a discussion of HHV, see Chapter 2.) A control volume analysis of the fuel reformer encapsulates chemical reactor 3 in Figure 10.15. The fuel processor’s efficiency (𝜀FP ) is described in similar terms, where 𝜀FP is the ratio of the HHV of H2 in the reformate (ΔH(HHV),H2 ) exiting the fuel processor compared with the HHV of the fuel entering the fuel processor (ΔH(HHV),fuel ), including any fuel that must be combusted to provide energy for the fuel processor itself: NEUTRAL SYSTEM WATER BALANCE Neutral water balance is achieved when all of the water that is consumed by system components is produced by other components internal to the system. In other words, no additional water needs to be added from an external source. For example, some parts of the fuel cell system may consume liquid water (such as the fuel processor) and other parts of the system may produce it (such as the fuel cell and the condenser). To achieve neutral water balance, water vapor in the fuel cell’s exhaust stream should be condensed. A fuel cell system can achieve neutral water balance if ∑ ∑ ṁ c ≥ 0 (10.15) ṁ p − ∑ ∑ where ṁ p is the sum of the mass flow rates of produced water and ṁ c is the sum of the mass flow rates of consumed water. To achieve neutral water balance, the system ∑ ∑ needs the sum of condensed water, ṁ CD , to equal ṁ c , or ∑ where

∑

ṁ p =

ṁ CD = ∑

∑

ṁ CD +

ṁ c

∑

ṁ NCD

(10.16)

(10.17)

∑ and ṁ NCD is the sum of the mass flow rates of noncondensed water, that is, water that ∑ leaves the system as a vapor. The quantity of noncondensed water ( ṁ NCD ) depends primarily on the outlet temperature of the condenser or gas stream. In some cases, the inlet air stream contains water vapor from natural humidity that must be accounted for in the system water balance.

𝜀FP =

ΔH(HHV),H2 ΔH(HHV),fuel

(10.18)

A control volume analysis of the fuel processor may encapsulate all chemical reactors (nos. 1–6) in Figure 10.15. In both cases, the denominator typically incorporates all energy inputs to the fuel reforming and/or fuel processing stages. A realistic 𝜀FP for an efficient natural gas fuel processor is 85%. The primary source of efficiency loss in fuel reformers and fuel processors is heat loss. The fuel processing/reformer subsystem is discussed in more detail in Chapter 11.

375

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OVERVIEW OF FUEL CELL SYSTEMS

10.5.2

Fuel Cell Subsystem

The fuel cell subsystem converts the H2 -rich fuel stream to DC electrical power. As shown in Figure 10.16, a H2 -rich fuel stream and water are fed to the fuel cell’s anode. This stream is often intentionally humidified for PEMFC systems so as to maintain electrolyte hydration. Simultaneously, compressed air is fed to the fuel cell’s cathode. Figure 10.17 shows that the gross fuel cell stack electrical efficiency differs from the net fuel cell subsystem electrical efficiency. The difference between these efficiencies is due to the parasitic power required to run pumps, compressors, and other system devices. This parasitic power is drawn from the fuel cell stack itself and thus reduces the net electrical power truly available from the system. Figure 10.17 shows that, for a fuel cell stack, the maximum electrical efficiency occurs at the minimum electric power draw when all other variables are held constant. By contrast, for the complete fuel cell subsystem, the electrical efficiency is very low at low rates of electric power draw, because ancillary loads (like pumps and compressors) draw all or most of their electric power off of the fuel cell stack at low powers just to run at a low output level. The fuel cell subsystem’s net electrical efficiency (𝜀R,SUB ) can be described in terms of the net electrical power of a fuel cell subsystem (Pe,SUB ) and the HHV of H2 in the inlet gas (ΔḢ (HHV),H2 ), 𝜀R,SUB =

Pe,SUB ΔḢ (HHV),H

(10.19)

2

A realistic 𝜀R,SUB is 42%. Example 10.3 The net electrical power of a fuel cell subsystem (Pe,SUB ) can be expressed as (10.20) Pe,SUB = Pe − Pe,P where Pe is the gross electrical power output of the stack and Pe,p is the electrical parasitic power. Based on Figure 10.17, develop an equation to approximate the behavior of Pe,p . Solution: One possible solution is an equation of the form Pe,p = 𝛼 + 𝛽Pe , where α represents a fixed parasitic power load (such as 1 kW) and 𝛽Pe is a variable parasitic power that scales with a percentage of the fuel cell power output (such as β = 0.10). Here, α represents the “upfront energy cost” of operating the system while 𝛽 accounts for the extra marginal energy cost to operate the system at higher and higher power. The term α is likely to refer to the minimum power draw required to turn on components like pumps and compressors, while 𝛽 accounts for the additional power draw of the pumps and compressors when flow rates are increased to accommodate higher power fuel cell operation.

Heat stream

condenser

DC electricity

Cathode exhaust N2 O2

Liquid H 2 O H2-rich fuel

Anode exhaust

compressor

Water pump

H2

Steam generator

Preheater

N 2 H 2O

Fuel cell

Fuel cell

anode

cathode

H2-rich fuel +H2 O

Air compressor Figure 10.16. Fuel cell subsystem.

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OVERVIEW OF FUEL CELL SYSTEMS

0.9 0.8

Parasitic power

0.7

Electrical efficiency

378

Fuel cell stack 0.6 0.5 0.4

Net fuel cell subsystem

0.3 0.2 0.1 0 0

10

20

30

40

50

60

70

80

90 100%

Relative electrical power output as a percentage of maximum fuel cell stack power output

Figure 10.17. Gross and net efficiency of a fuel cell subsystem.

10.5.3

Power Electronics Subsystem

The power electronics subsystem, detailed in Figure 10.18, incorporates both (1) power conditioning (discussed in Section 10.4) and (2) supply management. 1. Power conditioning devices convert a fuel cell’s low-voltage DC power to highquality DC or AC power (normally 120 V and 60 Hz single-phase for U.S. domestic applications and three-phase for commercial and industrial applications). A fuel cell

Electricity storage

DC/AC inverter AC electric grid

Boost regulator

DC electricity

Figure 10.18. Power electronics subsystem.

CASE STUDY OF FUEL CELL SYSTEM DESIGN: STATIONARY COMBINED HEAT AND POWER SYSTEMS

subsystem produces DC electricity at a voltage that varies with power output level. As you learned in Chapter 1, and as shown in Figure 1.12, a single fuel cell’s voltage declines at higher currents, potentially by as much as a factor of 2. A fuel cell stack’s voltage follows the same pattern, as shown in Figure 10.17, and also may deteriorate with time. To compensate for these changes in fuel cell stack voltage, a step-up converter (boost regulator) may be used, as shown in Figure 10.18. The boost regulator matches the fuel cell stack’s output voltage with the inverter’s input voltage by compensating for voltage fluctuations. The inverter then converts the fuel cell stack’s DC power into AC power, which also may be filtered to enhance its quality. 2. Supply management matches the instantaneous supply of electricity with that demanded through electrical storage buffers and/or power from the surrounding utility grid (the network of electricity lines that provide buildings with electric power). To ensure that the electricity demanded by the load can be supplied, a fuel cell system may rely on an electricity storage device such as a battery or capacitor for back-up power. The fuel cell system may charge the storage device, as shown in Figure 10.18, when electricity demand is low. Alternatively, the fuel cell system may rely on the surrounding AC electricity grid to make up for any additional power needed, as shown in Figure 10.18. Also, a fuel cell system may sell excess electricity back to the surrounding grid. The power electronics subsystem net electrical efficiency (𝜀R,PE ) compares the net electrical power of the fuel cell subsystem (Pe,SUB ) with that of the fuel cell system (Pe,SYS ): 𝜀R,PE =

Pe,SYS Pe,SUB

(10.21)

If the power electronics subsystem is simplified to include only a boost regulator (a type of DC–DC converter) in series with a DC–AC converter, 𝜀R,PE is also 𝜀R,PE = 𝜀R,DC−DC × 𝜀R,DC−AC

(10.22)

where 𝜀R,DC-DC is the electrical efficiency of the DC–DC converter and 𝜀R,DC-AC is the electrical efficiency of the DC–AC converter. If 𝜀R,DC-DC = 𝜀R,DC-AC = 96%, a realistic 𝜀R,PE is 92%.

10.5.4

Thermal Management Subsystem

The thermal management subsystem, shown in Figure 10.19, recovers waste heat from the system for both internal system use and external use, such as for heating a building’s air space and hot water. The thermal management subsystem manages heat flows from both the fuel processing subsystem and the fuel cell subsystem. For the thermal management subsystem shown in Figure 10.19, heat is recovered from (1) the catalytic fuel reformer (if it operates exothermically or exhibits heat losses), (2) the fuel cell stack, (3) the catalytic afterburner, and (4) the condenser. Heat is delivered to (1) the steam generator, (2) the preheater,

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OVERVIEW OF FUEL CELL SYSTEMS

Water heating system

Space heating system

Fuel cell anode

condenser

Fuel cell cathode

Catalytic afterburner

Steam generator

Preheater

Catalytic fuel reformer

Figure 10.19. Thermal management subsystem.

(3) the building’s hot-water heating system, and (4) the building’s space heating system. All of these streams are shown in Figure 10.19. Heat can be transferred within the system via both direct and indirect heat transfer. For example, in some fuel processor designs, upstream exothermic processes directly supply heat to downstream endothermic processes. For example, this approach is implemented when heat output from the catalytic afterburner warms the steam generator, as shown in Figure 10.19. The heat recovery efficiency of the thermal management system depends on the design and control of the heat exchangers and the integration of heating of cooling streams within the overall system design. The heat recovery subsystem efficiency can be described in terms of the heat recovery efficiency of the fuel processor subsystem (𝜀FP,H ) and the heat recovery efficiency of the fuel cell subsystem (𝜀SUB,H ), according to 𝜀FP,H = 𝜀TM (1 − 𝜀FP ) 𝜀SUB,H = 𝜀TM (1 − 𝜀R,SUB )

(10.23) (10.24)

where 𝜀TM is the thermal management subsystem efficiency, the percentage of heat successfully recovered for a useful purpose compared with the heat available. Well-designed systems of heat exchangers may capture 80% of available heat (𝜀TM = 80%). The thermal management subsystem is discussed in more detail in Chapter 12.

CASE STUDY OF FUEL CELL SYSTEM DESIGN: STATIONARY COMBINED HEAT AND POWER SYSTEMS

10.5.5

Net Electrical and Heat Recovery Efficiencies

CHP fuel cell systems can achieve high overall efficiencies (𝜀O ), where 𝜀O = 𝜀R + 𝜀H

(10.25)

The fuel cell system’s electrical efficiency (𝜀R ) compares the net electrical output of the system with the HHV of the fuel input: 𝜀R =

Pe,SYS ̇ ΔH(HHV),fuel

(10.26)

where 𝜀R = 𝜀FP × 𝜀R,SUB × 𝜀R,PE =

ΔḢ (HHV),H2

Pe,SUB Pe,SYS ΔḢ (HHV),fuel ΔḢ (HHV),H2 Pe,SUB

(10.27) (10.28)

The fuel cell system’s heat recovery efficiency 𝜀H is the sum of the heat recovery efficiency of the fuel cell system in terms of the original fuel input (𝜀SUB,H,fuel ) and the heat recovery efficiency of the fuel processor (𝜀FP,H ). This can be expressed by 𝜀SUB,H,fuel = 𝜀FP × 𝜀TM × (1 − 𝜀R,SUB )

(10.29)

𝜀H = 𝜀SUB,H,fuel + 𝜀FP,H

(10.30)

and

Example 10.4 The text above gives realistic efficiency values for the various subsystems of the stationary fuel cell system shown in Figure 10.14. Based on these efficiencies, calculate (1) the fuel cell system’s electrical efficiency, (2) the system’s heat recovery efficiency, and (3) the system’s overall efficiency and (4) report the H∕P. Solution: 1. For the four subsystems discussed above, Table 10.5 summarizes the efficiencies for the four individual subsystems, along with the system’s net electrical efficiency (𝜀R ), calculated as 𝜀R = 𝜀FP × 𝜀R,SUB × 𝜀R,PE = 0.85 × 0.42 × 0.92 = 0.328 = 33%

(10.31) (10.32)

2. Table 10.5 also summarizes the thermal recovery efficiencies for subsystems along with the overall system heat recovery efficiency (𝜀H ). A thermal

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OVERVIEW OF FUEL CELL SYSTEMS

management system that is 80% efficient can recover 80% of available heat from the fuel processor subsystem and the fuel cell subsystem, according to 𝜀FP,H = 𝜀TM × (1 − 𝜀FP ) = 0.80(1 − 0.85) = 0.12 𝜀SUB,H = 𝜀TM × (1 − 𝜀R,SUB ) = 0.80(1 − 0.42) = 0.46 𝜀SUB,H,fuel = 𝜀FP × 𝜀TM × (1 − 𝜀R,SUB ) = 0.85[0.80(1 − 0.42)] = 0.39 𝜀H = 𝜀SUB,H,fuel + 𝜀FP,H = 0.12 + 0.39 = 0.51 = 51% (10.33) 3. 𝜀O = 𝜀R + 𝜀H = 0.33 + 0.51 = 84%. 4. H∕P = 𝜀H ∕𝜀R = 0.51∕0.33 = 1.55. TABLE 10.5. Electrical Efficiency and Heat Recovery Efficiency for Four Main Subsystems

Electrical efficiency Heat recovery efficiency

Fuel Processing Subsystem

Fuel Cell Subsystem

Power Electronics Subsystem

Thermal Management Subsystem

Overall System

85%

42%

92%

NA

33%

12%

46%

NA

80%

51%

Example 10.5 Combined cooling, heating, and electric power (CCHP) fuel cell systems couple recoverable heat in the fuel cell system with an absorption chiller to produce a stream of cooling power for building space cooling or industrial cooling processes. An absorption chiller converts heat directly into cooling power. Chiller efficiency can be quantified with the term coefficient of performance (COP). The COP for an absorption chiller is defined as the amount of cooling power output to heat input. Chiller COP depends on the heat source temperature. Higher temperature heat can be coupled with higher effect absorption chillers to achieve higher COPs. Table 10.6 shows the impact of heat source temperature on COP. Single-effect lithium bromide (LiBr)–water chillers provide cooling power at temperatures cold enough for space cooling and refrigeration (but not freezing). As the heat source temperature increases, higher-effect chillers can be used and the COP generally increases. Assume that the fuel cell system in Example 10.4 produces 100 kW of net electrical power. Calculate (1) the recoverable heat from this system in kW, (2) the cooling power available in kW if 100% of this heat was captured in a single-effect LiBr–water absorption chiller with a COP of 0.7, and (3) the overall system efficiency.

CASE STUDY OF FUEL CELL SYSTEM DESIGN: SIZING A PORTABLE FUEL CELL

1. The recoverable heat is 0.51 ∕ 0.33 × 100kWe = ∼155 kW. 2. The cooling power is 0.7 × 155 kW = ∼108 kW. 3. The overall efficiency is 0.7 × 0.51 + 0.33 = ∼0.69. TABLE 10.6. Coefficient of Performance for Single-, Double-, and Triple-Effect LiBr–Water Absorption Chillers as a Function of Heat Source Temperature Absorption Chiller Type

Heat Source Temperature Range

COP

Single-effect LiBr–water

70–120∘ C

0.4–0.7

Double-effect LiBr–water

120–160∘ C

0.7–1.2

Triple-effect LiBr–water

160–200∘ C

1.2–1.5

10.6 CASE STUDY OF FUEL CELL SYSTEM DESIGN: SIZING A PORTABLE FUEL CELL Portable fuel cell systems are subject to several important constraints not faced by stationary fuel cell systems. When designing portable power systems, two critical constraints are the electric power and lifetime energy requirements of the application. For example, a laptop computer might require 10 W of power (power requirement) and need to run for 3 h (energy requirement). Given fuel cell power density information, it is relatively straightforward to size a fuel cell system that will produce 10 W of power. Given fuel energy density information, it is also straightforward to size a fuel reservoir that will supply the system sufficiently for 3 h of use. However, a more difficult task is to determine the optimal ratio between the fuel cell size and fuel reservoir size such that the power and energy requirements of the application are met with minimum possible volume or weight. This optimization is an exercise in fuel cell sizing and illustrates the complex trade-offs between energy density and power density in portable fuel cell systems. As an example of the subtleties of system sizing, consider a hypothetical fuel cell system consisting of a 99-L fuel reservoir and a 1-L fuel cell. Suppose that this fuel cell system must deliver 100 W of power. The 1-L fuel cell must therefore obtain a power density of 100 W/L to provide the required power. At 100 W/L, we will assume that the fuel cell is 40% efficient. Thus, the 99-L fuel reservoir, when used at 40% efficiency, effectively provides 39.6 L of extractable fuel energy. Now, suppose that we resize the system such that the fuel reservoir is 98 L and the fuel cell is 2 L. To deliver 100 W of power, the fuel cell must now obtain a power density of 50 W∕L. At this reduced power density, the electrical efficiency of the fuel cell most likely will be greater. (As shown in Figure 10.17, this increase in efficiency is likely because the fuel cell system can run at a lower current density and a higher voltage point and still meet the reduced power density requirements.) Assume that the fuel cell is 50% efficient at a power density of 50 W∕L. In this case, the 98-L fuel reservoir used at 50% efficiency effectively provides 49 L of extractable fuel energy. By changing the size of the fuel cell

383

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OVERVIEW OF FUEL CELL SYSTEMS

relative to the fuel reservoir, we have greatly extended the lifetime of this system without increasing its total volume! Essentially, we have sacrificed a small amount of the fuel reservoir volume to provide room for a larger fuel cell, but this sacrifice is more than compensated for by the fact that we are using the remaining fuel more efficiently (due to the reduced power density demands on the fuel cell). At the same time, however, as the size of the fuel cell increases, the capital cost of the system also increases. There are trade-offs among capital cost, system sizing, system net electrical efficiency, system volume, and system mass. Continuing the above example, if we sacrifice even more of the fuel reservoir to further increase the efficiency of the fuel cell, we can generate still greater system lifetimes. At some point, however, an optimum will be reached. How can we determine this optimum? Essentially, given a fixed system volume and a fixed power requirement, we want to maximize the “in-use” time of the system. The following text box describes how this optimum can be calculated given the properties of the fuel cell, fuel reservoir, and volume and power requirements of the system. By calculating this optimum over a range of system sizes and power requirements, a Ragone plot may be generated. OPTIMIZING A PORTABLE FUEL CELL SYSTEM Optimizing a portable fuel cell system essentially involves the following problem: For a given system volume and power requirement, what is the best ratio between the volume of the fuel cell stack and the volume of the fuel reservoir to maximize the lifetime of the system? (This optimization exercise also can be worked out on a gravimetric basis.) Figure 10.20 illustrates the key terms: pFC = power density of the fuel cell unit x= volume fraction taken up by the fuel cell unit eF = energy density of the fuel reservoir 1 − x= volume fraction taken up by the fuel reservoir V= total volume of the system P= total system power requirement Fuel cell

Fuel reservoir eF

pFC

VFC = xV

VF = (1-x)V

Entire system: V, P, E P = xVp FC , E = (1– x) Ve F ε

Figure 10.20. Optimizing a portable fuel cell system’s design involves finding the best ratio between fuel cell stack size and fuel reservoir size so that the system provides the required electric power for the longest possible time.

CASE STUDY OF FUEL CELL SYSTEM DESIGN: SIZING A PORTABLE FUEL CELL

Maximizing the system’s in-use time means maximizing E, the total extractable energy from the fuel reservoir. The system power, P, and the total system volume, V, are the constraints on the maximization. The power density of the fuel cell unit (pFC ) and the energy density of the fuel reservoir (eF ) are the knowns, and the volume fraction taken up by the fuel cell unit relative to the fuel reservoir (x) is the unknown. This problem can be solved in the following manner. First, construct an expression for the total extractable energy from the fuel reservoir (E), since this is what we are trying to maximize: (10.34) E = (1 − x)VeF 𝜀 In this expression, 𝜀 gives the efficiency at which the fuel contained in the fuel reservoir is utilized by the fuel cell and will be a function of the power density of the fuel cell (pFC ). In other words, 𝜀 = 𝜀(pFC ). As shown in Figure 10.17, at high power densities, the fuel cell subsystem generally will be less electrically efficient; at low power densities, the fuel cell subsystem generally will be more electrically efficient. The functional dependence between the fuel cell power density and electrical efficiency must be estimated or determined. (It can be calculated from the fuel cell’s i–V curve, ancillary load information, and stack volume information.) After explicitly acknowledging the functional dependence of 𝜀, Equation 10.34 becomes E = (1 − x)VeF 𝜀(pFC )

(10.35)

The system must attain a total power given by P. This constrains pFC such that xVpFC = P. Introducing this constraint into our optimization equation gives E = (1 − x)VeF 𝜀

(

P xV

) (10.36)

The volume fraction x that maximizes E can then be determined by setting the derivative of this expression with respect to x equal to zero and solving for x. Inserting x back into Equation 10.36 determines the optimal value of E. A Ragone plot nicely summarizes the trade-offs between energy density and power density and allows a designer to compare the maximum design limits for a set of different power systems. NASA engineers designing a portable power source for a space mission (where weight is critical) might pore over a gravimetric Ragone plot like the one shown in Figure 10.21. This plot displays the relationship between gravimetric power density and gravimetric energy density for a variety of portable power systems. A Ragone plot for volumetric power and energy density would likely look similar. A curve on the Ragone plot represents the locus of power density/energy density design points available to a designer using a particular technology. For example, consider the design of a 10-kg portable fuel cell system that needs to deliver 100 W of power (net system power density 10 W∕kg). A glance at Figure 10.21 indicates that such a system will provide an energy density of around 250 Wh∕kg, and we can thus expect its lifetime to be about 25 h. If the system instead needs a power of 200 W (increasing the net system power density to 20 W∕kg),

385

OVERVIEW OF FUEL CELL SYSTEMS

1000

Power density (W/kg)

386

c se 36 Electrochemical supercapacitors

in 6m

r

ou

1h

Portable microdiesel generator 100 Portable fuel cell w/compressed H2

s

ur

10

Li ion battery

ho

10 Lead-acid battery

rs

ou

h 00

1

1 1

10

100

1000

Energy density (Wh/kg) Figure 10.21. Gravimetric Ragone plots for a variety of portable power solutions showing trade-offs between system power density and system energy density. The dashed diagonal lines indicate contours of constant lifetime for various power density/energy density ratios.

then the energy density of the system will fall to about 150 Wh∕kg, and we can expect its lifetime to fall to about 8 h. This trade-off occurs because to increase the power of the fuel cell system, we have to devote more of the system mass to the fuel cell itself. This restructuring leaves less mass available for fuel. In the extreme, we could imagine designing a fuel cell system where 100% of the system weight is taken by the fuel cell (leaving 0% available for fuel). The power density of such a system would simply correspond to the power density of the fuel cell itself. The energy density of the system would be zero. This design point corresponds to the power density axis intercept of the fuel cell Ragone curve. At the other extreme, a fuel cell system that is 100% fuel would have a power density of zero and an energy density that corresponds to the energy density of the fuel itself. This design point corresponds to the energy density axis intercept of the fuel cell Ragone curve. Fuel cell systems are fully scalable; their Ragone curves extend fully across the energy density/power density space. In batteries and capacitors, power and capacity are convoluted; their Ragone curves cannot extend over the full energy density/power density space. A further difference between fuel cell and battery systems is illustrated in Figure 10.22. This figure illustrates how liquid-fueled portable fuel cell systems tend to outperform batteries when long operating lifetimes are needed but tend to underperform batteries when short operating lifetimes are needed. As we have just discussed, the overall size of a fuel cell system is determined by the size of the fuel cell itself and the fuel reservoir. The “upfront size cost” associated with the fuel cell is appreciable, and must be “paid,” even for very short operating lifetimes. However, this upfront cost is recouped at longer operating lifetimes,

CHAPTER SUMMARY

System size

Battery

Battery is better Fuel cell

Fuel cell is better Operating lifetime

Figure 10.22. System size versus operating lifetime comparison of a liquid-fueled portable power fuel cell system versus a battery system. The large upfront size cost of the portable fuel cell system is recouped for long operating missions by the higher energy density of the fuel cell’s liquid fuel.

where the fuel cell benefits from the much higher energy density of its liquid fuel reservoir compared to batteries. 10.7

CHAPTER SUMMARY

• A fuel cell system generally consists of a set of fuel cells combined with a suite of other system components. A set of fuel cells is required to meet the voltage requirements of real-world applications. The suite of system components typically includes devices to provide cooling, fueling, monitoring, power conditioning, and control for the fuel cell device. • Fuel cell system design is strongly application dependent. For example, in portable applications, where mobility and energy density are at a premium, there is an incentive to minimize system ancillaries. • Fuel cell stacking refers to the combination of multiple fuel cells in series to build voltage. The most common stacking arrangements include the vertical (bipolar) configuration, the planar banded configuration, the planar flip-flop configuration, and the tubular configuration. • As stack size and power density increase, stack cooling becomes more and more essential. Internal air or water cooling channels can be integrated into fuel cell stack designs to provide effective cooling. • Stack cooling is used to prevent (1) overheating and (2) thermal gradients within the stack. • Heat released by the stack can be recovered for (1) internal system heating and/or (2) external heating of a source of thermal demand (such as a building’s heating loop).

387

388

OVERVIEW OF FUEL CELL SYSTEMS

• A cooling system’s effectiveness can be computed by comparing the rate of cooling accomplished versus the electric power consumed by the cooling system. Good designs attain effectiveness ratios of 20–40. • Fuel candidates for stationary power applications should be evaluated primarily on their availability for fuel cell use. Fuel system candidates for mobile applications should be additionally evaluated on gravimetric and volumetric storage energy density metrics. • There are two primary fueling options for fuel cells: direct hydrogen or a hydrogen carrier. • Advantages of direct hydrogen include high performance, simplicity, and the elimination of impurity concerns. Unfortunately, hydrogen is not a widely available fuel and current hydrogen storage solutions are suboptimal. • The major direct hydrogen storage solutions include compressed gas storage, cryogenic liquid storage, and reversible metal hydride storage. • Hydrogen carriers are often far more widely available than hydrogen gas fuel and can greatly facilitate storage. • Hydrogen carriers can either be directly electro-oxidized in the fuel cell to produce electricity or reformed to produce H2 gas, which is then electro-oxidized by the fuel cell to produce electricity. • Other than H2 , only a few simple fuels can be directly electro-oxidized. Direct electro-oxidation assures a simple fuel cell system but often dramatically lowers fuel cell performance. • Fuel reforming processes produce hydrogen from the carrier stream. Impurities and poisons may also be generated. Depending on the fuel cell, these contaminants may need to be removed from the fuel prior to use. In high-temperature fuel cells, the reforming process can occur inside the fuel cell (internal reforming) rather than in a separate chemical reactor (external reforming). • For portable applications, direct or reformed methanol fuel systems may provide energy density improvements compared to direct hydrogen storage solutions. • For stationary applications, reformed natural gas (mostly methane) and biogas are the leading fuel solutions due to their greater availability and low cost compared to hydrogen. • The electric power delivered by a fuel cell must be conditioned to ensure a stable, reliable electrical output. • Power conditioning includes power regulation and power inversion. Power regulation uses DC–DC converters to step up or step down the variable voltage of a fuel cell stack to a predetermined, fixed output. Power inversion is used to transform the DC power provided by a fuel cell into AC power. (Power inversion is not needed in all cases.) • In both power regulation and power inversion, total electric power is conserved (minus some losses). DC–DC converters and DC–AC inverters are typically in the range of 85–98% efficient.

CHAPTER EXERCISES

• The fuel cell control unit is the “brain” of the fuel cell system. Control units use feedback loops between system-monitoring elements (sensors) and system actuation elements (valves, switches, fans) to maintain operation within a desired range. • Power supply management matches the fuel cell system’s electrical output with that electric power demanded by the load through the use of energy buffers and special controls. • The overall efficiency 𝜀O of a combined heat and power (CHP) fuel cell system is the sum of its net system electrical efficiency, 𝜀R , and its heat recovery efficiency, 𝜀H . • Combined cooling, heating, and electric power (CCHP) fuel cell systems couple recoverable heat from the fuel cell with an absorption chiller that converts heat into cooling power. • An absorption chiller’s efficiency can be described by its coefficient of performance (COP), which is its cooling output divided by its heat input. • Portable fuel cell sizing involves trade-offs between the size of the fuel cell unit and the size of the fuel reservoir unit. Correctly evaluating this sizing trade-off requires a careful optimization. • These kinds of design trade-offs can be analyzed with Ragone plots, which allow the power density/energy density limitations of multiple energy technologies to be compared against one another visually.

CHAPTER EXERCISES Review Questions 10.1

Imagine a combination of the vertical and tubular stacking configurations. Draw a possible stacking arrangement involving a series of stacked donut-shaped cells where H2 is provided to the stack up the central tubelike core and air is provided around the outside. Do not forget about sealing!

10.2

Which direct hydrogen storage systems tend to have the highest gravimetric storage energy density? Which direct hydrogen storage systems tend to have the highest volumetric storage energy density? Please see Table 10.1.

10.3

Identify fuels that can undergo direct electro-oxidation. Describe the reactions that take place with these fuels.

10.4

Identify fuels that are more typically associated with internal fuel reforming. Describe the reactions that take place with these fuels.

10.5

Identify fuels that are more typically associated with external fuel reforming. Describe the reactions that take place with these fuels.

10.6

What are the four primary subsystems of a fuel cell system? Give examples of subsystem components that depend on the operation of other subsystem components. How might these subsystem components be integrated?

389

390

OVERVIEW OF FUEL CELL SYSTEMS

10.7

Sketch out a process diagram for a fuel cell system for a scooter. Some primary components of the system include a PEMFC stack, a hydrogen tank, an electrical storage device such as a battery or capacitor for buffering load, and an electric motor that fits into the hub of the scooter’s wheel. Draw the primary system components, stream flows, and heat flows. Label the four subsystems. (One way to approach this problem is to begin with the process diagram shown in Figure 10.14 and decide which components are not needed.)

Calculations 10.8

(a) Assuming STP conditions, what is the rate of heat generation from a 1000-W hydrogen/air-fueled PEM running at 0.7 V (assume 𝜀fuel = 1)? (b) The fuel cell in part (a) is equipped with a cooling system that has an effectiveness rating of 25. To maintain a steady-state operating temperature, assuming no other sources of cooling, what is the parasitic power consumption of the cooling system?

10.9

(a) An automotive PEMFC stack produces 88 kWe of gross electrical power and operates with a gross stack electrical efficiency of roughly 65%. The stack contains 380 active electrochemical cells and uses a ratio of one cooling cell per electrochemical cell. As described in Section 10.2, the cooling cells use the flow channels in a bipolar plate for circulating cooling fluids rather than for delivering and extracting reactant and product gases. What is the approximate cooling load per cooling cell in this stack in units of watts per cell? (b) The PEMFC stack operating temperature is 90∘ C. The volume available for cooling fluid in each cooling cell is 1 cm3 . Use the volumetric heat capacities of water and air referenced in Section 10.2, phase change temperatures, and other information to compare the use of deionized water, air, and a water–glycol mixture as a cooling fluid for this application. What cooling fluids are appropriate for this application?

10.10 (a) Using the reference values in Table 10.1, identify the two direct H2 storage systems with the highest and the lowest gravimetric storage energy densities. Calculate how many times more energy dense the high-energy-density system is compared with the low one. Discuss the implications for system design and choice of application. (b) Using the reference values in Table 10.1, identify the two direct H2 storage systems with the highest and the lowest volumetric storage energy densities. Calculate how many times more energy dense the high-energy-density system is compared with the low one. Discuss the implications for system design and choice of application. 10.11 In Section 10.3.2, it was stated that a fuel system consisting of a 1-L reformer plus a 1-L fuel reservoir containing a 50:50 molar mix of methanol and water had a net energy density of 1.72 kWh∕L (in terms of the heating value of the fuel). Derive this value. Assume STP and use the HHV enthalpy for methanol. Assume that the density

CHAPTER EXERCISES

of water is 1.0 g∕cm3 and that the density of methanol is 0.79 g∕cm3 . Clearly show all steps. 10.12 We would like to compute the carrier system effectiveness of a fuel cell operating on reformed natural gas. Since the reforming process is not perfectly efficient, in this example, we assume that the enthalpy content of H2 provided to the fuel cell amounts to only 75% of the original enthalpy content of the natural gas. Furthermore, we recognize that the H2 supplied by the reformer will be diluted with CO2 , other inert gases, and perhaps even some poisons. We assume that these diluents lower the efficiency of the fuel cell by 20% compared to operation on pure H2 . What is the total net effectiveness of this reformed natural gas system? 10.13 Assume that the functional relationship between the power density of a fuel cell unit and the electrical efficiency of fuel utilization can be described as 𝜀(pFC ) = A − BpFC

(10.37)

In this equation, as the volumetric power density (pFC ) of the fuel cell goes up, the energy efficiency 𝜀 goes down (for A and B positive). (a) Using the procedure outlined in the optimization text box, derive the expression for the optimal value of X (the volume fraction occupied by the fuel cell unit) given a system volume of V and a power requirement of P. (b) Calculate X if V = 100 L, P = 500 W, A = 0.7, and B = 0.003 L∕W. Check to make sure that the fuel cell power density required by your solution is reasonable. 10.14 As discussed in Section 10.6, liquid-fuel-based portable fuel cells tend to make more sense than batteries for long-operating-lifetime applications. Consider a 20-W laptop system based on a direct methanol fuel cell. In order to produce 20W, this system requires a 400-cm3 DMFC. The DMFC is supplied by a 50%–50% methanol–water fuel reservoir with an energy density of 3400 Wh/L and can convert 20% of this fuel energy into electricity. In contrast, a lithium-ion battery system alternative provides an energy density of 200 Wh∕L, 100% of which can be converted to electricity. Based on these specifications, calculate the minimum operating lifetime for which the fuel cell system will deliver greater volumetric energy density than the lithium ion battery system. Draw a graph, similar to the one shown in Figure 10.22, which quantitatively compares the size versus operating lifetime characteristics of the fuel cell and battery systems.

391

CHAPTER 11

FUEL PROCESSING SUBSYSTEM DESIGN

Having introduced the four main subsystems of the fuel cell system in Chapter 10, we now look in greater detail at one of the subsystems, the fuel processing subsystem. In the context of the stationary fuel cell system example presented in Chapter 10, we will explore the details of fuel processor subsystem design. The fuel processing subsystem is a miniature chemical plant. Its primary purpose is to chemically convert a readily available fuel such as a hydrocarbon (HC) fuel into a hydrogen-rich fluid that can be oxidized at the fuel cell’s anode. It also serves to convert fuel or oxidant not consumed at the fuel cell’s anode and cathode into useful energy. Not all fuel cells require sophisticated fuel processing subsystems and, rather, simply use a fuel delivery subsystem. At the same time, when running on a hydrocarbon fuel, some level of fuel processing is typically required. The complexity of the fuel processing subsystem depends on the type of fuel cell it serves and the type of fuel it is processing. A fuel processing subsystem consists of a series of catalytic chemical reactors that convert hydrocarbon fuel into a low-impurity, high-hydrogen-content gas. Both the PEMFC and PAFC are sensitive to impurities in their feed gases, which might otherwise poison (i.e., block) catalyst sites for electrochemical reactions. Therefore, PEMFC and PAFC systems generally require extensive fuel processing systems that employ multiple stages. By contrast, MCFCs and SOFCs operate at high enough temperatures that they may be able to implement internal reforming, whereby the fuel mixture can be fed directly to the fuel cell’s anode, and fuel reforming reactions occur at the anode catalyst surface. The anode catalyst facilitates not only electrochemical oxidation reactions but also fuel reforming reactions. Chapter 10, Section 10.3.2, first introduced the concept of internal reforming. Limitations to internal reforming may include coking (deposition of carbon) on the anode’s surface that reduces performance and less precise control of

393

394

FUEL PROCESSING SUBSYSTEM DESIGN

reaction processes. As a result, in practice, most commercially deployed MCFC and SOFC systems today incorporate at least some external reforming. Since low-temperature fuel cells have the most stringent fuel processing requirements, we will take a look at a typical fuel processing subsystem for a PEMFC or PAFC. As previously discussed in Chapter 10, such a subsystem will probably consist of at least three primary reactor processes (see Figure 11.1): • Fuel reforming (labeled no. 3) • Water gas shift reaction (labeled no. 4) • Carbon monoxide clean-up (labeled no. 5) Although outside the scope of this discussion, the sulfur in natural gas fuel and other fuels also typically must be removed in an upstream processing step. For now, let’s examine the three main fuel processing stages. 11.1

FUEL REFORMING OVERVIEW

The overall goal of fuel reforming is to convert a HC fuel into a hydrogen-rich gas. The primary conversion may be accomplished with or without a catalyst via one of five major types of fuel reforming processes: • • • • •

Steam reforming (SR) Partial oxidation (POX) reforming Autothermal reforming (AR) Gasification Anaerobic digestion (AD)

To compare the effectiveness of various fuel reforming processes, we introduce the concept of H2 yield ( yH2 ), which represents the molar percentage of H2 in the reformate stream at the outlet of the fuel reformer: nH (11.1) yH2 = 2 n In this equation, nH2 is the number of moles of H2 produced by the fuel reformer and n is the total number of moles of all gases at the outlet. In a similar manner, we introduce the concept of a steam-to-carbon ratio (S∕C ), which represents the ratio of the number of moles of molecular water (nH2 O ) to the moles of atomic carbon (nc ) in a fuel (such as methane, CH4 ) in a chemical stream: nH O S (11.2) = 2 C nc Each of the reforming processes produces varying H2 yields, requires different steam-to-carbon ratios, and possesses unique advantages and disadvantages. The major characteristics of the first three reforming processes are described in Tables 11.1 and 11.2.

Cathode exhaust

System exhaust N 2 CO2 H 2O condenser

H 2O Catalytic afterburner 6

Liquid H 2 O

N2 O2

Anode exhaust

H2 N 2 CO2 H 2O

Fuel cell Fuel cell anode cathode

Natural gas compressor Water

Steam generator 1

pump

Preheater 2

CO2 H2 N 2

Catalytic fuel reformer 3

Water gas H 2 N 2 CO shift reactor CO2 H 2O 4

CO clean-up 5

Air

compressor Cathode exhaust Air stream Water line

Stream splitter Natural gas stream Anode exhaust

Figure 11.1. Fuel processing subsystem. Repeated from Chapter 10 for clarity.

395

H2 O

396 9%

CO

47%

3%

41% 19%

76%

H2

Trace NH3 , CH4 , SOx

Other

39% Some NH3 , CH4 , SOx , HC

0%

N2

15% 34% Trace NH3 , CH4 , SOx , HC

1%

15%

CO2

Neutral

Exothermic

Endothermic

Exothermic or Endothermic?

Note: For the three primary fuel reforming reactions, the table shows examples of outlet gas compositions on a dry, molar basis. The steam reforming reaction produces the highest H2 yield and the cleanest exhaust. The low H2 yield for the partial oxidation and autothermal reforming reactions is a result of their intake of air; the O2 in air partially oxidizes the fuel while the N2 in air dilutes the H2 composition in the outlet gas. For all three reactions, the H2 yield can be increased by downstream use of the water gas shift reaction. In the chemical reaction for steam reforming, the first line shows the typical reactants and products in their correct molar ratios. The second line below this shows the full range of products for an actual reactor, which may include not only CO and H2 but also CO2 and H2 O. The chemical reaction for autothermal reforming is shown in a similar manner. Concentrations are noted on a dry, molar basis (i.e., no water vapor in gas stream).

600–900

>1000

Cx Hy + 12 xO2 ↔ xCO + 12 yH2

Partial oxidation

( ) Autothermal reforming Cx Hy + zH2 O(g) + x − 12 z O2 ( ) ↔ xCO2 + z + 12 y H2 ⇒ CO, CO2 , H2 , H2 O

700–1000

( ) Cx Hy + xH2 O(g) ↔ xCO + 12 y + x H2 ⇒ CO, CO2 , H2 , H2 O

Steam reforming

Temperature Range (∘ C)

Chemical Reaction

Type

Gas Composition of Hydrogen Outlet Stream on a Dry, Molar Basis (with Natural Gas Fuel Input)

TABLE 11.1. Comparison of Chemical Reaction Characteristics of Three Primary Fuel Reforming Reactions

397

Quick to start and respond because reaction is exothermic Quick dynamic response Less careful thermal management required Works on many fuels Simplification of thermal management by combining exothermic and endothermic reactions in same process Compact due to reduction in heat exchangers

Partial oxidation

Requires careful control system design to balance exothermic and endothermic processes during load changes and start-up

Low H2 yield

Requires careful thermal management to provide heat for reaction, especially for (a) start-up and (b) dynamic response Only works on certain fuels Lowest H2 yield Highest pollutant emissions (HCs, CO)

Disadvantages

Note: Autothermal reforming combines steam reforming and partial oxidation to achieve some of the benefits of both, including simple heat management and quick response. Partial oxidation provides the greatest fuel-type flexibility.

Quick to start

Highest H2 yield

Steam reforming

Autothermal reforming

Advantages

Type

TABLE 11.2. Advantages and Disadvantages of Three Primary H2 Production Methods

398

FUEL PROCESSING SUBSYSTEM DESIGN

To further compare different fuel reforming processes, we can evaluate a reforming process’ fuel reformer efficiency and fuel processor subsystem efficiency, which are concepts discussed in Chapter 10, Section 10.5.1. As discussed in Chapter 10, the fuel reformer’s efficiency is often described as the ratio of H2 energy [based the higher heating value (HHV) of H2 ] in the reformate stream exiting the fuel reformer divided by the fuel energy (based on the HHV of the fuel) entering the fuel reformer, including any fuel that must be combusted to provide energy for the reformer itself. (For a discussion of HHV, please see Chapter 2.) The equation for fuel reformer efficiency is Equation 10.14. The term for fuel reformer efficiency applies to the control volume only around the fuel reforming unit. By comparison, fuel processor subsystem efficiency can be defined as the ratio of H2 energy (HHV of H2 ) in the reformate stream exiting the fuel processor divided by the fuel energy (based on the HHV of the fuel) entering the fuel processor, including any fuel that must be combusted to provide energy for the fuel processor itself. The equation for fuel processor subsystem efficiency is Equation 10.18. The term for fuel processor efficiency is generally applied to a control volume that includes the entire fuel processor, which may include a fuel reformer, a water gas shift reaction, carbon monoxide clean-up processes, afterburner treatment of anode and cathode off-gases, and/or other processes. In the following sections, all five reforming processes are discussed in greater detail.

11.1.1

Steam Reforming

Steam reforming (SR) is an endothermic reaction that combines a HC fuel with steam over a catalyst at high temperature, according to Cx Hy + xH2 O(g) ↔ xCO +

(

1 y 2

) + x H2 ⇒ CO, CO2 , H2 , H2 O

(11.3)

As we discussed in Chapter 10, endothermic reactions consume energy and exothermic reactions release energy. The SR of natural gas typically has a H2 yield of 76% on a dry, molar basis (i.e., no water vapor is in the outlet gas stream) [123]. Because no oxygen in air is involved in the reaction, the outlet stream is not diluted by N2 in air, and, therefore, the H2 yield is the highest of the first three reforming approaches discussed. Chapter 2, Section 2.4.2, first introduced Le Chatelier’s principle. To increase the H2 yield, Le Chatelier’s principle tells us that operating the reaction with excess water vapor would help shift the reaction’s equilibrium to favor H2 production. To further increase the H2 yield, the CO in the outlet of the SR reactor can be “shifted” to H2 via a second reaction, the water gas shift (WGS) reaction: (11.4) CO + H2 O(g) ↔ CO2 + H2 Chapter 10, Section 10.3.2, first introduced the WGS reaction. The WGS reaction can increase the H2 yield by about 5%. The primary SR reactions for methane are summarized in Table 11.3. [In this table and throughout the chapter, enthalpies of reaction are reported at standard temperature and pressure (STP). We will use these STP values for back-of-the-envelope calculations in the chapter. For a discussion on enthalpy of reaction, please see Chapter 2.]

FUEL REFORMING OVERVIEW

TABLE 11.3. Steam Reforming Reactions Reaction Number

Reaction Type

1

Stoichiometric Formula

Δĥ 0rxn (kJ/mol)

Steam reforming

CH4 + H2 O(g) → CO + 3H2

+206.4

2

Water gas shift reaction

CO + H2 O(g) → CO2 + H2

−41.2

3

Evaporation

H2 O(l) → H2 O(g)

+44.1

Note: The main steam reforming reaction is endothermic. Vaporized water (steam) is a reactant. The water gas shift reaction increases H2 yield.

A steam reformer must be designed to capture heat to sustain its endothermic reaction. A common steam reformer design is a tubular reformer. A tubular reformer consists of a furnace that contains tubes filled with catalysts through which the SR reactants pass. When operated on natural gas fuel and other sulfur-containing fuels, SR catalysts can be gradually poisoned by sulfur compounds in the fuel. To address this, many fuel processor subsystem designs include a sulfur removal bed upstream of the fuel reformer to clean the fuel to low sulfur levels [10–15 parts per million (ppm)]. The endothermic SR reaction takes place inside the tubes. Often, the tubes are heated by the combustion of some of the input fuel. Alternatively, within a fuel cell system, the heat for the endothermic SR reaction can be provided by combusting the anode exhaust gas (the unconsumed fuel exiting the fuel cell’s anode) in a catalytic afterburner, such as the one labeled 6 in Figure 11.1. If the SR is coupled to a SOFC or MCFC stack, recoverable heat from the fuel cell stack itself may be high enough in temperature to provide heat to the SR. Example 11.1 (1) For an idealized reformer consuming methane (CH4 ) fuel and operating with combined SR and WGS reactions, what is the maximum H2 yield? (2) What is the steam-to-carbon ratio for the combined reactions? (3) In a real fuel reformer, why might you want to operate the reactor with a higher steam-to-carbon ratio? (4) What quantity of heat is consumed by the reaction, assuming, for simplicity, that the reactants and products enter and leave the reactor at STP? Solution: 1. For SR of CH4 , we have CH4 + H2 O(g) ↔ CO + 3H2

(11.5)

and for the WGS reaction, we have CO + H2 O(g) ↔ CO2 + H2

(11.6)

For the two combined reactions, we have CH4 + 2H2 O(g) ↔ CO2 + 4H2

(11.7)

399

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FUEL PROCESSING SUBSYSTEM DESIGN

which is the sum of reactions 1 and 2 shown in Table 11.3. This combined reaction has a hydrogen yield of yH2 =

4 mol H2 = 0.80 4 mol H2 + 1 mol CO2

(11.8)

or 80%. 2. The steam-to-carbon ratio is nH O S = 2 =2 C nC

(11.9)

3. You might want to operate with a higher steam-to-carbon ratio to reduce carbon deposition and to increase the H2 yield, according to Le Chatelier’s principle. Carbon deposition occurs due to reaction 3 (thermal decomposition) from Table 11.4. Typically, a S∕C ratio of 3.5–4.0 can prevent carbon formation. 4. According to Table 11.3, 165.2 kJ∕mol of CH4 must be provided to drive the combined SR+WGS reactions (206.4 kJ∕mol–41.2 kJ∕mol = 165.2 kJ∕mol) if water is in a vapor state. This is the enthalpy of reaction for Equation 11.7. If water enters in a liquid state, an additional 44.1 kJ∕mol of H2 O or an additional 88.2 kJ∕mol of CH4 is required. In total, if water enters in a liquid state, 253.4 kJ∕mol of CH4 must be provided to drive the combined SR+WGS reactions if they were to take place at STP. TABLE 11.4. Partial Oxidation Reactions Reaction Number

Reaction Type

Stoichiometric Formula

Δĥ 0rxn (kJ/mol)

1

Partial oxidation

CH4 + 12 O2 → CO + 2H2

−35.7

2

Partial oxidation

CH4 + O2 → CO2 + 2H2

−319.1

3

Thermal decomposition

CH4 → C + 2H2

+75.0

4

Methane combustion

CH4 + 2O2 → CO2 + 2H2 O(l)

−890

5

CO combustion

−283.4

6

Hydrogen combustion

CO + 12 O2 → CO2 H2 + 12 O2 → H2 O(l)

−286

Note: Autothermal reforming reactions include these and the steam reforming reactions in Table 11.3.

11.1.2

Partial Oxidation Reforming

Partial oxidation reforming is an exothermic reaction that combines a HC fuel with some oxygen to partially oxidize (or partially combust) the fuel into a mixture of CO and H2 ,

FUEL REFORMING OVERVIEW

usually in the presence of a catalyst. In complete combustion, a HC fuel combines with sufficient oxygen (O2 ) to completely oxidize all products to CO2 and H2 O. In complete combustion, the product stream contains no H2 , CO, O2 , or fuel. For example, the complete combustion of propane (C3 H8 ) is C3 H8 + xO2 ↔ yCO2 + zH2 O

(11.10)

No H2 , CO, O2 , or C3 H8 is produced. According to the conservation of mass, the number of moles of H, C, and O must be equal on both sides of the equation. Then, we obtain C3 H8 + 5O2 ↔ 3CO2 + 4H2 O

(11.11)

The minimum quantity of O2 required is 5 mol O2 ∕mol C3 H8 . This minimum quantity of O2 required for complete combustion is called the stoichiometric amount of O2 . In POX (or partial combustion), a HC fuel combines with less than the stoichiometric amount of O2 such that the incomplete combustion products CO and H2 are formed. For example, the incomplete combustion of propane (C3 H8 ) is C3 H8 + xO2 ↔ yCO + zH2

(11.12)

According to the conservation of mass, we then obtain C3 H8 + 1.5O2 ↔ 3CO + 4H2

(11.13)

The quantity of O2 required is 1.5 mol O2 ∕mol C3 H8 , far less than the stoichiometric amount. Operating with less than the stoichiometric amount of O2 is also called operating fuel rich or O2 deficient. More generally, for any HC fuel, POX is defined as Cx Hy + 12 xO2 ↔ xCO + 12 yH2

(11.14)

As with SR, the H2 yield can then be further increased by shifting the CO in the outlet to H2 via the WGS reaction: CO + H2 O(g) ↔ CO2 + H2

(11.15)

The primary reactions of the POX reforming process for methane gas are listed in Table 11.4. Example 11.2 An idealized POX fuel reformer consumes methane (CH4 ) and air. (1) What is the maximum H2 yield? (2) What quantity of heat is released by the reaction if it were to take place at STP? (3) Using Equation 10.14 from Chapter 10, what is the fuel reformer efficiency? The HHV of methane is 55.5 MJ∕kg (890 MJ∕kmol) and the HHV of H2 is 142 MJ∕kg (286 MJ∕kmol) at STP.

401

402

FUEL PROCESSING SUBSYSTEM DESIGN

Solution: 1. Operating on air, for every mole of O2 we have 3.76 mol N2 , such that Cx Hy + 2x (O2 + 3.76N2 ) ↔ xCO + 12 yH2 + 1.88xN2

(11.16)

Then, for methane, CH4 + 12 (O2 + 3.76N2 ) ↔ CO + 2H2 + 1.88N2

(11.17)

Then, the reaction has a hydrogen yield of yH2 =

2 mol H2 = 0.41 2 mol H2 + 1 mol CO + 1.88 mol N2

(11.18)

or 41%. Because O2 in air is involved in the reaction, the outlet stream is diluted by N2 in air, and therefore the H2 yield is the lowest of the first three reforming types discussed. 2. According to Table 11.4, 35.7 kJ∕mol CH4 is released by the exothermic reaction at STP. 3. The fuel reformer efficiency in terms of HHV is 𝜀FR =

ΔH(HHV),H2 ΔH(HHV),fuel

=

2 kmol H2 (286 MJ∕kmol H2 ) = 0.64 1 kmol CH4 (890 MJ∕kmol CH4 )

(11.19)

or about 64%. 11.1.3

Autothermal Reforming (AR)

Autothermal reforming combines (1) the SR reaction, (2) the POX reaction, and (3) the WGS reaction in a single process. Autothermal reforming combines these reactions such that (1) they proceed in the same chemical reactor and (2) the heat required by the endothermic SR reaction and the WGS reaction is exactly provided by the exothermic POX reaction. Autothermal reforming incorporates SR by including steam as a reactant. Similarly, it incorporates POX by including a substoichiometric amount of O2 as a reactant. The AR reaction is ( ) ( ) Cx Hy + zH2 O + x − 12 z O2 ↔ xCO2 + z + 12 y H2 ⇒ CO, CO2 , H2 , H2 O

(11.20)

The value for the steam-to-carbon ratio, here shown as z∕x, should be chosen such that the reaction is energy neutral, neither exothermic nor endothermic.

FUEL REFORMING OVERVIEW

Example 11.3 (1) For methane (CH4 ), estimate the steam-to-carbon ratio that enables the AR reaction to be energy neutral. Assume that H2 O enters as a liquid and the only products are CO2 and H2 . For simplicity, assume that the reactants and products enter and leave the reactor at STP. (2) What is the H2 yield? (3) What is the reformer efficiency? Solution: 1. As shown in Example 11.1, for the endothermic SR+WGS reaction, we have CH4 + 2H2 O(1) ↔ CO2 + 4H2 + 253.4 kJ∕mol CH4

(11.21)

As shown in Table 11.4, for the exothermic POX reaction, we have CH4 + 12 O2 ↔ CO + 2H2 − 35.7 kJ∕mol CH4

(11.22)

For the products of these combined reactions to produce only CO2 and H2 , the CO in the POX reaction must be shifted to H2 via the WGS reaction. Table 11.5 shows the solution to this problem. Table 11.5 shows the SR+WGS (1), POX (2), and WGS (3) reactions and the heat of reaction for each. By adding reaction 2 (POX) to reaction 3 (WGS), we get reaction 4, in which the CO is removed so that only CO2 and H2 are products. The enthalpy of reaction for each reaction also adds. We calculate that reaction 4 would have to take place 7.73 times for the energy it releases to equal the energy consumed by reaction 1. This is shown as reaction 5. We add reactions 5 and 1 to attain reaction 6, which has an enthalpy of reaction of zero. We normalize reaction 6 by dividing by the number of moles of CH4 to attain reaction 7. According to reaction 7, the steam-to-carbon ratio is nH2 O S = 1.115 = C nc

(11.23)

z = 1.115

(11.24)

and

2. What is the H2 yield? Operating on air, for every mole of O2 , we have 3.76 mol N2 . For the 0.44 mol O2 at the intake, we must also have 1.66 mol N2 . Then, the reaction has a H2 yield of yH2 =

3.11 mol H2 = 0.54 3.11 mol H2 + 1 mol CO2 + 1.66 mol N2

(11.25)

or 54%. Because oxygen in air is involved in the reaction, the outlet stream is diluted by N2 from the air. The presence of N2 decreases the H2 yield. However,

403

404

FUEL PROCESSING SUBSYSTEM DESIGN

the presence of water vapor as a reactant increases the H2 yield. As a result, the H2 yield is lower than for SR but higher than for POX. This result can be quantified by comparing the hydrogen yields calculated in Examples 11.1, 11.2, and 11.3. 3. The fuel reformer efficiency in terms of HHV is 𝜀FR =

ΔH(HHV),H2 ΔH(HHV),fuel

=

3.11 kmol H2 (286 MJ∕kmol H2 ) = ∼1 1 kmol CH4 (890 MJ∕kmol CH4 )

(11.26)

or about 100%. TABLE 11.5. Solution for Example 11.3 Reaction Number Reaction Type

Chemical Formula

1

SR+WGS

1CH4 + 2H2 O(l) → 1CO2 + 4H2

2

POX

1CH4 + 0.5O2 → 2H2 + 1CO

3

WGS

1H2 O(l) + 1CO → 1CO2 + 1H2

4

POX + WGS

1CH4 + 1H2 O(l) + 0.5O2 → 1CO2

Δĥ 0rxn (kJ/mol) +253.4 −35.7 +2.9 −32.8

+ 3H2 5

(POX + WGS) × 7.73

7.73CH4 + 7.73H2 O(l) + 3.86O2 → 7.73CO2 + 23.2H2

−253.4

6

(POX + WGS) × 7.73 + (SR + WGS)

8.73CH4 + 9.73H2 O(l) + 3.86O2 → 8.73CO2 + 27.2H2

0.0

7

[(POX + WGS) × 7.73 + (SR + WGS)]∕8.73

1CH4 + 1.115H2 O(l) + 0.44O2 → 1CO2 + 3.11H2

0.0

Note: Calculation of the appropriate steam-to-carbon ratio for autothermal reforming of methane. Autothermal reforming combines steam reforming (SR), partial oxidation (POX), and the water gas shift (WGS) reactions to achieve neutral energy balance.

Example 11.4 You are designing a hydrogen generator to supply fuel cell vehicles with gaseous hydrogen. You want to use methane from nearby pipelines and liquid water from the utility as inputs. You choose the SR reaction as your primary fuel reforming method because of its high hydrogen yield. However, the endothermic SR reaction requires heat. To supply this heat, you design your steam reformer to burn some methane fuel. (1) Perform a back-of-the-envelope calculation to estimate the minimum quantity of methane fuel you must burn to provide enough heat for the steam reformer. Assume that heat transfer between your methane burner and the steam reformer is 100% efficient. Assume that the SR reactions achieve

FUEL REFORMING OVERVIEW

maximum H2 yield, as in Example 11.1. Assume complete combustion of CH4 with the stoichiometric quantity of O2 . For simplicity, we assume that the reactions take place at STP. The HHV of methane is 55.5 MJ∕kg (890 MJ∕kmol), and the HHV of H2 is 142 MJ∕kg (286 MJ∕kmol) at STP. (2) Calculate the reformer efficiency (𝜀FR ) in terms of HHV. Solution 1. Assuming perfect heat transfer, based on the conservation of energy, the heat released by the exothermic reaction (Qout ) will equal the heat absorbed by the endothermic reaction (Qin ), (11.27) Qin = Qout whereby

nCH4 ,SR (Δĥ 0rxn )SR = nCH4 ,C (Δĥ 0rxn )C

(11.28)

where nCH4 ,SR is the number of moles of CH4 consumed by the SR reaction, (Δĥ 0rxn )SR is the heat of reaction for the SR reaction, nCH4 ,C is the moles of CH4 consumed by the combustion reaction, and (Δĥ 0rxn )C is the heat of reaction for the combustion of CH4 . Then, the ratio of nCH4 ,C to nCH4 ,SR is nCH4 ,C nCH4 ,SR

=

(Δĥ 0rxn )SR (Δĥ 0rxn )C

(11.29)

Therefore, the ratio of the masses depends on the heats of reaction. According to Table 11.3, the SR reaction CH4 + 2H2 O(g) ↔ CO2 + 4H2

(11.30)

requires 165.2 kJ energy∕mol CH4 at STP. However, this reaction also assumes that H2 O is in vapor form [as indicated by the (g) for gas]. Because we are obtaining our H2 O in liquid form, we need to raise liquid H2 O to steam, according to the phase change reaction H2 O(1) → H2 O(g)

(11.31)

which requires +44.1 kJ energy∕mol H2 O. Therefore, in total, for every mole of CH4 reformed, we need to supply (Δĥ 0rxn )SR = 165.2 kJ∕mol CH4 + 44.1 kJ∕mol H2 O × 2 mol H2 O ∕ mol CH4 = 253.4 kJ∕mol CH4

(11.32)

405

406

FUEL PROCESSING SUBSYSTEM DESIGN

for the reaction, as also shown in Example 11.1. According to Table 11.4, the combustion of CH4 is CH4 + 2O2 ↔ CO2 + 2H2 O

(11.33)

which releases –890 kJ/mol CH4 = (Δĥ 0rxn )C if water is produced in the liquid state and reactants and products are at STP. This value is the same value as the HHV for methane. Therefore, nCH4 ,C nCH4 ,SR

=

253.4 kJ∕mol CH4 = ∼ 0.285 −890 kJ∕mol CH4

(11.34)

The moles, mass, or volume of CH4 needed for combustion is at a minimum about 28.5% of the moles, mass, or volume of CH4 consumed by the steam reformer. 2. The fuel reformer efficiency in terms of HHV is 𝜀FR =

ΔH(HHV),H2 ΔH(HHV),fuel

=

4 mol H2 (286 kJ∕mol H2 ) = ∼1 1.285 mol CH4 (890 kJ∕mol CH4 )

(11.35)

or about 100%. Example 11.5 In reference to Example 11.4, design engineers wish to avoid consuming 28.5% additional CH4 for combustion to provide heat to the SR reaction. (1) What other approaches might be considered to avoid this additional fuel consumption and to lower the net carbon dioxide (i.e., greenhouse gas) emissions released? Chapter 14, Section 14.3, discusses the impact of greenhouse gases on the environment. (2) At what temperature does heat need to be provided to heat the steam reformer? (3) What types of fuel cells may produce heat at high enough temperature and in great enough quantity to displace a CH4 combustor? Solution: 1. To avoid this additional fuel consumption, heat could be provided by other sources. A few heat source examples include (a) recoverable heat from an industrial process in close proximity to the SR that would otherwise be dissipated to the environment, (2) solar thermal heat from concentrating solar collectors, and (3) recoverable heat from a high-temperature fuel cell stack within a fuel cell system. All of these options would reduce net carbon dioxide emissions compared with burning 28.5% additional CH4 . 2. The steam reforming reaction’s operating temperature range is 700–1000∘ C, as shown in Table 11.1. At least a portion of the heat for the steam reformer needs to be provided at a temperature that is slightly above the SR’s operating temperature, due to the second law of thermodynamics, which was discussed in Chapter 2, Section 2.1.4.

FUEL REFORMING OVERVIEW

3. MCFCs and SOFCs. Indeed, this approach can be one of the primary sources of overall system efficiency gain for MCFC and SOFC systems, compared with PAFC or PEMFC systems which have stacks that operate at temperatures much lower than that required for SR. 11.1.4

Gasification

Stationary fuel cell systems may also utilize fuel gases produced from solid fuels through a process known as gasification. The process of gasification typically reacts a solid fuel containing carbon (such as coal) at high temperature (700–1400∘ C) under pressure with O2 and H2 O to produce H2 , CO2 , CO and other gases. For a fuel containing carbon (C), the overall (unbalanced) gasification reaction is C + aO2 + bH2 O ↔ cCO2 + dCO + eH2 + other species

(11.36)

The carbon fuel first undergoes devolatilization, a process by which a portion of the original fuel thermally decomposes into a complex gaseous mixture, with a porous solid char residue. The gaseous mixture then undergoes a combination of partial oxidation, steam reforming, and water gas shift reactions as discussed previously. The char particles are gasified to CO through partial oxidation of carbon, C + 12 O2 ↔ CO

(11.37)

C + H2 O ↔ CO + H2

(11.38)

and steam reforming of carbon,

Some of the CO further reacts through the water gas shift reaction, CO + H2 O ↔ CO2 + H2

(11.39)

The energy required to break the O–H bonds in H2 O for the endothermic steam reforming is typically provided by the energy released from the exothermic partial oxidation reaction of carbon in the fuel. For coal fuel (Cx H0.93x N0.02x O0.14x S0.01x ), the overall gasification reaction is Cx H0.93x N0.02x O0.14x S0.01x + (0.955x − 0.5z − r)O2 + zH2 O ↔ (x − r)CO2 + (0.465x + z)H2 + 0.02xNO2 + 0.01xSO2 + rC(s)

(11.40)

after all CO has been shifted to CO2 through the water gas shift reaction. The term r is the moles of solid carbon char produced and the term z∕x is the steam-to-carbon ratio chosen for this process. Because a significant percentage of the product H2 is derived from the H in reacting H2 O, the steam-to-carbon ratio chosen for operation can highly influence that particular coal gasification plant’s carbon dioxide (i.e., greenhouse gas) emissions, especially

407

408

FUEL PROCESSING SUBSYSTEM DESIGN

CO2 per unit of H2 produced. Chapter 14, Table 14.1, quantifies the carbon content of common fuels, including coal, which has the highest carbon content per unit energy and the highest carbon content per unit mass of atomic hydrogen. At the same time, the more H2 O added to the coal gasification process, the more energy needed to raise the liquid H2 O to steam and to break the O–H bonds. This energy can be provided by the partial oxidation of coal or by heating from an external source. To attenuate CO2 emissions, some of this energy may be able to be provided by (1) recovered heat from a fuel cell stack in an upstream fuel cell system, (2) recovered heat from industrial processes, (3) solar thermal devices, or (4) geothermal heating, especially if used in conjunction with lower temperature gasifiers. In practice, z/x may be higher than the stoichiometric amount needed for H2 production so as to supply an excess amount of unreacted water for cooling. This excess water is used mainly to moderate the temperature of the gasifier at a high enough oxygen-to-carbon ratio (O∕C) to obtain a reasonably high conversion of the feed (>95%). A cleaned gas stream from a coal gasification process can be consumed as fuel by a fuel cell system. Gasifiers can achieve high efficiencies. Gasification efficiency is defined in the same way as fuel processor subsystem efficiency, the quotient of the HHV of H2 in the output gas (ΔH(HHV),H2 ) over the HHV of input fuel (ΔH(HHV),fuel ), including fuel consumed to provide energy for the gasification process itself. According to this definition, state-of-the-art coal gasifiers produce H2 with an efficiency of 75%, with the remainder of the energy converted to heat. These coal gasification plants also have an efficiency loss associated with their electrical power consumption, mainly due to the air separation unit, equal to about 6% of the HHV of the inlet fuel (about 5% is due to air separation). This ancillary load efficiency loss does not include the additional efficiency lost during electric power generation.

11.1.5

Anaerobic Digestion (AD)

Stationary fuel cell systems may also consume anaerobic digester gas (ADG), commonly considered a renewable fuel. ADG is primarily a mixture of CH4 and CO2 that results from bacteria feeding off biodegradable feedstock such as livestock manure, sewage, municipal waste, biomass, energy crops, or food-processing waste. Anaerobic (meaning oxygen-free) digestion (AD) is the process that converts these biodegradable materials into a gaseous mixture in the absence of gaseous oxygen, at ambient or slightly elevated temperatures (70∘ C). First developed by a leper colony in Bombay, India, in 1859, AD facilities are today installed at modern dairy farms, which have a large cattle manure supply, and wastewater treatment plants, which coalesce and treat human waste. AD consists of a series of chemical reactions that progressively break down the plant or animal matter. First, through a process known as hydrolysis, carbohydrates, fats, and proteins in the biological matter chemically react with water and decompose into shorter chain molecules such as sugars, fatty acids, and amino acids. Then, different types of bacteria progressively decompose these molecules into even shorter chained acids, alcohols, and gases. These reactions can be represented by an overall reaction for the breakdown of glucose (C6 H12 O6 ) into CH4 and CO2 : C6 H12 O6 ↔ 3CO2 + 3CH4

(11.41)

WATER GAS SHIFT REACTORS

The typical gas output of AD may include not only CH4 and CO2 but also N2 , H2 , hydrogen sulfide (H2 S), and O2 . ADG can vary widely in composition, as the feedstock composition changes. A typical ADG composition can be CH4 (56%), CO2 (36%), N2 (5%), H2 (0.5%), H2 S (1.5%), and O2 (1%). Once contaminants (such as H2 S) are removed (scrubbed), ADG can be consumed directly in high-temperature fuel cell systems. In low-temperature fuel cell systems, additional chemical conversion is needed. The CH4 in the scrubbed ADG can be directly consumed in a SOFC or MCFC. In these fuel cells, the CH4 may undergo steam reforming into H2 and CO at the anode’s catalytic surface, with primarily the H2 subsequently undergoing electrochemical oxidation at the same surface. Alternatively, the scrubbed ADG could be fed to a PAFC or PEMFC system. This gas must undergo fuel reforming via SR, POX, or AR, as described in previous sections, with the resulting H2 -rich gas then consumed by the PAFC or PEMFC. Because the recoverable heat from PAFC and PEMFC stacks are typically too low in temperature to provide heat for these reforming processes, additional high-temperature heat would need to be added with these systems and this could lower the overall efficiency of energy conversion. By contrast, the recoverable heat of MCFC and SOFC systems is sufficiently high in temperature to provide heat for internal or external reforming. An additional advantage of using the ADG with a MCFC is that the large concentration of CO2 diluting the anode’s supply gas is balanced by a large concentration of CO2 at the cathode, such that these high CO2 concentrations on either side of the electrolyte offset each other and some performance losses are avoided. ADG is considered a renewable fuel for several reasons. First, the feedstock sources for ADG are typically (1) human bodily waste, (2) agricultural waste, or (3) food-processing waste. Second, if the biodegradable feedstock decays on its own, it can release CH4 into the atmosphere. As will be discussed in Chapter 14, CH4 is a greenhouse gas with 23 times the global warming impact as CO2 over a 100-year period. If the CH4 is not released but rather converted into CO2 via aerobic digestion, combustion, or electrochemical oxidation in a fuel cell, the released gas will have roughly 23 times less global warming impact over a 100-year period. Third, ADG is considered a renewable fuel also because it can replace fossil fuels in energy conversion devices (power plants, etc.) and therefore displace fossil-fuel-derived greenhouse gas emissions as well. Finally, the solid residue from ADG can be used as fertilizer and, in so doing, can displace the energy and greenhouse gas emissions associated with the highly energy-intensive process of manufacturing fertilizer.

11.2

WATER GAS SHIFT REACTORS

After bulk conversion of H2 in the fuel reforming stage, the reformate is usually sent through a WGS reactor. For example, in the fuel processor subsystem design shown in Figure 11.1, after the catalytic fuel reformer (labeled 3), the reformate enters a WGS reactor (4). The overall goals for the WGS reactor are to (1) increase the H2 yield in the reformate stream and (2) decrease the CO yield. (Even small CO levels can damage certain types of fuel cells, such as PEMFCs, which tolerate less than 10 ppm of CO.) We have already seen how WGS can increase H2 yields. We now examine the WGS reaction in more detail and discuss how it can also lower the CO yield in the reformate stream.

409

410

FUEL PROCESSING SUBSYSTEM DESIGN

The WGS reaction reduces the CO yield in the reformate stream by the same percentage that it increases the H2 yield. The CO yield (yCO ) is the molar percentage of CO in the reformate stream, n yCO = CO (11.42) n where nCO is the number of moles of CO in the reformate stream and n is the total number of moles in the reformate stream. The WGS reaction can reduce the CO yield to a range of 0.2–1.0% molar concentration, typically in the presence of a catalyst. CATALYST DEACTIVATION Catalysts can deactivate via several methods, including sintering and poisoning, both of which are a concern in WGS reactors. 1. Sintering is a process in which the surface area of a catalyst decreases under the influence of high temperatures. Exposed to high temperatures, catalyst particles will achieve a lower energy state by merging together to reduce their surface area. Over time, the reactor’s catalyst will therefore lose activity. For example, a WGS reactor may use a copper and zinc oxide catalyst supported on alumina. The zinc oxide molecules create a physical barrier that impedes the copper molecules from merging together. However, if the temperature is too high, the copper molecules can merge anyway. Thus, even a single high-temperature event can inactivate a reactor. For example, exposed to operating temperatures of 700∘ C, a catalyst’s active surface area can decrease by a factor of 20 within the first few days of operation. Lower temperature operation reduces sintering because the copper molecules are less mobile. 2. Poisoning is essentially the chemical deactivation of a catalyst surface. For example, chemical impurities like sulfur can aggregate onto catalyst particles and deactivate them by blocking reaction sites. Poisoning reduces the activity of the catalysts at the front of the reactor first. The WGS reactor is particularly susceptible to sulfur poisoning.

If water enters as a vapor, the WGS reaction is slightly exothermic: CO + H2 O(g) ↔ CO2 + H2

Δĥ r (25∘ C) = −41.2 kJ∕mol

(11.43)

According to Le Chatelier’s principle, because the WGS reaction is exothermic, at high temperatures, the balance is skewed towards the reactants (CO and H2 O). At low temperatures, the balance is skewed towards the products (CO2 and H2 ). Therefore, at low temperatures, the reaction increases its H2 yield. However, at high temperatures, the reaction rate

CARBON MONOXIDE CLEAN-UP

is higher. Chapter 3 discusses reaction kinetics in greater detail. To achieve the benefits of both a high H2 yield at equilibrium and fast kinetics, the WGS process may be designed to proceed in two or more stages. First, the WGS reaction proceeds at high temperature in one reactor to achieve a high reaction rate. Second, in a second reactor downstream of the first, the WGS reaction proceeds at low temperatures to increase the H2 yield. Also according to Le Chatelier’s principle, excess water vapor in the inlet shifts the reaction equilibrium to favor a higher H2 yield. (Chapter 2, Section 2.4.2, first introduced Le Chatelier’s principle.) 11.3

CARBON MONOXIDE CLEAN-UP

Even after high- and low-temperature WGS processing, the amount of CO in the reformate stream is still too high for some low-temperature fuel cells. For example, the most advanced PEMFC catalysts can withstand a CO yield of only 100 ppm or less, while WGS will typical leave 0.2% (2000 ppm) or more CO in the reformate stream. As a result, in fuel processor subsystem designs like the one shown in Figure 11.1, the reformate stream must often pass through a “CO clean-up reactor” (labeled 5). The overall goal of this CO clean-up process is to reduce the CO yield to extremely low levels. This goal can be achieved by either (1) chemical reaction or (2) physical separation. In chemical reaction processes, another species reacts with CO to remove it. Two such processes are 1. Selective methanation of CO 2. Selective oxidation of CO In both cases, the term selective means that a catalyst is used to promote one reaction that removes CO and to suppress another reaction that would otherwise consume H2 . In physical separation processes, either CO or H2 is physically removed from the gas stream by selective adsorption or selective diffusion. Two such processes are 1. Pressure swing absorption 2. Palladium membrane separation These four CO clean-up processes are explained in the next four sections.

11.3.1

Selective Methanation of Carbon Monoxide to Methane

In selective methanation, a catalyst selectively promotes one reaction that removes CO over another that might otherwise consume H2 . Selective methanation promotes the CO methanation reaction, CO + 3H2 ↔ CH4 + H2 O

Δĥ r (25∘ C) = −206.1 kJ∕mol

(11.44)

411

412

FUEL PROCESSING SUBSYSTEM DESIGN

TABLE 11.6. Chemical Removal of CO from Reformate Stream

Reaction Type

Chemical Reaction

Δĥ 0rxn (kJ/mol)

1. Selective methanation

CO + 3H2 ↔ CH4 + H2 O

–206.1

✓

CO2 + 4H2 ↔ CH4 + 2H2 O

–165.2

x

CO + 0.5O2 ↔ CO2

–284.0

✓

H2 + 0.5O2 ↔ H2 O

–286.0

x

2. Selective oxidation

Catalyst Promotes (✓) or Suppresses (x) Reaction?

Note: Catalysts selectively promote the consumption of CO over the consumption of H2 .

over the CO2 methanation reaction, CO2 + 4H2 ↔ CH4 + 2H2 O Δĥ r (25∘ C) = −165.2 kJ∕mol

(11.45)

The first reaction reduces the CO yield and the H2 yield. The second reaction consumes even more H2 while not reducing the CO yield. Therefore, a selective methane catalyst tries to promote the first reaction while suppressing the second. This relationship is summarized in Table 11.6. Selective methanation is only an option when the CO concentration in the reformate stream is low, because even the promoted reaction consumes H2 . 11.3.2

Selective Oxidation of Carbon Monoxide to Carbon Dioxide

In selective oxidation, a catalyst selectively promotes a reaction that removes CO over another that consumes H2 . Selective oxidation promotes the CO oxidation reaction, CO + 0.5O2 ↔ CO2 Δĥ r (25o C) = −284 kJ∕mol

(11.46)

over the H2 oxidation reaction, H2 + 0.5O2 ↔ H2 O Δĥ r (25∘ C) = −286 kJ∕mol

(11.47)

The first reaction decreases the CO yield while the second decreases the H2 yield. Chapter 2, Section 2.1.5, first introduced the concept of the change in Gibbs free energy. The change in Gibbs free energy (ΔGrxn ) for the CO reaction is increasingly more negative at lower temperatures, indicating a stronger driving force for that reaction at lower temperatures. Consequently, at lower temperatures, a higher percentage of CO adsorbs onto the catalyst surface. There, the CO blocks H2 adsorption and oxidation. According to Le Chatelier’s principle, more CO adsorbs at higher CO concentrations. As a result, CO is typically removed via a series of consecutive selective oxidation catalyst beds, each of which operates at increasingly lower temperatures and lower CO concentrations. The decrease in CO adsorption due to lower concentrations in the later catalytic reactors is offset by the increasing effectiveness of CO adsorption from lower temperature operation.

CARBON MONOXIDE CLEAN-UP

Example 11.6 You need to remove 0.2% CO molar concentration from your reformate stream. (1) You decide to use the methanation process. You have developed a catalyst that is 100% selective for the methanation of CO reaction. How much H2 is consumed? (2) To remove the same CO, you decide to try the selective oxidation process and can use a catalyst that is 100% selective for the oxidation reaction of CO. How much H2 is consumed? Solution: 1. For a methanation catalyst with 100% selectivity for CO, the removal of each molecule of CO still consumes three H2 molecules, a process that wastes desired hydrogen. For the 0.2% of CO removed from the stream, the H2 that is also removed is 0.6% of the total mixture. 2. For an oxidation catalyst with 100% selectivity for CO, all 0.2% of CO can be removed while no H2 is removed. 11.3.3

Pressure Swing Adsorption

Pressure swing adsorption (PSA) is a physical CO separation process. PSA removes not only CO, but also all other species except H2 . It can produce a 99.99% pure H2 stream. In a PSA system, all of the non-H2 species in the reformate stream (such as HCs, CO, CO2 , and N2 ) preferentially adsorb onto a high-surface-area adsorbent bed composed of zeolites, carbons, or silicas. The heat of adsorption characterizes the strength of surface–solute interactions, which are driven in part by the molecular weights of the adsorbing species. Only hydrogen passes through the bed unadsorbed due to its low molecular weight compared with all other species; the molecular weight of H2 is 2.016 g/mol whereas all other molecules have a higher molecular weight. As a result, these beds adsorb most other species compared with H2 . Secondary determinants of adsorption include the molecule’s polarity and shape. A PSA unit operates with at least two such adsorption beds. Each adsorption process is a batch process. As a result, to have a continuous flow of reformate purified, at least two beds must operate in parallel: While one adsorbs impurities, the other desorbs. After one bed is saturated with all non-H2 species, this saturated bed is isolated from fresh reformate by closing the entrance valve. Fresh reformate is diverted to a second, unsaturated adsorbent bed, where the same adsorption process occurs. At the same time, non-H2 species are removed from the first (saturated) bed via three regeneration steps: (1) depressurization, (2) purging, and (3) repressurization. The first step (depressurization) releases the non-H2 species, because the adsorbent bed holds less material at lower pressures. The second step (purging) removes the non-H2 species from the adsorbent vessel. The third step (repressurization) ensures that the bed will be ready for the next batch of reformate. The two beds oscillate between adsorption and desorption such that reformate can be continuously purified [124]. The process of reducing the pressure of the bed to reduce its adsorptive ability and then repressurizing it is called the pressure swing mechanism [125]. Parasitic power for the PSA includes electrical power needed to run compressors to pressurize inlet gases to the PSA. However, in most cases, the parasitic power required to operate a PSA is negligible; the PSA’s control system consumes only a small fraction of the fuel processor subsystem’s electrical load.

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11.3.4

Palladium Membrane Separation

Palladium–silver alloy membranes filter out pure H2 . Different species in a gas can permeate a membrane at different rates. The H2 molecules can diffuse through a palladium membrane at a faster rate than other species, such as CO, N2 , and CH4 , due to the lattice structure of palladium metal [126]. The H2 yield from a palladium membrane depends on its (1) pressure differential, (2) operating temperature, and (3) thickness: 1. The hydrogen flux through the membrane can be increased by increasing the pressure drop across the membrane such that a higher density of hydrogen molecules permeates the membrane. A high pressure drop drives H2 molecules through the membrane and produces low-pressure H2 . 2. Hydrogen flux also can be increased by increasing the operating temperature. Higher temperatures increase the permeation kinetics, because the rates of processes governed by activation energies change exponentially with temperature. Chapter 3, Section 3.1.7, first introduces the concept of activation energies. The kinetics of permeation is controlled by bulk diffusion at low temperatures and surface chemisorption at high temperatures [127]. At higher temperatures, the palladium material changes to the 𝛼 phase, which has a substantially higher hydrogen solubility and therefore permits a higher amount of hydrogen molecules to permeate. 3. In addition to the pressure differential and the operating temperature, the thickness of the membrane affects its performance. Hydrogen molecules need to do less work to diffuse through a thin membrane, although thinner membranes may be more delicate and susceptible to leaks. According to Sievert’s law, which describes the bulk diffusion of species across a pressure differential through a thickness, the normalized flux (the product of the flux and the thickness) should be independent of the thickness if processes are controlled by bulk diffusion. In practice, Sievert’s law does not usually hold. High H2 yield is limited by (1) purging and (2) leaks. Hydrogen yield is limited by the need to purge the gas stream, which releases some H2 . As the palladium membrane allows H2 gas to filter through it, non-H2 species that have not passed through the membrane build up at its surface. As a result, the concentration of H2 at the surface declines. In most designs, to increase the concentration of H2 at the surface, this gas stream is periodically purged just for a moment; both H2 and non-H2 species are intentionally released from the system. Periodic purging of the gas stream increases the concentration of H2 at the palladium membrane’s surface and therefore the partial pressure of H2 and the hydrogen flux through it. Hydrogen yield is also limited by pinhole leaks in the membrane that reduce gas purity. 11.4

REFORMER AND PROCESSOR EFFICIENCY LOSSES

The primary source of efficiency loss in fuel reformers and fuel processors is heat loss. Heat is lost partly through radiative, conductive, and convective heat transfer from the reactors

REFORMER AND PROCESSOR EFFICIENCY LOSSES

to the surrounding environment [128]. Heat is also lost via unrecovered heat in the thermal mass of the exiting product gas stream. A secondary source of efficiency loss is associated with incomplete chemical conversion. For higher temperature reformers, one of the most important sources of efficiency loss is radiative heat transfer (qR ), described by qR = F𝜀 FG 𝜎A(T1 4 − T2 4 )

(11.48)

where F𝜀 is the emissivity (the degree to which an emitting/receiving surface resembles an ideal black body surface), FG is the geometric view factor between the surfaces, 𝜎 is the Stefan–Boltzmann constant (5.669 × 10–8 W∕m2 ⋅ K4 ), and T1 and T2 are the temperatures of the two surfaces. Reformers are typically enveloped in insulation to reduce heat loss via conduction (qC ) through reactor walls and piping and via free convection (qV ) from the reactor’s outer surface to the ambient environment. Heat loss via conduction (qC ) can be described by 𝜕T (11.49) qC = −kA 𝜕x where k is the thermal conductivity of the reactor or piping material, A is the cross-sectional area perpendicular to the direction in which heat is transferred, and 𝜕T∕𝜕x is the temperature gradient in the direction of heat flow. Heat loss via free convection (qV ) can be described by qV = hA(T𝑤 − T∞ )

(11.50)

where h is the convective heat transfer coefficient, A is the surface area in contact with the convective fluid (typically air), and T𝑤 − T∞ is the temperature difference between the reactor wall and the fluid. A significant source of fuel reformer efficiency loss also can be due to unrecovered heat from the exiting product gas. In the absence of preheating, the incoming reactants will enter a fuel reformer at a much lower temperature than the outgoing products. This temperature difference between the inlet fuel and the outlet products can be a source of significant thermal losses if this heat is not recovered and reused either internally within the fuel cell system or externally to supply heat to some useful purpose (such as space or hot water heating for a building). An economic trade-off exists between the cost of the additional heat exchangers needed to capture this available heat and the financial value of the heat itself. The crucial aspects associated with effective heat management will be discussed in detail in Chapter 12. A secondary source of efficiency loss is attributable to incomplete chemical conversion. Incomplete conversion refers to the fact that all hydrogen atoms in a reactant fuel may not be converted to molecular H2 . Incomplete chemical conversion can take place if any of a fuel processor’s reactors are poorly designed or operated. A reactor may be poorly designed if it does not contain enough catalyst surface area to allow a reaction to proceed to completion. A well-designed reactor should include some extra catalyst to mitigate the effects of catalyst sintering and loss of active surface area with time. A reactor also may be poorly designed if the thermal management around it cannot maintain the reactor’s temperature within its design range. Temperature excursions above design points increase the rate

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of catalyst sintering and may render the reactor entirely dysfunctional. A reactor may be poorly operated if the temperature, pressure, and inlet compositions required for complete fuel conversion are not maintained. Operation at off-design-points is more likely to happen if the fuel processor is rapidly cycled between high and low throughput levels or dynamically operated at different output rates. In fuel processors that are well designed and that are operated carefully at a steady throughput rate, incomplete chemical conversion is not a major efficiency loss. Example 11.7 A catalytic partial oxidation reactor is 5 cm in diameter and 40 cm2 in its outer surface area. The reactor’s peak internal temperature is 1100∘ C, its wall temperature is 1000∘ C, and the surrounding air is 30∘ C. No insulation is covering the walls of the reactor. The convective heat transfer coefficient for a 5-cm-diameter horizontal cylinder in air for free convection is 0.00065 W∕cm2 ∘ C. (1) Using Equation 11.73, calculate the heat loss via free convection. Solution: 1. According to Equation 11.50, the heat loss via free convection (qV ) is 0.00065 W∕cm2 ⋅ ∘ C [40 cm2 (1000∘ C – 30∘ C)] = ∼ 25 W. 11.5

REACTOR DESIGN FOR FUEL REFORMERS AND PROCESSORS

Experimental data on catalyst performance can be used to appropriately size chemical reactors. When a fluid passes through a catalyst bed, the fluid requires a certain residence time (𝜏) to react with the catalyst. As the activity of a catalyst increases, the gas needs a lower 𝜏 to react to completion. Experimental data can indicate an appropriate range of 𝜏 for a certain percentage of reactant conversion. This data is specific to a certain reactor type, reactant phase, catalyst type, and operating temperature and pressure. Based on the required 𝜏 for ̇ of fluid passing through the chemical conversion and the desired volumetric flow rate (V) reactor (or volumetric throughput), the desired reactor volume V is V = 𝜏 V̇ =

V̇ SV

(11.51)

where the inverse of 𝜏 is also referred to as the space velocity (SV). For example, experimental data show that gaseous methane and steam can be steam reformed into a hydrogen-rich gas with 100% conversion in a multitubular reactor over a nickel catalyst at 790∘ C and 13 atm with a 𝜏 of 5.4 s. Example 11.8 Experimental data show that methane and steam can be steam reformed into a hydrogen-rich gas with 100% conversion in a catalytic steam reforming reactor within a residence time (𝜏) of 4 s. (1) Calculate the space velocity (SV) associated with this reactor in units of s–1 . (2) The reactor is being designed for a maximum volumetric flow rate of 0.02 L∕s at the reactor’s operating temperature and pressure. What is the minimum reactor volume needed for 100% conversion

CHAPTER SUMMARY

in units of liters? (3) Catalysts degrade over time. Catalyst replacement can be expensive, especially in terms of labor time. The steam reforming section of a fuel processor subsystem may be difficult to access to change the catalyst bed. At the same time, catalyst materials may also be expensive, depending on the catalyst type. After taking into account these considerations, designers choose a safety factor of 3 for the volume of this catalyst bed. To accommodate for catalyst degradation and maintenance, what is the reactor volume design specification in liters? Solution: 1. The space velocity (SV) is 1∕𝜏 or 1∕4 s = 0.25 s–1 . 2. The minimum reactor volume, V, is 4 s (0.02 L∕s) = 0.08 L. 3. The reactor volume design specification is 3 (0.08 L) = 0.24 L. 11.6

CHAPTER SUMMARY

In this chapter, you learned in detail about one of the main subsystems for fuel cell systems, the fuel processing subsystem. • As discussed in Chapter 10 and reemphasized in this chapter, the term fuel processor efficiency applies to a control volume that encompasses the entire fuel processor subsystem, which may include a fuel reformer, a water gas shift reactor, carbon monoxide clean-up processes, afterburner treatment of anode and cathode off-gases, and/or other processes. By contrast, the term fuel reformer efficiency applies to a control volume that includes only the fuel reformer. • As discussed in Chapter 10 and reemphasized in this chapter, the efficiency equation for both fuel processor and fuel reformer efficiency is mathematically similar, with the control volumes being drawn around different sets of equipment. The fuel processor efficiency (𝜀FP ) is the ratio of the H2 energy based on the HHV of H2 in the output gas (ΔH(HHV),H2 ) compared with the fuel energy based on the HHV of fuel (ΔH(HHV),fuel ) in the input, including fuel consumed to provide energy for the fuel processor itself, or 𝜀FP =

ΔH(HHV),H2 ΔH(HHV),fuel

(11.52)

• Fuel reformer efficiency is defined similarly, but based on a control volume only surrounding the reformer and not the entire subsystem. • Exothermic reactions release energy; endothermic ones consume it. • Hydrogen yield yH2 is the molar percentage of H2 in a chemical stream: yH2 =

nH2 n

(11.53)

where nH2 is the number of moles of H2 and n is the total number of moles of all gases in the stream.

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• Hydrogen can be produced from a hydrocarbon (HC) fuel via five main processes: (1) steam reforming, (2) partial oxidation, (3) autothermal reforming, (4) gasification, and (5) anaerobic digestion. • Steam reforming is an endothermic reaction that combines a HC fuel with steam: Cx Hy + xH2 O(g) ↔ xCO +

(

1 y 2

) + x H2

(11.54)

• Partial oxidation is an exothermic reaction that combines a HC fuel with deficient O2 : Cx Hy + 12 xO2 ↔ xCO + 12 yH2

(11.55)

• Autothermal reforming is energy neutral and combines a HC fuel with H2 O and O2 : ( ) ( ) Cx Hy + zH2 O(1) + x − 12 z O2 ↔ xCO + z + 12 y H2

(11.56)

• In general, the hydrogen yield tends to be the highest for steam reforming, the second highest for autothermal reforming, and lowest for partial oxidation, when comparing these three fuel reforming approaches. • For a fuel containing carbon (C), the unbalanced gasification reaction is C + aO2 + bH2 O ↔ cCO2 + dCO + eH2 + other species

(11.57)

• For coal fuel (Cx H0.93x N0.02x O0.14x S0.01x ), the overall gasification reaction is Cx H0.93x N0.02x O0.14x S0.01x + (0.955x − 0.5z − r)O2 + zH2 O ↔ (x − r)CO2 + (0.465x + z)H2 + 0.02xNO2 + 0.01xSO2 + rC(s)

(11.58)

where z∕x is the steam-to-carbon ratio. • The anaerobic digestion of biodegradable materials can be approximated by the overall reaction for the breakdown of glucose (C6 H12 O6 ) into CO2 and CH4 : C6 H12 O6 ↔ 3CO2 + 3CH4

(11.59)

• The water gas shift reaction (1) increases H2 yield and (2) decreases CO yield: CO + H2 O(g) ↔ CO2 + H2

Δĥ r (25∘ C) = −42.1 kJ∕mol

(11.60)

• Based on the required residence time (𝜏) for chemical conversion and the desired ̇ of fuel passing through the reactor, the desired reactor volume volumetric flow rate (V) V is V̇ V = 𝜏 V̇ = (11.61) SV

CHAPTER EXERCISES

CHAPTER EXERCISES Review Questions 11.1

Describe five major fuel reforming processes. Discuss the benefits and limitations of each.

11.2

Compare and contrast gasification and anaerobic digestion processes. What are the typical operating temperatures, pressures, fuels, products, and energy requirements for each?

11.3

If water enters as a liquid and the heat of reaction is calculated at STP, is the water gas shift reaction considered endothermic or exothermic? If water enters as steam and the heat of reaction is calculated at STP, is the water gas shift reaction considered endothermic or exothermic?

11.4

Describe two chemical reaction processes for carbon monoxide clean-up. Discuss the benefits and limitations of each.

11.5

Describe two physical separation processes for carbon monoxide clean-up. Discuss the benefits and limitations of each.

11.6

Explain the purpose and operation of the pressure swing absorption (PSA) unit, including the reason for its name.

11.7

Label the following processes as endothermic, exothermic, or neither: (1) oxidation of hydrogen fuel in a fuel cell, (2) steam reforming, (3) partial oxidation, (4) autothermal reforming, (5) the water gas shift reaction with water entering as steam and the heat of reaction calculated at STP, (6) selective methanation, (7) selective oxidation, (8) hydrogen separation via palladium membranes, (9) pressure swing adsorption, (10) combustion of fuel cell exhaust gases, (11) condensing water vapor to liquid, (12) compression of natural gas, and (13) expansion of hydrogen gas.

11.8

Describe four sources of heat loss in fuel processor subsystems. Delineate equations for each.

11.9

Describe the impact of incomplete chemical conversion on fuel processor subsystem efficiency.

11.10 Describe the term space velocity with an equation and explain how this term can be used to design chemical reactors. Calculations 11.11 Liquid petroleum gas (LPG) is a mixture of gases primarily composed of propane (C3 H8(g) ), butane (C4 H10(g) ), or both. LPG is a common fuel for remote locations and for back-up energy systems. (a) What is the overall steam reforming equation for propane fuel? What is the steam-to-carbon ratio? What is the maximum hydrogen yield? If the water gas shift reaction followed the steam reforming reaction in series, what would be the maximum hydrogen yield?

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FUEL PROCESSING SUBSYSTEM DESIGN

(b) How does this value for maximum hydrogen yield compare with the maximum hydrogen yield for the partial oxidation of propane in air reaction that is similar to the reaction shown in Equation 11.13? If the water gas shift reaction followed this partial oxidation reaction in series, what would be the maximum hydrogen yield and how would this value compare to the other maximum hydrogen yield values calculated here? 11.12 (a) What is the overall steam reforming equation for butane (C4 H10(g) ) fuel? What is the steam-to-carbon ratio? What is the maximum hydrogen yield? If the water gas shift reaction followed the steam reforming reaction in series, what would be the maximum hydrogen yield? (b) LPG fuel can vary in gas composition. The molar concentrations of propane and butane in LPG can change over time, by location, and by fuel source. Please reference information from problem 11.11. What design features might be included in a fuel processor subsystem design based on steam reforming to accommodate varying molar compositions of propane and butane in LPG fuel? 11.13 An idealized partial oxidation fuel reformer consumes isooctane fuel (C8 H18(1) ), which is similar to gasoline, and air. What is the maximum H2 yield? 11.14 (a) What is the partial oxidation equation for ethane fuel (C2 H6 ) with air? What is the maximum hydrogen yield? If the water gas shift reaction followed the partial oxidation reaction in series, what would be the maximum hydrogen yield? (b) What is the autothermal reforming equation for ethane fuel and air? What is the steam-to-carbon ratio? What is the maximum hydrogen yield? 11.15 (a) Based on Examples 11.1 and 11.4, what is the minimum quantity of methane fuel you must burn to provide enough heat for the steam reformer assuming that the efficiency of heat exchange is only 72%? (b) Referencing Equation 10.14 from Chapter 10 (also Equation 11.19), calculate the fuel reformer efficiency 𝜀FR in terms of HHV. (c) Recalculate the fuel reformer efficiency 𝜀FR using another method. Assume the same number of moles of additional methane fuel must be burned as in (a). However, calculate the fuel reformer efficiency based on the enthalpies of combustion of methane and hydrogen at 1000 K and 1 atm (not based on their HHV at STP). The enthalpy of combustion is the difference between the enthalpy of the products and the enthalpy of the reactants, on a per mole basis, under conditions of complete combustion and constant temperature and pressure of reactants and products. Calculate these values based on water leaving as a vapor. 11.16 Building on Examples 11.4 and 11.5, calculate the potential efficiency gains from coupling a steam reformer to a SOFC or MCFC system, compared with coupling the same steam reformer to a PAFC or PEMFC system; calculate potential efficiency gains from the opportunity to reuse high-temperature heat from the SOFC or MCFC stack in place of natural gas combustion for providing heat for the steam reformer. 11.17 You would like a hydrogen generator similar to the one discussed in Example 11.4 to operate on emergency back-up fuels such as propane (C3 H8(g) ) and to use an

CHAPTER EXERCISES

autothermal reformer (not a steam reformer). For a 100% efficient reformer, specify a reasonable steam-to-carbon ratio (S∕C) and the quantity of hydrogen the reformer would produce per unit of fuel consumed. Assume that the reactants and products enter and leave the reformer at 1000 K. 11.18 Assuming that the endothermic steam reformer attains its heat from the combustion of methane, compare the ratio of hydrogen produced per unit of methane consumed for (1) a steam reformer, (2) a partial oxidation reformer, and (3) an autothermal reformer. Assume, in all three cases, that the reactants enter the reactor at 1000 K, having been preheated, and the products leave at 1000 K. 11.19 Calculate the enthalpy of reaction (at STP) for coal gasification with a S∕C of 3 using Equation 11.40 and assuming no solid carbon is formed. Calculate the S∕C for which the reaction is neither endothermic nor exothermic. Calculate the enthalpy of reaction (at STP) for the anaerobic digestion of glucose using Equation 11.41. Report the yH2 for all three. 11.20 A PEM fuel cell system produces 1.5 kWe at 32% overall net system electrical efficiency (HHV). Its cylindrical autothermal reformer is 12 cm long by 8 cm in diameter, is metallic black in color, has a surface temperature of 700∘ C, is not insulated, and is completely enclosed by a very large room with walls, ceiling, and floor maintained at 25∘ C. Assume that the autothermal reformer acts like a black body and its inlet and outlet channels are infinitely small. Ignore all other components of the fuel processor. Calculate the radiative heat losses from the reformer as a percentage of the available fuel energy. 11.21 Estimate the convective heat losses from the reformer in problem 11.20 as a percentage of the available fuel energy. Air at 25∘ C convects heat from the entire surface area of the cylinder with a heat transfer coefficient of 15 W∕m2 ⋅ K. 11.22 Estimate the reactor volume for a steam reformer serving a 5-kWe PEM fuel cell. Methane and steam react in a multitubular reactor over the same nickel catalyst described in Section 11.5 operating at 790∘ C and 13 atm, according to the steam reforming reaction in Equation 11.7. Referencing Equation 10.19 in Chapter 10, assume 𝜀R,SUB is 42%.

421

CHAPTER 12

THERMAL MANAGEMENT SUBSYSTEM DESIGN

Having learned about important components of the fuel processing subsystem in Chapter 11, we now look in detail at a second primary subsystem, the thermal management subsystem. This subsystem is used to manage heat among the fuel cell stacks, the chemical reactors in the fuel processing subsystem, and any source of thermal demand or supply internal or external to the system. The thermal management subsystem incorporates a system of heat exchangers to heat or cool system components, channeling recoverable heat from exothermic reactors (such as the fuel cell and afterburner) to endothermic ones (such as a steam generator) and to external sinks (such as a CHP fuel cell system providing heat to a building for space and hot water heating). As we discussed in Chapter 10, endothermic reactors consume energy and exothermic reactors release energy. A CHP fuel cell system with optimized heat recovery can achieve an overall efficiency 𝜀O of 80% of the fuel energy, as defined in Equation 10.12. In this section, we will learn about a methodology for managing heat within a fuel cell system so as to maximize heat recovery and meet the operating temperature ranges required by different parts of the system. We will learn about managing heat in a fuel cell system using the technique of pinch point analysis [129, 130]. The primary goal of pinch point analysis is to optimize the overall heat recovery within a process plant by minimizing the need to supply additional heating and/or cooling [131, 132]. In an ideal pinch point analysis solution, hot streams are used to heat cold ones with a minimum amount of additional heat transfer from

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an external source. Unnecessary external heat transfer increases fuel consumption and thereby decreases overall energy efficiency (𝜀O ) and profitability. The goals of maximum heat recovery and minimum supplemental energy can be met by designing a network of heat exchangers. Various permutations of these heat exchanger networks can be tested using scenario analysis and chemical engineering process plant models of the fuel cell system.

12.1

OVERVIEW OF PINCH POINT ANALYSIS STEPS

Pinch point analysis is a heat transfer analysis methodology that follows several steps: 1. Identify hot and cold streams in the system. 2. Determine thermal data for these streams. 3. Select a minimum acceptable temperature difference (dTmin,set ) between hot and cold streams. Acceptable ranges tend to vary between 3 and 40∘ C. 4. Construct temperature–enthalpy diagrams and check that the pinch point temperature, i.e., the minimum temperature difference observed between hot and cold streams (dTmin ), ≥ dTmin,set . 5. If dTmin < dTmin,set , change heat exchanger orientation. 6. Conduct scenario analysis of heat exchanger orientation until dTmin ≥ dTmin,set . These steps are illustrated below using the fuel cell system design shown in Chapter 10, Figure 10.14, as an example. This figure is repeated here for ease of reference.

12.1.1

Step One: Identify Hot and Cold Streams

1. Identify Hot and Cold Streams. A hot stream is a flowing fluid that needs to be cooled (or can be cooled). A cold stream is one that needs to be heated. In reference to the system design of Figure 12.1, we will investigate three important hot streams that require cooling: (a) The hot reformate stream exiting the water gas shift (WGS) reactor and eventually entering the fuel cell’s anode (labeled 4 through 2) (b) The cooling loop for the fuel cell stack (labeled 1) (c) The hot anode and cathode exhaust stream exiting the afterburner and entering the condenser (labeled 5)

k

Stream splitter Natural gas stream Anode exhaust Cathode exhaust Heat stream Air stream Electricity line Water line

Electricity storage

DC/AC inverter AC electric grid

6 Water heating system

Space heating system

Boost regulator

DC electricity

System exhaust N 2 CO 2 H 2O

k

Liquid H 2 O Natural gas

Cathode exhaust H 2O N 2 O2

condenser 5

Catalytic afterburner

Anode exhaust H 2 N 2 CO 2 H 2O

Fuel cell anode

compressor

1

Fuel cell cathode

2

H2 N 2 Water pump

Steam generator

Preheater

Catalytic fuel reformer

Water gas H N CO 2 2 4 shift CO2 H 2 O reactor

H 2O CO clean-up CO 2 3

Air compressor

1

2

3

4

5

6 Reference Figure 12.6 and Table 12.1

Figure 12.1. Process diagram of CHP fuel cell system. Repeated from Chapter 10 for reference.

425

k

k

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THERMAL MANAGEMENT SUBSYSTEM DESIGN

HOW HEAT EXCHANGERS WORK Due to the second law of thermodynamics, which was discussed in Chapter 2, Section 2.1.4, heat only flows from hot to cold. A heat exchanger is a mechanical device that conveys thermal energy or heat (Q) from a hot fluid stream on one side of a barrier to a cold fluid stream on the other side without allowing the fluids to directly mix. An example of a heat exchanger is a car’s radiator, which conveys heat from fluids inside the engine to the surrounding air by forced air convection. Figure 12.2 illustrates one type of heat exchanger, a counter-flow heat exchanger, in which the hot fluid flows in one horizontal direction and the cold fluid flows in the reverse horizontal direction. As the hot fluid flows across the top of the plate, heat (Q) is transferred through the conductive plate to the cold fluid below. As a result, the hot stream temperature declines along the length of the plate, from its inlet (TH , IN ) to its outlet (TH , OUT ). This decline in temperature over the length of the heat exchanger is shown by a nonlinear temperature profile. Over the same length of heat exchanger, the cold stream temperature increases from its inlet (TC , IN ) to its outlet (TC , OUT ), also shown by a temperature profile. The temperature difference between the hot and cold streams is dT. The pinch point (dTmin ) is the minimum temperature difference between hot and cold streams across the length (L) of the heat exchanger, located at L = 0 in Figure 12.2. In a co-flow or parallel-flow heat exchanger, the fluids flow in the same direction, and their outlet temperatures converge. The pinch point is often at the outlet of a co-flow or counter-flow exchanger. Temperature profile along length of heat exchanger TH, IN

Pinch point temperature

Tempe

rature p

Hot fluid

rofile

dT TC, OUT

Q

Temp eratur e profi le

TH, OUT Heat transfer

Cold fluid TC, IN

L Length along heat exchanger

Figure 12.2. Temperature profiles of hot and cold streams in counter-flow heat exchanger.

WHY IS HEAT RECOVERY IMPORTANT FOR FUEL CELLS? We will now touch on two heat transfer design problems for fuel cells: • External Heat Transfer. External heat transfer can signify using excess heat from the fuel cell system for heating a source of thermal demand outside of the fuel cell system, such as a building. An aim of pinch point analysis is to reserve this type of

OVERVIEW OF PINCH POINT ANALYSIS STEPS

heat transfer to include primarily the heat that first cannot be recovered for internal use within the fuel cell system, for example, because the temperature of that heat is not high enough to serve an internal source of heat demand. • External heat transfer also can signify the need to bring additional heat or fuel energy in from outside of the fuel cell system to provide for internal heating needs of the fuel cell system. A goal of pinch point analysis is to minimize this type of external heat transfer. • Say that you have a 70∘ C PEMFC stack producing 6 kW of electricity and 9 kW of heat. Given the large percentage of heat released, you want to use this heat to heat water for a building up to 90∘ C. Because heat only flows from hot streams to cold streams, you might initially assume that the heat from the fuel cell stack is NOT transferable to the building. However, some of it is. You will see this in our example problems. Effective heat recovery becomes more challenging the smaller the difference between the hot (TH ) and cold temperature (TC ) streams and the lower the temperature of the hot stream (TH ). Because low-temperature fuel cell systems (such as PEMFCs and PAFCs) produce heat at low hot stream temperatures (TH ), it is even more important to design their heat exchanger network carefully to capture this heat [133]. • Internal Heat Transfer. Internal heat transfer signifies recovering heat from a heat source, i.e., a device or stream, within a fuel cell system and redirecting that heat to a heat sink, i.e., another device or stream that is operating at a lower temperature within the same fuel cell system. A goal of pinch point analysis is to maximize this type of internal heat transfer. • Say that you are operating the fuel cell system shown in Figure 12.1. You want to design a heat exchanger system to extract heat dissipated by the fuel cell at 150∘ C (shown by 1 in Figure 12.1) and from the afterburner at 600∘ C (shown in Figure 12.1). You would like to use this heat for an upstream endothermic steam reformer that operates at 800∘ C and for the inlet gas preheater that operates at 500∘ C. What is the optimal design? Pinch point analysis can help answer these questions. As you will learn, designing a heat exchanger network is especially important for fuel cell systems that integrate multiple devices, chemical reactors, complex fuel processors, energy storage devices, etc., where different components may produce or consume heat.

Heat management along each of these streams is important. (a) The hot reformate stream must remain within certain temperature ranges to avoid sintering the catalysts in the CO cleanup reactor and at the fuel cell’s anode. (b) The fuel cell stack also operates most effectively within a certain temperature range. Also, quite importantly, the stack produces a large portion of the recoverable heat from the system. (c) The condenser also releases a large portion of the recoverable heat from the system over a wide temperature. We also will

427

k

428

k Condenser

Building heat loop

5

6

Domestic water-cooling loop exchanging heat between fuel cell system and building

Heat extracted from condensing water from anode and cathode exhaust

Heat extracted from reformate stream after shift reactor

Heat extracted from reformate stream at exothermic selective oxidation reactor

Heat extracted from reformate stream after selective oxidation reactor

Heat extracted from fuel cell stack

Stream Description 70 110

120

260 219

25

Hot

Hot

Hot Hot

Cold

Supply Temperature, Tin (∘ C)

Hot

Hot or Cold?

80

65

120

110

70

60

Target Temperature, TOUT (∘ C)

143

200/9.5

6

6

276/6

276

Heat Flow Capacity, mc ̇ p (W/K)

7890

3370

840

60

860

2760

Heat Flow Q̇ (W)

Note: Stream numbers refer to labeled streams in Figure 12.1. Data were used to construct T–H diagrams. Streams 1–5 refer to hot streams within the fuel cell system. Stream 1 is the cooling stream for the fuel cell stack. Stream 2 is the reformate stream before it enters the fuel cell. Stream 3 is the reformate passing through a selective oxidation chemical reactor. Stream 4 is the reformate stream passing through the WGS reactor. Streams 2–4 are essentially the same contiguous streams passing through different stages. Stream 5 is the anode and cathode exhaust stream passing through a condenser. Stream 6 refers to a building’s cold stream. This stream requires heating to provide hot water and space heating for the building. For each stream, thermodynamic data are listed, including (1) inlet and (2) outlet temperatures, (3) the heat flow capacity, and (4) the change in enthalpy or heat flow within the stream. Heat flow capacity is the product of the stream’s mass flow rate (m) ̇ and its heat capacity (cp ).

Post-WGS reactor

4

Aftercooler

2

Selective oxidation

Fuel cell stack

1

3

Source of Heat or Cooling

Stream Number

TABLE 12.1. Thermodynamic Data for Hot and Cold Streams in Fuel Cell System Design Shown in Figure

k

k

OVERVIEW OF PINCH POINT ANALYSIS STEPS

investigate the coldest stream that requires heating: the building’s heating loop (labeled 6). This loop provides heating for the air space in the building and for hot water.

12.1.2

Step Two: Identify Thermal Data

2. Determine Thermal Data for These Streams. For each hot and cold stream identified, thermal data must be compiled. These data include the following: (a) The supply temperature Tin , the initial temperature of the stream before entering a heat exchanger (b) The target temperature Tout , the desired outlet temperature for the stream upon exiting a heat exchanger (c) The heat capacity flow rate mc ̇ p , the product of the stream’s mass flow rate ṁ (in kg∕s) and the specific heat of the fluid in the stream, cp (in kJ∕kg ⋅ ∘ C), whereby the specific heat of the stream may be assumed constant over the temperature range in many cases (except where a phase change occurs) (d) The change in enthalpy per unit time dḢ in the stream passing through the heat exchanger The dots above variables like ṁ and dḢ indicate a flow rate, i.e., that the variable is per unit time. These two variables indicate mass and energy flow rates, respectively. As discussed in Chapter 2, according to the first law of thermodynamics, at constant pressure, dḢ = Q̇ + Ẇ (12.1) Since a heat exchanger performs no mechanical work (Ẇ = 0), dḢ = Q̇ = mc ̇ p (Tin − Tout ), where Q̇ represents the flow of heat into or out of a stream and dḢ represents the change in enthalpy flow from the stream per unit time. The supply temperature data may be measured from an operating system or may be calculated by heat transfer calculations or chemical engineering modeling of reactors. Target temperatures (the desired outlet temperatures) may be determined this way or can be based on other system constraints. For the hot and cold streams identified in step 1, data are tabulated in Table 12.1. Example 12.1 Recoverable heat from a SOFC system is used to provide space heating to a building. A counter-flow double-pipe heat exchanger, similar to that described in Figure 12.2, is used to exchange heat between hot oil from the fuel cell system and water sent to the building. The hot stream conveys 140 kW of heat to the cold stream that is flowing at 0.5 kg∕s and entering the heat exchanger at 25∘ C. The heat capacity of water is 4.19 kJ∕kg ⋅ ∘ C. (1) Calculate the outlet temperature of the water. (2) The first heat exchanger unexpectedly breaks due to a manufacturing defect, and this heat exchanger is replaced with one that is immediately available and in stock, a parallel-flow double-pipe heat exchanger. With the change in heat exchanger type, the minimum temperature difference between hot and cold streams in the heat exchanger (dTmin ), that is, the pinch point temperature, decreases. To avoid exceeding a set point

429

430

THERMAL MANAGEMENT SUBSYSTEM DESIGN

for the pinch point temperature (dTmin,set ), the heat transferred to the cold stream declines by 20 kW. If the flow rate of water is held constant, what would be the new water supply temperature to the building? (3) To achieve the same higher outlet temperature as in 1, the flow rate of water can be decreased to what value? Solution: 1. According to the first law of thermodynamics, the heat transferred to the stream ̇ is equal to the product of the mass flow rate of the stream (ṁ ), the specific (Q) heat of the fluid in the stream (cp ), and the temperature difference between the outlet and inlet of the stream Tout − Tin , or Q̇ = mc ̇ p (Tout − Tin ) which is also Tout = Tin +

Q̇ mc ̇ p

(12.2)

(12.3)

where Tout = 25∘ C + 140 kJ∕s∕[(4.19 kJ∕kg ⋅ ∘ C)(0.5 kg∕s)] = 91.8∘ C 2. Tout = 25∘ C + 140 kJ∕s∕[(4.19 kJ∕kg ⋅ ∘ C)(0.5 kg∕s)] = 82.3∘ C 3. From the same conservation-of-energy equation, ṁ =

Q̇ [cp (Tout − Tin )]

(12.4)

where ṁ = 120 kJ∕s∕[(4.19 kJ∕kg ⋅ ∘ C)(91.8∘ C–25∘ C)] = ∼ 0.43 kg∕s. Example 12.2 The fuel cell system in Figure 12.1 produces 6 kW of net electricity with a net electrical efficiency of 34% based on the HHV of the original natural gas fuel consumed by the system. To simplify the calculation, assume that parasitic electric power draw is zero. (1) Estimate the maximum quantity of heat available from the system for heating the building. (2) Based on the first law of thermodynamics only, if it were possible to transfer all of the energy available in the hot streams of this fuel cell system to a cold stream heating the building via radiators, estimate the maximum flow rate of water for this stream. Assume that this cold water stream circulating through the building has a supply temperature of 25∘ C and a target temperature of 80∘ C. (3) Assume 100% efficient heat transfer exists between the building’s closed-loop hot water heating loop and its open loop for potable hot water. If the heat calculated in 2 was used for a building’s potable hot water, how many hot showers could it provide? Solution: 1. As you learned in Chapter 2, Section 2.5.2, and in Chapter 10, Equation 10.26, the net electrical efficiency of the fuel cell stack can be described by 𝜀R =

Pe,SYS ΔḢ (HHV),fuel

(12.5)

OVERVIEW OF PINCH POINT ANALYSIS STEPS

where Pe,SYS is the net electrical power output of the fuel cell stack. Assuming that the parasitic power draw from pumps and compressors is negligible and referencing Chapter 10, Equation 10.25, the maximum heat recovery efficiency, 𝜀H , is given as 𝜀H = 1 − 𝜀R (12.6) The maximum quantity of heat recoverable (dḢ MAX ) from the system is then dḢ MAX = =

(1 − 𝜀R )Pe 𝜀R

(12.7)

(1 − 0.34) 6 kW = 11.6 kW 0.34

(12.8)

2. Assuming perfect heat exchange, the mass flow rate of water is ṁ =

Q̇ cp (Tin − Tout )

(12.9)

The heat capacity of water is 4.19 kJ∕kg ⋅ ∘ C over this ΔT, such that ṁ =

11.6 kW = 0.05 kg∕s 4.19 kJ∕kg ⋅ ∘ C (80∘ C − 25∘ C)

(12.10)

3. The flow rate of hot water from a shower is estimated to be 0.20 kg∕s for maximum flow. With a 100-L(100-kg) hot-water storage tank on site, this flow rate would be enough for a single 8-min shower every 30 min.

12.1.3

Step Three: Select Minimum Temperature Difference

3. Select a Minimum Temperature Difference (dTmin,set ) between Hot and Cold Streams. As discussed in Chapter 2, while the first law of thermodynamics describes the conservation-of-energy equation for calculating changes in enthalpy, the second law of thermodynamics describes the direction of heat flow. Heat may only flow from hot streams to cold streams. As a result, for example, within a heat exchanger, it is not possible for the temperature of the hot stream to dip below the temperature of the cold stream at the same length along the heat exchanger, and the cold stream cannot be heated to a temperature higher than the supply temperature of the hot stream. A minimum temperature difference dTmin must exist between the streams to drive heat transfer such that, for the hot-stream temperature TH and the cold-stream temperature TC , TH − TC ≥ dTmin

(12.11)

431

432

THERMAL MANAGEMENT SUBSYSTEM DESIGN

at any and all points along the length of the heat exchanger. For a set of streams, the minimum temperature difference observed between the streams at any length along the heat exchanger is referred to as the pinch point temperature. In the heat exchanger shown in Figure 12.2, the temperature difference between the hot and cold streams (dT) changes along the length of the heat exchanger, as shown by the difference in the hot and cold temperature profiles over its length L. In this heat exchanger, the minimum temperature difference, dTmin , is at L = 0, at the inlet of the hot fluid stream and the outlet of the cold fluid stream. For the purposes of pinch point analysis, dTmin is often set at a desired value, between 3 and 40∘ C, depending on the type of heat exchanger and the application. For example, while shell-and-tube heat exchangers require, dTmin,set , of 5∘ C or more, compact heat exchangers can achieve higher heat transfer rates due to their larger effective surface areas and may require dTmin,set of only 3∘ C. For our analysis of the heat streams in Figure 12.1, we select dTmin,set = 20∘ C. 12.1.4

Step Four: Evaluate Thermodynamic Plots

4. Construct Temperature–Enthalpy Diagrams and CheckTmin ≥ dTmin,set . Temperature–enthalpy diagrams (T–H) show the change in temperature versus the change in enthalpy for hot and cold streams. On a T–H diagram, any stream with a constant cp should be represented by a straight line from Tin to Tout . We use the thermal data we gathered in Table 12.1 to make the T–H plots. Given the large quantity of energy available from the condenser (stream 5 in Table 12.1), we plot its data on a T–H diagram. The data plotted in Figure 12.3 are based on the condenser’s Tin = 219∘ C, Tout = 65∘ C, and Q̇ = 3370 W. We also plot data for the building’s cold stream loop (stream 6 in Table 12.1), based on its Tin = 25∘ C and Tout = 80∘ C. We assume that this loop could absorb up to 3370 W from the condenser, which represents a considerable portion of the 7890 W of total heat it could absorb from all five hot streams. We plot these data on a T–H diagram. Figure 12.3 shows a T–H diagram that illustrates our best understanding of the hot stream of the condenser and the cold stream of the building’s heating loop assuming that these loops are separated (i.e., not connected by a heat exchanger). Note the schematic at the bottom of the diagram, which indicates how the hot and cold streams are separated in different pipes that do not intersect. If the two streams are separated, each must rely on an external energy source for heat transfer from outside the fuel cell system to provide cooling or heating. The approach of using external heat in this scenario is wasteful. For example, in an external arrangement, the building’s heating loop might have to rely on burning additional natural gas as a source of heat instead of utilizing the heat from the condenser. This approach unnecessarily consumes additional fuel and releases additional, harmful greenhouse gas and air pollution emissions, which are discussed in greater detail in Chapter 14. Figure 12.4 shows the effect of incorporating a heat exchanger between the two streams in order to thermally connect them. For heat exchange to take place between the hot stream and the cold stream, the hot stream T–H curve must lie above the cold stream T–H curve. The T–H diagram for the cold stream has been shifted to the left such that the cold stream

OVERVIEW OF PINCH POINT ANALYSIS STEPS

Temperature (°C) 260 240

(3370 W, 219°C)

220

Inlet

TH, IN

200

tre

am

180

Ho ts

160 140 120 100

(6740 W, 80°C)

80 Outlet

TC, IN

20 0

tre ld s Co

Inlet

TH, OUT

40

Outlet

am

(0 W, 65°C)

60

TC, OUT

(3370 W, 25°C) 0

2000

1000

3000

4000

5000

6000

7000

8000 Enthalpy (W)

Inlet

Outlet

Inlet

Hot stream Q Q

= 3370 W

H, ext

F, ext

=

Q

C, ext

Cold stream

TC, IN

+ Q

H, ext

Outlet

TC, OUT

Q

= 3370 W

C, ext

= 3370W + 3370W = 6740W

Figure 12.3. Temperature–enthalpy diagram for a hot stream and a cold stream not connected by a heat exchanger. External heat transfer is maximal. The hot stream rejects 3370 W to the environment. The cold stream absorbs 3370 W from an external heat source. Arrow heads on the T–H plots indicate the direction of stream flow. The schematic at the bottom illustrates the processes occurring by showing pipes carrying fluid and the heat transfer through these pipes. The change in enthalpy is roughly commensurate with the change in the length along a pipe.

now cools the hot stream and the hot stream heats the cold one. Less external heat transfer is necessary. The heat recovery efficiency (𝜀H ) of the system increases and, therefore, according to (12.12) 𝜀O = 𝜀R + 𝜀H which references Equation 10.25 in Chapter 10, the overall efficiency of the system increases. When a hot T–H diagram and a cold T–H diagram are horizontally shifted on top of each other in these diagrams, the change in enthalpy along the x-axis can be thought of in terms of the change in length along the heat exchanger. At a given length along a heat exchanger,

433

434

THERMAL MANAGEMENT SUBSYSTEM DESIGN

Temperature (°C) 260 240

(3370 W, 219°C)

220

Inlet TH, IN

200

Hot stream

180

Pinch point temperature dTMIN = 40°C

160 140 120 Outlet

TH, OUT

100 80

(3370 W, 80°C)

(0 W, 65°C)

(6740 W, 80°C)

Outlet TC, OUT

60 40

Cold stream

20

Inlet TC, IN

(0 W, 25°C)

(3370 W, 25°C)

0 0

1000

2000

3000 Outlet T

Inlet T

Q

F, ext

Cold stream

5000 = Q

C, ext

6000 + Q

H, ext

Q H, INT

Hot stream Cold stream Outlet T

4000

=

7000

8000 Enthalpy (W)

=0W + 0W = 0W

Q C, INT

Q IN = QOUT = Q = 3370 W Inlet T

Figure 12.4. Temperature–enthalpy diagram for hot and cold streams in Figure 12.3 but connected by a heat exchanger (shown at bottom). External heat transfer Q̇ ext is zero. The hot stream rejects 3370 W to the cold stream. The pinch point, the minimum temperature difference between hot and cold streams, appears to be at the entrance to the cold stream and has a value of 40∘ C, based on our available data. The figure at the bottom depicts the combined streams in a counter-flow double-pipe heat exchanger.

the quantity of heat that leaves the hot stream to enter the cold stream (the cumulative change in enthalpy of the hot stream) must be equal to the quantity of heat absorbed by the cold stream from the hot stream (the cumulative change in enthalpy of the cold stream), assuming no losses to the surrounding. The bottom of Figure 12.4 shows that the hot and cold streams of the two separate pipes have been merged together as two concentric pipes, i.e., a counter-flow, double-pipe heat exchanger, which is the device discussed in Example 12.1. In this way, the length along the heat exchanger is analogous to the cumulative change in enthalpy of the streams.

OVERVIEW OF PINCH POINT ANALYSIS STEPS

Example 12.3 You would like to use the heat of the condenser (stream 5 in Table 12.1) to warm a cold stream of utility water from 25 to 80∘ C for a building’s heat. Build on data available in the table and from Example 12.2. (1) Report the quantity of heat available from this component as a percentage of the HHV fuel energy input. (2) Construct the appropriate T–H diagrams and check the pinch point temperature. Ensure that Tmin ≥ Tmin,set = 20∘ C. Solution: 1. Based on data in Example 12.2 and on Equation 10.26, ΔḢ (HHV),fuel =

Pe,SYS 𝜀R

=

6kWe = 17.6 kW 0.34

(12.13)

The maximum quantity of heat available from the hot stream is dḢ MAX = 3370 W dḢ MAX = 19% ΔḢ (HHV),fuel

(12.14) (12.15)

Almost 20% of the energy in the fuel is available as heat for recovery from this single component. 2. The T–H diagrams are shown in Figures 12.3 and 12.4 for these assumptions: dTmin = 40∘ C > dTmin,set = 20∘ C

(12.16)

Example 12.4 Given the large quantity of heat available from the condenser, you improve your understanding of it. You realize that the condenser’s stream changes phase during the heat exchange process, as the water vapor condenses to liquid water. Because the heat capacity cp of the stream changes between gas and liquid phases, mc ̇ p is not constant across the heat exchanger. You more carefully measure the thermodynamic properties of the stream. You measure the mass flow rate and estimate the heat capacity for the vapor and gas phases based on the stream’s constituent species. ̇ p,vap = 9.5 W/∘ C. For the liquid phase mc ̇ p,liq = 200 W/∘ C, and for the vapor phase mc (1) Calculate the temperature at which the stream changes phase. (2) Reconstruct the appropriate T–H diagram and check the pinch point temperature. Solution: 1. Using

̇ p,vap (Tin − Tcond ) Q̇ = mc ̇ p,liq (Tcond − Tout ) + mc

(12.17)

we have 3370 W = (200 W∕∘ C)(Tcond − 65∘ C) + (9.5 W∕∘ C)(219∘ C − Tcond ) (12.18) or

Tcond = 75∘ C

(12.19)

435

THERMAL MANAGEMENT SUBSYSTEM DESIGN

2. Figure 12.5 shows the appropriate T–H diagram. In this condenser example, the pinch point does not occur at either the entrance or the exit of the heat exchanger but rather occurs within the heat exchanger. The pinch point occurs during the phase change from gas to liquid and is 17∘ C. Because dTmin = 17∘ C is not greater than or equal to dTmin,set = 20∘ C, we need to reconfigure the heat exchangers to meet the set pinch point temperature. To do this, for example, after partially heating the utility water cooling loop, we may have the condenser heat a colder stream in the system and have a hotter stream heat the utility water loop the rest of the way.

Temperature (°C) 260 240

(3370 W, 219°C)

220

Inlet

200

TH_IN

180

Pinch point temperature dTMIN= 18°C

Hot stre am

436

160 140 120

Outlet TH, OUT

100

(3370 W, 80°C)

(0 W, 65°C)

80

Outlet T C_OUT

60

am tre ld s o C Inlet

40 20 0

(0 W, 25°C)

T C_IN

1000

2000

0

3000

4000

5000

6000

7000

Enthalpy (W) 8000

Outlet

Inlet

Cold stream

Hot stream Cold stream Outlet

Inlet

Figure 12.5. Temperature–enthalpy diagram for a hot and a cold stream connected by a heat exchanger, with the hot stream changing phase from gas to liquid in the middle. The change in phase is marked by the hot stream’s abrupt change in slope, where slope is the inverse of the heat flow capacity mc ̇ p . The change in phase causes a pinch point. Aggregate conservation-of-energy calculations would not have detected the pinch.

OVERVIEW OF PINCH POINT ANALYSIS STEPS

This example is extremely important because a significant portion of the total recoverable heat is available at the condenser. This example is also very important because all fuel cell systems produce water vapor in the product stream and most will use cold streams from other components to condense the water for heat recovery and water balance. Also, a pinch point frequently occurs in components that contain a liquid–gas phase change, at the point at which the mixture changes phase. Within fuel cell systems, components such as low temperature fuel cell stacks, condensers, and low temperature heat exchangers often experience a liquid–gas phase change. 12.1.5

Step Five: Redesign Heat Exchanger Network

5. If dTmin < dTmin,set , Change Heat Exchanger Orientation. If the actual pinch point temperature is less than the set minimum pinch point temperature, the hot and cold streams must be reoriented. For the new orientation, a new T–H diagram is developed and the pinch point temperature and location within the heating network are recalculated. Additional streams may be included in the analysis to increase the number of options available.

12.1.6

Step Six: Evaluate Multiple Scenarios

6. Conduct Scenario Analysis of Heat Exchanger Orientation Until dTmin ≥ dTmin,set . Different orientations of streams and heat exchangers can be evaluated using scenario analysis. In scenario analysis, different network designs and orientations of heat exchangers are postulated, and then this network is analyzed with T–H diagrams. The analysis identifies the minimum temperature difference between hot and cold streams in each heat exchanger and then also the network-wide pinch point among all heat exchangers in the network. If a network design is identified to have a pinch point above the set point, the design process may converge on that particular heat exchanger network design. Alternatively, the design processes may continue, and network designs may be iterated upon to find the heat exchanger network design with the highest pinch point temperature. Scenario analysis is greatly aided by computer software that incorporates the chemical engineering process plant descriptions and pinch point temperature analysis capability. Although beyond the scope of this brief introduction to pinch point analysis, these programs can be used to investigate better heat exchanger network designs. From these analyses, one can determine the number of heat exchangers required and conduct a cost–benefit analysis to compare the cost of different heat exchanger network scenarios with the financial benefits of higher fuel efficiency and heat recovery.

437

438

THERMAL MANAGEMENT SUBSYSTEM DESIGN

Example 12.5 You are designing the thermal management subsystem for the fuel cell system shown in Figure 12.1. You plan to capture heat from the fuel cell system to heat a building. Table 12.1 provides the thermal characteristics of some of the most important hot streams within the fuel cell system (streams 1–5). Figure 12.1 shows the arrangement of these five hot streams within the system (numbered 1–5). You plan to use heat from these five streams (a total of 7890 W) to heat the building. Table 12.1 also shows the thermal characteristics of the single stream you want to heat, the building’s cold stream (steam 6). You would like to heat this cold stream from 25 to 80∘ C, as shown in Table 12.1. You would like to capture every single watt of heat from the five hot streams to warm the building. Capturing this heat will give the fuel cell system a very high heat recovery efficiency and therefore a high overall efficiency. Conduct a pinch point analysis on one possible heating loop design. Assume the building’s cold stream exchanges heat with the hot streams placed in series in this order: (1) the fuel cell stack, (2) the aftercooler, (3) the selective oxidation reactor, (4) the post-WGS reactor, and (5) the condenser. 1. Plot these hot and cold streams on a T–H diagram and identify the location of the pinch point. 2. Calculate the pinch point temperature dTmin . 3. If dTmin < dTmin.set = 10∘ C, suggest another heating loop design to increase the pinch. For the aftercooler, the heat flow capacity mc ̇ p,liq,aft for the liquid portion of the stream is 276 W∕∘ C. The heat flow capacity mc ̇ p,vap,aft for the vapor portion of the stream is 6 W∕∘ C. Solution 1. To do this analysis, we realize that the heat capacity cp of the condenser’s stream does not remain constant. The same is true for the aftercooler’s stream. In both of these streams, water condenses from a vapor to a liquid midstream. Using Q̇ = mcp,liq,aft (Tcond,aft − Tout ) + mc ̇ p,vap,aft (Tin − Tcond,aft )

(12.20)

we have 860 W = (276 W∕∘ C)(Tcond,aft − 70∘ C) + (6 W∕∘ C)(100∘ C − Tcond,aft ) (12.21) Thus, the aftercooler stream condenses at Tcond,aft = 72.3∘ C

(12.22)

From Example 12.4, we know the fluid in the condenser will condense at Tcond = 75∘ C. The thermodynamic characteristics listed in Table 12.1 gives us ̇ and the change in temperature (dT) for each the change in enthalpy (dḢ = Q)

OVERVIEW OF PINCH POINT ANALYSIS STEPS

4

5

Condenser

3

Postshift

2

Aftercooler

250

1

Seletive oxidation

Temperature (°C) 300

Fuel cell

of the five streams. We plot a curve of dH versus dTfor each of the five stages consecutively from coldest to hottest, resulting in the T–H curve in Figure 12.6, which shows us that the pinch point occurs in the condenser.

Hot s

150

Ho t st

tream

rea m

200

100 Hot stream

m

Hot strea

tream

50 tream

tream Cold s

Cold s

Cold s

0 0

2000 4000 6000 Cumulative enthalpy transfer (heat load) (W)

8000

Figure 12.6. Temperature–enthalpy diagram for hot and cold streams from fuel cell system of Figure 12.1. The two separate hot streams are from two different parts of the system. They heat the cold stream in series. First, the cold stream absorbs heat from (1) the fuel cell stack, (2) the aftercooler, (3) a selective oxidation reactor, and (4) the reformate leaving the water gas shift reactor. Second, the cold stream absorbs heat from a condenser. The dT–dH curves were plotted using the data from Table 12.1.

2. To find the value of the pinch point temperature dTmin at the condenser, we observe that the pinch point occurs just as the vapor condenses, at Tcond = 75∘ C. Also, dTmin = Tcond − Tb , where Tb is the building loop temperature. At Tcond = 75∘ C, we want to know the cumulative enthalpy transfer dḢ cum , the value on the x-axis: dḢ cum = Q̇ FC + Q̇ AC + Q̇ SO + Q̇ PS + Q̇ cond, liq, A

(12.23)

where Q̇ FC is the heat flow at the fuel cell, Q̇ AC is the heat flow at the aftercooler, Q̇ SO is the heat flow at the selective oxidation reactor, Q̇ PS is the heat flow at the post-WGS

439

440

THERMAL MANAGEMENT SUBSYSTEM DESIGN

reactor, and Q̇ cond liq,A is the heat flow in the cold stage (liquid) of the condenser. From Example 12.4, Q̇ cond liq,A = mc ̇ p,liq (Tcond − Tout ) = (200 W∕∘ C)(75∘ C − 65∘ C) = 2000 W (12.24) dḢ cum = 2760 W + 860 W + 60 W + 840 W + 2000 W = 6520 W

(12.25)

where dḢ cum = 6520 W is the value on the x-axis where the pinch occurs. For the building’s heating loop, the relationship between Tb and dḢ can be described by (

) 80∘ C − 25∘ C dḢ + 25∘ C Tb = 7890 W ( ∘ ) 80 C − 25∘ C = 6520 W + 25∘ C 7890 W = 70.5∘ C dTmin = Tcond − Tb = 75o C − 70.5o C = 4.5o C < dTmin,set = 10o C

(12.26) (12.27) (12.28) (12.29)

This pinch point temperature is extremely low. By employing the approach of scenario analysis, we will propose another heat exchanger network design to try to increase the pinch. 3. One option is to split the building’s cooling stream into two separate but parallel streams. One stream extracts heat from the first four heat sources in series: (1) the fuel cell stack, (2) the aftercooler, (3) the selective oxidation reactor, and (4) the post-WGS reactor. The second stream extracts heat from the fifth heat source, the condenser. The ratio of flow rates between the building loop’s two parallel streams could be optimized to maximize the pinch. Such a detailed analysis, performed by computer simulations, leads to a pinch greater than dTmin,set (10∘ C) over a range of molar flow ratios.

12.2

CHAPTER SUMMARY

In this chapter, we learned in detail about one of the four primary fuel cell subsystems, the thermal management subsystem. We learned how to design effective thermal management subsystems for fuel cell systems using pinch point analysis. • Fuel cell systems are composed of different subcomponents with different heating and cooling requirements. Fuel cell stacks and condensing heat exchangers often need to be cooled. Chemical reactors for fuel reforming often need to be heated. • The primary goal of pinch point analysis is to optimize the overall heat recovery within a fuel cell system by minimizing the need to supply additional heating and/or cooling.

CHAPTER EXERCISES

• In an ideal pinch point analysis solution, hot streams (such as from a partial oxidation chemical reactor) are used to heat cold ones (such as inlet fuel, air, and water at ambient temperature) with a minimum amount of additional heat transfer from an external source (such as a dedicated electric heater). • Temperature–enthalpy diagrams are constructed to locate the pinch point temperature dTmin , the minimum temperature difference between hot and cold streams. • Heat exchangers are arranged to maximize (1) internal use of heating and (2) dTmin . • Pinch point analysis can be broken down into six main steps: 1. Identify hot and cold streams in the system. 2. Determine thermal data for these streams. 3. Select a minimum acceptable temperature difference (dTmin,set ) between hot and cold streams. 4. Construct temperature–enthalpy diagrams and check dTmin > dTmin,set . 5. If dTmin < Tmin,set , change heat exchanger orientation. 6. Conduct scenario analysis of heat exchanger orientation until dTmin > dTmin,set . • A pinch point may be likely to arise in components that contain a liquid–gas phase change, at the point at which the mixture changes phase. Within fuel cell systems, components experiencing such a phase change may include low-temperature fuel cell stacks, condensers, and heat exchangers. • Fuel cell system designers may need to balance the goals of higher heat recovery efficiency and neutral/positive water balance with the additional expense and complexity of including condensing heat exchangers. CHAPTER EXERCISES Review Questions 12.1

What types of streams may need heating or cooling in a fuel cell system?

12.2

Considering prior discussions of fuel cell types in Chapter 8 and fuel cell system design in Chapter 10, provide some examples of how heating and cooling needs change with fuel cell type and with system design. Identify streams that must be heated or cooled in each system design considered.

12.3

How does a heat exchanger work?

12.4

What does the temperature profile along the length of the pipe look like for both hot and cold streams in a double-pipe counter-flow heat exchanger?

12.5

What does the temperature profile along the length of the pipe look like for both hot and cold streams in a double-pipe, co-flow or parallel-flow heat exchanger?

12.6

What is meant by internal vs. external heat transfer in this chapter’s discussion?

12.7

What is the pinch point?

12.8

Why is pinch point analysis so important for fuel cell system design?

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THERMAL MANAGEMENT SUBSYSTEM DESIGN

12.9

Pinch point analysis requires what types of thermal data on streams?

12.10 What equation(s) can be used to describe the flow of heat into or out of a stream? 12.11 Based on material in this chapter and in Chapter 11 in Sections 11.2 and 11.4, please comment on the impact that temperature excursions in a fuel cell system’s chemical reactors can have on catalysts. What is happening to the materials at a molecular level? What are the long-term effects of this on reactor performance? 12.12 How are temperature–enthalpy (T–H) diagrams a useful tool? A phase change from gas to liquid is marked by what feature on a T–H diagram? 12.13 What types of fuel cell and fuel processing subsystem components are likely to be more challenging for effective heat recovery? 12.14 What is a reasonable range for the minimum temperature difference (dTmin,set ) between hot and cold streams within a heat exchanger and within a pinch point analysis? For a particular application, what does this range depend on? 12.15 Based on material in this chapter and in Chapter 11 in Sections 11.2 and 11.4, what types of operating approaches (steady-state vs. dynamic, etc.) are likely to be more challenging for effective heat recovery and thermal management of reactors and devices within a fuel cell system? 12.16 What fuel cell system components may have a large quantity of available heat but at a low temperature? 12.17 At a given length along a heat exchanger, the cumulative change in enthalpy of the hot stream must be equal to what thermodynamic value, assuming no losses to the surrounding? 12.18 What fuel cell system components may experience phase changes in their streams? Why should a heat transfer analysis carefully focus on these components? 12.19 Based on material in this chapter and in Chapter 11 in Section 11.3, which of the following are likely to be heat sources and which are likely to be heat sinks? (a) selective methanation reactor, (b) selective oxidation reactor, (c) PSA unit, and (d) palladium membrane unit. Which are likely to benefit most from a pinch point analysis that also facilitates operating these units within a narrow operating temperature range? Calculations 12.20 We reconsider the combined cooling, heating, and electric power (CCHP) fuel cell system design discussed in Chapter 10, Example 10.5. Recoverable heat from the fuel cell system is used to provide heating to an absorption chiller. A counterflow double-pipe heat exchanger, similar to that described in Figure 12.2, is used to exchange heat between hot oil from the fuel cell system’s cooling loop and a fluid in the absorption chiller unit. The hot stream conveys 90% of the recoverable heat delineated in Example 10.5 to the chiller’s fluid, which is flowing at a rate of 0.8 kg∕s and entering the heat exchanger at 60∘ C. The heat capacity of the

CHAPTER EXERCISES

chiller’s fluid is 4 kJ∕kg ⋅ ∘ C. (a) Calculate the outlet temperature of the chiller’s fluid. (b) There is a change in system design and heat exchanger configuration. To avoid exceeding a set point for the pinch point temperature (dTmin,set ), the heat transferred to the chiller fluid declines by 10 kW. If the flow rate of the working fluid is held constant, what would be the new outlet temperature? (c) To achieve the same higher outlet temperature as in (a), the flow rate of water can be decreased to what value? 12.21 Consider both the first and second laws of thermodynamics, as first discussed in Chapter 2. Design a stationary SOFC system that uses an upstream steam methane reformer to convert methane fuel into a hydrogen-rich gas for a SOFC stack and is optimally thermally integrated for minimum heat transfer from the external environment into the fuel cell system. Design the system to ensure that recoverable heat from the SOFC stack heats the endothermic steam reforming reaction. Perform a set of calculations to show that the recoverable heat from the SOFC stack and system is large enough in quantity (first law of thermodynamics) and high enough in temperature (second law of thermodynamics) to serve all of the heating needs for the upstream steam methane reformer, under certain design and operating conditions. Specify these design and operating conditions. Perform a pinch point analysis to demonstrate the validity of your design. 12.22 Perform the same analysis as described in the prior question but for MCFCs. 12.23 Consider both the first and second laws of thermodynamics, as first discussed in Chapter 2. Design a stationary PAFC system that uses an upstream steam methane reformer and water gas shift reactor to convert methane fuel into a hydrogen-rich gas for a PAFC stack and is optimally thermally integrated for minimum heat transfer from the external environment into the fuel cell system. Design the system to ensure that recoverable heat from the PAFC stack heats as much of the cold inlet gases, the water gas shift reactor, and the steam reformer as may be feasible. Perform a set of calculations to show that the recoverable heat from the PAFC stack may be large enough in quantity (first law of thermodynamics) but not high enough in temperature (second law of thermodynamics) to serve all of the heating needs for the upstream steam methane reformer and water gas shift. Specific design and operating conditions for minimizing external heat transfer into the device and minimizing the methane that must be combusted to provide high enough temperature heat for endothermic steam reforming. Perform a pinch point analysis to demonstrate the validity of your design approach. 12.24 If the fuel cell system described in Example 12.2 was used for space heating, estimate the air space it could heat during winter with an outside temperature of 0∘ C and a desired indoor temperature of 23∘ C. Assume a radiative heating system that is closed loop, based on heating a fluid that circulates in the building. How many rooms in a building can be heated? Assume a log cabin structure made of 5-cm-thick wood with a thermal conductivity of 0.17 W∕m ⋅ ∘ C, no windows, and no free convection of air along the outside.

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12.25 You are designing a PEMFC scooter for use in a developing country where water resources are scarce. You design your fuel cell system to condense the product water in the outlet stream for reuse, taking advantage of the PEMFC stack’s relatively low operating temperature. Sketch T–H diagrams for capturing heat from such a condenser. Determine how the forced convection of air against the scooter could provide enough cooling for the condenser, such that no additional air pumps or blowers would be needed. The stack’s maximum electrical power output is 1 kW. Estimate the volume and mass of the onboard water tank. Assume half of the waste heat from the fuel cell system exits via the cathode exhaust gas, a 40% efficient fuel cell system, and the scooter stores enough hydrogen at minimum for a 2-h ride. 12.26 Continue the analysis of Example 12.5. Develop alternative heat exchanger network designs that increase the pinch. If the parallel stream network is implemented, calculate a range of mass flow rate ratios over which dTmin ≥ dTmin,set = 10∘ C. 12.27 In Example 12.5, locate and determine the value of the pinch point considering all hot streams except the condenser. 12.28 Chapter 10, Section 10.3.1, introduces metal hydride operation. Resketch the process diagram from Chapter 10 in homework problem 10.7 assuming hydrogen is stored on the bike in a metal hydride that requires heating and cooling for hydrogen storage and release. Sketch T–H diagrams for managing heat. Discuss important thermodynamic characteristics of metal hydrides. 12.29 Consider the design of a thermal management subsystem for a PEM fuel cell vehicle using reversible metal hydride storage. (a) Referencing Equation 10.19 in Chapter 10, estimate the rate of heating needed to release the hydrogen from the metal hydride to power the fuel cell subsystem at a rate of 40 kWe for 𝜀R,SUB = 60%. (b) Identify a potential source of internal heat transfer to provide this heat. Assume the metal hydride is sodium alanate catalyzed with titanium dopants that follows this two-step reaction: NaAlH4 ⇐⇒ 1∕3Na3 AlH6 + 2∕3Al + H2 Na3 AlH6 ⇐⇒ 3NaH + Al + 3∕2H2

(12.30) (12.31)

The first reaction takes place at 1 atm at 130∘ C and releases 3.7 weight percent (wt.%). The second reaction proceeds at 1 atm at 130∘ C and releases 1.8wt.% H2 . Assume that the enthalpies of reaction are +36 kJ∕mol of H2 produced (not per mole of reactant) for the first reaction and +47 kJ∕mol H2 for the second reaction at the reaction temperatures. For a discussion on enthalpy of reaction, please see Chapter 2. Both reactions are endothermic, as defined in Chapter 10. Assume 100% efficient heat transfer. 12.30 Consider the same vehicle system as in the previous homework problem. (a) Estimate the rate of cooling needed to refill hydrogen back into the metal hydride with

CHAPTER EXERCISES

2 min of refueling time. The tank-to-wheel efficiency of the vehicle (𝜀R, ) is 52%, or 2.9 L of H2 ∕100 km, and its range is 400 km. (b) Which of these design requirements (the heating requirement from problem or the cooling requirement from this problem) poses a greater constraint? 12.31 Consider the same vehicle system as in the previous homework problem. Develop a configuration of heat exchangers that could provide at least a portion of the required metal hydride heating using internal heat transfer. Evaluate different heat exchanger configurations using pinch point analysis.

445

CHAPTER 13

FUEL CELL SYSTEM DESIGN

In the last few chapters, we have discussed in detail most of the major subsystems relevant to fuel cells. In this chapter, we now turn our attention to the integrated design of a complete fuel cell system. Designing a complete fuel cell system can be a complex process. Put simply, however, the overall object of the design process is to construct a system that meets certain principal design goals (or specifications). Common specifications often include target power, weight, volume, cost, reliability, lifetime, and maintenance criteria. These target specifications, and therefore the design process, will change dramatically depending on the specific fuel cell application (e.g., portable systems vs. distributed power generation systems). In this chapter, you will learn basic tools and procedures that are commonly used to design fuel cell systems.

13.1

FUEL CELL DESIGN VIA COMPUTATIONAL FLUID DYNAMICS

The first step in the design of a complete fuel cell system is to design the fuel cell itself. When designing a fuel cell, there are literally dozens of different parameters to consider. Some parameters, like the size of the fuel cell, flow channel configuration, and the optimum thickness of electrode, electrolyte, and catalyst layers, must be chosen before the fuel cell is even built. Other parameters, like the operating temperature, the fuel and oxidant stoichiometry, the humidification level of the fuel or oxidant streams, and the operating voltage, can be tuned (to some extent) after the fuel cell is built. As you learned in Chapter 6, you can use simplified fuel cell models to explore some of these parameters for fuel cell design. However, numerous assumptions are made in these simplified fuel cell models; these simplifications lessen the accuracy of these models for

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design purposes. Also, if we want to examine geometric parameters like flow channel optimization and inlet/outlet positioning, the 1D models presented in Chapter 6 are insufficient. In order to extract detailed geometric design guidelines from fuel cell models, we must employ more sophisticated modeling techniques. Because solving these sophisticated models is not a trivial issue, we must rely on the use of computers, specifically computational fluid dynamics (CFD) software, to help us. As we will see, the use of CFD enables more accurate estimates of fuel cell performance. More importantly, CFD can provide three-dimensional localized information inside a fuel cell. For example, we can find out which parts of the fuel cell might be starved for hydrogen (or air) or which parts of the fuel cell might reach unacceptably high temperatures. This information provides opportunities to redesign the fuel cell (for example, by changing the flow channel dimensions or patterns), thereby improving the reactant distribution or the cooling capacity. In the first portion of this chapter, we will introduce a popular CFD fuel cell model and learn how to use it. Fuel cell models are implemented in a variety of commercially available CFD software packages, including ANSYS Fluent®, STAR-CCM+®, CFD-ACE+®, and COMSOL Multyphysics®. These software packages provide intuitive interfaces that allow designers to build prospective fuel cell geometries, establish boundary and volume conditions, solve complicated governing equations, and then properly visualize the results. In the following sections, we explain the design process of a fuel cell model assuming that you have access to one of these commercial software packages or at least an equivalent CFD code. (Writing your own CFD code is out of the scope of this textbook!)

13.1.1

Governing Equations

Because they can employ an extensive set of governing equations, CFD methods offer extensive computational flexibility. This provides greater realism to fuel cell models. The governing equations employed by CFD models start with the conservation laws. In Section 6.2, we used the concept of “flux balance” to develop a simplified model. The flux balance concept we presented in Chapter 6 is actually a simplified form of the “mass conservation,” “species conservation,” and “charge conservation” equations. For a more complete fuel cell model, however, we need a more complete set of conservation equations. These equations can then be coupled to each other (and solved together) to calculate a variety of fuel cell performance parameters, including fluid pressure, velocity, temperature, current density, overpotential distribution, and so on, in three dimensions. Even though we will briefly describe each conservation equation in the text below (see Table 13.1), we will not provide the derivations of these equations. Instead, we will focus on the meaning and typical values of the principal physical variables in these equations that are relevant to fuel cell design. In this CFD model development, we do not make any of the assumptions that we made for the 1D model in Chapter 6 except one: we retain the single-phase flow assumption (no liquid water) to avoid the complexity of dealing with liquid water. Also, please note that the model presented here is just one out of several popular (and similar) models that have been employed by fuel cell researchers [134–139].

FUEL CELL DESIGN VIA COMPUTATIONAL FLUID DYNAMICS

TABLE 13.1. Governing Equations of CFD Fuel Cell Models Category

Conservation Equations

Mass conservation Momentum conservation Species conservation Energy conservation Charge conservation Electrochemical reaction

𝜕 (𝜀𝜌) + ∇ • (𝜀𝜌U) = 0 𝜕t 𝜀2 𝜇U 𝜕 (𝜀𝜌U) + ∇ • (𝜀𝜌UU) = −𝜀∇p + ∇ • (𝜀𝜍) + 𝜕t 𝜅 𝜕 ̇ (𝜀𝜌Xi ) + ∇ • (𝜀𝜌UXi ) = ∇ • (𝜌Ddff ,i ∇Xi ) + Si 𝜕t dp i•i ̇ 𝜕 (𝜀𝜌h) + ∇ • (𝜀𝜌Uh) = ∇ • q + 𝜀 − j𝜂 + + Sh 𝜕t dt 𝜎 ∇ • ielec = −∇ • iion = j { }] N ( )𝛽i [ } { ∏ Xi −n (1 − 𝛼) F n𝛼F j = j0 exp 𝜂 − exp 𝜂 RT RT Xi0 i=1

Note: Symbols in boldface represent vectors.

Mass Conservation. Mass conservation equations (or continuity equations) simply require that the rate of mass change in a unit volume must be equal to the sum of all the species entering (exiting) the volume in a given time period. Equation 13.1 formulates the concept mathematically: 𝜕 (𝜀𝜌) 𝜕t rate of mass change per unit volume

+

∇ ⋅ (𝜀𝜌U) = 0 net rate of mass change per unit volume by convection

(13.1)

Here, 𝜌 and U stand for density and the velocity vector of the fluid in the fuel cell, respectively. Please note that the porosity 𝜀 is implemented in this equation to account for porous domains such as electrode and catalyst layers. By setting the correct value for porosity in each domain, this equation is globally applicable over the entire fuel cell structure. For example, within the fuel cell electrode, we can choose 𝜀 = 0.4, as this is a typical value for porosity in fuel cell electrodes. In contrast, we would choose 𝜀 = 1 for the flow channels since they are fully empty. For the electrolyte, on the other hand, we may choose 𝜀 = 0 (or a very small value near zero) because the electrolyte is fully dense (or nearly so). Porosity is similarly incorporated into all the other conservation equations as well. After solving the mass conservation equation, we obtain the density (𝜌) and the velocity profiles (U) of the fluids flowing through our fuel cell.1 1 Implicitly,

calculation of fluid density requires an extra equation describing the state of the fluid. A good example would be the ideal gas law (p = 𝜌Rm T). Also, most of the variables (including 𝜌 and U) in the governing equations (see Table 13.1) can be obtained by solving them together as they appear in multiple equations.

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Momentum Conservation. Similar to the mass conservation equation, we can set up an equation for momentum conservation as 𝜀2 𝜇U 𝜕 (𝜀𝜌U) + ∇ ⋅ (𝜀𝜌UU) = −𝜀∇p + ∇ ⋅ (𝜀𝜁 ) + 𝜕t 𝜅 rate of momentum net rate of change momentum change per unit per unit 𝑣olume 𝑣iscous pore 𝑣olume con𝑣ection by pressure frinction structure

(13.2)

Here, 𝜁 and 𝜇 stand for the shear stress tensor and the fluid viscosity, respectively. Please note that the last term on the right-hand side (RHS) is known as “Darcy’s law,” which quantifies the viscous drag of fluids in porous media. Permeability, 𝜅 [m−2 ], quantifies the strength of this viscous drag interaction and depends on the pore structure. A low permeability indicates greater interaction. Obviously, we may use an extremely large value of 𝜅 (105 m–2 or larger) in the flow channels as viscous drag is typically negligible there. The second to last term on the RHS accounts for fluid–fluid interactions. Solving the momentum conservation equation permits us to obtain the pressure (p) distribution of the fluids flowing through our fuel cell. Species Conservation. The mass conservation and momentum conservation equations discussed above are used to describe the overall bulk motion of a fluid mixture (such as humidified hydrogen = H2(g) + H2 O(g) or humidified air = N2(g) + O2(g) + H2 O(g) ). In contrast, the species conservation equation describes the differential movement (or production/consumption) of each individual species (e.g., H2(g) only or H2 O(g) only) within the fluid mixture. 𝜕 (𝜀𝜌Xi ) 𝜕t

+

∇ • (𝜀𝜌UXi )

=

∇ • (𝜌Deff i ∇Xi )

+

Ṡ i

net rate of a species mass change per unit 𝑣olume by electrochemical con𝑣ection diffusion reaction

rate of a species mass change per unit 𝑣olume

(13.3)

eff

Here, Xi and Di stand for species mass fraction and effective diffusivity of each species i. For simplicity, we use Fick’s diffusion equation (the first term on the RHS) to account for the diffusive mass flux. However, this term can be easily replaced with the Maxwell–Stefan equation for a more precise description of diffusion. Conventionally, Ṡ i stands for a species source or sink. In fuel cells, electrochemical reactions act as species sources and sinks (e.g., hydrogen and oxygen consumption or water generation). As we have seen before, because of the direct correspondence between fuel cell current (j) and species consumption/production (Ṡ i ), we can write j Ṡ i = Mi ni F

(13.4)

FUEL CELL DESIGN VIA COMPUTATIONAL FLUID DYNAMICS

where ni is the charge carried by the species i and Mi is the molecular weight of the species i. The molecular weight term allows us to convert from molar flux rate (mol∕cm2 ⋅ s) to mass flux rate (g∕cm2 ⋅ s). Solving the species conservation equation allows us to obtain species mass fraction (Xi ) and current density (j) throughout our fuel cell. Energy Conservation. The energy conservation equation describes the thermal balance within the fuel cell: 𝜕 (𝜀𝜌h) 𝜕t

+ ∇ • (𝜀𝜌Uh) = ∇ • keff ∇T

+ 𝜀

dp dt

−

j𝜂

+

rate of energy

net rate of energy change per unit volume by

activation loss +

change per unit volume

mechanical work

concentration loss

convection

conduction

i•i 𝜎

+ Ṡ h

electroohmic loss

chemical reaction

(13.5) Here, h and keff stand for the enthalpy of the fluid flowing through the fuel cell and its effective thermal conductivity, respectively. The fluid enthalpy may be calculated based on the species present in the fluid and the fluid temperature, T. (These enthalpy calculations are analogous to those discussed in Section 2.2.2.) The first term on the RHS accounts for the rate of energy change due to thermal conduction. We use an effective thermal conductivity (keff ) to account for heat conduction through porous domains such as the electrode. The second term on the RHS accounts for the rate of energy change due to the mechanical work of the fluids. This term may generally be ignored in fuel cells, since very little pressure–volume work is done. In the last three terms on the RHS, 𝜂, i, 𝜎, and Ṡ h stand for activation + concentration overvoltage, current flux vector, electric conductivity, and heat sources (or sinks) due to reaction entropy, respectively. These terms are important as they account for heat generation due to electrochemical losses in the fuel cell. Specifically, the third term on the (RHS ) (j𝜂) describes the heat generation due to charge transfer. The fourth represents joule heating due to ohmic losses. Finally, we use the term term on the RHS i•i 𝜎 ̇Sh to account for entropy losses associated with the electrochemical reaction Ṡ h = Δ⌢ s rxn nFj . Solving the energy conservation equation permits us to obtain the temperature profile (T), activation and concentration overpotentials (𝜂), and current flux vector (i) throughout our fuel cell model. Charge Conservation. From the continuity of current in a conducting material, ∇•i=0

(13.6)

Here, i stands for the current flux vector. Two types of charges are present in fuel cell systems—electrons and ions. Since both types of charge are generated from originally neutral species (hydrogen and/or oxygen), overall charge neutrality must be conserved, ∇ • ielec + ∇ • iion = 0

(13.7)

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where iion stands for the ionic current through an ion conducting phase such as the catalyst layer or membrane and ielec stands for the electronic current in an electron conducting phase such as a catalyst layer or electrode. We rearrange Equation 13.7 and relate it to local current density as (13.8) − ∇ • iion = ∇ • ielec = j By incorporating Ohm’s law into Equation 13.8, we get ∇ • (𝜎ion ∇Φion ) = −∇ • (𝜎elec ∇Φelec ) = j

(13.9)

where Φion and Φelec are the electric potential in the ion conductor and the electronic conductor, respectively, and 𝜎ion and 𝜎elec are the conductivities. Please note that this equation can be universally applied to all domains in a fuel cell by simply setting a proper value for 𝜎 in each domain. For example, we may use 𝜎ion = 𝜎elec = 0 in the flow channels and 𝜎elec = 0 in the membrane (no electronic conduction). The catalyst layer has both ionic and electronic conduction and so both conductivities may be considered. Electrochemical Reaction. As explained in Section 3.7, the Butler–Volmer (BV) equation describes the change transfer reaction process in the catalyst layer of a fuel cell. As a reminder, the full BV equation can be written as [ ∗ { }] { } c∗ cR −n (1 − 𝛼) F n𝛼F P j = j0 0∗ exp (13.10) 𝜂 − 0∗ exp 𝜂 RT RT c c R

P

To maintain consistency with our prior conservation equations, we can replace the concentration ratios that appear in Equation 13.10 with mass fraction ratios instead. Also, we must modify Equation 13.10 somewhat in order to account for more general electrochemical reactions where multiple species may be involved. In this case, the equation becomes [ j = j0 exp

{

{ }] N } ∏ −n (1 − 𝛼) F n𝛼F 𝜂 − exp 𝜂 RT RT i=1

(

Xi Xi0

)𝛽i (13.11)

The product symbol at the end of the equation allows us to treat reactions involving multiple species. Each species i participating in the reaction may have a different exponent 𝛽i associated with it. Recall that the activation overpotential, 𝜂, represents the potential difference that develops between the ionic and electron conducting phases during an electrochemical reaction (see Figure 3.8). In Equation 13.9, we introduced Φion and Φelec to represent the potentials in the ionic and electron conducting phases, respectively. Thus, the difference between these two potential is the overpotential (𝜂 = Φion − Φelec ), and so we have { }] N ( )𝛽i [ { ∏ Xi )} −n (1 − 𝛼) F n𝛼F ( j = j0 exp Φion − Φelec − exp (Φion − Φelec ) RT RT Xi0 i=1 (13.12)

FUEL CELL DESIGN VIA COMPUTATIONAL FLUID DYNAMICS

Solving this equation in concert with the other conservation equations allows us to calculate the overpotential (𝜂 = Φion − Φelec ), current density (j), and species mass fractions (Xi ) in our fuel cell. Example 13.1 Based on Equation 13.12, establish the electrochemical reaction governing equations for a hydrogen/oxygen fuel cell model. Solution: In a hydrogen/oxygen fuel cell, two electrochemical reactions occur: hydrogen dissociation (at the anode) and oxygen reduction (at the cathode). First, we write the governing equation for the anode reaction: H2 ↔ 2H+ + 2e− [ j = jA0

{ exp

}] { ( ) } XH2 −2 1 − 𝛼 A F ) 2𝛼 A F ( Φion − Φelec − exp (Φion − Φelec ) RT RT X0 H2

(13.13) Here, symbols marked with superscript A are model constants required for the anodic reaction. For the cathode, we have 2H+ + 2e− + 12 O2 ↔ H2 O [ j = jC0

{ exp

(

XO2

}] { ( ) } −2 1 − 𝛼 C F ) 2𝛼 C F ( − exp Φion − Φelec (Φion − Φelec ) RT RT

)1 2

XO0

(13.14)

2

In these equations, we ignore concentration term contributions from protons and electrons, since we assume that they do not limit the reaction rate compared to hydrogen and oxygen reactants. Please note that the temperature T in these two equations is actually unknown and must be found by solving the energy conservation equation. 13.1.2

Building a Fuel Cell Model Geometry

When building a CFD fuel cell model, it is important to create a computational geometry that represents the physical geometry of the real fuel cell as closely as possible. At the same time, however, we may be able exclude or neglect certain portions of physical geometry without impairing the validity of the model, thereby conserving computational resources. The flow structure provides a good example. Typically, the materials used for the flow structure (e.g., graphite or metal) have high thermal and electrical conductivities. Thus, we can often assume that the temperature and electric potential profile in these structures is more or less uniform. By making this assumption, we can then neglect the bulk of the flow structure and incorporate only its surface in our computational geometry. (This is

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done by imposing proper boundary conditions along the periphery of the flow structure.) Take a look at the fuel cell geometry shown in Figure 6.8 as an example. This computational geometry does not include the flow plate structure. This simplified geometry is useful as it saves significant modeling and calculation time. When building a model fuel cell geometry, the CFD software is used to define the various physical domains of the fuel cell. Each domain corresponds to a different physical portion of the fuel cell—for example, the bipolar plate, flow channels, electrode layers, catalyst layers, the electrolyte, and so on. Each domain receives its own version of the governing equations as well as specific boundary and volume conditions. The parameters governing each domain will vary according to the physical nature of each domain. For example, the porosity and electronic conductivity of the bipolar plate domain will be different from the electrode domain. The next step in building the fuel cell geometry is to populate each domain in the model with a “grid.” The purpose of grid generation is to divide the model into a three-dimensional set of discrete elements, each of which will be evaluated numerically to provide discrete solution values to the governing equations we have discussed above. Even though the governing equations in Table 13.1 are mathematically continuous differential equations, we cannot solve these equations analytically. Therefore, the CFD code uses a discretized geometry to solve these equations numerically. The smoothness and “accuracy” of the numerical solution depend strongly on the refinement of the grid. However, as we divide the grid into finer and finer elements, we also increase the computation time. So, we often must carefully balance trade-offs between solution accuracy and computation time. Proper grid refinement is usually informed by prior experience based on past solution profiles from similar geometries. For example, a relatively coarse grid can usually be deployed in the flow channels, but a much finer grid is typically required in the catalyst and electrode layers. An example fuel cell geometry and grid are shown in Figure 13.1. When we generate a grid, we usually have the option to choose an unstructured grid or a structured grid. In an unstructured grid, the CFD software automatically fills the domain space with an array of elements of predefined shape (such as tetrahedra, hexahedra, prisms, pyramids, etc.). Grid density is controlled by changing the allowed size of these shapes. Unstructured grid deployment is a fast, easy, automated process in most CFD software packages. The drawback with unstructured grids is that it does not allow complete control over element shape or placement. In contrast, a structured grid permits precise control but can be a painful process, as the grid and element sizes must be defined manually. However, a strategically defined structured grid can greatly reduce the overall number of elements required and thereby save significant computation time. For example, in fuel cells we know that the change of fluid concentration is less severe along the flow channel (x direction) compared to out of the plane of the electrode (y direction), since reactant depletion is driven by the electrochemical reaction taking place near the electrode–electrolyte interface. Therefore, a structured grid that employs flat tetragonal shape elements (coarse in the x direction, fine in the y direction) could significantly reduce computation time without degrading the accuracy or resolution of the solution within the electrode (see Figure 13.1). Most CFD software packages include various features to expedite structured grid generation. In fuel cell models, structured grid approaches are often favored to save calculation time while still enabling the visualization of changes occurring in thin layers like the electrode and electrolyte.

FUEL CELL DESIGN VIA COMPUTATIONAL FLUID DYNAMICS

Wall Fuel outlet (current collector) (fixed gas pressure) Wall (adiabatic) Fuel inlet (fixed gas velocity Anode Electrolyte Cathode

Air outlet (fixed gas pressure) y x

z

Wall (symetric) Wall (adiabatic) Air inlet

Figure 13.1. A single-channel fuel cell geometry, including computational grid and boundary conditions. A fine grid structure is deployed in the thin electrode layer to monitor the steep changes in gas concentration, temperature, and voltage that are expected in this domain. In the flow plates, a coarse grid is deployed, since steep changes in the physical variables are not expected here. The grid associated with the flow channels has been removed to distinguish the fluid domain from the solid domain. This model is used to investigate a “counterflow” arrangement, where the flow of fuel and air are in opposite directions.

Usually, model geometry and grid generation is accomplished within the CFD software environment. However, if the model geometry is exceptionally complex, professional CAD software programs can be used for geometry generation. Most CFD software packages are able to import model geometries from CAD software programs.

13.1.3

Boundary and Volume Conditions

After building the fuel cell geometry and grid, volume conditions and boundary conditions must be assigned to each of the model domains (e.g., the flow channel domain, the electrode domain, and so on). Volume conditions are physical properties that are defined for each of the physical domains in the model geometry. These physical properties are specifically called out in the governing equations for each domain and are required to solve them. For example, the anode catalyst layer domain in the fuel cell model consists of a mixture of fluid phase (hydrogen and possibly water vapor), electron conducting phase, ionic conduction phase, and reaction sites. Thus, all six governing equations in Table 13.1 apply to this domain. We must enter all the physical properties (porosity, permeability, electrical conductivity, ionic conductivity, etc.) associated with this domain that are required to solve the governing equations within this domain. Tables 13.2a and 13.2b summarize volume conditions appropriate for the various physical properties in the flow channel, anode, electrolyte, cathode, and flow plate domains for both SOFC and PEMFC models. We will briefly review these properties.

455

456 Kinetic theory Kinetic theory Kinetic theory

Viscosity

Thermal conductivity

Diffusivity

1 — — — 0 0 — —

Porosity

Permeability

Effective thermal conductivity

Effective diffusivity

Tortuosity

Electrical conductivity

Transfer coefficient

Exchange current density

−12

m

2

10 A∕m3

14

0.5

100, 000 S∕m

1.5

Bruggeman model

11 W∕m K

1.523 × 10

0.4

10 S∕m

Kinetic Theory

Kinetic Theory

Kinetic theory

Ideal gas law

Anode

m

2

—

—

10

−20

1.5 S∕m

Bruggeman model

2.7 W∕m K

10

−18

0.001

10 S∕m

Kinetic theory

Kinetic theory

Kinetic theory

Ideal gas law

Electrolyte

−12

m

2

10 A∕m3

10

0.5

2512 S∕m

1.5

Bruggeman model

6 W∕m K

2.67 × 10

0.4

10 S∕m

Kinetic theory

Kinetic theory

Kinetic theory

Ideal gas law

Cathode

—

—

—

—

—

—

0

0

10−20 S∕m

—

33.443 W∕m K

—

7780 kg∕m3

Flow Plate

Note: The values in this table are average values that may be employed to simulate typical fuel cell behavior. More accurate values can be obtained from experiments or literature sources.

10

Ionic conductivity

S∕m

Ideal gas law

Gas property

−20

Channels

Property

TABLE 13.2. (a) Typical SOFC Volume Conditions

457

Kinetic theory 4.2 S∕m

Ideal gas law Kinetic theory Kinetic theory Kinetic theory 10−20 S∕m 1 — — — 0 0 — —

Gas property

Viscosity

Thermal conductivity

Diffusivity

Ionic conductivity

Porosity

Permeability

Effective thermal conductivity

Effective diffusivity

Tortuosity

Electrical conductivity

Transfer coefficient

Exchange current density

m

2

—

m

2

m

—

3

10 A∕m

6

0.5

53 S∕m

10−20 S∕m

Bruggeman model 1.5

—

2

200 W∕m ⋅ K

10

−11

0.4

4.2 S∕m

Kinetic theory

Kinetic theory

Mix kinetic theory

Ideal gas law

Catalyst

5

Bruggeman model

200 W∕m ⋅ K

10

−18

0.28

Nafion model

Kinetic theory

Kinetic theory

Mix kinetic theory

Ideal gas law

Cathode Membrane

m

2

—

—

53 S∕m

1.5

Bruggeman model

200W∕m ⋅ K

10

−11

0.4

10−20 S∕m

Kinetic theory

Kinetic theory

Mix kinetic theory

Ideal gas law

GDL

—

—

—

—

—

—

0

0

0.00027 S∕m

—

210 W∕m ⋅ K

—

2698.9 kg∕m3

Flow Plate

Note: The values in this table are average values that may be employed to simulate typical fuel cell behavior. More accurate values can be obtained from experiments or literature sources.

10 A∕m

—

53 S∕m

1.5

Bruggeman model

8

3

m

2

200W∕m ⋅ K

10

−11

0.4

10−20 S∕m

Kinetic theory

Kinetic theory

Kinetic theory

Ideal gas law

GDL

0.5

53 S∕m

1.5

Bruggeman model

200 W∕m ⋅ K

10

−11

0.4

Kinetic theory

Kinetic theory

Ideal gas law

Channels

Property

Anode Catalyst

TABLE 13.2. (b) Typical PEMFC Volume Conditions

458

FUEL CELL SYSTEM DESIGN

Volume Conditions. Porosity (𝜀). Porosity in the flow channels equals 1 since no pore structure exists. In solid structures such as the flow plate, we set property equal to zero. Porosity values for the electrode and catalyst layers may be obtained from the fuel cell literature. Typical values are 0.3–0.6. Permeability (𝜅). In the porous media domains (such as the electrode and catalyst layers), we must specify typical permeability values in addition to porosity values. For solid phases such as electrolyte and flow plate domains we can assign a very low (almost zero) permeability, while the flow channel is assigned a very large permeability value. Exchange current density (j0 ). The anode and the cathode domains require separate exchange current density values to describe the electrochemical reaction kinetics for each. Please note that j0 values with units of current per volume (A∕m3 ) must be used for 3D fuel cell models. These units allow the catalyst layer to be treated more realistically as a volume, rather than a surface. In our simplified fuel cell model from Chapter 6, we assumed an extremely thin catalyst layer using units of current per area (A∕m2 ) for j0 . Transfer coefficient (𝛼). Like j0 , this parameter is also used to describe the electrochemical reaction kinetics in the anode and cathode. In the ideal case, the transfer coefficient value should be equal to 0.5 (Section 3.7). This value agrees well with experimental observations in case of hydrogen dissociation in the anode. For the cathode, smaller values of 0.2–0.5 are in better agreement with experimental observations. Electronic conductivity (𝜎elec ). In the flow channel and electrolyte domains, we can set electronic conductivity equal to zero. Values for the other domains are usually set according to the experimental measurements provided by the fuel cell literature. Ionic conductivity (𝜎ion ). Typically, we can set ionic conductivity to zero in all the domains except the catalyst layer and the electrolyte. For the electrolyte and catalyst layers, it is important to incorporate the Arrhenius equation for ionic conductivity (Equation 4.32) as a volume condition rather than a constant number for better accuracy. Use of this equation allows us to account for the fact that ionic conductivity will change locally based on the local temperature. Deployment of the Arrhenius conductivity equation still requires specification of two parameters: the reference conductivity and the activation energy. The local temperature is calculated as part of the model solution. Tortuosity (𝜏). In the porous media domains (such as the electrode and catalyst layers), nominal fluid diffusivities must be corrected by the tortuosity of the pore structure (recall Section 5.2.1). Typical tortuosity values in porous fuel cell media vary from ∼1 to 4. Thermal conductivity (k). Thermal conductivity values should be assigned for all domains. For the fluid mixture in flow channels, thermal conductivity can be calculated based on the kinetic theory of gases. Most CFD programs support this option. Density (𝜌). In the gas-phase regions, a volume condition based on the ideal gas law is typically used. Viscosity (𝜇) and diffusivity (D). Like thermal conductivity, viscosity and diffusivity in the fluid regions are commonly calculated from the kinetic theory of gases (Sections 5.2.1 and 5.3.1). Several different equation-based approximations are available in most CFD programs. Effective diffusivity (Deff ). Most CFD programs provide several equations that allow the calculation of effective diffusivity based on nominal diffusivity, porosity, and tortuosity. The most popular equations have previously been presented in Section 5.2.1.

FUEL CELL DESIGN VIA COMPUTATIONAL FLUID DYNAMICS

Boundary Conditions. Boundary conditions are required to define the relationship between the outer surfaces of the model with the surrounding physical environment. In fuel cell models, the following boundary conditions are commonly employed: Inlet condition. The inlet condition is applied to the flow inlet face of the fuel cell geometry. In order to define the inlet condition, we must stipulate the composition, velocity, and temperature of the fluid entering into the fuel cell. Inlet fluid velocity is often determined based on the desired fuel and oxidant stoichiometry numbers. Outlet condition. The outlet condition is applied to the flow outlet face of the fuel cell geometry. The outlet condition is typically based on pressure. The most common outlet condition is to assume that the fuel cell outlet is exposed to atmospheric pressure. In this case, we set the outlet pressure equal to 1 atm. Wall condition. Aside from the fuel cell inlet and outlet, most other exterior surfaces in our fuel cell model are walls—meaning that no fluid can go in or out. There are two wall conditions which are critical for fuel cell models—thermal wall conditions and electric wall conditions. The two most common thermal wall conditions are adiabatic or isothermal. The adiabatic condition applies to well-insulated walls, while the isothermal condition applies to uninsulated walls. Electric (potential) wall conditions are applied to the exterior surface of the anode and cathode current collector plates (see Figure 13.1). The difference in voltage applied to the anode versus the cathode walls represents the overpotential driving the fuel cell. It is important to reinforce this point: We control 𝜂, the overpotential (voltage loss) applied to our fuel cell through the electric wall conditions, not V, the fuel cell output voltage. The fuel cell output voltage must be calculated after the fact as the difference between the reversible voltage (obtained from thermodynamics, Chapter 2) and the imposed overpotential (i.e., V = Ethermo –𝜂). The higher the overpotential condition between the anode and the cathode walls, the higher the calculated current density from the fuel cell. Solving the fuel cell model for a set of overpotential conditions allows the calculation of a complete model j–V curve. Symmetry condition. Symmetry conditions are used to reduce model construction and calculation time. If a fuel cell has identical structural and physical model geometry along a certain plane, we can split the model along this “symmetry plane,” establish a symmetry boundary condition on this plane, and then simulate only one-half of the model. Because of the symmetry, the solution we obtain for one-half of the model can simply be mirrored to provide the solution for the other half of the model. In Figure 13.1, for example, we have used a symmetry condition to split our fuel cell flow channel in half down its long axis. By making use of this symmetry condition, we save time and computational resources. 13.1.4

Solution Process and Results Analysis

Most CFD packages numerically solve the complex, coupled set of governing equations regulating a fuel cell model through an iterative process. This iterative process is started by assuming (or guessing) an initial solution to the governing equations. This initial solution is usually quite unrealistic (for example, zero values for all the physical parameters). After each successive iteration step, the CFD algorithm calculates approximate solutions, which move closer and closer to the real solution. The iteration process stops when the normalized

459

460

FUEL CELL SYSTEM DESIGN

difference between the solution from the previous iteration step and current iteration step is acceptably small. (How small is “acceptable” is defined by an error-range input that must be specified by the user before starting the solution process.) This iterative solution process can take a long time—hours, days, or even weeks depending on whether the model has tens of thousands, hundreds of thousands, or millions of grid elements. Increasing the “acceptable” iteration error range in the CFD code from 0.01% (10−4 ) to 1% (10−2 ) can often significantly reduce calculation time by sacrificing a small amount of solution accuracy. Solution convergence rate can also be improved by adjusting the CFD “relaxation parameters.” Essentially, these relaxation parameters decide how rapidly the CFD algorithm adjusts successive solution iterations. Small relaxation parameter values result in a slow but stable iteration process. With high relaxation parameter values, the iteration process may be faster but can be unstable, since the iterated solution may overshoot or diverge from the real solution. When iteration is complete, the next step is to visualize the solution. Most CFD packages provide programs that facilitate solution visualization. These programs interpolate the discrete solution values provided by the CFD solver to generate and display smooth solution profiles. Any number of model output properties, including temperature, current density, fluid flow, reactant/product concentration, voltage, and so on, can be visualized. Figure 13.2 shows a few examples of output properties obtained from a solid-oxide fuel cell model. Often, the most important physical property to calculate is the predicted current output of the fuel cell. The predicted current output of the fuel cell may be obtained by integrating the current density profile along the length of the current collecting wall. Using this calculated current value together with the overpotential difference imposed by the electrical wall condition provides one model data point for the fuel cell’s j–V curve. Please note, we obtain only one point on our model fuel cell’s j–V curve from our CFD solution! In order to generate a complete j–V curve, we need to go back and solve the model again at a number of different voltages (by changing the electric wall conditions). CFD model solutions provide an enormous amount of information and insight about the electrochemical phenomena occurring inside a fuel cell. Figure 13.2 provides example solutions for the hydrogen, oxygen, temperature, and current density distributions within a single-channel SOFC model. As shown in Figures 13.2a and b, the hydrogen and oxygen concentration profiles within this model SOFC channel decrease from inlet to outlet as these species are consumed. While fuel must be used efficiently, air can be provided in large excess quantities. This means that fuel is typically supplied to fuel cells with stoichiometry values between 1.1 and 2, while air stoichiometry values can be as high as 8–10. Accordingly, air starvation is substantially reduced. This effect is also seen in Figure 13.2b, where the oxygen concentration drop along the channel is minimized because air is supplied at 8 times stoichiometric excess. The model temperature and current density profiles are shown in Figures 13.2c and d, respectively. Since this fuel cell model was implemented with a counterflow configuration (recall Figure 13.1), both ends of the fuel cell show relatively low temperature due to the introduction of the reactant gases. In the center of the fuel cell, the temperature increases significantly, due to the generation of heat from the electrochemical reactions occurring within the fuel cell. Like temperature, current density also decreases at both ends of the

FUEL CELL DESIGN VIA COMPUTATIONAL FLUID DYNAMICS

H2 0.49 0.45 0.4 0.35 0.3 0.25 0.21

(a) O2 0.233 0.23 0.22 0.21 0.2 0.19 0.18 0.17

(b) Temperature [K] 1316 1300 1280 1260 1240 1220 1200 1180 1173

(c)

Current density [A/m2] 0 –500 –1000 –1500 –2000 –2500 –3000 –3500 –4000 –4286

(d)

Figure 13.2. Solutions obtained from a solid-oxide fuel cell model: (a) hydrogen concentration profile; (b) oxygen concentration profile; (c) temperature profile; (d) current density profile. The total overpotential is 0.3 V and the inlet gas temperatures are 900∘ C.

461

462

FUEL CELL SYSTEM DESIGN

fuel cell. Decreased temperature combined with hydrogen depletion near the fuel outlet and oxygen depletion near the air outlet lead to the decreased current density near the two ends. Although CFD fuel cell models are much more resource intensive compared to the simple 1D fuel cell models discussed in Chapter 6, they provide detailed 3D information about fuel depletion regions, hot spots, and other geometric effects that are crucial for optimizing fuel cell design. Some of this information is very difficult or even impossible to measure experimentally. This makes CFD modeling a compelling and powerful tool in the arsenal of any fuel cell designer. Before we move to system-level design, a final word of warning about CFD fuel cell modeling is warranted. Like any other model, the validity of a CFD fuel cell model depends crucially on the validity of the original assumptions and physical properties (e.g., governing equations, volume conditions, and boundary conditions) that were supplied to it. The old modeling adage “junk in leads to junk out” is highly appropriate. If the original data or assumptions grounding the CFD model are inadequate or even incorrect, the solutions obtained will be meaningless! 13.2

FUEL CELL SYSTEM DESIGN: A CASE STUDY

Now that we have examined model-based design of fuel cells, in this section you will learn how to design a complete fuel cell system. As a case study, we will design a portable solid-oxide fuel cell system. Our portable SOFC system will be required to deliver 20 W power at 12 V. Portable SOFC system design proves to be particularly challenging because of the difficulties associated with thermal management and packaging. Therefore, this case study serves as an excellent example to demonstrate the finer points of thermal and mass balance bookkeeping in fuel cell system design. At the same time, this case study is also small enough and simple enough for demonstration purposes as opposed to more complicated stationary or transportation systems designs. A complete fuel cell system includes not only the fuel cell itself but also a number of ancillary components that are collectively referred to as the balance of plant (BOP). Common BOP components include power converters, heat exchangers, air blowers, fuel-processing units, and so on. Many of these components were briefly reviewed in Chapter 10. Designing a complete fuel cell system involves not only designing and sizing the fuel cell properly but also selecting the right BOP components as well. Fuel cell system design should be approached as an iterative process that is repeated until the desired design goals are reached. We will employ the following iteration process in this case study: 1. Construct a reasonable system configuration and make a good guess on the specifications necessary for each of the various system components. 2. Calculate the thermal and mass balance of the complete system based on the starting component parameters guessed in 1. 3. Refine the choice and specification of system components according to the thermal and mass balance calculated in 2. For coupled components, verify compatibility based on the expected magnitude and rate of mass, heat or current transfer between them.

FUEL CELL SYSTEM DESIGN: A CASE STUDY

4. Review the system’s performance considering the original design goals. If system refinement is required, decide which components or parameters should be changed and repeat the design process. 13.2.1

Design of a Portable Solid Oxide Fuel Cell System

We have been commissioned by a highly motivated (and deep-pocketed) sponsor to design a portable SOFC system that is capable of providing DC power to drive a suite of small electronic devices. Currently, there is significant interest in fuel cell–based portable power systems to overcome critical limitations associated with traditional battery technology. For example, the military is interested in small-scale portable power fuel cell systems for soldier field missions; in the commercial sector, portable fuel cell systems might be ideal for scientific field workers in remote or environmentally sensitive locations. Our sponsor has provided us with the following design requirements: The fuel cell system should be able to deliver 20 W at 12 V. Other than this overall power requirement, we have complete freedom on how to design our system. Obviously, there are many different SOFC system configurations that can achieve this design goal. For simplicity, however, we will restrict ourselves to a relatively simple SOFC system that contains only a few essential components, including: • Fuel Cell Stack. Our target design goal is a fuel cell system with a net power output of 20 W. The fuel cell stack, then, must be designed to produce an output power significantly larger than 20 W. This is because BOP components, such as the air blower and the DC–DC converter, will consume part of the fuel cell stack power. In portable systems, it is not uncommon for these BOP components to drain as much as 50% of the stack power. Therefore, we may need to choose a 40–50 W fuel cell stack in order to ensure that we can produce 20 W net. The second specification is that our system should deliver power at 12 V. To obtain a stack voltage of 12 V, approximately 17 cells will be required if each cell operates at 0.7 V (this is a typical per-cell operating voltage). Because designing and fabricating a 17-cell stack is both difficult and expensive, a better option may be to use fewer cells (for example 6–8 cells) and then include a 12 V DC–DC converter in the system to boost the output voltage. • Hydrogen Supply. Our next design decision is to choose a fuel supply system. Because we have been given complete freedom here, we will use a metal hydride cylinder for our hydrogen supply system. Metal hydrides provide good volumetric storage capability, which can be advantageous for portable systems. They are also simple and safe and can deliver the hydrogen at relatively high pressures/flow rates without the requirement for pumps or blowers. Alternatively, we could choose to design a hydrocarbon-fuel-based reformer system (recall Chapter 11), but for our portable system we will avoid this option due to its complexity. • Air Supply. Air must be delivered to our fuel cell stack both to feed the cells and to cool them. Relatively large flow rates will be needed; therefore, a compressor, fan, or blower will be required. We will use an air blower. Because the fuel cell system will be generating DC power at 12 V, the air blower should also be specified to operate using 12 V DC power.

463

464

FUEL CELL SYSTEM DESIGN

• Heat Management. Our SOFC system will be generating significant amounts of heat. Rather than wasting this heat, we will probably want to recycle it using a heat exchanger. We can use a heat exchanger to warm up the cold fuel cell inlet gases using the hot fuel cell exhaust gases. This will minimize the temperature differences within the SOFC stack and significantly improve performance. • DC–DC Converter. Because we have made the strategic decision to use only a 6–8 cell stack, we will need a DC–DC converter to boost up the output voltage from the fuel cell to 12 V. Fuel cell output voltages tend to fluctuate somewhat in time. Therefore, the DC–DC converter also serves a second role by stabilizing the output voltage to the external load. Based on this analysis, we can come up with a potential system configuration as shown in Figure 13.3. Here is a brief description of our initial system design. Hydrogen from the hydride canister and air from the blower are first sent through a heat exchanger, where both gases are preheated from ambient temperature before entering the SOFC. Preheating ensures that the inlet gases do not “quench” the SOFC, which must sustain a high operating temperature. After flowing through the SOFC, the now very hot exhaust gases pass through the heat exchanger, releasing their heat to the inlet gases and cooling to acceptable levels before being vented to the environment. Electric power from the fuel cell is delivered to the DC–DC converter, where it is boosted to 12 V. A portion of this electric power is used to drive the air blower, while the rest (hopefully at least 20 W!) is supplied to the external load. The system configuration we have chosen is in fact quite simple. In future design iterations, we may want to think about adding additional components, like a tail-gas

Exhaust gas Air Heat exchanger Blower Fuel feed (hydrogen storage) Cathode

Fuel cell stack Anode Packaging

DC/DC converter

Packaging

Net power (20W, 12V)

Figure 13.3. Schematic of a simple portable SOFC system.

FUEL CELL SYSTEM DESIGN: A CASE STUDY

combustor or fuel recirculator, to utilize wasted fuel in the exhaust. In a practical system, we would also need to integrate sensors, valves, and controllers to regulate flow rates, temperatures, and power output. Additionally, fuel cell start-up and shutdown must be dealt with. For example, it might be necessary to incorporate a small combustion heater to warm up the cell from a cold start. For simplicity, we will not consider these issues in this case study. Now that we have decided on a basic system configuration, the next step is to make some preliminary estimates for our fuel cell stack requirements. Guessing the appropriate specifications for our fuel cell stack is difficult. The power density we can extract from our SOFC will depend strongly on the operating temperature. However, as the operating temperature increases, the need for cooling also increases, which means that more power will be sacrificed to power our air blower. We don’t know how large to make the fuel cell stack, because we don’t know how much power the blower will consume. However, we don’t know how much power the blower will consume until we set the size and air stoichiometry requirements for the fuel cell stack! It’s almost a classic chicken-vs.-egg problem. The various parameters in our system are strongly coupled (usually nonlinearly) and therefore cannot be solved explicitly. So how do we start, then? We are forced to take a guess at a set of initial fuel cell stack parameters based on our intuition and experience. After the design is done, we can then go back and update the fuel cell stack parameters with more suitable values. After several design iterations, we may reach a design that is close to optimum. Let’s take a look at Figure 13.4, which provides performance information on our SOFC, to start guessing these values. 1.2

2.0 Cell voltage Power density

1.8

Cell voltage (V)

800˚C

1.4

750˚C 0.8

700˚C

1.2

650˚C

1.0

600˚C 0.6

550˚C

0.8 0.6

0.4

Power density (W/cm2)

1.6

1.0

0.4 0.2

0.2 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0 4.0

Current density (A/cm2)

Figure 13.4. Polarization curve of SOFC obtained from a CFD solid-oxide fuel cell model. The model calculation is based on a stoichiometry of 1.2 for hydrogen and 8.0 for air at 1 atm.

465

466

FUEL CELL SYSTEM DESIGN

Figure 13.4 shows a set of j–V performance curves for the SOFC that we will use for our system. These j–V performance curves were likely obtained in one of two ways. They were either calculated from a CFD fuel cell model, as we discussed in the previous portion of this chapter (section 13.1), or they were obtained from direct single-cell experimental measurements in a fuel cell laboratory. For the j–V curves presented in Figure 13.4, the stoichiometry number for hydrogen was set at 1.2, and the stoichiometry number for air was set at 8.0. We assume that the fuel cell outlet is exposed to atmospheric pressure. You may notice that these data were obtained with a fairly high air stoichiometry number. There are two reasons for that. High airflow rates are necessary to cool the stack and can also help improve fuel cell performance. In SOFC systems, operating with large (>5) air stoichiometry values is often a wise design choice. Sizing the fuel cell stack involves a trade-off between efficiency and power density. If we choose to operate the fuel cell at maximum power, we can minimize the size of the fuel cell. However, fuel cell efficiency is low at high power density, and so we will need a larger hydrogen tank or we will sacrifice system lifetime. Also, operating at maximum power density generates more heat, which requires increased cooling and therefore increased parasitic power consumption from the air blower. On the other hand, if we choose to operate the fuel cell at high voltage, the fuel cell efficiency goes up, but the fuel cell power density goes down. In this case, we would need a larger stack, which would increase system size and cost. Based on the trade-offs discussed above, a frequent design target for many fuel cell systems is a cell voltage of around 0.7 V, which combines reasonable efficiency (∼ 50%) with reasonable power density. Our choice of operating temperature also involves a compromise, this time between performance and cooling requirements. Again, an intermediate temperature probably makes a good first guess. Let’s choose to operate at 700∘ C. Finally, we need to choose a power output for our fuel cell stack. Since we know the blower and DC–DC converter might consume a significant fraction of the total power, let’s be conservative and specify a stack output power of 50 W. Based on these considerations the initial values for our fuel cell stack operation are set as shown in Table 13.3. Finally, we also make some initial guesses about the efficiency parameters for several of our other system components. Based on our previous discussions on DC–DC converters and heat exchangers (see Sections 10.5.3 and 10.5.4, respectively), we will assume design efficiencies for these components as shown in Table 13.4. TABLE 13.3. Initial Values of Design Parameters for the SOFC Stack Stack Design Parameters

Value

Fuel cell operating temperature, Tfc

700 ∘ C

Hydrogen and air pressure

1 atm

Hydrogen stoichiometry, 𝜆H2

1.2

Air stoichiometry, 𝜆O2

8

Fuel cell operating voltage, Voper

0.7 V

Fuel cell output power, Pfc

50 W

FUEL CELL SYSTEM DESIGN: A CASE STUDY

TABLE 13.4. Initial Values of Design Parameters for the SOFC BOP Components

13.2.2

System Component Design Parameters

Value

DC-DC converter efficiency, 𝜀DC-DC

90%

Heat exchanger efficiency, 𝜀HX

90%

Thermal and Mass Balance

Now that we have settled on initial design specifications, the next step is to conduct a complete mass and heat balance for our fuel cell system. The mass and heat balance calculations will help us to properly size the various components in our system because we will know the size of the heat flow rates and gas flow rates passing through our system. Mass Balance. From Figure 13.4, fuel cell operation at 0.7V generates joper = 1.5A∕cm2 at 700∘ C. For 50 W total power generation, we will therefore need a total fuel cell area of Afc = 47.62 cm2 [= 50W∕(0.7 V × 1.5A∕cm2 )]. The total current generation from this fuel cell is then itotal = 71.43A(= 50W∕0.7V). We can find the required hydrogen supply rate (taking into account the fuel stoichiometry, 𝜆H2 = 1.2) as itotal 71.43 A × 𝜆H2 = × 1.2 nF 2 × 96, 485 C∕mol = 4.442 × 10−4 mol∕s = 0.02665 mol∕min

𝑣H2 ,supply =

(13.15)

Similarly, taking into account the air stoichiometry of 8, we can find the oxygen supply rate: i 71.42 A 𝑣O2 ,supply = total × 𝜆O2 = ×8 nF 4 × 96, 485 C∕mol (13.16) = 0.001481 mol∕s = 0.08884 mol∕min We then calculate the nitrogen supply rate as 𝑣N2 ,supply = 𝑣O2 ,supply × 𝜔 = 1.481 × 10−3 mol∕s ×

0.79 0.21

(13.17)

= 0.005571 mol∕s = 0.3342 mol∕min where 𝜔 stands for the molar ratio of nitrogen versus oxygen in air. The total air supply rate is simply the sum of the oxygen plus nitrogen supply rates: 𝑣air,supply = 𝑣N2 ,supply + 𝑣O2 ,supply = 0.3342 mol∕min + 0.08884 mol∕min = 0.4230 mol∕min

(13.18)

Using the ideal gas law, we can convert this molar supply rate into a volume supply rate at STP: 25C V̇ air,supply =

𝑣air,supply RT

p = 10.35 LPM

=

0.4230 mol∕min × 0.0820578 atm × L∕mol × K × 298.15 K 1 atm (13.19)

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This is the volumetric air flow rate that our air blower will need to supply. Based on our fuel cell current, we can calculate the water production rate from the fuel cell as itotal 71.42 A = = 3.702 × 10−4 mol∕s = 0.0222 mol∕min nF 2 × 96, 400 C∕mol (13.20) This is the same as the hydrogen consumption rate (𝑣H2 ,cons = 3.702 × 10−4 mol∕s) and is equal to twice the oxygen consumption rate (𝑣O2 ,cons = 1.851 × 10−4 mol∕s). Using these values, we can then find the flow rates at the exhaust of the fuel cell: 𝑣H2 O,prod =

𝑣H2 ,exhaust = 𝑣H2 ,supply − 𝑣H2 ,cons = 7.403 × 10−5 mol∕s 𝑣O2 ,exhaust = 𝑣O2 ,supply − 𝑣O2 ,cons = 0.001296 mol∕s 𝑣N2 ,exhaust = 𝑣N2 ,supply = 0.005571 mol∕s 𝑣H2 O,exhaust = 𝑣H2 O,prod = 3.702 × 10−4 mol∕s

(13.21)

Table 13.5 summarizes various flow rate values that we have calculated. Thermal Balance. Compared to the mass balance, the thermal balance calculation is a bit more complex. For our mass balance calculation, we can guarantee that no mass disappears in the system (unless there are leaking components within the system). For thermal balance calculation, however, it is likely that some heat will dissipate, or “leak,” to the environment from many of our system components, especially hot components like the fuel cell stack and heat exchanger. Good packaging with thermal insulation will reduce the heat dissipation, but we will not be able to stop it entirely. However, for simplicity in this case study, we will ignore all heat dissipation in our fuel cell system and assume adiabatic conditions. The enthalpy of the hydrogen–oxygen reaction is –247.7 kJ∕mol at 700∘ C. Thus, the equivalent voltage for this reaction enthalpy is EH =

̂ 247, 700 J∕mol |Δh| = = 1.28 V nF 2 × 96, 400 C∕mol

(see Section 2.6). The heat generation rate from the fuel cell can thus be calculated as Pheat = (EH − Voper ) × itotal = (1.28 V − 0.7 V) × 71.43 = 41.71 W TABLE 13.5. Various Flow Rates in the SOFC System Flow Rates

Value (mol/s)

𝑣H2 ,supply

4.442 × 10−4

𝑣O2 ,supply

0.001481

𝑣N2 ,supply = 𝑣N2 ,exhaust

0.005571

𝑣H2 ,exhaust 𝑣O2 ,exhaust 𝑣H2 O,exhaust = 𝑣H2 ,cons = 2𝑣O2 ,cons

7.403 × 10−5 0.001296 3.702 × 10−4

(13.22)

FUEL CELL SYSTEM DESIGN: A CASE STUDY

Our goal is to maintain our fuel cell stack at the designed operating temperature point of 700∘ C. The stack will maintain this constant operating temperature only if all the generated heat can be removed from the fuel cell. Since we have assumed adiabatic conditions for the fuel cell, the only way for this heat to be removed is if it is carried out of the fuel cell by the exhaust gas stream. Accordingly, we can set up the following equation: ) ) ( ( ∑ ∑ cp,i 𝑣i,exhaust ΔTfc = cp,i 𝑣i,exhaust (Tfc,out − Tfc,in ) (13.23) Pheat = i

i

where cp,i (J∕mol ⋅ K) and ΔTfc (K) stand for the heat capacity of species i and the temperature difference between the inlet gas and outlet gas of the fuel cell, respectively. For convenience, we will assume that cp,i is constant (i.e., that it does not change with temperature) throughout these calculations. Representative cp,i values for the various gases involved in our system are provided in Table 13.6. We will assume that our inlet fuel cell gases have been properly preheated to avoid thermal quenching of our fuel cell. Therefore, we set the inlet gas temperature Tfc,in to be same as the fuel cell operating temperature Tfc in Equation 13.23. Plugging the values from Tables 13.5 and 13.6 into Equation 13.23 allows us to solve for the outlet gas temperature of the fuel cell: Pheat = 41.71 W [ ] = cp,H2 𝑣H2 ,exhaust + cp,O2 𝑣O2 ,exhaust + cp,N2 𝑣N2 ,exhaust + cp,H2 O 𝑣H2 O,exhaust (Tfc,out − Tfc,in ) = (30.116 J∕mol ⋅ K × 0.00007403 mol∕s + 34.366 J∕mol ⋅ K × 0.001296 mol∕s + 32.409 J∕mol ⋅ K × 0.005571 mol∕s + 40.924 J∕mol ⋅ K × 0.00037 mol∕s)(Tfc,out − 973.15) ∴Tfc,out = 1145 K

(13.24)

TABLE 13.6. Heat Capacity of Various Gases at 700∘ C ∘ Gas cp700 C (J/mol⋅K) H2

30.116

O2

34.366

N2

32.409

H2 O

40.924

Note: We assume that these heat capacity values are independent of temperature and that they can therefore apply over a relatively large temperature range around 700∘ C. Recall from homework problem 2.9 that this is a reasonable assumption.

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According to our system design, the hot exhaust gases from the fuel cell pass through a heat exchanger where they transfer heat to the cold inlet gases. Recall from Section 12.1 that the heat transfer rate (Q̇ HX ) in the heat exchanger can be described as ) ) ( ( ∑ ∑ ( ) Q̇ HX = cp,i 𝑣i ΔThot 𝜀HX = cp,i 𝑣i Thot,in − Thot,out 𝜀HX i

=

( ∑ i

) cp,i 𝑣i

ΔTcold =

( ∑

i

)

cp,i 𝑣i

( ) Tcold,in − Tcold,out

(13.25)

i

where cp,i (J∕mol ⋅ K) stands for the heat capacity of species i, ΔT (K) stands for the temperature difference experienced by either the hot or cold stream across the heat exchanger, and 𝜀HX is the efficiency of the heat exchanger. Since heat exchangers are not perfectly efficient, only a portion of the heat carried by the hot stream is transferred to the cold stream. This effect is accounted for by the heat exchanger efficiency, 𝜀HX . As discussed in Chapter 12, heat exchanger efficiencies close to 90% are not unreasonable. Based on our system design, we know that the supply gases must be heated up to the fuel cell operating temperature by the time they exit from the heat exchanger. Therefore, we set Tcold,out = Tfc,in = Tfc . For Tcold,in , we will assume that the supply gases enter the heat exchanger at ambient temperature (Tcold,in = 298.15 K). Plugging the relevant quantities into Equation 13.25 allows us to calculate the required Q̇ HX : [ ]( ) Q̇ HX = cp,H2 𝑣H2 ,exhaust + cp,O2 𝑣O2 ,exhaust + cp,N2 𝑣N2 ,exhaust Tcold,out − Tcold,in = (30.116 J∕mol ⋅ K × 0.000444 mol∕s + 34.366 J∕mol ⋅ K × 0.001481 mol∕s + 32.409 J∕mol ⋅ K × 0.005571 mol∕s)(973.15 K − 298.15 K) = 165.3 W

(13.26)

Again, we have assumed constant cp values using the information provided in Table 13.6. We know that all this heat must be provided by transfer from the hot stream. Therefore, we can calculate the outlet temperature of the hot stream exiting the heat exchanger. To make this calculation, we assume that Thot,in = Tfc,out since the hot stream entering into the heat exchanger comes directly from the outlet of the fuel cell stack: Q̇ HX = 165.3 W [ ] = cp,H2 𝑣H2 ,exhaust + cp,O2 𝑣O2 ,exhaust + cp,N2 𝑣N2 ,exhaust + cp,H2 O 𝑣H2 O,exhaust ( ) Thot,in − Thot,out 𝜀HX = (30.116 J∕mol ⋅ K × 0.00007403 mol∕s + 34.366 J∕mol ⋅ K × 0.001296 mol∕s + 32.409 J∕mol ⋅ K × 0.005571 mol∕s

∴Thot,out

( ) + 40.924 J∕mol ⋅ K × 0.00037 mol∕s) 1145 K − Thot,out × 0.9 = 388 K = 115∘ C

(13.27)

FUEL CELL SYSTEM DESIGN: A CASE STUDY

TABLE 13.7. Thermal Balance Parameters for the SOFC System Thermal Parameters

Values

PHeat

41.7 W

Q̇ HX

165.3 W

Tfc,in = Tcold,out

973.15 K

Tfc,out = Thot,in

1145 K

Tcold,in = Tambient

298.15 K

Thot,out

388 K

Thus, the fuel cell exhaust gases exit the heat exchanger with a temperature of 115∘ C. This temperature is still 90∘ C higher than the temperature of the cold supply gases entering the heat exchanger (Tcold,in = 25∘ C). Recall from Chapter 12 that this minimum temperature difference between the hot and cold streams represents the “pinch point” for this heat exchanger. Our pinch point is 90∘ C. Because this temperature difference is reasonably high, we can expect our exchanger to function very close to its rated efficiency of 90%. The hot stream exhaust gases leave the heat exchanger at a temperature of 115∘ C. This is probably almost ideal. We are above 100∘ C, so we don’t have to worry about liquid water condensation within the heat exchanger. At the same time, however, this temperature is likely cool enough to allow direct venting to the ambient. Table 13.7 summarizes our thermal balance calculations.

13.2.3

Specifying System Components

Now that we have worked out the thermal and mass balance analysis for our system, the next step is to choose system components that will be able to properly handle our calculated mass and heat flows. In particular, we will need to specify our air blower, our heat exchanger, our DC–DC converter, and our metal hydride tank. We start with the air blower. Based on our air supply mass balance calculation from Equation 13.19, the blower must be able to supply at least 10.35 LPM. Furthermore, since the fuel cell system delivers 12 V DC power to the external load, the air blower should operate at 12 V DC (otherwise, a second DC–DC converter for the blower would be required). Table 13.8 shows example specifications for an air blower (taken from a real catalogue) that satisfies these requirements. Next, we will specify the heat exchanger. Based on our thermal balance calculations, the heat exchanger must be able to accept gas temperatures as high as 872∘ C (= Thot, in ). Based on our mass balance calculations, the heat exchanger must also be sized properly to handle flow rates on the order of 12–15 LPM. The heat exchanger will need four paths—two “hot paths” for the (initially hot) anode and cathode fuel cell exhaust gases and two “cold paths” for the (initially cold) hydrogen and air supply gases. Table 13.9 shows an example specification for a heat exchanger that is suitable for this system.

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TABLE 13.8. Example Specification of an Air Blower Specification

Value

Maximum flow rate

15.5 LPM

Maximum current

3.2 A

Operating voltage

12 V

Max pressure

1200 mbars

Power vs. flow rate

Linear (Pblower ∝ V̇ air,supply )

TABLE 13.9. Example Specification of a Heat Exchanger Specification

Values

Maximum fluid temperature

900∘ C

Rated flow range

5–20 LPM

Number of paths

4

Thermal efficiency

90%

TABLE 13.10. Example Specifications for the DC–DC Converter Specification

Values

Input voltage

Min 4.0 V, Max 25 V

Output voltage (adjustable)

Min 1.5, Max 15 V

Maximum input current

20 A

Maximum output current

5A

Efficiency

90%

Our DC–DC converter must deliver an output voltage of 12 V. Standard converters allow voltage multiplication typically up to a factor of 2–3. Greater voltage multiplication decreases converter efficiency and increases converter costs. Table 13.10 describes example specifications for a suitable DC–DC converter. This converter provides up to three times voltage multiplication. Thus, to generate an output voltage of 12 V, a minimum input voltage of 4 V or greater is required. Since our fuel cell operation voltage is 0.7 V, our fuel cell stack needs to have at least six cells (0.7 V × 6 = 4.2 V). With a stack voltage of 4.2 V, the current output will be 11.9 A (istack = stack power∕output voltage = 50 W∕4.2 V). Our DC–DC converter can handle up to 20 A input current, so this is OK. We must also check the maximum power output rating for the converter. Our converter has a maximum output

FUEL CELL SYSTEM DESIGN: A CASE STUDY

TABLE 13.11. Example Specification for the Metal Hydride Cylinder Specification

Values

Dimension

D 6.4 cm, H 26.5 cm

Weight

2.2 kg

Hydrogen capacity

250 L

Internal pressure

17 atm

current of 5 A. At 12 V, this leads to a maximum power output of 60 W. Since our fuel cell stack delivers 50 W, we should be OK here as well. We now turn our attention to the metal hydride cylinder. Our choice for tank size depends on our desired operational lifetime and/or system size constraints. Because neither of these metrics were part of our design guidelines, we have great latitude to choose our cylinder. Table 13.11 shows an example specification of a relatively small metal hydride cylinder (about the size of two 12-ounce soda cans stacked end to end). Although the cylinder is less than 1 L in size, it holds 250 standard liters of hydrogen! The output pressure from the hydride cylinder is 17 atm, so we will need a pressure regulator to reduce the outlet pressure to something more manageable. From our mass balance analysis, we know that our fuel cell stack requires 0.02665 mol/min of hydrogen (see Equation 13.15). This corresponds to 0.652 LPM at STP. Therefore, this cylinder will provide enough hydrogen for about 6.4 h of operation (= 250 L∕0.652 LPM). If a longer runtime is desired, a larger hydride tank can be specified. Now that we have specified all the components of our system, we can calculate net power output and efficiency of our system design. In order to calculate net power output in this simple example, we need to take into account losses due to the DC–DC converter efficiency and the power consumption of the blower. Thus, the net output power can be written as Pnet = Pfc × 𝜀DC-DC − Pblower

(13.28)

where Pfc , 𝜀DC-DC , and Pblower stand for the power output of the fuel cell stack, the DC–DC conversion efficiency, and the power consumption of the air blower, respectively. Based on the blower specifications (see Table 13.7), we know that the power consumed by the blower scales linearly with the amount of air it is required to blow. The blower operates at 12 V and consumes 3.2 A to blow 15.5 LPM of air. Since our system requires 10.35 LPM of air, we can estimate the power consumption of the blower as max current × actual flow rate × Vblower max flow rate 3.2 A = × 10.35 LPM × 12 V = 25.64 W 15.5 LPM

Pblower =

(13.29)

The net power from the fuel cell system is therefore Pnet = 50 W × 0.9 − 25.64 W = 19.36 W This is slightly smaller but very close to our original design goal of 20 W.

(13.30)

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Finally, we calculate the net efficiency of our system. We define net efficiency as the power delivered to the external load versus the incoming enthalpy of the hydrogen fuel. Using this definition, our net system efficiency is 𝜀net = =

Pnet Pnet = |Δḣ HHV | |ΔhHHV | × 𝑣H2 ,supply 19.36 W mol H2 J 247, 700 × 4.442 × 10−4 mol H2 s

=

19.36 W = 0.176 110 W

(13.31)

On a net basis, our fuel cell system is only 17.6% efficient. This very poor efficiency is mainly due to the fact that more than half of the stack power is consumed by the balance of plant components. If we considered only the fuel cell stack itself, the efficiency would be 50 W∕110 W = 45.5%. 13.2.4

Design Review

Table 13.12 summarizes the design specifications that we have developed for our portable SOFC system. Let’s review and discuss some of the key points of our current design. • The blower consumes more than half of the overall system power. The blower requires a lot of power because of the large air stoichiometry number (8) that we have chosen for our design. Significant airflow is needed to cool the fuel cell stack. Even with this large airflow, the temperature difference between the inlet and outlet of the fuel cell is still 172∘ C (= 872 − 700∘ C). If we were to reduce the air stoichiometry (to reduce air blower power consumption), the temperature difference would be even bigger. Large temperature differences can lead to severe thermal stress in the fuel cell stack; therefore, it may be difficult to reduce blower power. In reality, however, some heat will dissipate from the fuel cell stack into the surrounding environment (in a sense, our adiabatic assumption represented a “worst-case” scenario). In reality, therefore, TABLE 13.12. Final Specifications for the SOFC Stack and System Specification

Value

System net power

19.36 W

Fuel cell power

50 W

Fuel cell voltage

4.2 V

Number of cells in the fuel cell stack

6

Temperature range of the fuel cell

700–872∘ C

Temperature range of the heat exchanger

25–872∘ C

System operation time

6.4 h

Air blower power consumption

25.64 W

FUEL CELL SYSTEM DESIGN: A CASE STUDY

•

•

•

•

•

•

the cooling requirements would likely be somewhat less stringent, and we might be able to lower the air stoichiometry a bit. In addition, we could supply gases to the fuel cell stack that are only partially preheated (e.g., to 600∘ C) and allow final heating of these gases within the fuel cell stack to act as part of the cooling load. Our fuel cell stack generates a relatively low voltage of 4.2 V. If we choose a stack that has more cells, and hence a higher output voltage, we could use a more efficient DC–DC converter, thereby improving net system efficiency. However, increasing the number of cells in our stack will make the stack fabrication process more difficult and costly. The temperature range in the heat exchanger is quite high. Since the heat exchanger will experience gas temperatures up to 872∘ C, it will need to be constructed from special (and perhaps costly) high-temperature materials. Similarly, other tubing, valves, and connectors in the system may also need special materials consideration. We have not evaluated the pressure resistance of our system. The air blower must be able to apply sufficient pressure to work against the total pressure resistance caused by the gas lines, fuel cell stack, and heat exchanger. The blower can apply up to 1.2 bars (see Table 13.8). This value will likely be adequate, but it will need to be verified. As mentioned previously, our design assumption of an ideal adiabatic system is probably unrealistic. In a real fuel cell system, heat will dissipate from many of the system components and gas lines. Therefore, the actual temperatures in various parts of the system will be lower than what we have calculated. In some cases, this may help (for example, by decreasing our air cooling requirements). In other cases, this may cause problems (for example, by lowering fuel cell or heat exchanger performance). An estimation of heat dissipation effects could be taken into consideration during a second design iteration. Our system design did not consider weight, volume, efficiency, or cost. Trade-offs become considerably more complex when these criteria are also added to the equation. Thermal packaging, which we also did not consider, would probably significantly increase the weight and volume of our SOFC system. Our design was based on the j–V performance curves provided in Figure 13.4, which were measured (or modeled) from a single cell. However, the fuel cell stack in our design consists of six cells connected in series. We have assumed that the performance of each cell in the six-cell stack will be identical to the performance of a single cell measured alone. This is probably not a good assumption. When multiple cells are stacked together, the flow distribution and temperature distribution in each cell will not be perfectly identical. Accordingly, stack performance changes significantly compared to the single-cell prediction. Usually, stack performance is worse (typically, by 5–20%) than the performance obtained from single-cell measurements. Better design data could be obtained by actually constructing and measuring (or modeling) a complete six-cell stack.

Fortunately, our initial design parameter guesses (see Table 13.3) brought us very close to our design goal of 20 W net power. If this had not been the case, we would need to go back and change the initial design parameters (or even redesign the fuel cell) until the desired system goals were obtained.

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13.3

CHAPTER SUMMARY

The purpose of this chapter is to explain how to model and design fuel cells and fuel cell systems. At the single-cell level, we have learned how CFD can be a convenient and powerful tool for design. While it can provide great insight, building, solving, and analyzing CFD fuel cell models require significant time and computational resources. CFD modeling can also provide misleading or meaningless information if the grounding assumptions or input parameters provided to the model are incorrect. Designing a complete fuel cell system is an iterative process. The design process involves choosing an initial system configuration based on intuition and experience, analyzing the mass, thermal, and power balances, specifying the system components, and then evaluating the system results. The components in a fuel cell system are often implicitly coupled to one another through the heat, mass, and power flows. Good fuel cell system design depends crucially on properly choosing, sizing, and matching system components to meet the overall design goals. • CFD-based fuel cell models involve the numerical solution of mass, momentum, energy, and charge conservation equations across complex (often 2D or 3D) geometries. These equations are coupled to the Butler–Volmer equation to describe fuel cell electrochemistry in addition to the mass, heat, and charge transport. • In a CFD fuel cell model, the complex 2D or 3D fuel cell geometry is divided into a 2D or 3D grid structure. Each element in the grid represents a discretized entity associated with physical property values. • The CFD model geometry may be discretized using either a structured or an unstructured grid. Structured grids can reduce computation requirements and are often useful for fuel cell models but can take more time to construct than unstructured grids. • CFD fuel cell models require various physical property inputs as boundary and volume conditions. The reliability of the solution depends on the proper choice and setup of these conditions. • Solving a CFD fuel cell model can require significant computational resources. Solution accuracy, number of grid elements, and the relaxation parameter conditions can affect the computation time. • The solution from a CFD fuel cell model contains unorganized but detailed information on the electrochemical processes occurring in the fuel cell. Extraction, visualization, and analysis of this information are crucial and important steps in order to gain full insight from the fuel cell model results. • A complete fuel cell system is composed of a fuel cell stack plus ancillary components that are collectively referred to as the balance of plant (BOP). • Fuel cell system design is an iterative process. Starting the design process with a good initial guess on the system configuration and critical system parameters can reduce the number of iterations required. • Many components in a fuel cell system are coupled to one another through mass, heat, and power balances. It is important to match the size and specifications of coupled components by carefully considering each of these balances.

CHAPTER EXERCISES

• Not only are system components coupled to one another via the mass, heat, and power balances, but the mass, heat, and power balances are coupled to each other as well. Understanding how changes can simultaneous affect all three is helpful for improving system design.

CHAPTER EXERCISES Review Questions 13.1

List all the variables in the conservation equations (see Table 13.1) that must be calculated (in other words, they are solution outputs) when these equations are solved.

13.2

You are constructing a structured grid for a CFD solid-oxide fuel cell model. You are especially interested in monitoring the current density profile through the electrode (out-of-plane direction) in detail. You divide the out-of-plane electrode direction for the cathode into 10 evenly spaced grid elements. Will you use the same grid structure in the anode as well? If not, how would you define the anode grid?

13.3

The anode reaction for a direct methanol fuel cell is: CH3 OH + H2 O ↔ CO2 + 6H+ + 6e− Write the Butler–Volmer for this reaction in the form of Equation 13.12.

Calculations 13.4

You have constructed a 3D PEMFC CFD model of a serpentine flow channel-based fuel cell using an evenly spaced grid. You then decide to increase the number of grid elements in the U-shaped regions of the serpentine flow channels (where flow abruptly changes direction). Within these U-shaped regions (which account for 1/10 of the total volume of the model), you have decreased the dimensions of the individual grid elements by 1/2 in all three dimensions (x, y, and z). If the calculation time of the model is proportional to the square of the number of grid elements, how much more calculation time will your refined model take?

13.5

According to Table 6.1, the typical exchange current density for the anode catalyst layer of a PEMFC is 0.1A∕cm2 . This is the “per-area” exchange current density. If the thickness of the catalyst layer is 10 μm, what is the “per-volume” (A∕cm3 ) exchange current density?

13.6

Find the power of the fuel cell stack that would be required to generate exactly 20 W net power for the case study in Section 13.2. Assume that we keep the same system components and configuration (although the heat, mass, and power balances will change).

13.7

Design a portable SOFC system delivering net power of 10 W and make a table of overall system parameters for your design similar to Table 13.12. You can use the

477

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specifications for the SOFC and BOP components given in Section 13.2. Use the system parameters shown in Tables 13.3 and 13.4, except that the design goal is now 10 W rather than 20 W. 13.8

In this problem, we will attempt to account for heat dissipation in the SOFC system case study presented in Section 13.2. We will assume that our fuel cell stack dissipates heat to the environment. The heat dissipation rate, Q̇ diss (W) is proportional to the difference between the fuel cell temperature, Tfc and the ambient temperature, Tamb = 298 K. The output power of the fuel cell also affects the heat dissipation rate, since a bigger fuel cell stack will have more surface area. We therefore assume that the heat dissipation rate can be represented as Q̇ disspation = k(Tfc − Tamb )Pfc,0.7V where Pfc,0.7V (W) is the power of the fuel cell at 0.7 V and k (K-1 ) is a proportionality constant. As another important design constraint, we must account for the fact that our heat exchanger will not function properly if the “pinch point” becomes too small. We will assume that our heat exchanger requires a minimum temperature difference between hot and cold streams of 20∘ C. Using the design parameters from the case study in Section 13.2, find the maximum acceptable value of k for a 20 W SOFC system. (You may have to redesign the system since the SOFC system in the text delivers 19.36 W.) Assume the heat is dissipated from the stock according to the equation provided and that the rest of the heat is contained in the exhaust.

13.9

The solid oxide fuel cells that we used in the Section 13.2 case study tend to break due to thermal stress if the temperature difference between the inlet and outlet gases is more than 150∘ C. We want to resolve this issue by increasing the air stoichiometry number in our system. We will assume that increasing the air stoichiometry does not affect the SOFC polarization curves (see Figure 13.4). Determine the required air stoichiometry and fuel cell stack power necessary to still deliver 20 W net to the external load when subject to this stack temperature constraint. In this scenario, how much power is our air blower going to consume?

13.10 The SOFC system discussed in Section 13.2 generates 19.36 W when each cell is operating at 0.7 V and 700∘ C. Suppose that the power demanded by the external load decreases such that now each cell is operating at 0.8 V. Assuming that all other conditions remain the same (such as operating temperature and stoichiometry numbers), recalculate the parameter values shown in Table 13.7 as well as the blower power consumption and the fuel cell power at this new operating voltage point. Hint: Although the stoichiometry numbers remain constant, because we have changed the operating voltage (and hence the current), the hydrogen and air flow rates have changed considerably! 13.11 Suppose that the electric wall boundary conditions for a 1D PEMFC model impose a 0.6 V difference between the anode and the cathode. The solution of this model indicates that the ohmic overvoltage is 0.1 V. The entire fuel cell is at an isothermal temperature of 25∘ C. The fuel cell is supplied with pure hydrogen and oxygen, both at 1 atm. Electronic resistance is ignored (= 0) in the catalyst layers and

CHAPTER EXERCISES

in the electrodes. Assuming that the overpotentials at the interface of the anode catalyst/anode electrode and cathode catalyst/cathode electrode are zero, answer the following questions: (a) Sketch the electronic potential profile across the anode, electrolyte, and the cathode. (b) Derive an equation for the exchange current density profile across the catalyst layers assuming constant gas concentration profiles across the catalyst layers and electrodes. (Hint: Use the equations from Example 13.1, ignoring the backward reaction term.) (c) Calculate the anodic and cathodic overpotentials. What is the current density? Both catalyst layers are 10 μm thick. Use parameter values from Table 13.2b if necessary. (d) Sketch the ionic potential profile across the anode, electrolyte, and cathode.

479

CHAPTER 14

ENVIRONMENTAL IMPACT OF FUEL CELLS

In this chapter, you will learn how to quantify the potential environmental impact of fuel cells. You will calculate potential changes in emissions from their use and how these changes in emissions affect global warming, air pollution, and human health. You will learn how to evaluate these changes not just at the vehicle or power plant level but also across the entire supply chain, from raw material extraction to end use. First, you will learn a tool called life cycle assessment (LCA), which we can use to evaluate how a new energy technology (such as fuel cells) affects energy use, energy efficiency, and emissions. Second, to conduct an LCA thoroughly, we will need to quantify the most important global warming and air pollution emissions. Therefore, we will briefly discuss the theory behind global warming and detail the primary global warming emissions from conventional vehicles, power plants, and fuel cell systems. Third, we will review the primary air pollutants from fossil fuel combustion devices and fuel cell systems and their effects on human health. Finally, using LCA and our knowledge of emissions impacts, we will develop a complete “what-if” scenario to look at how fuel cell implementation can change the global environmental context. After learning these tools and following these examples, you will be equipped to quantify the impact of future fuel cell scenarios.

14.1

LIFE CYCLE ASSESSMENT

Life cycle assessment is a methodology for systematically analyzing the effect of changes in the implementation and use of energy-related technologies.1 With a change in energy 1 Life cycle assessment also may be referred to as well-to-wheel analysis, process chain analysis, or supply chain analysis, depending on the emphasis of the analysis, and it can include either environmental or economic considerations, or both.

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482

ENVIRONMENTAL IMPACT OF FUEL CELLS

technology, LCA helps us evaluate changes in efficiency, emissions, and other environmental consequences [140, 141]. These environmental consequences include the economic costs of global warming and the human health impacts of air pollution. 14.1.1

Life Cycle Assessment as a Tool

Life cycle assessment consists of three primary stages: 1. Analyze the relevant energy and material inputs and outputs associated with the change in energy technology along the entire supply chain. The supply chain begins with raw material extraction, continues to processing, then to production and end use, and finally to waste management. Within this chain, it is important to focus on the most energy- and emission-intensive processes, the “process bottlenecks” [142]. 2. Quantify the environmental impacts associated with these energy and material changes. 3. Rate the proposed change in energy technology against other scenarios. Figure 14.1 shows an example of a supply chain for today’s conventional gasoline internal combustion engine (ICE) vehicles. The figure shows primary energy and pollutant flows during petroleum fuel extraction, production, transport, processing, delivery, storage, and use on a vehicle. Processes are depicted via boxes, emissions via wavy arrows at the top of the boxes, fuel flow via small arrows between boxes, and energy flows via thick arrows at the bottoms of the boxes. This supply chain could serve as a base case for comparing alternative vehicle supply chains. Now that we understand the concepts of the supply chain and process bottlenecks, we will dig deeper into a detailed methodology for LCA. A useful methodology for LCA follows these steps: 1. Research and develop an understanding of the supply chain from raw material production to end use.

Petroleum oil exploration 1

Crude oil production from fields

Crude oil transport

2

Crude oil process stream Energy input

Gasoline process stream

3

Centralized crude oil processing

Gasoline transport

4

Petroleum-based emission leakage Gasoline-based emission leakage

Gasoline storage

5

Gasoline ICE vehicle use

6

CO emissions CO2 emissions

7

H2O vapor emissions Other pollutants

Figure 14.1. Supply chain for today’s conventional gasoline internal combustion engine vehicles. Energy is consumed (bottom arrows) and emissions are produced (top arrows) during the primary processes (represented as boxes) from petroleum fuel extraction to its use on a vehicle.

LIFE CYCLE ASSESSMENT

2. Sketch a supply chain showing important processes and primary mass and energy flows. Examples of processes include chemical and energy conversion, production and transport of fuels, and fuel storage. Mass flows include the flow of raw materials, fuels, waste products, and emissions. Energy flows include the use of electric power, additional chemical energy consumed in a process, and work done on a process. 3. Identify the bottleneck processes, which consume the largest amounts of energy or produce the largest quantities of harmful emissions (or both). 4. Analyze the energy and mass flows in the supply chain using a control volume analysis and the principles of conservation of mass and energy. A control volume is a volume of space into which (and from which) mass flows. The boundaries of the control volume are shown by a control surface. Draw a control surface around individual processes in the supply chain, with particular focus on bottleneck processes. Analyze the mass and energy flows entering and exiting these processes. Employ the conservation-of-mass equation m1 − m2 = Δm

(14.1)

where m1 is the mass entering the control volume, m2 is the mass leaving the control volume, and Δm is the mass accumulating within the control volume. (An application of the principle of conservation of mass was previously highlighted in Chapter 6, Section 6.2.1.) Employ the conservation-of-energy equation for steady flow assuming Δm = 0, [ ( ) ( )] (14.2) Q̇ − Ẇ = ṁ h2 − h1 + g z2 − z1 + 12 V22 − V12

5.

6.

7. 8. 9.

where Q̇ is the heat flow into the process, Ẇ is rate of work done by the process, ṁ is the mass flow rate, h2 – h1 is the change in enthalpy between outgoing and incoming streams, g is the acceleration of gravity, z2 – z1 is the change in height, and V22 − V12 is the change in the square of the velocity. The last three terms refer to the change in the internal energy, potential energy, and kinetic energy of a flowing stream, respectively. (For a discussion of the conservation of energy, please see Chapter 2, Section 2.1.3.) Having analyzed the individual processes within the supply chain, evaluate the entire supply chain as a single control volume. Aggregate net energy and emission flows for the chain. Quantify the environmental impacts of these net flows, for example, in terms of human health impacts, external costs, and potential for global warming. We will discuss definitions of these terms and methods for conducting this analysis in subsequent sections. Compare the net change in energy flows, emissions, and environmental impacts of one supply chain with another. Rate the environmental performance of each supply chain against the others. Repeat the analysis for an expanded, more detailed number of processes in the supply chain.

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ENVIRONMENTAL IMPACT OF FUEL CELLS

Each of these steps is expanded on throughout the rest of the chapter through examples and explanations with a particular focus on fuel cell technologies. Additional attention is given to methods for quantifying environmental impacts.

14.1.2

Life Cycle Assessment Applied to Fuel Cells

Using the first three steps in this methodology for LCA, we will build and analyze a potential supply chain for fuel cell vehicles: 1. Research and develop an understanding of the supply chain from raw material production to end use. Using our knowledge from Chapter 11, we know that we can chemically process natural gas into a H2 -rich gas. Assume that we will fuel our fuel cell vehicles with H2 derived from steam reforming of natural gas. These steam reformers could be placed at similar locations as conventional gasoline refueling stations and could consume natural gas fuel piped in through the existing natural gas pipeline network. During these processes some methane (CH4 ) in the natural gas could leak into the surrounding environment. Hydrogen produced at the fuel processor could then be compressed into high-pressure tanks, stored at the station to buffer supply, and finally used to refuel high-pressure tanks onboard the vehicle. During these processes, some H2 could leak into the environment. 2. Sketch a supply chain showing important processes and primary mass and energy flows. Figure 14.2 shows a sketch of this potential fuel cell vehicle supply chain. Processes include natural gas exploration (box 1); production from gas fields (box 2); storage in underground tanks and reservoirs (box 3); chemical processing into a refined gas, including the addition of sulfur as an odorant (box 4); and transmission through pipelines (box 5). Up to this point, this part of the chain is identical to the supply chain already in existence for natural gas used to supply homes and buildings with fuel for heating and gas turbine power plants with fuel for generating electric power. Remaining processes include the conversion of natural gas to H2 at the fuel processor (box 6), H2 compression (box 7), storage (box 8), and use onboard

Natural gas exploration 1

Natural gas production from fields 2

Natural gas storage

Natural gas processing

3

Natural gas process stream Energy input

H2 gas process stream

4

Natural gas pipeline transmission 5

CH4 leakage H2 gas leakage

Fuel processor operation 6

Hydrogen compression

Hydrogen storage

7

8

CO emissions CO2 emissions

Hydrogen fuel cell vehicle use 9

H2O vapor emissions Other pollutants

Figure 14.2. Supply chain for hydrogen fuel cell vehicle fleet that obtains its hydrogen fuel from steam reforming of natural gas. Approximately 30% of the HHV of natural gas is needed for the operation of the steam reformer (box 6). Approximately 10% of the HHV of H2 is required for H2 compression (box 7). These are the most energy-intensive links in the supply chain.

LIFE CYCLE ASSESSMENT

the vehicle (box 9). As shown in Figure 14.2, most of these processes require at least some additional energy or work input. The dark arrows show natural gas fuel flow and the light arrows show H2 fuel flow. Emissions include leaked CH4 in the natural gas stream; leaked H2 in the H2 stream; carbon dioxide (CO2 ), carbon monoxide (CO), and other emissions produced during fuel processing and electricity production for powering the compression of hydrogen; and water vapor emissions (H2 O) at the vehicle. 3. Identify the most energy-intensive and most polluting portions of the chain, that is, bottleneck processes. Think about the energy input arrows at the bottom of the process boxes. Approximately 0.7% of the higher heating value (HHV) of natural gas is required for its exploration (box 1), about 5.6% for production (box 2), 1.0% for storage and processing (boxes 3 and 4), and 2.7% for transmission (box 5). (Chapter 2, Section 2.5.1, introduces the concept of HHV.) Thus, about 10% of the HHV of natural gas is required to provide energy for the first five boxes in Figure 14.2. As shown in Chapter 11, approximately 30% of the HHV of natural gas is required for the operation of the fuel processor. As you learned in Chapter 10, the energy required to compress H2 is approximately 10% of the HHV of H2 . Storage energy is a fraction of this. Therefore, the two single most energy-intensive processes in the chain are (1) fuel processing of natural gas and (2) compression of H2 . The most energy-intensive processes are likely to produce the largest quantities of harmful emissions. Therefore, the most energy-intensive processes should be examined closely. At the same time, this relationship may not always hold. Different types of emissions are more harmful than others. Therefore, the most energy-intensive processes are an excellent starting point for determining the highest emitting processes, but other processes must also be investigated. Think about the emission arrows at the top of the process boxes, beginning with the most energy-intensive processes: (1) fuel processing of natural gas and (2) compression of H2 . Consider the first process bottleneck: fuel processing. Based on research of steam reformers used in conjunction with fuel cell systems, emission factors for a commercial natural gas steam reformer are shown in Table 14.1. For reference, Table 14.1 also benchmarks the steam reformer’s emissions against emissions from another type of hydrogen generator, a coal gasification plant, and against emissions from electric power plants fueled by natural gas and coal. The steam reformer’s emissions are quite low. For example, the steam reformer produces negligible SOx and particulate matter. Now consider the second process bottleneck: H2 compression. Hydrogen compressors run on electric power from the surrounding electric grid. Although the energy required to compress H2 is 10% of its HHV, this energy refers to the electric power drawn by the compressor. An additional energy penalty is paid due to the efficiency of the electric power plant. The average efficiency of all power plants connected to the grid is approximately 32% and their approximate distribution by fuel type is shown in Figure 14.3. Over half of U.S. electric power plants are coal plants, which produce the most harmful emissions of any power plant per unit of electricity produced. Considering the relatively low emissions from the natural gas steam reformer and the efficiency penalty of the power plants, emissions from the use of electric power for H2 compression may be the most significant contributor to air pollution.

485

486 Natural Gas Steam Reformer (kg Emission/kg Natural Gas Fuel) 2.6 0.000048 Negligible Negligible 0.00046 0.0000033 0.00000066

Emission

CO2

CH4

Particulate matter

SO2

NOx as NO2

CO

VOC

0

0.00734

0.000108

0.000762

0

Unknown

2.37

Coal Gasification (kg Emission/kg Coal Fuel)

Hydrogen Generator Emission Factors

0.016

0.33

0.70

0.27

0.074

1.5

390

0.00010

0.0021

0.0045

0.0017

0.00047

0.010

2.5

0.013

0.12

2.0

1.0

0.20

3.0

850

0.000038

0.00035

0.0056

0.0028

0.00056

0.0084

2.4

(kg Emission/kg Coal Fuel)

(g Emission/kWh Electricity)

(g Emission/kWh Electricity)

(kg Emission/kg Natural Gas Fuel)

Coal Combustion (Coal Boiler, Steam Turbine, low NOx )

Natural Gas Combustion (Combined-Cycle Gas Turbine, Low NOx )

Electricity Plant Emission Factors

TABLE 14.1. Emission Factors for Two Types of Hydrogen Generators and Two Types of Electricity Generators

LIFE CYCLE ASSESSMENT

Distribution of U.S. power plants by fuel type based on annual production

51.7%

Coal

19.8%

Nuclear

15.9%

Natural gas

2.0%

Non-hydro renewable

0.6%

Other fossil fuels

Natural

gas

c tri

Oil

Oil

2.8%

Coal

lec

Hydroelectric

N

oe dr Hy

7.2%

r

ea

l uc

Figure 14.3. Most U.S. electric power derives from conventional coal-fired power plants, which burn coal in a boiler to generate steam that runs through a steam turbine. The second largest portion of electric power comes from nuclear power plants, which extract heat from nuclear fission reactions to generate steam in a boiler that is then run through a steam turbine. The third most prevalent form of electric power production is from natural gas plants, which burn gas in a turbine.

Example 14.1 (1) Identify the bottleneck processes in the gasoline vehicle supply chain. (2) Estimate the energy required to complete some of the important processes in the chain from petroleum production from oil fields (box 2 in Figure 14.1) to the delivery of gasoline at the vehicle (box 6). Solution: 1. Bottleneck processes are those that consume the largest quantities of energy or that produce the largest quantities of harmful emissions in the supply chain. Based on background research on the petroleum industry and the supply chain shown in Figure 14.1, some of the bottleneck processes are (1) production of crude oil from fields (box 2), (2) centralized chemical processing of crude oil into gasoline (box 4), and (3) combustion of gasoline in the engine onboard the vehicle (box 7). Additional energy-intensive processes may include the transport of crude oil and gasoline (boxes 3 and 5), depending on the location of the vehicles relative to the oil fields. These bottleneck processes should be the focus of a further study of this supply chain via LCA. 2. Although estimates vary, approximately 12% of the HHV of gasoline fuel is required for its production, transport, and processing (boxes 2–5) [143]. The storage of gasoline (box 6) does not require a large quantity of energy because it remains a liquid at room temperature, with some evaporation. (Consider conducting additional research on the petroleum industry to quantify these estimates, which may vary by region because of differences in the distance to oil fields and in environmental legislation.)

487

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ENVIRONMENTAL IMPACT OF FUEL CELLS

Example 14.2 Having completed steps 1–3 of the LCA, we will now explore step 4 of LCA, analyzing the energy and mass flows in the supply chain using a control volume analysis and the principles of conservation of mass and energy. Imagine that the fuel cell vehicle fleet described in Figure 14.2 replaces the current on-road vehicle fleet shown in Figure 14.1. Emissions from this fuel cell fleet ultimately depend on the quantity of H2 fuel it consumes. Assume this fuel cell fleet requires the same propulsive power as the current fleet—the total mass of the vehicles, their aerodynamic drag, rolling resistance, frontal area, and inertia are the same [144]. Based on fuel tax revenue records, the U.S. Environmental Protection Agency (EPA) estimates that on-road vehicles traveled 2.68 trillion miles (2.68 × 1012 miles) in 1999, and the average mileage of this fleet was 17.11 miles per gallon. The HHV of gasoline fuel is 47.3 MJ∕kg and the HHV for H2 fuel is 142.0 MJ∕kg [145]. The density of gasoline is 750 kg∕m3 . Having reviewed the relevant literature, you estimate that for the current vehicle fleet the average gasoline vehicle’s efficiency (its motive energy to propel the vehicle/HHV of fuel) is 16%. Considering the performance of pre-commercial fuel cell vehicle prototypes, you estimate that the fuel cell vehicle’s efficiency is 41.5% [146, 147]. Build on the fuel cell system energy efficiency terms discussed in Chapter 10. Based on the conservation of energy, estimate the mass of H2 needed to fuel this fleet. Solution: We draw a control surface around box 7 in Figure 14.1 and box 9 in Figure 14.2 to compare mass and energy flows into and out of these processes. Based on the conservation of energy, we assume the work done by the current fleet (Ẇ c ) equals the work done by the fuel cell fleet (Ẇ f ), Ẇ c = Ẇ f . The required propulsive work of the average car in each fleet is the same. The propulsive work of the current fleet is (14.3) Ẇ c = ṁ g ΔH(HHV),g 𝜀g where ṁ g is the mass of gasoline fuel consumed by vehicles per year (kg∕yr), ΔH(HHV),g is the HHV of gasoline fuel (MJ∕kg), and 𝜀g is the gasoline vehicle’s efficiency. The mass of gasoline consumed per year is also ṁ g =

𝜌g VMT

(14.4)

M g𝑣f Vc

where 𝜌g is the density of gasoline (kg∕m3 ), VMT the vehicle miles traveled per year (106 miles), M g𝑣f the average mileage of the conventional fleet (miles∕gal), and Vc the volumetric conversion (264.17 gallons∕m3 ). The propulsive work of the fuel cell fleet is (14.5) Ẇ f = ṁ h ΔH(HHV),h 𝜀h where ṁ h is the mass of H2 consumed by vehicles per year (kg∕yr), ΔH(HHV),h is the HHV of H2 fuel (MJ∕kg), and 𝜀h is the fuel cell vehicle’s efficiency. Setting

LIFE CYCLE ASSESSMENT

Ẇ c = Ẇ f and combining the last three equations, the mass of hydrogen consumed by the fleet is V ṁ H2 ,C = MT (14.6) Fh where Fh =

M g𝑣f Vc ΔH(HHV),h 𝜀h 𝜌g ΔH(HHV),g 𝜀g

(14.7)

is the mileage of hydrogen fuel cell vehicles (miles∕kg H2 ). Based on the information in our example, Fh = ṁ H2 ,C =

(17.11 miles∕gal)(264 gal∕m3 )(142 MJ∕kg)(0.415) (750 kg∕m3 )(47.3 MJ∕kg)(0.16)

(14.8)

VMT 2.68 × 1013 miles∕yr = = 5.71 × 1010 kg H2 ∕year Fh 46.9 miles∕kg H2

(14.9)

Based on this derivation, a fuel cell vehicle fleet would consume 57 megatonnes (MT) of H2 ∕yr. Figure 14.4 shows a spatial distribution of hydrogen consumption by such a fuel cell fleet by county, based on gasoline consumption data by county recorded by the EPA [148].

Kilotonnes/year 640

Figure 14.4. Annual hydrogen consumption by fuel cell vehicles by county, plotted at the center of each U.S. county, assuming a complete switch of fleet from conventional vehicles to fuel cell vehicles.

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ENVIRONMENTAL IMPACT OF FUEL CELLS

14.2

IMPORTANT EMISSIONS FOR LCA

To conduct the next steps in LCA (especially steps 5 and 6), we first have to determine which types of emissions are important to evaluate in the supply chain. Important emissions fall into two categories: (1) those that influence global warming and (2) those that influence air pollution. In the two subsequent sections, we will discuss both of these. Emissions that influence global warming include CO2 and CH4 . Important emissions that lead to air pollution include ozone (O3 ),2 CO, nitrogen oxides (NOx ), particulate matter (PM), sulfur oxides (SOx ), and volatile organic compounds (VOCs). In the sections that follow, we will (a) discuss the importance of these emissions and (b) describe methods for quantifying their environmental impact.2 14.3 14.3.1

EMISSIONS RELATED TO GLOBAL WARMING Climate Change

Earth’s climate has changed over time. Earth’s average near-surface temperature is currently close to 15∘ C, but geological evidence suggests that in the past one million years it may have fluctuated to as high as 17∘ C and as low as 8∘ C. Climate scientists are now concerned that these natural fluctuations are being overtaken by warm-side temperature changes induced by human activity, specifically the combustion of fossil fuels that release gases and particles that have a warming effect on the atmosphere [149]. 14.3.2

Natural Greenhouse Effect

The natural greenhouse effect is the process by which gases normally contained in the atmosphere, such as CO2 and water vapor (H2 O), trap a portion of the sun’s energy in the form of infrared (IR) radiation. As a result, Earth’s temperature is high enough to support life as we know it. When the sun’s light hits Earth’s surface, some of this energy is absorbed and warms Earth. Earth’s surface then reemits some of this energy to the atmosphere as IR radiation or thermal energy. Greenhouse gases are special in that, unlike other molecules, they selectively absorb 80% of IR radiation and then reemit this radiation back up to space and back toward Earth’s surface. The left portion of Figure 14.5 shows the warming mechanism of greenhouse gases. In a process somewhat similar to heat trapping in a glass greenhouse, greenhouse gases absorb and reemit some IR radiation while remaining transparent to 50% of visible sunlight and other wavelengths. As a result, the more greenhouse gases present in the atmosphere, the more heat is trapped near Earth’s surface. The natural greenhouse effect contributes 33 K of Earth’s average near-surface air temperature of 288 K. Without this effect, Earth would be too cold to support life as we know it. 2 In the upper atmosphere, ozone creates a protective layer around the Earth by absorbing ultraviolet radiation that would otherwise harm life. However, ozone emitted at sea level causes smog and air pollution and damages human health.

EMISSIONS RELATED TO GLOBAL WARMING

Greenhouse effect Su

nli

nli

gh

t

gh

t

Sunlig

Su nli gh t

Su

M

d

Light-colored particles

Infrared

Infrared

Infr are

O

O

O

O

Dark-colored particles

O

O

Greenhouse gases

N

Black carbon

S

ht

Organic matter

O

M

Figure 14.5. Left: Sunlight hits Earth’s surface and is partly absorbed. Earth reemits some of this energy as IR radiation (thermal energy). Greenhouse gases, including H2 O, CH4 , CO2 , and N2 O, selectively absorb this IR radiation and reemit it out to space and back toward Earth’s surface and thereby warm Earth’s surface. Center: Sunlight hits dark-colored particles, such as black carbon, suspended in Earth’s atmosphere. These dark particles absorb the light and reemit this energy as IR radiation, some of which may reach Earth’s surface and may warm it. Organic matter focuses light onto black carbon, thereby enhancing black carbon’s warming effect. Right: Light-colored particles, including sulfates and nitrates, reflect sunlight and have a cooling effect.

14.3.3

Global Warming

Most climate scientists concur that an increase in anthropogenic (i.e., man made) emissions of greenhouse gases is contributing to an intensification of the greenhouse effect. Global warming refers to the increase in Earth’s temperature above that caused by the natural greenhouse effect as a result of the addition of anthropogenic greenhouse gases and certain particles. Anthropogenic greenhouse gases include CO2 , CH4 , H2 O, and nitrous oxide (N2 O). In addition to these gases, certain particles also have a warming effect on Earth but through a different mechanism. Dark-colored particles, such as soot, absorb sunlight, reemit this energy as IR radiation, and therefore also may warm Earth’s surface. Black carbon (BC) is a predominant global warming particle [150, 151]. The warming effect of black carbon is enhanced by organic matter (OM), which focuses additional light onto black carbon. The center portion of Figure 14.5 shows the warming mechanism of dark-colored particles. Figure 14.5 shows that these gases and particles reemit IR radiation toward Earth’s surface to cause warming; they also reemit IR radiation away from Earth. In contrast, light-colored particles reflect sunlight and have a cooling effect. Light-colored particles that cool Earth include sulfates (SULF) and nitrates (NIT). SULF also attract water,

491

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ENVIRONMENTAL IMPACT OF FUEL CELLS

which reflects light as well. Emitted gases that have a cooling effect include SOx , NOx , and non-methane organic compounds, or VOCs. These gases react in the atmosphere and convert to particles which are mostly light in color. Sulfur oxide converts to SULF, NOx converts to NIT, and VOCs convert to light-colored organics. The right portion of Figure 14.5 shows the cooling mechanism of light-colored particles.

14.3.4

Evidence of Global Warming

Since the 1860s, the concentration of primary greenhouse gases—CO2 , CH4 , and N2 O—in the lower atmosphere has increased by 30%, 143%, and 14%, respectively. Figure 14.6 shows the increase in CO2 and CH4 over the past 150 years. With the start of the Industrial Revolution 200 years ago, people began to combust fossil fuels to provide energy for industrial processes and began releasing much larger quantities of CO2 into the atmosphere than in previous times. At the start of the Industrial Revolution, CO2 concentrations were close to 280 parts per million by volume (ppmv). Currently, they are close to 380 ppmv and are increasing at a rate of 2 ppmv/yr. Over the same period, Figure 14.7 shows the change in Earth’s near-surface temperature, which has increased by 0.6∘ C ± 0.2∘ C between the years ∼1920 and ∼2000. Compared with historical records, this rate of temperature increase is unusually high. Further evidence of global warming includes the following: 1. An increase in temperature in the past four decades in the lowest 8 km of the atmosphere 2. A decrease in snow cover, ice extent, and glacier extent 3. A 40% reduction in the thickness of Arctic sea ice in summer and autumn in recent decades 4. An increase in global average sea level by 10–20 cm because of warmer oceans expanding 5. An increase in the heat content of the ocean Other pieces of evidence indicating anthropogenic climate change include flowers blooming earlier, birds hatching earlier, and a cooling of the middle portion of the atmosphere.

14.3.5

Hydrogen as a Potential Contributor to Global Warming

Since industrialization, the concentration of H2 in the atmosphere is estimated to have increased via 200 parts per billion by volume (ppbv) [152] to 530 ppbv [153]. The majority of H2 emissions originate from the oxidation of HCs, especially the incomplete combustion of gasoline and diesel fuels in automobiles, and the burning of biomass. When released, H2 most commonly does not combust with oxygen in the air because its concentration and its temperature are usually too low to facilitate the reaction. The self-ignition temperature of H2 is 858 K and its ignition limits in air are between 4 and 75%. Once released into the atmosphere, H2 is estimated to have a lifetime of between 2 and 10 years.

EMISSIONS RELATED TO GLOBAL WARMING

CH4 concentration (ppmv)

CO2 concentration (ppmv)

380 360 340 320 300 280 260 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 1860

1880

1900

1920

1940

1960

1980

Year 2000

Figure 14.6. Between the ∼1860s and recent times, concentrations of the primary greenhouse gases CO2 and CH4 in the lower atmosphere have increased by about ∼30% and ∼140%, respectively.

Temperature (°C) 1.0 0.8

Annual temperature change 5-year average temperature change

0.6 0.4 0.2 0 –0.2 –0.4 1860

Year

1880

1900

1920

1940

1960

1980

2000

Figure 14.7. Since the 1860s, Earth’s average, near-surface temperature has increased by over ∼0.6∘ C.

If fuel cells become widespread, H2 release will likely accelerate. As we saw in Figure 14.2, H2 may leak into the environment during its production, compression, storage, and use onboard vehicles (boxes 6–9). In addition, H2 may leak during transport, especially if transmitted over long distances through pipelines, in much the same way natural gas leaks today (box 5). Because H2 is one of the smallest molecules, it may be more likely than other fuels to escape from small openings. For example, the mass

493

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ENVIRONMENTAL IMPACT OF FUEL CELLS

diffusion coefficient of H2 is four times higher than that of natural gas. In addition to leakage, H2 may also be intentionally released into the environment. For example, some fuel cell systems are designed to purge anode exhaust gas (containing H2 ) from the stack periodically so as to prevent blockage of reaction sites at the anode by other species (such as water). Also, liquid H2 tanks require a periodic release of H2 to avoid pressure buildup. As a result, climate researchers are now trying to determine the potential effect of released H2 on global warming. One mechanism through which released H2 might increase global warming is by indirectly increasing the concentration of the greenhouse gas CH4 . In the troposphere (lower atmosphere), H2 reacts with the hydroxyl radical (OH) according to the reaction (14.10) H2 + OH → H + H2 O If H2 did not consume OH in this reaction, OH might otherwise reduce the presence of CH4 via the reaction (14.11) CH4 + OH → CH3 + H2 O However, numerous other chemical reactions must also be considered. The net effect of H2 on global warming is still the subject of research. Example 14.3 You read an article that claims that fuel cell vehicles might increase global warming as a result of the additional water vapor they will produce. You decide to invoke LCA to make your own determination. You decide to compare two different scenarios, one being the current vehicle fleet (shown in Figure 14.1) and the other being a fuel cell vehicle fleet (shown in Figure 14.2). You decide to calculate the water vapor emitted in each of these scenarios to compare them to see if there would be a genuine increase in water vapor emissions between a current fleet and a fuel cell fleet. The 1999 vehicle fleet consumed approximately 450 MT∕yr of combined gasoline (Cn H1.87n ) and light diesel fuel (Cn H1.8n ) [154]. Gasoline and light diesel fuels represented 78 and 22% of fuel consumption in vehicles [155], respectively. 1. Locate the sources of H2 O emission in each supply chain. 2. Identify the bottleneck processes for H2 O emission. 3. Based on the conservation-of-mass equation, calculate the quantity of water vapor emitted in the bottleneck processes. 4. Is the article’s assertion valid? Solution: 1. Locate the source of H2 O vapor emission in the supply chain. In the current fleet, water vapor is emitted as a product of combustion. As shown in Figure14.1, water vapor is emitted during the transport of petroleum fuel by truck, railroad, or ship (boxes 3 and 5) and during ICE vehicle use (box 7). As shown in Figure 14.2, in the fuel cell fleet scenario, water vapor is emitted as a product of the electrochemical oxidation of hydrogen at the exhaust of the fuel cell vehicle (box 9). Water vapor is also emitted indirectly because

EMISSIONS RELATED TO GLOBAL WARMING

hydrogen compressors consume electric power from power plants (box 7), and some of these power plants (coal and natural gas) produce water as a product of combustion. 2. Identify the bottleneck processes for H2 O emission. As a first approximation, we assume that the majority of H2 O emissions occur in the last step of each process chain (box 7 in Figure 14.1 and box 9 in Figure 14.2) during vehicle use. 3. Calculate the quantity of water vapor emitted in the bottleneck processes. Within an internal combustion engine, combustion can be described by CH1.85 + 1.4625O2 → CO2 + 0.925H2 O + work + heat

(14.12)

where CH1.85 is a chemical formula representing gasoline (Cn H1.87n ) and light diesel (Cn H1.8n ) fuels weighted by their consumption in the vehicle fleet (78 and 22%, respectively). The molecular weight of CH1.85 is 13.85 g∕mol. The molecular weight of H2 O is 18 g∕mol. Every kilogram of CH1.85 consumed produces 1.2 kg H2 O (18 kg∕mol H2 O × 0.925 mol H2 O∕13.85 g∕mol CH1.85 ). For every 450 MT/yr fuel consumed, approximately 540 MT H2 O/yr is produced. Within a fuel cell, every mole of hydrogen consumed produces 1 mol H2 O, according to H2 + 0.5O2 → H2 O + electricity + heat

(14.13)

The molecular weight of H2 is 2 g∕mol. Thus, every kilogram of H2 consumed produces 9 kg H2 O. In Example 14.2, we calculated that a fuel cell fleet would consume 57 MT H2 ∕yr. The fleet would then produce about 510 MT∕yr of H2 O. Based on these estimates, a fuel cell vehicle fleet would produce approximately the same quantity of water vapor as the current fleet [156]. (This calculation may overestimate the amount of water vapor produced by the fuel cell vehicles because it assumes that all water is emitted in vapor form, when it could actually condense as a liquid, especially given the low operating temperature of PEM fuel cells.) 4. Is the article’s assertion valid? The quantity of water vapor produced by either the current fleet or a fuel cell fleet is one million times smaller than the emission rate of water vapor from natural sources—5 × 108 MT∕yr. Based on these considerations, the water vapor emitted by either fleet will have a negligible effect on the atmosphere. Thus, the article’s assertion does not appear to be valid. 14.3.6

Mitigating Climate Change with Low Carbon Fuels and Fuel Cells

The rate of accrual of greenhouse gases in the atmosphere is equal to the emission rate minus the depletion rate. The emission rate can be expected to change slowly due to both time lags in the adoption of new, lower emission technologies and the high emission rate of incumbent technologies. If societies want to reduce the accrual rate of atmospheric greenhouse gases,

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one potential approach is to reduce the emission rate by switching to fuels that contain low (or even zero) levels of carbon and switch to more efficient energy conversion devices, including fuel cells. Low-carbon fuels can be consumed in fuel cell systems to power vehicles and to provide energy to buildings, including electricity, space heating, and cooling power. For example, in the United States, power plants expend approximately 1/5th of total U.S. energy consumption, or 21 quadrillion British thermal units (Btu) per year, as unrecovered waste heat. U.S. homes, commercial buildings, and industrial facilities regenerate approximately the same quantity of heat per year. Stationary distributed fuel cell systems that both provide electricity to buildings and recapture their waste heat for heating buildings can mitigate these large energy losses and their associated greenhouse gas emissions. In principle, an entire economy could be designed based on low-carbon fuels and including fuel cell systems such that much lower levels of CO2 emissions are released. All other things being equal, an energy process will produce less CO2 emissions if it consumes a fuel with less carbon (C) content per unit energy. The combination of a fuel’s carbon content and the efficiency with which the fuel is consumed determine the CO2 emissions per unit of useful output. Almost all commercial fuels, such as gasoline, natural gas, and coal, contain some carbon. Table 14.2 compares the carbon content of several fuels. The table shows each fuel’s molecular formula, the energy content of the fuel per unit mass in units of megajoules (MJ) per kilogram (kg) based on the lower heating value (LHV), the mass of carbon in the fuel per unit of energy in the fuel shown in kilograms of carbon per TABLE 14.2. Carbon Content of Various Fuels Carbon Content of Fuel Per Unit Energy of Fuel

Per Unit Mass of Hydrogen

Mass of Carbon Mass of Carbon per Unit Mass LHV Fuel per Unit of Fuel of Hydrogen Energy Energy (kg (kg carbon/kg (MJ/kg) carbon/GJ fuel) atomic hydrogen)

Fuel

Chemical Formula

Coal

CnH0.93nn0.02nO0.14nS0.01n(s)

26.7

28.5

12.8

Gasoline

Cn H1.87n (l)

44.0

19.6

6.5

Ethanol

C2 H6 O(l)

26.9

19.4

4.0

Methanol

CH4 O(l)

20.0

18.7

3.0

Natural gas Cn H3.8n N0,1n (g)

45.0

15.5

3.1

Methane

CH4 (g)

50.0

15.0

3.0

Hydrogen

H2 (g)

120.0

0.0

0.0

Note: Coal and gasoline have among the highest carbon contents per unit energy. Natural gas, methane, and hydrogen have among the lowest carbon contents per unit energy. Assuming a constant efficiency in energy conversion, the higher carbon content fuels will produce more CO2 emissions than the lower carbon content fuels:

EMISSIONS RELATED TO GLOBAL WARMING

gigajoule (GJ) of fuel energy, and the carbon content per unit of atomic hydrogen (H).3,4 Chapter 2, Section 2.5.1, introduces the concept of lower heating value (LHV). The fuels are ordered from top to bottom from the highest to the lowest carbon content per unit energy. As the table shows, coal has the highest carbon content per unit energy and the highest C/H ratio. By contrast, hydrogen (H2 ) fuel has a high energy content but contains no carbon. One approach to mitigating climate change is to switch the fuel mix to fuels with a lower carbon content, such as methane (CH4 ) and H2 , while minimizing their leakage, and to use these fuels in more efficient energy conversion devices, such as fuel cells. 14.3.7

Quantifying Environmental Impact—Carbon Dioxide Equivalent

One important method for quantifying the environmental impact of emissions related to global warming is the calculation of the carbon dioxide equivalent (CO2equivalent ) of a mixture of emitted gases and particles. To estimate the potential for a mixture of gases and particles to contribute to global warming, one can calculate the CO2equivalent of these gases. The CO2equivalent is the mass of CO2 gas that would have an equivalent warming effect on Earth as the mixture of different gases. The CO2equivalent helps us quantify and compare the warming effect of different types and quantities of emissions. One equation for measuring the CO2equivalent of gases over a 100-year period is [156, 157] CO2 equivalent = mCO2 + 23mCH4 + 296mN2 O + 𝛼(mOM,2.5 + mBC,2.5 ) − 𝛽(mSULF,2.5 + mNIT,2.5 + 0.40mSOx + 0.10mNOx + 0.05mVOC )

(14.14)

where m is the mass of each species emitted, with, for example, mOM,2.5 indicating the mass of organic matter 2.5 μm in diameter and less. The coefficient 𝛼 can range between 95 and 191. The coefficient 𝛽 can range between 19 and 39. The logic of this formula follows from our description of the various gases and particles that contribute to global warming or cooling, as shown in Figure 14.5. In the formula, gases or particles with a warming effect are preceded by a plus sign and those with a cooling effect are preceded by a minus sign. The coefficients in front of the masses (23, 296, 𝛼, and 𝛽, respectively) represent the global warming potential (GWP) of each of the species over a 100-year period. The GWP is an index for estimating the relative global warming contribution of a unit mass of a particular greenhouse gas or particle emitted compared to the emission of a unit mass of CO2 . For example, a GWP of 23 for CH4 indicates that it is 23 times more efficient at absorbing radiation than CO2 . The GWP for H2 is not included in the equation above because its value is still being determined by climate researchers. Values of the GWP are calculated for different time horizons due to the different lifetimes of gases in the atmosphere. Anthropogenic H2 O emission is not usually considered in CO2equivalent calculations because, as we learned in Example 14.3, natural sources of H2 O 3 Derived from (a) For all liquid fuels: Heywood, John B. Internal Combustion Engine Fundamentals (New York: McGraw-Hill, Inc., 1988), Table D.4 “Data on fuel properties,” p. 915; (b) For coal: Starkman, Ernest S. Combustion-Generated Air Pollution (New York-London: Plenum Press, 1971) via Ohio Supercomputer Center (OSC) website http://www.osc.edu/research/perm/emissions/coal.shtml. 4 Calculations are based on Lower Heating Values (LHV).

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TABLE 14.3. U.S. Emissions from All Man-Made Sources, 1999 (MT/year) Species Gases Carbon monoxide (CO) Nitrogen oxides (NOx ) as NO2 Sulfur oxides (SOx ) as SO2 Ammonia (NH3 ) Hydrogen (H2 ) Carbon dioxide (CO2 ) Water (H2 O) Organics Paraffins (PAR) Olefins (OLE) Ethylene (C2 H4 ) Formaldehyde (HCHO) Higher aldehydes (ALD2) Toluene (TOL) Xylene (XYL) Isoprene (ISOP) Total non-methane organics Methane (CH4 ) Particulate matter Organic matter (OM2.5 ) Black carbon (BC2.5 ) Sulfate (SULF2.5 ) Nitrate (NIT2.5 ) Other (OTH2.5 ) Total PM2.5 Organic matter (OM10 ) Black carbon (BC10 ) Sulfate (SULF10 ) Nitrate (NIT10 ) Other (OTH10 ) Total PM10

On-Road Vehiclesa

Total All Sourcesb

6.18 × 107 7.57 × 106 2.72 × 105 2.39 × 105 1.55 × 105 1.37 × 109 5.19 × 108

1.12 × 108 2.19 × 107 1.81 × 107 4.53 × 106 2.79 × 105 5.30 × 109 1.99 × 109

3.53 × 106 1.61 × 105 2.27 × 105 4.43 × 104 1.72 × 105 3.29 × 105 4.66 × 105 4.86 × 103 4.93 × 106 7.91 × 105

1.40 × 107 5.21 × 105 9.12 × 105 2.23 × 105 3.39 × 105 2.60 × 106 2.25 × 106 9.92 × 103 2.09 × 107 6.31 × 106

5.04 × 104 9.07 × 104 1.88 × 103 2.47 × 102 2.40 × 104 1.67 × 105 7.19 × 104 1.07 × 105 2.99 × 103 3.15 × 102 3.66 × 104 2.19 × 105

2.64 × 106 5.92 × 105 3.10 × 105 2.67 × 104 8.26 × 106 1.18 × 107 5.77 × 106 9.62 × 105 4.91 × 105 7.10 × 104 3.75 × 107 4.48 × 107

a Conventional b All

on-road fossil fuel vehicles. man-made sources including industrial facilities and power plants.

are five orders of magnitude higher than anthropogenic sources. The CO2equivalent equation above is only an estimate of the potential for global warming of some of the important gases and particles and must be periodically updated with further climate research findings. More accurate results than ones derived using the above equation can be obtained through the use of global-scale computer models of the atmosphere.

EMISSIONS RELATED TO GLOBAL WARMING

Example 14.4 (1) A California company sells a stationary solid-oxide fuel cell (SOFC) system that consumes natural gas fuel and produces electricity only (with no recoverable heat). The systems are said to operate at a net electrical efficiency of about 55%, based on the LHV. To simplify calculations, assume that the gas composition of natural gas is 100% methane, reactants and products enter and leave the system at STP, and the LHV of methane can be used. What is the average annual CO2 emission factor (𝛾E−CO2 ) for these systems in units of grams (g) of CO2 ∕kilowatt-hour of electricity (kWhe)? This term is sometimes referred to as the “carbon footprint” of a power plant. (2) Compare this emission factor/carbon footprint with those for natural gas combustion and coal combustion electric power plants. Solution: 1. Based on Table 14.2, the mass of carbon (C) per unit energy of CH4 is 15 kg of C∕GJ of CH4 , and the average annual CO2 emission factor is 𝛾E−CO2

15 kg C = GJ CH4

(

GJ CH4 0.55 GJe

)(

1 GJ 278 kWh

) ( 1000 g ) 1 kg

= 360g CO2 ∕kWhe

(

g

44 mol CO2

)

g

12 mol C (14.15)

2. The calculated emission factor is lower than 390g CO2 ∕kWhe, which is the natural gas combustion combined-cycle gas turbine (CCGT) plant CO2 emission factor listed in Table 14.1. It is also lower than 850g CO2 ∕kWhe, which is the coal combustion plant (a coal boiler coupled with a steam turbine) CO2 emission factor listed in Table 14.1. In practice, a SOFC system’s CO2 emission factor is likely to be lower than that of most coal plants. However, it may or may not be lower than that of a natural gas CCGT plant, depending on the plant’s electrical efficiency. This electrical efficiency can vary from ∼40% to 60%, depending on plant size, design, operating strategy, and application.

14.3.8 Quantifying Environmental Impact—External Costs of Global Warming A second important method for quantifying the environmental impact of emissions related to global warming is the calculation of external costs of global warming. The potential effects of global warming include the following: 1. An increase in sea level, resulting in flooding of some low-lying areas 2. An intensification of the hydrological cycle, resulting in both more drying and more flooding due to an increase in extreme precipitation events 3. Shifts in regions with arable land and changes in agricultural regions 4. Damage to ecosystems

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Researchers estimate the external cost of global warming at between $0.026 and $0.067 per kilogram of CO2equivalent emission in 2004 dollars [157, 158]. This external cost is the damage cost of an additional unit of mass of CO2 (or equivalent gas) into the atmosphere.5 An external cost arises when all of the costs of a good are not included into its free-market price [159]. An example of an externality is the cost of damage to a piece of real estate due to flooding resulting from a sea-level rise related to global warming. By definition, the external costs of global warming related to land use are not incorporated into the free-market prices for property. Researchers’ estimates of the economic value of externalities vary over a large range because these costs are difficult to quantify precisely. However, to ignore external costs is to incorrectly assume that their value is zero. Example 14.5 (1) The EPA tabulates emissions from vehicles, power plants, and all other sources in a National Emission Inventory (NEI). You check the NEI for emissions from on-road fossil fuel vehicles in 1999 and create Table 14.3. In Table 14.3, PM10 refers to particulate matter that is 10 μm in diameter and less; PM2.5 refers to matter 2.5 μm in diameter and less. Calculate the CO2equivalent of this fleet. Compare this with only the CO2 released by the fleet. (2) Now imagine instantaneously replacing this fossil fuel vehicle fleet with a hydrogen fuel cell vehicle fleet. Calculate the CO2equivalent of this fleet, considering only the change in vehicles. What is this percentage reduction in terms of total anthropogenic CO2equivalent in the United States? (3) To make this comparison more even-handed, what might you also consider? (4) What is the reduction in external costs (costs of the damage to society from global warming that is not incorporated into free-market prices)? Solution: 1. Based on the CO2equivalent formula and the data in Table 14.3, we can calculate high and low values for the range of CO2equivalent gases and particles emitted by on-road vehicles: CO2 equivalent, LOW = mCO2 + 23mCH4 + 296mN2 O + 95(mOM2.5 + mBC2.5 ) −39(mSULF2.5 + mNIT2.5 + 0.40mSOx + 0.10mNOx + 0.05mVOC ) (14.16) CO2 equivalent, HIGH = mCO2 + 23mCH4 + 296mN2 O + 191(mOM2.5 + mBC2.5 ) −19(mSULF2.5 + mNIT2.5 + 0.40mSOx + 0.10mNOx + 0.05mVOC ) (14.17) Because the NEI does not tabulate N2 O, as an estimate, consider only the other terms. This range is between 1.36 × 109 and 1.39 × 109 tonnes/year. These values differ from the total CO2 fleet emissions by0.87 and 1.75%. Thus, in this example, the primary contributor to CO2equivalent is CO2 itself. 5 External costs are referred to also as damage, societal, and/or environmental costs, depending on the source of the costs.

EMISSIONS RELATED TO GLOBAL WARMING

2. Considering only the change in the fleet and no upstream fuel production sources, the hydrogen fuel cell vehicle fleet would produce no CO2 . Its CO2equivalent would also be zero. Based on the CO2equivalent formula and the data in Table 14.3, we can calculate high and low values for the range of CO2equivalent gases and particles emitted by all sources in the United States, 5.33 × 109 and 5.86 × 109 tonnes/year. This change represents an approximate reduction in CO2equivalent in a range of 23.21–26.17%. 3. To make this analysis more even handed, one might also consider the change in CO2equivalent gases and particles from upstream sources, including fuel production in both the fossil fuel and hydrogen supply chain. 4. Based on a range of external costs of global warming of between $0.026 and $0.067 per kilogram of CO2equivalent , a reduction in CO2equivalent in a range of 1.36 × 109 – 1.39 × 109 tonnes/year translates to a reduction in external costs of between $35.3 and $93.5 billion/year due to global warming. 14.3.9 Quantifying Environmental Impact—Applying the Appropriate Emission Data To conduct an accurate LCA, emission data should be carefully verified and applied. We must check the sources of emission data, understand the methods used for gathering and categorizing these data, properly apply definitions, and benchmark estimates against independent sources. A common limitation of LCA is the misuse of emission data. Emissions of CO2 are often estimated from fuel consumption data. While air pollutant emissions are typically measured at the outlet of a process, CO2 emissions are typically estimated based on the total fuel entering a device and an estimated carbon content of that fuel. For electric power plants, the U.S. federal government legally requires that power plant operators over a certain size (1 MW) manually report their monthly and total annual fuel consumption (mF ). Based on the principle of conservation of mass, the government uses mF and estimates of the carbon content of different fuels to back-calculate CO2 emission from each plant. The government then calculates the total quantity of CO2 emissions from electricity generation (mCO2 ) as the summation of CO2 emissions from all plants. According to this method, mCO2 is the summation over all fuel types of the product of (1) mF and (2) an average annual emission factor per unit of fuel consumption for each electric power plant of a given fuel type (𝛾 F-CO2 ) mCO2 =

n ∑ ( ) mF 𝛾 F-CO2 i

(14.18)

i=m

This methodology assumes an average power plant efficiency and carbon content of fuel. For example, if the economy contained only natural gas (N) and coal (C) plants, this method would calculate mCO2 as the summation of emissions from natural gas and coal plants, or mCO2 =

n ∑ ( ) mF 𝛾 F-CO2 i = mN 𝛾 N-CO2 + mC 𝛾 C-CO2 i=m

(14.19)

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ENVIRONMENTAL IMPACT OF FUEL CELLS

Rather than using an average emission factor, other methods estimate CO2 emissions based on the marginal emissions from the next dispatched power plant. In the United States, these marginal emissions tend to be higher than the average, because at peaking power times when the electricity system is stressed, higher emission power plants are typically called on to meet the extra demand. To conduct a precise LCA, the control volume applied to the reported emission data must be very carefully understood in detail. For example, reported CO2 emissions from the electric power sector may include or exclude power plants of certain fuel types, sizes, and localities. These CO2 estimates may include or exclude imported and/or exported electric power into/from a region. In some cases, fuel consumption data may not be reported, especially for imported or exported power across regions. For example, the U.S. government currently does not report the fuel content of electricity exchanged between states, such that the state of California cannot rely on federal data alone to determine the CO2 emissions from the >30% of electricity that it imports from other states. For vehicles, the government applies a similar method. The government uses receipts from sales tax revenue on gasoline, diesel, and ethanol fuels to estimate total annual fuel consumption (mF ) of each type by vehicles. The government then estimates total annual CO2 emissions from vehicles (mCO2 ) based on (1) mF and (2) an average annual emission factor per unit of each fuel type (𝛾 F-CO2 ), or mCO2 =

n ∑ ( ) mF 𝛾 F-CO2 i

(14.20)

i=m

Again, the control volume applied to reported emission data must be very carefully understood. Transportation data may include or exclude passenger vehicles, heavy duty vehicles, shipping vessels, airplanes, and other vehicles, or any subset or combination of these. 14.4

EMISSIONS RELATED TO AIR POLLUTION

To conduct the later steps in LCA related to emissions, in addition to emissions that influence global warming, we have to determine which emissions in the supply chain influence air pollution. The primary source of air pollution is combustion in power plants, furnaces, and vehicles. This air pollution can harm the health of humans, animals, and vegetation and can damage materials. Six primary emissions that create air pollution are O3 , CO, NOx , PM, SOx , and VOCs. Volatile organic compounds are non-methane organic compounds, such as the higher HCs (Cx Hy ). Some of these compounds are air pollutants themselves. Others react with chemicals to produce air pollution. Effects of air pollution on human health can include respiratory illness, pulmonary illness, damage to the central nervous system, cancer, and increased mortality. 14.4.1

Hydrogen as a Potential Contributor to Air Pollution

Because an increase in the use of fuel cells might increase the quantity of H2 released into the atmosphere, climate researchers are now trying to determine the potential effect

EMISSIONS RELATED TO AIR POLLUTION

of released H2 on air pollution. One mechanism through which released H2 might increase one type of air pollutant is a series of chemical reactions that enhance the concentration of O3 . In the troposphere, H2 might increase O3 by increasing the concentration of atomic hydrogen (H). After several years in the atmosphere, molecular hydrogen decays to atomic hydrogen in the presence of the hydroxyl radical (OH), via the reaction H2 + OH → H2 O + H

(14.21)

Atomic hydrogen (H) could then react with oxygen (O2 ) in air in the presence of photon energy (h𝑣) from light to increase O3 through the following set of reactions: H + O2 + M → HO2 + M NO + HO2 → NO2 + OH

(14.22) (14.23)

NO2 + h𝑣 → NO + O

(14.24)

O + O2 + M → O3 + M

(14.25)

where M represents any molecule in the air that is neither created nor destroyed during the reaction but that absorbs energy from the reaction. However, other sets of reactions must also be considered, with a focus on their net effect on air pollution. The net effect of these reactions might be determined with computer simulations of chemical reactions in the atmosphere (atmospheric models). As you learned in LCA, to be accurate, these simulations should model, not the mere addition or subtraction of an individual chemical component, but rather the net change in emissions among different scenarios.

14.4.2

Quantifying Environmental Impact—Health Effects of Air Pollution

Table 14.4 summarizes some of the most important emissions and the ambient air pollutants that evolve from them via chemical reactions with other compounds [160]. The table also lists some important health effects from these pollutants. For example, emissions of both CO and PM increase the human death rate (mortality). Finally, the table shows estimates of the number of cases of each health effect per unit mass of ambient pollutant.6 The estimates in Table 14.4 primarily apply to vehicles rather than power plants; vehicles tend to be used in population centers where they are close to people. Therefore, their emissions have a stronger impact on human health per unit mass of emission than power plants, which tend to be located further from population centers. Table 14.4 lists incidents of health effects as a function of ambient pollutant levels. To calculate the health effects per tonne of emission, one can estimate that every tonne of VOC or NOx emitted yields, via chemical reaction, 1 tonne of O3 as an ambient pollutant, mO3 ,AMB = mVOC + mSOx

(14.26)

6 Estimates were derived from the number of U.S. cases of each health effect stemming from automotive pollution and the total U.S. emissions of each type from automobiles (based on the NEI).

503

504 O3

VOC+NOx

Asthma attacks Respiratory restricted activity days (RRAD) Chronic illness Mortality

Asthma attacks Eye irritation Low respiratory illness Upper respiratory illness Any symptom or condition (ARD2)

Sore throat Excess phlegm Eye irritation

Headache Hospitalization Mortality

0.147 4.33 0.00190 0.00391

0.155 5.87 0.00454 0.00669

−188 −5566 −2 −5

−199 −7540 −6 −9

−3.19 × 103 −1.04 × 104 −2.25 × 104 −6.85 × 103 −7.67 × 104 0.255 0.830 1.80 0.548 6.13

14.6 5.34 4.81

−1.01 × 103 −9.40 × 103 −1.35 × 104 −4.10 × 103 0 0.0811 0.752 1.08 0.328 0

14.5 5.26 4.73

−8.81 × 104 −4.04 × 104 −3.64 × 104

High

−8.68 × 104 −3.98 × 104 −3.58 × 104

Low

−8.95 × 104 −10.2 −0.663

High

Change in Health Effects (thousands of cases) with a Fleet Change from Conventional to Fuel Cell

1.22 1.45 −7.53 × 104 0.000572 0.000164 −3.54 0.00000357 0.0000107 −0.221

Low

Health Effect Factor (thousands of cases/ tonne ambient pollutant)

Note: Emissions from vehicles (column 1) evolve by chemical reaction to ambient pollutants (column 2). These ambient pollutants lead to various health effects in people (column 3). The health effects are estimated primarily for automotive pollutants in terms of the number of cases of each health effect per unit mass of ambient pollutant, with low values (column 4) and high values (column 5). An example is shown for the change in health effects cases with a switch in vehicle fleet from the conventional one to a hydrogen fuel cell fleet (columns 6 and 7).

PM10

NO2

NOx

PM10 , SO2 , NOx , VOC

CO

Ambient Pollutant Health Effect

CO

Emission

TABLE 14.4. Health Effects of Air Pollution

EMISSIONS RELATED TO AIR POLLUTION

TABLE 14.5. Financial Costs of Air Pollution

Health Cost of Air Pollution ($2004/tonne of emission)

Change in Health Costs Due to Air Pollution ($2004) with a Fleet Change from Conventional to Fuel Cell

Emission

Ambient Pollutant

Low

High

Low

High

CO

CO

12.7

114

−7.87 × 108

−7.08 × 109

NOx

Nitrate-PM10

1.30 × 103

2.11 × 104

−9.83 × 109

−1.60 × 1011

NO2

191

929

−1.45 × 109

−7.03 × 109

PM2.5

PM2.5

1.33 × 104

2.03 × 105

−2.22 × 109

−3.39 × 1010

PM2.5 -PM10

PM2.5 -PM10

8.52 × 103

2.25 × 104

−4.38 × 108

−1.16 × 109

SOx

Sulfate-PM10

8.78 × 103

8.33 × 104

−2.39 × 109

−2.27 × 1010

VOC

Organic-PM10

127

1.46 × 103

−6.27 × 108

−7.21 × 109

VOC + NOx

O3

12.7

140

−1.59 × 108

−1.75 × 109

−1.79 × 1010

−2.40 × 1011

Total

Note: Emissions from vehicles (column 1) evolve by chemical reaction to ambient pollutants (column 2). These ambient pollutants lead to health effects in people and therefore a human health cost to society (columns 3 and 4). An example is shown for the change in health costs with a switch in vehicle fleet from the conventional one to a hydrogen fuel cell fleet (columns 5 and 6).

where m is the mass of each type of emission, and that ambient pollution of PM10 can be calculated as mPM10,AMB = mPM2.5 + 0.1(mPM10 − mPM2.5 ) + 0.4mSO2 + 0.1mNO2 + 0.05mVOC (14.27) where the coefficients in front of the m refer to the percentage of emitted mass that converts to ambient PM10 pollution via reaction with other species [156]. 14.4.3

Quantifying Environmental Impact—External Costs of Air Pollution

If people are less healthy, they require more medical services and miss more productive working days. Additional medical services and a decrease in labor productivity incur a financial cost on society. Therefore, the health effects of air pollution can be quantified in financial terms. Based on the health effects data shown in Table 14.4, Table 14.5 estimates the financial costs of these and other emissions on human health [160]. Interestingly, the majority of the health costs in Table 14.5 are the result of automotive emissions. The health costs per unit mass of emissions are estimated to be about an order of magnitude lower for power plants because of their greater distance from people. The financial costs related to human health are the dominant source of external costs of air pollution. As with the external costs of global warming, the external costs of air pollution are not incorporated into free-market prices. Although these costs are difficult to quantify, ignoring them incorrectly assumes that their value is zero.

505

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ENVIRONMENTAL IMPACT OF FUEL CELLS

Example 14.6 (1) Identify a stationary fuel cell system that is available for purchase and fueled by natural gas. Investigate the air pollution emissions associated with this unit. Report these air pollution emissions in units of g of species emitted/kWhe. (2) Compare these emission factors with those for natural gas CCGT combustion and coal combustion electric power plants, shown in Table 14.1. (3) Comment on the significance of this comparison. Solution: 1. and 2. Table 14.6 reports NOx , SOx , and PM10 emission factors for a molten carbonate fuel cell (MCFC) system sold by a manufacturer in Connecticut. The emissions are extremely low. Table 14.6 also compares these emission factors with those of the CCGT and coal plants. The last two rows show what percentage the MCFC system emission factors are compared with the CCGT and coal plant emission factors. The MCFC system NOx emissions are less than 1% of the CCGT or coal plant emissions. The MCFC system SO2 and PM10 emissions are less than 0.02% of the CCGT or coal plant emissions. 3. A significance of this comparison is that it quantitatively shows that a main competitive advantage of the fuel cell system over other types of fossil-fuel power generation is its extremely low air pollution emissions. This point is further underscored by reflecting back on Example 14.4, which showed that the CO2 emission factors were the same order of magnitude for the three different power plants compared. By contrast, the air pollution emission factors for the fuel cell system are several orders of magnitude lower than for CCGT or coal plants. TABLE 14.6. Comparison of Air Pollution Emission Factors for Three Plants Emission factor (g/kWhe)

MCFC system CCGT Plant Coal Plant % of CCGT emissions % of coal emissions

NOx

SO2

PM10

0.00453592 0.70 2.0 0.65% 0.23%

4.5359E-05 0.27 1.0 0.017% 0.005%

9.0718E-06 0.074 0.2 0.012% 0.005%

Example 14.7 (1) Based on the scenario of fuel cell vehicle adoption outlined in Example 14.5, calculate the change in health effects for the replacement of conventional vehicles with fuel cell vehicles. For simplicity, in this LCA comparison, focus on the change in emissions at the vehicle, ignoring upstream changes in emissions. (2) Calculate the change in external costs (the financial costs of health damage born by society). (3) Compare the change in external costs due to air pollution with the change due to global warming.

ANALYZING ENTIRE SCENARIOS WITH LCA

Solution: 1. The change in health effects is shown in the last column of Table 14.4. Volatile organic compounds include all of the organics listed in Table 14.3 except methane. One can calculate the quantity of ambient ozone pollution from the emitted VOCs and NOx based on mO3 ,AMB = mVOC + mNOx and the quantity of ambient pollution of PM10 from several emissions based on mPM10,AMB = mPM2.5 + 0.1(mPM10 − mPM2.5 ) + 0.4mSOx + 0.1mNOx + 0.05mVOC

(14.28)

The reduction in health effects shown in Table 14.4 is an upper bound estimate. A more developed analysis takes into account the net change in emissions all along each supply chain. 2. The change in health costs is shown in the last column of Table 14.5. The external costs shown in Table 14.5 are per-unit mass of emission (not per unit mass of ambient pollutant as in Table 14.4.) With a switch in the vehicle fleet, health costs decrease by between $18 billion and $240 billion per year. 3. With a switch in fleet, we have seen that global warming costs decrease by between $35.3 billion and $93.5 billion per year. The reduction in health costs is in a similar range. 14.5

ANALYZING ENTIRE SCENARIOS WITH LCA

We have now seen several examples of different segments of LCA. We have also learned important tools for quantifying the environmental impact of different supply chains. We will now combine these tools to analyze an additional scenario on electric power production through the lens of energy efficiency. 14.5.1

Electric Power Scenario

Having read so much about fuel cells, you are interested in exploring the possibility of installing a fuel cell system on your local university’s campus. You would like this system to provide electricity to nearby buildings. Because you live in an area of the country rich in coal reserves, you would like to explore the possibility of using coal as the original fuel. Your local university currently gets most of its electricity from a nearby coal power plant. You decide to compare (1) the current scenario with electricity derived from a coal power plant against (2) a possible process chain of a fuel cell system fueled by hydrogen derived from coal. You would like to determine whether it would be more efficient to use a fuel cell system. You decide to compare the overall electrical efficiency across the process chain to see which scenario might be more efficient. 1. Research and develop an understanding of the supply chain. First, think about the current supply chain for electricity. Coal is extracted from coal mines, processed

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ENVIRONMENTAL IMPACT OF FUEL CELLS

from chunks into smaller pieces, and transported via railroad or barge to power plants that are usually within close proximity to the mine. The coal power plant produces electricity that is transmitted across high-voltage transmission lines long distances and later at reduced voltages over low-voltage distribution lines to the university’s buildings. Second, think about the potential H2 supply chain. Based on our knowledge of fuel processing from Chapter 11 and some additional reading, we learn how we can chemically process coal into an H2 -rich gas, a process called coal gasification. Coal gasification is a chemical conversion process that transforms solid coal and steam into a gaseous mixture of H2 and CO at elevated pressures and temperatures. Because coal contains little H2 , much of the H2 originates from the added steam. Emissions for a coal gasification plant optimized for H2 production are shown in Table 14.1. This plant has an HHV efficiency of 60%. Assume that our coal gasification plants are placed at similar locations as conventional coal power plants. They rely on the same upstream processes as traditional coal plants, including coal mining, processing, and transport. After H2 production, H2 is transmitted through large hydrogen transmission pipelines over long distances and then through smaller distribution lines to local areas. Then H2 is stored and consumed in fuel cell systems located throughout your university campus. Each fuel cell system provides electricity to one or more buildings. 2. Sketch a supply chain. Figures 14.8 and 14.9 describe these two separate supply chains. The first three boxes of Figure 14.8 are the same as for Figure 14.9. 3. Identify the “bottleneck” processes. In Figure 14.8, think about the energy input arrows at the bottom of the process boxes in terms of efficiency. The HHV efficiency of the first three combined processes—extraction, processing, and transport—is approximately 90%; about 10% of the original energy in the coal fuel is required for its combined mining (box 1), processing (box 2), and transport (box 3). The HHV efficiency of a typical coal plant (box 4) is approximately 32%; for every 100 units of coal energy entering the plant, 32 units leave as electricity and 68 leave

Coal extraction from mines 1

Coal processing 2

Coal process stream

Coal transport 3

4

CO emissions CO2 emissions

Energy input

Electricity stream

Electricity transmission

Coal plant electricity generation

Coal production pollution

Electricity distribution

5

SOx emissions NOx emissions CH4 emissions

6

VOC Particulate matter

Figure 14.8. Supply chain for conventional electricity generation from coal. The most energy and emission intensive process in the chain is electricity generation (box 4).

ANALYZING ENTIRE SCENARIOS WITH LCA

Coal extraction from mines 1

Coal processing

Coal transport

2

3

Coal gasification to produce H2 4

H2 pipeline transmission

H2 pipeline distribution

H2 storage

6

7

5

Stationary fuel cell system 8

Coal production pollution Coal process stream Energy input

H2 gas process stream

CO emissions CO2 emissions

SOX emissions NOX emissions

H2 gas leakage H2O vapor emissions

Electricity Heat

Figure 14.9. Supply chain for coal gasification plant. The most energy-intensive processes in the chain are coal gasification (box 4) and electricity generation at the stationary fuel cell system (box 8).

as heat dissipated to the environment. The efficiency of electricity transmission (box 5) is 97%; about 3% of the electricity transmitted over the high-voltage wires from the coal plant to urban areas is dissipated as heat. The efficiency of electricity distribution is about 93%; about 7% of electricity conveyed over the low-voltage wires around local areas is lost to the environment as heat. Therefore, for the scenario in Figure 14.8, the most energy-intensive process is by far electricity generation at the coal plant. In Figure 14.9, think about the energy input arrows at the bottom of the process boxes in terms of efficiency. The HHV efficiency of the first three combined processes (boxes 1, 2, and 3) is the same as in the supply chain of Figure 14.8, approximately 90%. The HHV efficiency of the coal gasification plant (box 4) is approximately 60%; that is, for every 100 units of coal energy entering the plant, 60 units leave as hydrogen energy. The efficiencies of hydrogen transmission (box 5) and distribution (box 5) are both 97%, similar to natural gas. The HHV efficiency of hydrogen storage not at pressure is about 100%. The HHV electrical efficiency of the fuel cell system is 50%. Therefore, for the scenario in Figure 14.9, the most energy-intensive processes are by far coal gasification and electricity generation at the fuel cell system. 4. Analyze the energy and mass flows in the supply chain. Focusing on the bottleneck processes, emissions for the coal plant and the coal gasification plant are shown in Table 14.1 Emissions at the fuel cell system are only water vapor. 5. Aggregate net energy and emission flows for the chain. The supply chain in Figure 14.8 has an overall efficiency across the entire chain of 26%. The supply chain in Figure 14.9 has an overall efficiency across the entire chain of 25%. Therefore, there might be no gain in overall efficiency from switching to fuel cell power in this scenario. However, a comparison of the emissions per unit mass of fuel in Table 14.1 shows a potential reduction in emissions with a switch to the supply chain of Figure 14.9. Therefore, you continue to think about how a fuel cell scenario might work for your campus. You realize that the fuel cell system you were interested in installing can also recover heat. The HHV heat recovery efficiency of the fuel cell system is 20%. Across the entire supply chain, the heat recovery efficiency (𝜀H,SC ) is then 10%; that is,

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ENVIRONMENTAL IMPACT OF FUEL CELLS

10% of the original energy in the coal mined can be used as heat on your university campus. Therefore, the overall (electrical and heat recovery) efficiency across the entire supply chain is 𝜀O,SC = 𝜀R,SC + 𝜀H,SC = 25% + 10% = 35%

(14.29)

in this scenario. To make a fair comparison, you also investigate heat recovery for the supply chain of Figure 14.8. You discover that coal plants are almost always located close to coal mines because of the high cost of transporting a solid fuel. As a result, coal plants are not often located near large population centers where there is a source of demand for electricity or heat. The coal plant that serves your university is no different; it is located 20 miles away from your university and 50 miles away from the nearest city. As a result, it would not be practical to try to recover heat from it. The practical heat recovery efficiency of this supply chain is zero. The overall electrical and thermal efficiency of the supply chain in Figure 14.8 is then 𝜀O,SC = 𝜀R,SC + 𝜀H,SC = 26% + 0% = 26%

(14.30)

You thus decide to investigate more seriously the prospect of installing a fuel cell system with heat recovery on your university campus.

14.6

CHAPTER SUMMARY

The purpose of this chapter was to understand the potential environmental impact of fuel cells by applying quantitative tools to help us calculate changes in emissions, energy use, and efficiency with their adoption. We learned a tool called life cycle assessment (LCA). • To compare a change in energy technology from one to another, the entire supply chain associated with each technology is considered. • The supply chain begins with the extraction of raw materials, continues on to the processing of materials, then on to energy production and end use, and finally to waste management. • Within a chain, attention focuses on the most energy- and emission-intensive processes, the process bottlenecks. • Scenarios are compared by analyzing the relevant energy and material inputs and outputs along the entire supply chain based on the conservation-of-mass equation m1 − m2 = Δm and the conservation-of-energy equation [ ( ) ( )] Q̇ − Ẇ = ṁ h2 − h1 + g z2 − z1 + 12 V22 − V12

(14.31)

• Aggregate emissions and energy use for one supply chain are compared with aggregate emissions and energy use for another.

CHAPTER EXERCISES

• All other things being equal, an energy process produces less carbon dioxide (CO2 ) emissions if that process consumes a fuel with less carbon (C) content per unit energy. The combination of a fuel’s carbon content and the efficiency with which the fuel is consumed determine its CO2 emissions per unit of useful output. • The environmental impact of emissions related to global warming is quantified by (1) calculating the CO2 equivalent of emitted gases and (2) the external costs of these emissions. • CO2 equivalent is the mass of CO2 gas that would have an equivalent warming effect on Earth as a mixture of different types of gases and particles. One equation for measuring the CO2 equivalent of gases and particles over a 100-year period is CO2 equivalent = mCO2 + 23mCH4 + 296mN2 O + 𝛼(mOM,2.5 + mBC,2.5 ) − 𝛽[mSULF,2.5 +mNIT,2.5 + 0.40mSOx + 0.10mNOx + 0.05mVOC ]

(14.32)

• An “external cost” is an economic term that refers to the cost of a good that is not included in its free-market price. • Annual CO2 emissions (mCO2 ) from a sector are often calculated as the summation over all fuel types of the product of (1) total annual fuel consumption (mF ) and (2) an average annual emission factor per unit of fuel consumption for each energy conversion device of a given fuel type (𝛾 F-CO2 ), or mCO2 =

n ∑ ( i=m

mF 𝛾 F-CO2

) i

(14.33)

• The environmental impact of emissions related to air pollution can be quantified by calculating (1) the impacts on human health and (2) the external costs of these emissions. • By comparing these quantities, the environmental performance of various supply chains can be rated against one another. • The analysis can be repeated to incorporate greater detail along the various segments of the chain. • Multiple chains can be evaluated against different energy and environmental metrics.

CHAPTER EXERCISES Review Questions 14.1

What are the primary steps of life cycle assessment (LCA)?

14.2

What are some of the gases and particles that have a warming effect on Earth? How? What are some of the gases and particles that have a cooling effect on Earth? How?

14.3

What are some of the most important air pollutants that affect human health?

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14.4

In the United States, what type of power plant provides more than 50% of all electric power? How does the U.S. distribution of electricity by power plant type compare with that of other countries? And with your region?

14.5

What is a national emissions inventory (NEI)? Describe the type of information it contains.

14.6

When might leaked hydrogen combust with oxygen in air?

14.7

How do the average annual CO2 emission factors (𝛾E−CO2 ) for fuel cell, coal, and CCGT plants compare, particularly on a relative-order-of-magnitude basis?

14.8

How do the air pollution emission factors for fuel cell, coal, and CCGT plants compare, on a relative-order-of-magnitude basis? Between the two types of emission factors (greenhouse gases in the prior question vs. air pollution emission factors), which highlights a true competitive advantage of the fuel cell system?

14.9

Develop an abstract for a research proposal to answer a question you feel is important that relates to the environmental impact of fuel cells. You plan to use data from a national emissions inventory (NEI) and to conduct an LCA.

14.10 Which fuels have the highest carbon content per unit of fuel energy and therefore may release the highest levels of CO2 emissions? Which fuels have the lowest carbon contents per unit of fuel energy? 14.11 Do emission inventories typically include direct measurements of air pollution emissions? What about greenhouse gas emissions? What methods may be used to estimate CO2 emissions from fuel consumption and financial data? Calculations 14.12 Example 14.4 discusses an SOFC system and its CO2 emission factor. Building on this example, assume that methane leaks out of the natural gas pipeline at a rate of 1% by mass. Considering only the CO2 emission factor calculated previously and this methane leakage rate, what is the average annual CO2equivalent emission factor associated with this SOFC system? Use the first three terms in the equation for measuring the CO2equivalent of gases over a 100-year period (Equation 14.14) and ignore all other species. 14.13 Revise the calculation shown in Example 14.4 assuming that the fuel composition is not 100% methane, but rather the fuel composition for natural gas delineated in Table 14.2. Use the carbon content per unit of fuel energy for natural gas, shown Table 14.2. (a) What is the average annual CO2 emission factor (𝛾E−CO2 ) for these systems in units of g of CO2 ∕kWhe? (b) Comment on how this emission factor compares with those listed for natural gas combustion and coal combustion electric power plants in Table 14.1. (c) Assume that methane leaks out of the natural gas pipeline at a rate of 1% by mass. Considering only the CO2 emission factor calculated here and this methane leakage rate, what is the average annual CO2equivalent emission factor associated with this SOFC system? Use the first three

CHAPTER EXERCISES

terms in the equation for measuring the CO2equivalent of gases over a 100-year period (Equation 14.14) and ignore all other species. 14.14 A Connecticut company sells a stationary combined heat and power (CHP) molten carbonate fuel cell (MCFC) system that consumes natural gas fuel and produces electricity and recoverable heat. CHP fuel cell systems were discussed in detail in Chapter 10. The systems are said to operate at a net electrical efficiency of about 47%, and a net heat recovery efficiency of up to 43%, based on the lower heating value (LHV). To simplify calculations, assume that the gas composition of natural gas is 100% methane, reactants and products enter and leave the system at STP, and the LHV of methane can be used. (a) What is the average annual CO2 emission factor (𝛾E−CO2 ) for these systems in units of g of CO2 ∕kWhe? (b) Is this emission factor a fair unit of comparison when comparing CHP plants with non-CHP plants? 14.15 (a) Building on the prior problem, develop and describe some approaches for “crediting” this CHP MCFC system for displacing heat as well as electricity in terms of its reported carbon footprint/CO2 emission factor. (b) The manufacturer reports the MCFC plant’s CO2 emission factor to be 426 g/kWhe, without considering the benefits of CHP. With considering the benefits of CHP, the manufacturer reports the MCFC plant’s CO2 emission factor to be between 236 and 308 g∕kWhe. How do your results compare with these values? Indicate potential sources of discrepancies between your calculations and the manufacturer’s calculations (such as assumptions about degradation over time, fuel composition, heating value basis, etc.). 14.16 Example 14.6 discusses a fuel cell system and its air pollution emission factors. Building on this example, (a) translate the fuel cell, CCGT, and coal plant emissions into ambient air pollutants. (b) Analyze the human health-related impacts and financial costs of the ambient pollutants from the fuel cell, CCGT, and coal plants. (c) Calculate the change in health costs due to air pollution from (i) switching from coal plants to fuel cell systems and (ii) switching from CCGT plants to fuel cell systems. 14.17 Identify a stationary fuel cell system that is available for purchase and fueled by natural gas, other than the one already discussed in Example 14.6. (a) Investigate the air pollution emissions associated with this unit. Report these air pollution emissions in units of grams of species emitted/kWhe. (b) Compare these emission factors with those for CCGT and coal plants, shown in Table 14.1. (c) Comment on the significance of this comparison. (d) Translate the fuel cell, CCGT, and coal plant emissions into ambient air pollutants. (e) Analyze the human health-related impacts and financial costs of the ambient pollutants from the fuel cell, CCGT, and coal plants. (f) Calculate the change in health costs due to air pollution from (i) switching from coal plants to fuel cell systems and (ii) switching from CCGT plants to fuel cell systems. 14.18 Estimate the CO2 equivalent of the following mixture of gases and particles: all organic gases and particulate matter from all sources listed in the 1999 NEI. 14.19 Based on Example 14.2, estimate the mass flow rate of natural gas that must be produced at the gas field to supply enough fuel to the downstream steam reformers.

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Assume the ratio of fuel cell vehicle efficiency to gasoline vehicle efficiency is 2, 2% of total hydrogen production is leaked in the supply chain, and 1% of methane in natural gas is leaked. How does this quantity of natural gas compare with current annual natural gas production as a percentage? Calculate the CO2 equivalent and the external cost of the leaked methane. 14.20 Based on U.S. emissions listed in Table 14.3 and Example 14.5, compare the CO2 equivalent emissions from the fossil fuel vehicle fleet with a hydrogen vehicle fleet taking into account changes in upstream emissions during the production of hydrogen and fossil fuels. Assume that all hydrogen is produced via a high-efficiency steam reformer. Assume that half of the U.S. total VOC emissions are related to the transportation sector and are emitted during gasoline and diesel production. Rely on the 1999 U.S. NEI, available at the EPA’s website, for additional data on emissions. 14.21 Imagine replacing current U.S. electrical power with stationary hydrogen fuel cell power plants. Conduct an LCA to evaluate the change in efficiency and emissions across the supply chain. 14.22 Imagine the same scenario as in problem 14.21 except that heat is also recovered from the fuel cell systems. Heat recovered from the fuel cell systems replaces heat that would otherwise be produced by combusting natural gas and oil in furnaces. Assume that, on average throughout the seasons, 30% of the HHV of natural gas fuel is recovered by the fuel cell systems as useful heat and consumed in surrounding buildings for space heating or industrial applications. Assume the same emissions profile as shown in Table 14.1 for a steam reformer matches that of a fuel cell system. The original emissions data shown in Table 14.1 are from a United Technologies Corporation PAFC 200-kWe system. Conduct an LCA to evaluate the change in efficiency and emissions across both the electricity supply chain and the heating supply chain. 14.23 Building on Examples 14.5 and 14.7, for the same LCA comparison recalculate the change in health effects and in external costs due to air pollution taking into account changes in upstream emissions. Also, for the entire supply chain, calculate the change in CO2 equivalent and in external costs due to global warming. Rely on the 1999 U.S. NEI, available at the EPA’s website, for additional data on emissions. 14.24 Conduct an LCA for a scenario in which hydrogen is derived from coal gasification. Assume that the coal gasification plant has the emissions profile shown in Table 14.1. 14.25 Building on Example 14.2, estimate the quantity of hydrogen leaked into the environment by a fuel cell vehicle fleet. Assume the hydrogen leakage rate is similar to that for natural gas (approximately 1% of production). How does this quantity of released hydrogen compare with the amount released by conventional on-road vehicles, shown in Table 14.3? 14.26 Estimate the expected minimum CO2 emissions per unit of fuel energy from consuming all of the fuels in Table 14.2. For simplicity, ignore all energy consumed and

CHAPTER EXERCISES

emissions released in the upstream processing of these fuels. Assuming these fuels are used to produce hydrogen fuel, estimate the expected minimum CO2 emissions per unit of hydrogen fuel (H2 ). To create a rough estimate, assume 100% efficient conversion processes with no losses, and ignore emissions from all upstream fuel processing. 14.27 Identify the sales tax revenues per year from transportation fuel consumption in your country or region. Using known tax rates, estimate the total fuel consumption per year per fuel type. These fuels may include diesel, gasoline, ethanol, and other transportation fuels. Be sure to carefully define your definition of the transportation sector and the vehicles included (passenger cars, heavy-duty trucks, shipping vessels, airplanes, etc.). Using Table 14.2 as a guide, estimate an emission factor for each of these fuels in terms of CO2 emitted per unit of fuel consumed. Calculate the total annual CO2 emissions from vehicles in your region per year per capita over time. Estimate the margin of error in your calculations. Cross-check your value against your region’s reported emissions. 14.28 Agency F and Agency S both report CO2 emission data for electric power plants in region C. Cogenerative power plants produce (A) electricity, (B) heat that is recovered for some useful purpose such as space heating for buildings, and (C) waste heat. Agency F reports CO2 emissions from these power plants based on the total fuel consumed at them for all three purposes (A + B + C). Agency S reports CO2 emissions from these cogenerative power plants as the summation of the fuel consumed only for electricity generation (A) and a random portion of the waste heat (a random portion of C). Agency S allocates the CO2 emissions associated with heat recovered (B) and the remaining portion of the waste heat (the remaining portion of C) with its manufacturing sector, not with its power plants. Discuss the pros and cons of each reporting method. You are developing a new power plant with low CO2 emissions. To demonstrate the reduction in CO2 emissions that your power plant would achieve relative to region C’s co-generative plants, which agency’s numbers would you rely on, and why? Which method provides the appropriate benchmark for your new technology?

515

APPENDIX A

CONSTANTS AND CONVERSIONS

Physical Constants Avogadro’s number Universal gas constant

NA R

Planck’s constant

h

Boltzmann’s constant

k

Electron mass Electron charge Faraday’s constant

me q F

6.02 × 1023 atoms∕mol 0.08205 L ⋅ atm∕mol ⋅ K 8.314 J∕mol ⋅ K 83.14 bars ⋅ cm3 ∕mol ⋅ K 8.314 Pa ⋅ m3 ∕mol ⋅ K 6.626 × 10−34 J ⋅ s 4.136 × 10−15 eV ⋅ s 1.38 × 10−23 J∕K 8.61 × 10−5 eV∕K 9.11 × 10−31 kg 1.60 × 10−19 C 96485.34 C∕mol

Conversions Weight Distance Volume

2.20 lb = 1 kg 0.622 mile = 1 km 3.28 × 10−2 ft = 1 cm 1000 L = 1 m3 0.264 gal = 1 L 3.53 × 10−2 ft3 = 1 L 517

518

APPENDIX A: CONSTANTS AND CONVERSIONS

Conversions (cont.) Pressure

Energy

Power

1.013250 × 105 Pa = 1 atm 1.013250 bars = 1 atm 105 Pa = 1 bar 14.7 psi = 1 atm 6.241506 × 1018 eV = 1 J 1 calorie = 4.184 J 9.478134 × 10−4 Btu = 1 J 2.777778 × 10−7 kWh = 1 J 1 J∕s = 1 W 1.34 ⋅ 10−3 horsepower = 1 W 3.415 Btu∕h = 1 W

APPENDIX B

THERMODYNAMIC DATA

This appendix lists thermodynamic data for H2 , O2 , H2 O(g) , H2 O(l) , CO, CO2 , CH4 , N2 , CH3 OH(g) , and CH3 OH(l) as a function of temperature at P = 1 bar.

519

520

APPENDIX B: THERMODYNAMIC DATA

TABLE B.1. H2(g) Thermodynamic Data T (K) 200 220 240 260 280 298.15 300 320 340 360 380 400 420 440 460 480 500 520 540 560 580 600 620 640 660 680 700 720 740 760 780 800 820 840 860 880 900 920 940 960 980 1000

̂ G(T) (kJ/mol) −26.66 −29.07 −31.54 −34.05 −36.61 −38.96 −39.20 −41.84 −44.51 −47.22 −49.96 −52.73 −55.53 −58.35 −61.21 −64.08 −66.99 −69.91 −72.86 −75.83 −78.82 −81.84 −84.87 −87.92 −90.99 −94.07 −97.18 −100.30 −103.43 −106.59 −109.75 −112.94 −116.14 −119.35 −122.58 −125.82 −129.07 −132.34 −135.62 −138.91 −142.22 −145.54

̂ H(T) (kJ/mol)

̂ S(T) (J/mol ⋅ K)

Cp (T) (J/mol ⋅ K)

−2.77 −2.22 −1.66 −1.09 −0.52 0.00 0.05 0.63 1.21 1.79 2.38 2.96 3.54 4.13 4.71 5.30 5.88 6.47 7.05 7.64 8.22 8.81 9.40 9.98 10.57 11.16 11.75 12.34 12.93 13.52 14.11 14.70 15.29 15.89 16.48 17.08 17.68 18.27 18.87 19.47 20.08 20.68

119.42 122.05 124.48 126.75 128.87 130.68 130.86 132.72 134.48 136.14 137.72 139.22 140.64 142.00 143.30 144.54 145.74 146.89 147.99 149.06 150.08 151.08 152.04 152.97 153.87 154.75 155.61 156.44 157.24 158.03 158.80 159.55 160.28 161.00 161.70 162.38 163.05 163.71 164.35 164.99 165.61 166.22

27.26 27.81 28.21 28.49 28.70 28.84 28.85 28.96 29.04 29.10 29.15 29.18 29.21 29.22 29.24 29.25 29.26 29.27 29.28 29.30 29.31 29.32 29.34 29.36 29.39 29.41 29.44 29.47 29.50 29.54 29.58 29.62 29.67 29.72 29.77 29.83 29.88 29.94 30.00 30.07 30.14 30.20

APPENDIX B: THERMODYNAMIC DATA

TABLE B.2. O2(g) Thermodynamic Data T (K) 200 220 240 260 280 298.15 300 320 340 360 380 400 420 440 460 480 500 520 540 560 580 600 620 640 660 680 700 720 740 760 780 800 820 840 860 880 900 920 940 960 980 1000

̂ G(T) (kJ/mol) −41.54 −45.45 −49.41 −53.41 −57.45 −61.12 −61.54 −65.66 −69.82 −74.02 −78.25 −82.51 −86.80 −91.12 −95.47 −99.85 −104.25 −108.68 −113.13 −117.61 −122.10 −126.62 −131.17 −135.73 −140.31 −144.92 −149.54 −154.18 −158.84 −163.52 −168.21 −172.93 −177.66 −182.40 −187.16 −191.94 −196.73 −201.54 −206.36 −211.20 −216.05 −220.92

̂ H(T) (kJ/mol)

̂ S(T) (J/mol ⋅ K)

Cp (T) (J/mol ⋅ K)

−2.71 −2.19 −1.66 −1.10 −0.54 0.00 0.03 0.62 1.21 1.81 2.41 3.02 3.63 4.25 4.87 5.50 6.13 6.76 7.40 8.04 8.68 9.32 9.97 10.62 11.27 11.93 12.59 13.25 13.91 14.58 15.24 15.91 16.58 17.26 17.93 18.61 19.29 19.97 20.65 21.34 22.03 22.71

194.16 196.63 198.97 201.18 203.27 205.00 205.25 207.13 208.92 210.63 212.26 213.82 215.32 216.75 218.14 219.47 220.75 221.99 223.20 224.36 225.48 226.58 227.64 228.67 229.68 230.66 231.61 232.54 233.45 234.33 235.20 236.05 236.88 237.69 238.48 239.26 240.02 240.77 241.51 242.23 242.94 243.63

25.35 26.41 27.25 27.93 28.48 28.91 28.96 29.36 29.71 30.02 30.30 30.56 30.79 31.00 31.20 31.39 31.56 31.73 31.89 32.04 32.19 32.32 32.46 32.59 32.72 32.84 32.96 33.07 33.19 33.30 33.41 33.52 33.62 33.72 33.82 33.92 34.02 34.12 34.21 34.30 34.40 34.49

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APPENDIX B: THERMODYNAMIC DATA

TABLE B.3. H2 O(l) Thermodynamic Data T (K) 273 280 298.15 300 320 340 360 373

̂ G(T) (kJ/mol) −305.01 −305.46 −306.69 −306.82 −308.27 −309.82 −311.46 −312.58

̂ H(T) (kJ/mol) −287.73 −287.20 −285.83 −285.69 −284.18 −282.68 −281.17 −280.18

̂ S(T) (J/mol ⋅ K)

Cp (T) (J/mol ⋅ K)

63.28 65.21 69.95 70.42 75.28 79.85 84.16 86.85

76.10 75.81 75.37 75.35 75.27 75.41 75.72 75.99

APPENDIX B: THERMODYNAMIC DATA

TABLE B.4. H2 O(g) Thermodynamic Data T (K) 280 298.15 300 320 340 360 380 400 420 440 460 480 500 520 540 560 580 600 620 640 660 680 700 720 740 760 780 800 820 840 860 880 900 920 940 960 980 1000

̂ G(T) (kJ/mol) −294.72 −298.13 −298.48 −302.28 −306.13 −310.01 −313.94 −317.89 −321.89 −325.91 −329.97 −334.06 −338.17 −342.32 −346.49 −350.69 −354.91 −359.16 −363.43 −367.73 −372.05 −376.39 −380.76 −385.14 −389.55 −393.97 −398.42 −402.89 −407.37 −411.88 −416.40 −420.94 −425.51 −430.08 −434.68 −439.29 −443.92 −448.57

̂ H(T) (kJ/mol) −242.44 −241.83 −241.77 −241.09 −240.42 −239.74 −239.06 −238.38 −237.69 −237.00 −236.31 −235.61 −234.91 −234.20 −233.49 −232.77 −232.05 −231.33 −230.60 −229.87 −229.13 −228.39 −227.64 −226.89 −226.13 −225.37 −224.60 −223.83 −223.05 −222.27 −221.48 −220.69 −219.89 −219.09 −218.28 −217.47 −216.65 −215.83

̂ S(T) (J/mol ⋅ K)

Cp (T) (J/mol ⋅ K)

186.73 188.84 189.04 191.21 193.26 195.20 197.04 198.79 200.47 202.07 203.61 205.10 206.53 207.92 209.26 210.56 211.82 213.05 214.25 215.41 216.54 217.65 218.74 219.80 220.83 221.85 222.85 223.83 224.78 225.73 226.65 227.56 228.46 229.34 230.21 231.07 231.91 232.74

33.53 33.59 33.60 33.69 33.81 33.95 34.10 34.26 34.44 34.62 34.81 35.01 35.22 35.43 35.65 35.87 36.09 36.32 36.55 36.78 37.02 37.26 37.50 37.75 37.99 38.24 38.49 38.74 38.99 39.24 39.49 39.74 40.00 40.25 40.51 40.76 41.01 41.27

523

524

APPENDIX B: THERMODYNAMIC DATA

TABLE B.5. CO(g) Thermodynamic Data T (K) 200 220 240 260 280 298.15 300 320 340 360 380 400 420 440 460 480 500 520 540 560 580 600 620 640 660 680 700 720 740 760 780 800 820 840 860 880 900 920 940 960 980 1000

̂ G(T)(kJ/mol) −150.60 −154.34 −158.14 −161.99 −165.89 −169.46 −169.83 −173.80 −177.81 −181.86 −185.94 −190.05 −194.19 −198.36 −202.55 −206.77 −211.01 −215.28 −219.57 −223.89 −228.22 −232.58 −236.95 −241.35 −245.77 −250.20 −254.65 −259.12 −263.61 −268.12 −272.64 −277.17 −281.73 −286.30 −290.88 −295.48 −300.09 −304.72 −309.37 −314.02 −318.69 −323.38

̂ H(T) (kJ/mol) −113.42 −112.82 −112.23 −111.64 −111.06 −110.53 −110.47 −109.89 −109.31 −108.72 −108.14 −107.56 −106.97 −106.38 −105.79 −105.20 −104.60 −104.00 −103.40 −102.80 −102.19 −101.59 −100.98 −100.36 −99.75 −99.13 −98.50 −97.88 −97.25 −96.62 −95.99 −95.35 −94.71 −94.07 −93.43 −92.78 −92.13 −91.48 −90.82 −90.17 −89.51 −88.84

̂ S(T) (J/mol ⋅ K)

Cp (T) (J/mol ⋅ K)

185.87 188.73 191.31 193.66 195.83 197.66 197.84 199.72 201.49 203.16 204.73 206.24 207.67 209.04 210.35 211.61 212.83 214.00 215.13 216.23 217.29 218.32 219.32 220.29 221.24 222.17 223.07 223.95 224.81 225.65 226.47 227.28 228.07 228.84 229.60 230.34 231.07 231.79 232.49 233.18 233.86 234.53

30.20 29.78 29.50 29.32 29.20 29.15 29.15 29.13 29.14 29.17 29.23 29.30 29.39 29.48 29.59 29.70 29.82 29.94 30.07 30.20 30.34 30.47 30.61 30.75 30.89 31.03 31.17 31.31 31.46 31.60 31.74 31.88 32.01 32.15 32.29 32.42 32.55 32.68 32.81 32.94 33.06 33.18

APPENDIX B: THERMODYNAMIC DATA

TABLE B.6. CO2(g) Thermodynamic Data T (K) 200 220 240 260 280 298.15 300 320 340 360 380 400 420 440 460 480 500 520 540 560 580 600 620 640 660 680 700 720 740 760 780 800 820 840 860 880 900 920 940 960 980 1000

̂ G(T) (kJ/mol) −436.93 −440.95 −445.04 −449.19 −453.39 −457.25 −457.65 −461.95 −466.31 −470.71 −475.15 −479.63 −484.16 −488.73 −493.33 −497.98 −502.66 −507.37 −512.12 −516.91 −521.72 −526.59 −531.46 −536.37 −541.31 −546.28 −551.29 −556.31 −561.37 −566.45 −571.56 −576.71 −581.86 −587.05 −592.26 −597.50 −602.76 −608.05 −613.35 −618.68 −624.04 −629.41

̂ H(T) (kJ/mol) −396.90 −396.25 −395.59 −394.89 −394.18 −393.51 −393.44 −392.69 −391.92 −391.13 −390.33 −389.51 −388.67 −387.83 −386.96 −386.09 −385.20 −384.31 −383.40 −382.48 −381.54 −380.60 −379.65 −378.69 −377.72 −376.74 −375.76 −374.76 −373.76 −372.75 −371.73 −370.70 −369.67 −368.63 −367.59 −366.54 −365.48 −364.42 −363.35 −362.27 −361.19 −360.11

̂ S(T) (J/mol ⋅ K)

Cp (T) (J/mol ⋅ K)

200.10 203.16 206.07 208.84 211.48 213.79 214.02 216.45 218.79 221.04 223.21 225.31 227.35 229.32 231.23 233.09 234.90 236.66 238.38 240.05 241.69 243.28 244.84 246.37 247.86 249.32 250.75 252.15 253.53 254.88 256.20 257.50 258.77 260.02 261.25 262.46 263.65 264.82 265.97 267.10 268.21 269.30

31.33 32.77 34.04 35.19 36.24 37.13 37.22 38.13 39.00 39.81 40.59 41.34 42.05 42.73 43.38 44.01 44.61 45.20 45.76 46.30 46.82 47.32 47.80 48.27 48.72 49.15 49.57 49.97 50.36 50.73 51.09 51.44 51.78 52.10 52.41 52.71 53.00 53.28 53.55 53.81 54.06 54.30

525

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APPENDIX B: THERMODYNAMIC DATA

TABLE B.7. CH4(g) Thermodynamic Data T (K) 200 220 240 260 280 298.15 300 320 340 360 380 400 420 440 460 480 500 520 540 560 580 600 620 640 660 680 700 720 740 760 780 800 820 840 860 880 900 920 940 960 980 1000

̂ G(T)(kJ/mol) −112.69 −116.17 −119.71 −123.32 −126.97 −130.33 −130.68 −134.43 −138.23 −142.07 −145.95 −149.88 −153.85 −157.86 −161.90 −165.99 −170.11 −174.27 −178.46 −182.69 −186.96 −191.26 −195.60 −199.97 −204.38 −208.82 −213.29 −217.79 −222.33 −226.90 −231.51 −236.14 −240.81 −245.50 −250.23 −254.99 −259.78 −264.60 −269.45 −274.33 −279.23 −284.17

̂ H(T) (kJ/mol)

̂ S(T) (J/mol ⋅ K)

Cp (T) (J/mol ⋅ K)

−78.25 −77.53 −76.83 −76.14 −75.44 −74.80 −74.73 −74.01 −73.27 −72.52 −71.74 −70.94 −70.12 −69.27 −68.41 −67.51 −66.60 −65.66 −64.70 −63.71 −62.70 −61.67 −60.61 −59.53 −58.43 −57.31 −56.16 −55.00 −53.81 −52.60 −51.37 −50.13 −48.86 −47.57 −46.26 −44.94 −43.60 −42.23 −40.86 −39.46 −38.05 −36.62

172.23 175.63 178.67 181.45 184.03 186.25 186.48 188.80 191.04 193.20 195.31 197.35 199.36 201.32 203.25 205.15 207.01 208.86 210.67 212.47 214.24 215.99 217.72 219.43 221.13 222.80 224.46 226.10 227.73 229.34 230.94 232.52 234.08 235.64 237.17 238.70 240.20 241.70 243.18 244.65 246.11 247.55

36.30 35.19 34.74 34.77 35.12 35.65 35.71 36.47 37.36 38.35 39.40 40.50 41.64 42.80 43.98 45.16 46.35 47.54 48.73 49.90 51.07 52.23 53.37 54.50 55.61 56.71 57.79 58.85 59.90 60.93 61.94 62.93 63.90 64.85 65.79 66.70 67.60 68.47 69.33 70.17 70.99 71.79

APPENDIX B: THERMODYNAMIC DATA

TABLE B.8. N2(g) Thermodynamic Data T (K) 200 220 240 260 280 298.15 300 320 340 360 380 400 420 440 460 480 500 520 540 560 580 600 620 640 660 680 700 720 740 760 780 800 820 840 860 880 900 920 940 960 980 1000

̂ G(T) (kJ/mol) −38.85 −42.48 −46.16 −49.89 −53.66 −57.11 −57.48 −61.33 −65.22 −69.15 −73.10 −77.09 −81.11 −85.15 −89.22 −93.32 −97.44 −101.59 −105.76 −109.95 −114.16 −118.40 −122.65 −126.92 −131.22 −135.53 −139.86 −144.21 −148.57 −152.96 −157.35 −161.77 −166.20 −170.64 −175.11 −179.58 −184.07 −188.58 −193.10 −197.63 −202.17 −206.73

̂ H(T) (kJ/mol)

̂ S(T) (J/mol ⋅ K)

Cp (T) (J/mol ⋅ K)

−2.83 −2.26 −1.68 −1.11 −0.53 0.00 0.04 0.62 1.20 1.78 2.37 2.95 3.54 4.13 4.72 5.32 5.92 6.51 7.12 7.72 8.33 8.93 9.54 10.16 10.77 11.39 12.01 12.63 13.25 13.88 14.51 15.14 15.77 16.40 17.04 17.68 18.32 18.96 19.61 20.25 20.90 21.55

180.08 182.82 185.31 187.61 189.75 191.56 191.74 193.60 195.36 197.02 198.60 200.11 201.54 202.91 204.23 205.50 206.71 207.89 209.02 210.12 211.19 212.22 213.22 214.19 215.14 216.06 216.96 217.83 218.69 219.52 220.34 221.13 221.91 222.68 223.43 224.16 224.88 225.58 226.28 226.96 227.63 228.28

28.77 28.72 28.72 28.76 28.81 28.87 28.88 28.96 29.05 29.14 29.25 29.35 29.46 29.57 29.68 29.79 29.91 30.02 30.13 30.24 30.36 30.47 30.58 30.69 30.80 30.91 31.02 31.13 31.24 31.34 31.45 31.55 31.66 31.76 31.86 31.96 32.06 32.16 32.25 32.35 32.44 32.54

527

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APPENDIX B: THERMODYNAMIC DATA

TABLE B.9. CH3 OH(g) Thermodynamic Data T (K) 280 298.15 300 320 340 360 380 400 420 440 460 480 500 520 540 560 580 600 620 640 660 680 700 720 740 760 780 800 820 840 860 880 900 920 940 960 980 1000

̂ G(T) (kJ/mol) −268.11 −272.44 −272.88 −277.71 −282.60 −287.54 −292.54 −297.59 −302.69 −307.84 −313.04 −318.29 −323.59 −328.93 −334.32 −339.76 −345.25 −350.78 −356.35 −361.97 −367.64 −373.34 −379.09 −384.88 −390.72 −396.59 −402.51 −408.46 −414.46 −420.50 −426.57 −432.68 −438.84 −445.02 −451.25 −457.51 −463.81 −470.15

̂ H(T) (kJ/mol) −201.73 −200.94 −200.86 −199.96 −199.04 −198.09 −197.11 −196.09 −195.05 −193.97 −192.86 −191.72 −190.55 −189.34 −188.10 −186.83 −185.53 −184.20 −182.84 −181.45 −180.03 −178.59 −177.12 −175.62 −174.10 −172.56 −170.99 −169.41 −167.80 −166.18 −164.53 −162.87 −161.18 −159.48 −157.76 −156.01 −154.25 −152.47

̂ S(T) (J/mol ⋅ K)

Cp (T) (J/mol ⋅ K)

237.08 239.81 240.08 242.97 245.77 248.49 251.14 253.74 256.28 258.79 261.25 263.68 266.08 268.44 270.78 273.09 275.37 277.63 279.86 282.07 284.25 286.41 288.54 290.64 292.72 294.78 296.81 298.82 300.80 302.76 304.70 306.61 308.50 310.38 312.23 314.06 315.88 317.68

42.95 44.04 44.15 45.46 46.85 48.31 49.83 51.40 53.00 54.62 56.26 57.90 59.53 61.14 62.74 64.30 65.84 67.33 68.79 70.20 71.56 72.88 74.15 75.37 76.54 77.67 78.76 79.81 80.82 81.81 82.78 83.73 84.68 85.63 86.59 87.58 88.59 89.66

̂ S(T) (J/mol ⋅ K)

Cp (T) (J/mol ⋅ K)

127.19 127.28 150.75

81.59 81.59 81.59

TABLE B.10. CH3 OH(l) Thermodynamic Data T (K) 298.15 300 400

̂ G(T) (kJ/mol) −276.37 −276.61 −290.56

̂ H(T) (kJ/mol) −238.5 −238.42 −230.26

APPENDIX C

STANDARD ELECTRODE POTENTIALS AT 25∘C

E0

Electrochemical Half Reaction Li+ + e− 2H2 O + 2e− Fe2+ + 2e− CO2 + 2H+ + 2e− 2H+ + 2e− CO2 + 6H+ + 6e− 1 O + H2 O + 2e− 2 2 O2 + 4H+ + 4e− H2 O2 + 2H+ + 2e− O3 + 2H+ + 2e− F2 + 2e−

→ Li → H2 + 2OH− → Fe → CHOOH(aq) → H2 → CH3 OH + H2 O → 2OH− → 2H2 O → 2H2 O → O2 + H2 O → 2F−

−3.04 −0.83 −0.440 −0.196 +0.00 +0.03 +0.40 +1.23 +1.78 +2.07 +2.87

529

APPENDIX D

QUANTUM MECHANICS

A number of key discoveries in the early part of the twentieth century led to the foundation of modern quantum mechanics. We will highlight some of these discoveries and describe a few of the underlying assumptions in mostly qualitative terms. Readers are encouraged to broaden their knowledge in this area by studying relevant quantum mechanics and chemistry texts [161,162]. Before the emergence of modern quantum mechanics, Bohr [163], an early pioneer in atom physics, proposed in 1913 a model for the hydrogen atom in which the electron encircles the nucleus in only one of a number of allowed orbits. He assumed that the energy of the electron is quantized and that the change in energy of the electron, associated with transitioning from one orbit to the other, is accompanied by the absorption or emission of discrete light quanta. The Bohr model was able to predict the radius of the hydrogen atom quite accurately as 0.529 × 10−10 m. Nevertheless, Bohr’s model is fundamentally based on Newtonian mechanics for which the quantization of energy levels does not occur naturally. About a decade later, de Broglie [164] was the first to propose that electrons have both a particle and a wave nature. The electron diffraction experiments in atomic crystal structures of Davisson and Germer [165] in 1928 confirmed de Broglie’s view that electrons may be indeed assigned a wavelength. Schrödinger was able create the formalism of modern quantum mechanics by combining the wave nature of electrons following de Broglie and their quantized energy states in hydrogen according to Bohr. In 1926 Schrödinger [166] wrote in the journal Annalen der Physik1 :

1 Translation

from German appears in Ref. [167].

531

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APPENDIX D: QUANTUM MECHANICS

The usual rule of quantization can be replaced by another postulate, in which there occurs no mention of whole numbers. Instead, the introduction of integers arises in the same natural way as, for example, in a vibrating string, for which the number of nodes is integral. The new conception can be generalized, and I believe that it penetrates deeply into the true nature of quantum rules.

In vibrating strings with fixed ends, the location of nodes does not change over time. More importantly, the number of nodes in vibrating strings with stationary ends can only be changed in discrete steps, that is, integer numbers (1, 2, 3, … , n). In other words, one cannot add a portion of a wave to a vibrating string with given length and fixed ends; only whole waves can be added. In analogy to string waves, quantum mechanics assumes that matter can be described with wave functions of amplitude 𝛙 (t, x, y, z). These are “material” waves rather than electromagnetic waves. The wave function 𝜓 cannot be directly observed or measured. But one can measure |𝜓 (t, x, y, z)|2 , which corresponds to the probability of finding the particle in (t, x, y, z), that is, the density of the material at a specific location and time. It is important to realize that quantum mechanics is based on a number of postulates, such as: there exists a wave function that contains all possible information about the system considered. (A more detailed description of the postulates is given later in this appendix.) Postulates or axioms are underlying assumptions that cannot be further explained and cannot be further questioned. Their justification stems from the practicality of their results. The wave function cannot be measured; however, the absolute square can be. If experimental results are consistent with the assumptions of the theory, the theory is considered valid, at least until proven wrong. Let’s ask how one can calculate 𝜓 (t, x, y, z) for a given atomic structure. One of the postulates of quantum mechanics is that 𝜓 (t, x, y, z) can be obtained by solving the Schrödinger equation. The Schrödinger equation describes the evolution of a particle (wave function) over time. In classical mechanics, the time evolution of any particle system is described by its kinetic and potential energy. Similarly, the Schrödinger equation involves the kinetic and potential energy of the particles involved. In fact, it is a further postulate in quantum mechanics that the kinetic and potential energy in the Schrödinger equation are similar to that of the particles in classical mechanics. For the present considerations, we are interested in stationary waves only. In stationary waves, the nodes do not change as a function time; stationary waves depend on spatial coordinates only. We define the amplitudes of stationary waves as 𝜓 (x, y, z)—this is the so-called time-independent wave function. The time-independent wave function is useful for examining electrons with stationary boundaries such as an electron in a box, or an electron wrapped around a positively charged nucleus, or electrons in an array of positively charged atoms, as found in any crystal structure. In solving the time-independent part of the Schrödinger equation, all terms dependent on time are constant, like the nodes in stationary waves. If we take the absolute square of the stationary, or time-independent, solutions of the Schrödinger equation, we obtain a picture of the location and shape of the particles (in our case the electrons) and how they rearrange during different stages of a chemical reaction. Following decades of research in quantum mechanics and the availability of modern numerical methods, a broad community of scientists and engineers is now able to study and

APPENDIX D: QUANTUM MECHANICS

visualize electron densities, quantify chemical bond formation, charge transfer reactions, and diffusion phenomena. For example, the quantum simulation figures in Chapter 3 used a commercially available tool called Gaussian,2 which is capable of determining the electron density and the minimum energy of the quantum system considered. Gaussian is based on density functional theory (DFT). Kohn [168], a pioneer of the DFT method, helped initiate a revolution that made quantum mechanical tools available for routine research in chemistry, electrochemistry, and physics.

D.1 ATOMIC ORBITALS Using Gaussian we can illustrate the shape of an electron by considering the simplest atom there is: the hydrogen atom. Figure D.1a shows the hydrogen atom from Bohr’s perspective, a proton being encircled by an electron; Figure D.1b describes the same atom by plotting |𝜓 2 |, the proton surrounded by a stationary electron cloud, spherically symmetric but with varying electron density along the radius r. It just so happens that the radius of the electron orbit in the Bohr model turns out to be the same as the location of maximum electron density calculated by the time-independent Schrödinger equation. The space in which the electron may reside is called the orbital. The more electrons there are in an atom, the more orbitals exist. Orbital geometry is not easy to visualize. We can comfortably imagine stationary waves of a string since deflections occur in one dimension. We can also imagine that in a string with fixed ends the number of waves can be increased in incremental steps of whole numbers only (compare above remarks by Schrödinger). Yet, we have a hard time imagining 3D waves, especially 3D waves of higher order, interacting with electrically charged nuclei. Computer tools such as Gaussian help in visualizing the complexity of 3D orbitals. Analogies to mechanical scenarios such as the buckling of a column also help our intuition. In fact, the 1D Schrödinger equation of a 1D particle in a box is identical to the differential equation leading to the calculation of the Euler buckling load. Engineers know there is a first-, second-, and higher-order buckling load. Due to the 3D nature of orbitals,

e-

high Probability of finding electron on controur line

r p+

low

(a)

(b)

(c)

Figure D.1. (a) Electron circling proton according to Bohr, (b) stationary electron density (1s) around proton, and (c) (2p) electrons in oxygen. Note that (b) and (c) are not drawn to the same scale. 2 Gaussian is a computational tool predicting energies, molecular structures, and vibrational frequencies of molecular systems by Gaussian Inc.

533

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APPENDIX D: QUANTUM MECHANICS

not only one quantum number n (as in buckling) exists to describe the possible states of an electron in an atom. Instead, there are several quantum numbers describing the possible solutions of the Schrödinger equation and their respective energy levels. The quantum numbers commonly used in the solution for the Schrödinger equation are called n (principal quantum number), l (angular momentum number), and m (quantum number of z component of angular momentum). The following relations hold between the integers l, m, and ∑ n∶ 0 ≤ l ≤ n − 1, −l ≤ m ≤ l, and for a given n there are (2l + 1) = n2 different states which happen to have the same energy. Two electrons (one with spin up, the other with spin down) may occupy the same set of quantum numbers (n, m, l). It is helpful to make the link to the popular notation for electrons in most periodic systems: s, p, d, f . Historically, this notation came from the optical spectroscopy literature and means s (sharp), p (principal), d (diffuse), and f (fundamental). The orbital s corresponds to l = 0, p to l = 1, d to l = 2, and f to l = 3. Optical spectroscopy led to the first observations that electrons reside in discrete orbital states around the nucleus and was crucial for the earlier mentioned atomic hydrogen model of Bohr. Figure D.1b shows the 1s electrons in hydrogen, and Figure D.1c illustrates 2p electrons in oxygen. Overlapping of orbitals between different atoms causes the formation of chemical bonds. Examples are hydrogen (H–H) or oxygen (O–O) or the formation of bonds between a catalyst and H2 or O2 . Needless to say, these “molecular orbitals” may be quite complex. Only numerical tools can provide quantitative insight into the strength of chemical bonds.

D.2

POSTULATES OF QUANTUM MECHANICS

The postulates, also referred to as axioms, of quantum mechanics were articulated by generations of physicists after Schrödinger’s initial paper. Postulates or axioms are assumptions that cannot be further explained. They should be accepted as stated since they were shown to be useful and practical, but they sure sound abstract and are not necessarily intuitive. However, they allow for the derivation of results that can be experimentally verified. In that sense, they can be indirectly checked for their truth and practicality. 1. The first axiom in quantum mechanics says that there exists a wave function 𝜓 depending on time and space that contains all possible information about the system considered. In this book we consider the wave functions for electrons only. 2. The wave function 𝜓 has certain mathematical properties: It is differentiable, finite, unique, and continuous. It is also important to realize that 𝜓 is complex, and it can be separated into a product of functions depending on time and space: 𝜓(t, x, y, z) = f (t)𝜓(x, y, z)

(D.1)

3. The wave function 𝜓 cannot be measured. Only the function |𝜓|2 can be observed, and it represents the probability of the particle to be in the location (x, y, z) at time t. For electrons the expression |𝜓|2 is a measure of the electron density that can be observed in a variety of ways. Given the fact that the electron exists somewhere,

APPENDIX D: QUANTUM MECHANICS

it is reasonable to assume that the probability to find it in space is equal to 1. In equation form ∫

|𝜓|2 dV = 1

(D.2)

This property of the wave function is referred to as being normalizable. 4. An operator exists, the so-called Hamiltonian H, which, when applied to the wave function, describes the change of the wave function over time: 𝜕 H𝜓 = −iℏ 𝜓 𝜕

(D.3)

This equation is called the Schrödinger equation, where ℏ = h∕2𝜋 (h = Planck’s constant). For the steady-state case, or the time-independent case, the Schrödinger equation can be reduced to (D.4) H𝜓n = 𝜀n 𝜓n where 𝜀n represents the energy of the system in state n. The 𝜓n are the eigenfunctions of the operator H and 𝜓n the corresponding eigenvalue.3 5. The Hamiltonian H is equivalent to the energy of classical mechanics, that is, H = T + V, kinetic energy plus potential energy. More specifically, the kinetic energy is T = 12 m𝑣2 =

p2 2m

(D.5)

and m is the mass of the electron. The linear momentum p is, in contrast to classical mechanics, now an operator. In one dimension, 𝜕 𝜕x

(D.6)

𝜕 𝜕 𝜕 p = −iℏ p = −iℏ 𝜕x y 𝜕y z 𝜕z

(D.7)

p = −iℏ and for three dimensions. px = −iℏ

For convenience the gradient vector is frequently defined as ( ) 𝜕 𝜕 𝜕 ∇= , , 𝜕x 𝜕y 𝜕z

(D.8)

The potential energy is a function of the three dimensions V = V (x, y, z). One should not attempt to understand these axioms but rather should become familiar with them or, better yet, memorize them. We need to mention that the axiom list as stated above is not quite complete but captures the essence of what we need for the present section. 3 For further information about operators, eigenfunctions, and eigenvalues: http://hyperphysics.phy-astr.gsu .edu/hbase/quantum/eigen.html].

535

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APPENDIX D: QUANTUM MECHANICS

D.3

ONE-DIMENSIONAL ELECTRON GAS

We will illustrate the quantum mechanical axioms by describing the behavior of the simplest system: one “free” electron in a 1D box of length L. Free means that there is no potential acting on the electron. Consequently, the Schrödinger equation for the free electron (only kinetic energy) reads H𝜓 = −

ℏ2 d 2 𝜓 = 𝜀n 𝜓 2m dx2

(D.9)

The “box” of length L means that the wave function of the electrons is constrained at either end of the box. In other words, 𝜓n (0) = 0 𝜓n (L) = 0

(D.10)

A solution to this equation is obviously harmonic in nature. We guess the solution 𝜓n = A sin

(

n𝜋 x L

) (D.11)

To check the guess, we take derivatives with respect to x of Equation D.11, yielding ( ( ) ) d𝜓n n𝜋 n𝜋 cos =A x dx L L

(D.12)

( )2 ( ) d 2 𝜓n n𝜋 n𝜋 = A sin x L L dx2

(D.13)

The resulting levels for the energy are 𝜀n =

( ) ℏ2 n𝜋 2 2m L

(D.14)

The wave functions 𝜓n are referred to as orbitals. The electron can be in any of the n orbitals. The important insight we obtain from this solution is that there are discrete, time-independent “stationary” states in which the electron may reside. The energy levels change incrementally; they are proportional to n2 . Transitions of the electron from one orbital to the other are accompanied by the emission or absorption of light quanta. Clearly, multiple electrons may be in the same box and may reside in available orbitals. Following the Pauli principle, which we accept without further explanation, a maximum of only two electrons may have the same orbital number n. However, the two electrons with the same n will differ in their spin, one is to be spin “up,” the other one “down.” In addition, the presence of multiple electrons in the same system (box) will modify the Hamiltonian in the Schrödinger equation, since the presence of one electron will influence the others in the form of a nonzero potential energy term. The details of this problem go well beyond the introductory nature of this appendix and we refer to other texts [169].

APPENDIX D: QUANTUM MECHANICS

P

x y L

Figure D.2. Pinned column of length l subjected to force p buckles according to discrete modes.

D.4 ANALOGY TO COLUMN BUCKLING Since this book is largely targeted for the engineering audience, we would like to draw attention to an analogy between the Schrödinger equation of the 1D electron gas and the mechanics of a buckling column. Consider a simple column of length L with pinned ends (see Figure D.2) subject to an applied force P. The differential equation describing the bending moment in a column is formally identical to that of the Schrödinger equation of the electron in the box: 𝜕2y EI 2 = −Py (D.15) 𝜕x Where E stands for Young’s modulus, I is the cross-sectional moment of inertia, and y is the lateral deflection of the beam from the neutral position. The boundary conditions for the column and the solution for y are the same as the ones for the wave function; so are the solutions yn (x): ( ) n𝜋 x (D.16) yn (0) = 0 yn (L) = 0 yn = A sin L Interestingly, the discrete levels of energy 𝜓n resulting from the Schrödinger equation can now be interpreted as the discrete loads for column buckling, also called Euler buckling load: ( )2 n𝜋 Pn = EI (D.17) L We know that Euler buckling only happens above a critical threshold load, in analogy to the discrete levels of energy required to move an electron from one shell to another. The mathematical expressions in both cases are the same.

537

538

APPENDIX D: QUANTUM MECHANICS

D.5

HYDROGEN ATOM

The hydrogen atom is the only physical quantum mechanical system for which an analytical solution can be found. It consists of the nucleus, that is, one proton, and one electron surrounding the nucleus. The earlier discussed 1D free-electron gas is hypothetical in nature, but it gives insight into the methodology used below for the hydrogen atom. The solution of the Schrödinger equation for hydrogen is of significant historical importance since it shaped the thinking of generations of physicists. It provides qualitative insights into the behavior of more complex, multielectron systems for which analytical solutions are not available. The Schrödinger equation of hydrogen can be established as follows. We are interested in the position of the electron relative to the proton only. Hence, the motion of the entire atom is unimportant. Establishing the Hamiltonian is the crucial step. The rest is mathematics and algebra. The kinetic energy in quantum mechanics is the square of the momentum divided by the mass (axiom 5). From axiom 5 we also know that the momentum is a differential operator acting on the wave function. For the electron in the box there was no potential energy. For the interaction between the two electrically charged particles, the proton and electron, we know from classical electrostatics that there exists an attractive force of interaction that is inversely proportional to the distance square. Accordingly, the potential energy is inversely proportional to the distance between the particles: e2 (D.18) V(r) = − 4𝜋𝜀0 r with 𝜀0 = 8.854 × 10−12 C∕V ⋅ m. Since e2 ∕4𝜋𝜀0 has the dimension of action times velocity (the units of action are energy times time), we can rewrite this term by incorporating Planck’s constant, which has the dimension of action and the speed of light c. In other words, e2 ∕4𝜋𝜀0 = 𝛼ℏc with 𝛼 ≈ 1∕137. We can now write the Schrödinger equation for hydrogen as ) ( ℏ 𝛼 𝜓 = E𝜓 (D.19) − ∇2 − ℏc 2m r A hydrogen atom is completely spherical; there is no preferred orientation. Therefore, it is convenient to express all functions in spherical coordinates: ( ) ℏ 𝛼 − ∇2 − ℏc 𝜓(r, 𝜃, 𝜑) = E𝜓(r, 𝜃, 𝜑) 2m r

(D.20)

Partial differential equations like this one are frequently solved by a separation “Ansatz”: 𝜓(r, 𝜃, 𝜑) = R(r)Θ(𝜃)Φ(𝜑)

(D.21)

This separation leads to three differential equations. The discrete energy levels En [eigenvalues of R(r)] can be found as 𝛼2 (D.22) En = − 12 Mc2 2 n

APPENDIX D: QUANTUM MECHANICS

Without proof we give the solutions of these three differential equations as [( )3 ]1∕2 (n − l − 1)! 2 2l+1 Rnl (r) = − d−𝜌∕2 𝜌l Ln+l (𝜌) na 2n[(n + l)!]3 [ ]1∕2 (2l + 1) (l − |m|)! Θlm (𝜃) = Pl|m| (cos 𝜃) 2(l + |m|)! 1 Φm (𝜑) = √ eim𝜑 2𝜋

(D.23) (D.24) (D.25)

By solving the three differential equations, one finds that, similar to the case of column buckling, there are discrete solutions or modes that we assign the indices (l, m, n). Accordingly, the stationary solution of the Schrödinger equation is of the form 1 𝜙m (𝜑) = √ eim𝜑 2𝜋

(D.26)

The following polynomial expressions were used: L and P. The so-called Laguerre polynomial used by Schrödinger can be expressed as ∑

n−l−1 2l+1 (𝜌) = Ln+l

(−1)k+1

k=0

[(n + l)!]2 𝜌k (n − l − 1 − k)!(2l + 1 + k)!k!

(D.27)

and the associated Legendre function P is recursively defined as Pl|m| (cos 𝜃) = (1 − cos2 𝜃)|m|∕2

d|m| Pl (cos 𝜃) dz|m|

(D.28)

The Legendre polynomial Pl is given by Pl (x) =

1 dl 2 (x − 1)l 2l l! dxl

(D.29)

Furthermore, we used the notation p = [2∕(na)]r and, more importantly, a=

ℏ2 = 0.5292 × 10−10 m 𝛼me2

(D.30)

which is the radius of the innermost orbital of the electron in hydrogen, which coincides with Bohr’s calculation of the size of the hydrogen atom. We provided the solutions of the Schrödinger equation for hydrogen to give the students a perspective of the mathematical complexity of a relatively simple quantum system. From that sheer complexity, it appears obvious that more comprehensive systems than hydrogen can be solved with numerical means only by using computer tools.

539

540

APPENDIX D: QUANTUM MECHANICS

D.6

MULTIELECTRON SYSTEMS

Understanding the hydrogen atom was a key step in solidifying the foundations of modern quantum mechanics. Scientists could relate mathematical solutions of the Schrödinger equation to observations of the optical spectra of hydrogen gas. Such spectra showed remarkable agreement with the predicted energy difference between discrete electronic states. In particular, energy levels of light absorption spectra could be directly related to the electronic transitions between atomic orbitals. From a fuel cell perspective, modern quantum mechanics is important to understand and select the best catalyst materials for enhancing the hydrogen evolution and the oxygen reduction reaction. In Chapter 3 we provided examples of how a platinum surface consisting of a few atoms may facilitate the splitting of hydrogen and oxygen. For that purpose, we need to gain insight into the quantum mechanics beyond just a single hydrogen atom. At least a qualitative insight into the quantum mechanics of multiple atoms and electrons is needed for designing next-generation catalysts. The Schrödinger equation is valid not only for individual atoms but for ensembles of atoms that may condense in crystalline form. For that purpose we generalize the Hamiltonian beyond a single atom. We need to establish the framework for applying the Schrödinger equation to trillions of atoms. We can do this by generalizing equation D.19. The Schrödinger equation for a multiple atom and electron system reads as follows: ⎡ ∑ h2 ∇2 ∑ h2 ∇2 ∑ Z e2 ∑ 2 ∑ ZA ZB e2 ⎤ e i i A ⎥ Ψ(x, y, z) = EΨ(x, y, z) ⎢− − − + + ⎥ ⎢ i 2mi 2M r r R A i,A ij A,B A i,A i,j>i A,B>A ⎦ ⎣ (D.31) Similar to the Schrödinger equation of the hydrogen, the Hamilton operator acts on a wave function, the multielectron wavefunction, depending on the spatial coordinates x, y, z, with corresponding eigenfunction Ψ(x, y, z), and eigenvalue E, both scalar quantities. Contrary to the hydrogen atom, crystal structures are not spherically symmetric. This is the reason why the multielectron wavefunction is described in terms of Cartesian coordinates. The first two terms in D.31 describe the kinetic energy of the electrons and the nuclei, respectively. The third term accounts for the attraction between electrons and nuclei, the fourth term is representative of the electron–electron interaction, and the last term accounts for the repulsion between nuclei. The indices A and B stand for the number of nuclei that carry charge, i.e., protons. The indices i and j represent the electrons. We must rely on substantial simplifications to solve this equation as any crystal, consisting of trillions of atoms, cannot be treated, not even with the most powerful computers today and in the foreseeable future.

D.7

DENSITY FUNCTIONAL THEORY

A very important simplification of solving the Schrödinger equation can be accomplished by expressing the Schrödinger equation in terms of electron density rather than wave functions. The relation between wave function and electron density is mentioned in D.2.

APPENDIX D: QUANTUM MECHANICS

The electron density is the product of the wave function and its complex conjugate, integrated over the crystal volume. More importantly, the entire kinetic and potential energy of electrons and nuclei may be expressed as a function of electron density. Next, the electron density may be varied until the total energy is a minimum, resulting in the equilibrium charge distribution. This is a difficult task as the electron density is an unknown function. To overcome this challenge Hohenberg and W. Kohn [169a] used a functional rather than an explicit function for the electron density. A functional is a function of functions. Functional variables are changed numerically until a global minimum of the electronic and nucleic energy is obtained. This method is referred to as density functional theory (DFT). DFT algorithms were shown to have lower computational complexity compared to wave-function-based methods. Complexity describes how the number of atoms scales with computational effort. Doubling the size of an atomic system may require eight times the computational resources if the order of complexity is n3 . DFT algorithms typically scale with the cube power of the number of atoms while wave-based methods scale with a higher power, relative to DFT. A practical study of catalytic reactions requires at least tens, if not hundreds, of atoms. Computer clusters available at the beginning of the twenty-first century may take days or longer to solve the Schrödinger equation for hundreds or more atoms. Beyond the inherent computational challenge of any quantum mechanical calculation, a key limitation of DFT methods is that they can only determine minimum energy states. The electron density distribution with minimum energy is only representative of the lowest energy state, also called the ground state. Therefore, excited states, including semiconductor bandgaps, cannot be determined with any degree of confidence when using DFT methods. However, for fuel cells, DFT methods have been shown to be highly effective in exploring catalyst alternatives for both anode and cathode reactions and this is currently an active area of research and development. In the future, improved catalyst materials and catalyst structures are likely to be inspired by the use of DFT calculations. Beyond catalysis, DFT has also shown to be effective in estimating ion diffusivity in crystal structures by calculating the energy barriers or so-called saddle points that ions must overcome as they make the transition from one lattice position to an adjacent one. In addition to adopting DFT methods, further simplifications can be accomplished in determining electronic charge distributions. Considering that atomic crystal structures are inherently periodic, we can restrict quantum computations to a small or even a smallest repetitive unit of the crystal structure. One needs to assure that certain continuity constraints are met at all interfaces between the unit cell and its adjacent counterparts. For example, the wavefunction must be continuous across such interfaces. With interfacial constraints across boundaries, calculations may be reduced to a few atoms, yet the results of such computations can deliver properties that are representative of the bulk behavior. We may apply periodic boundary conditions for 2D structures or for 2D slabs. The latter is important for catalysis. On the surface of a slab, for example, one can study the adhesion strength between reactants and products, thereby evaluating and comparing catalytic performance of one catalytic material versus another.

541

APPENDIX E

PERIODIC TABLE OF THE ELEMENTS KEY

MAIN-GROUP ELEMENTS

Atomic number Symbol Element Atomic mass

20

Ca IA(1)

Calcium 40.078

1

1

2

Period

3

H 1.00794 Hydrogen

IIA(2)

3

4

Li

Be

6.941 Lithium 11

9.01218 Beryllium 12

Na

Mg

4

5

6

7

K Potassium 39.0983

VIII(18) 2

He IIIA(13) IVA(14) VA(15) VIA(16) VIIA(17)

TRANSITION ELEMENTS

IIIB(3) IVB(4)

20

21

Ca

Sc

VB(5) VIB(6) VIIB(7)

22

Ti

Calcium Scandium 40.078 44.95591

Titanium 47.867

5

6

7

8

9

10

C

N

O

F

Ne Neon 20.1797

Boron 10.811

Carbon 12.0107

Nitrogen 14.00674

Oxygen 15.9994

Fluorine 18.99840

(VIII)

13

14

15

16

17

18

Si

P

S

Cl

Ar

Aluminum 26.98153

Chlorine 35.4527

Argon 39.948

IB(11) IIB(12)

(8)

(9)

(10)

23

24

25

26

27

28

V

Cr

Mn

Fe

Co

Ni

Cu

Zn

Iron 55.845

Cobalt 58.93320

Nickel 58.6934

Copper 63.546

Zinc 65.39

Vanadium Chromium Manganese 50.9415 51.9961 54.93809

29

37

38

39

40

41

42

43

44

45

46

47

Rb

Sr

Y

Zr

Nb

Mo

Tc

Ru

Rh

Pd

Ag

Rubidium Strontium Yttrium Zirconium 85.4678 87.62 88.90585 91.224

Helium 4.002602

B

Al

Sodium Magnesium 22.98977 24.3050 19

MAIN-GROUP ELEMENTS

Metals (main group) Metals (transition) Metals (inner transition) Nonmetals Metalloids

30

31

32

33

Ga

Ge

As

34

Gallium Germanium Arsenic 69.723 72.61 74.92160

48

Niobium MolybenumTechnetium Ruthenium Rhodium Palladium Silver 95.94 92.90638 (98) 101.07 102.9055 106.42 107.8682

Silicon Phosphorus Sulfur 28.0855 30.973761 32.066

49

50

Cd

In

Sn

Sb

Cadmium 112.411

Indium 114.818

Tin 118.710

Antimony 121.760

35

36

Se

Br

Kr

Selenium 78.96

Bromine 79.904

Krypton 83.80

52

53

Te

I

51

Tellurium Iodine 127.60 126.90447

54

Xe Xenon 131.29

55

56

57

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

Cs

Ba

La

Hf

Ta

W

Re

Os

Ir

Pt

Au

Hg

Tl

Pb

Bi

Po

At

Rn

Cesium 132.9054

Barium 137.327

Lanthanum 138.9055

Hafnium 178.49

Tantalum 180.9479

Osmium 190.23

Iridium 192.217

Thallium 204.3833

Lead 207.2

Astatine (210)

Radon (222)

87

88

89

104

105

106

107

108

109

Fr

Ra

Ac

Unq

Unp

Unh

Uns

Uno

Une

Tungsten Rhenium 183.84 186.207

Mercury Platinum Gold 195.078 196.96655 200.59

Bismuth Polonium (209) 208.98038

Francium Radium Actinium Unnilquadium Unnilpentium Unnilhexium Unnilseptium Unniloctium Unnilennium (265) (265) [223.0197] [226.0254] [227.0278] [261.11] [262.114] [263.118] [262.12]

INNER TRANSITION ELEMENTS

6 Lathanides

7 Actinides

58

59

60

61

62

63

64

65

66

67

68

69

70

71

Ce

Pr

Nd

Pm

Sm

Eu

Gd

Tb

Dy

Ho

Er

Tm

Yb

Lu

Cerium 140.116

Praseodymium

90

91

92

93

94

95

96

97

98

99

100

101

102

103

Th

Pa

U

Np

Pu

Am

Cm

Bk

Cf

Es

Fm

Md

No

Lr

140.90765

Neodymium Promethium Samarium Europium Gadolinium Terbium Dysprosium Holmium 150.36 151.964 144.24 (145) 157.25 158.92534 162.50 164.93032

Thorium Protactinium Uranium Neptunium Plutonium Americium (244) [243.0614] 232.0381 231.03588 238.0289 [237.0482]

Curium (247)

Erbium 167.26

Thulium Ytterbium 168.93421 173.04

Lutetium 174.967

Berkelium Californium Einsteinium Fermium Mendelevium Nobelium Lawrencium (251) [252.083] [257.0951] (258) (247) 259.1009 [262.11]

Figure E.1. Periodic table of the elements. 543

APPENDIX F

SUGGESTED FURTHER READING

The following references are suggested for further reading on the subject of fuel cells or electrochemistry (please see the bibliography for the detailed citations): Fuel Cells: • • • •

Fuel Cell Handbook [170] Fuel Cell Systems Explained [171] Handbook of Fuel Cell Technology [5] Springer Model of the PEMFC [8]

Electrochemistry: • Electrochemical Methods [7] • Electrochemistry [172] Other: • • • •

Basic Research Needs for the Hydrogen Economy [173] Transport Phenomena [12] Flow and Transport in Porous Formations [174] CFD Research Corporation User Manual [138]

545

APPENDIX G

IMPORTANT EQUATIONS Thermodynamics dU = dQ − dW = dQ − p dV dS = k ln Ω =

dQ T

H = U + pV G = H − TS ΔG = ΔH − TΔS (isothermal process) ΔG = −nFE 𝜇 = 𝜇 0 + RT ln a

∏ Vi aprod ΔS RT E=E + (T − T0 ) − ln ∏ V i nF nF areact 0

𝜀real = 𝜀thermo 𝜀voltage 𝜀fuel ΔG ΔH V = E i∕nF = 𝑣fuel

𝜀thermo, fc = 𝜀voltage 𝜀fuel

𝜀thermo,electrolyzer =

ΔH ΔG

547

548

APPENDIX G: IMPORTANT EQUATIONS

Reaction Kinetics + +

j0 = nFC∗ f e−ΔG ∕(RT) ( ∗ ) CR 𝛼nF𝜂∕(RT) CP∗ −(1−𝛼)nF𝜂∕(RT) 0 j = j0 e − 0∗ e CR0∗ CP nF𝜂act RT j RT = ln 𝛼nF j0

j = j0 𝜂act

(small overpotential∕current) (large overpotential∕current)

Charge Transport L 𝜎 L = 𝜎

𝜂ohmic = j(ASRohmic ) = j ASRohmic = Afuelcell Rohmic 𝜎 = |z|Fcu u=

|z|FD RT

D = D0 e−ΔG∕(RT)

Mass Transport jL = nFDeff 𝜂conc

c0R

𝛿 j j RT = ln L = c ln L 𝛼nF jL − j jL − j

Modeling V = Ethermo − 𝜂act − 𝜂ohmic − 𝜂conc V = Ethermo − [aA + bA ln(j + jleak )] − [aC + bC ln(j + jleak )] ( − (jASRohmic ) −

c ln

jL

( ) jL − j + jleak

)

APPENDIX G: IMPORTANT EQUATIONS

Characterization ZΩ = RΩ ZC =

1 j𝜔C

Zseries = Z1 + Z2 −1 Zparallel = Z1−1 + Z2−1

Zinfinite

Zfinite

Warburg

Warburg

𝜎 = √1 (1 − j) 𝜔

( √ ) 𝜎1 j𝜔 = √ (1 − j) tanh 𝛿 Di 𝜔

Ac =

Qh Qm ∗ Ageometric

Systems stored enthalpy of fuel system mass stored enthalpy of fuel Volumetric energy storage density = system volume

Gravimetric energy storage density =

Carrier system effectiveness =

% conversion of carrier to electricity % conversion of neat H2 to electricity

Fuel Cell Systems 𝜀0 = 𝜀 R + 𝜀 H 𝜀R = 𝜀FP × 𝜀R, SUB × 𝜀R, PE = dḢ H = P Pe, SYS nH yH2 = 2 n nH2 O S = C nC

ΔḢ (HHV), H2

ΔḢ (HHV), fuel

×

Pe, SUB Pe, SYS × Pe, SUB ΔḢ (HHV), H 2

549

550

APPENDIX G: IMPORTANT EQUATIONS

Environmental Impact

[ ] ( ) Q − W = ṁ h2 − h1 + g z2 − z1 + 12 (V22 − V12 ) CO2 = mCO2 + 23mCH4 + 296mN2 O + 𝛼(mOM2.5 + mBC2.5 ) − 𝛽[mSULF2.5 + mNIT2.5 + 0.40mNOX + 0.05mVOC ]

APPENDIX H

ANSWERS TO SELECTED CHAPTER EXERCISES

Chapter 1 1.7 –241 kJ/mol 1.8 386 L and 229 kg 1.10 (d)

Chapter 2 2.3 2.7 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16

Cannot determine Yes, 𝜀 can be greater than 1 if ΔS< 0. Consider ΔG, ΔH, and ΔS for a C/CO fuel cell T = 1010 K = 747∘ C (b) T2 = 254 K PH2 = 5.4×10–42 atm 𝜀 = 0.46 (46%) (c) Ph = 942 W (b) (c) (b)

551

552

APPENDIX H: ANSWERS TO SELECTED CHAPTER EXERCISES

Chapter 3 3.11 Reaction A has a higher reaction rate 3.12 (a) 2.5 W, (b) 5 cells, (c) 18.7 g H2 , (d) 457 cm3 (compressed gas), 37.3 cm3 (hydride), (e) 99 g CH3 OH = 78.2 cm3 CH3 OH 3.13 45.9 kJ/mol 3.14 (b) ΔT = 121 K

Chapter 4 4.2 4.8 4.10 4.12 4.13 4.14 4.15 4.16

Increase 10 nm 75 mV aw = 2.44 nohmic,FCa = 0.176 V, nohmic,FCb = 0.220 V. Anode humidification is more effective (a) 2.46 ×10–7 cm2 /s, (b) 1.1 ×10–3 mol/cm3 , (c) 9.65 ×10-4 (Ω ⋅ cm)–1 ΔGact = 89.4 kJ/mol, D0 = 3.9 ×10-2 cm2 /s (d)

Chapter 5 5.1 5.4 5.6 5.7 5.11 5.12 5.13

Lower (DO2 ∕He is higher than DO2 ∕N2 ) 3.44 A/cm2 272 m/s 34 cm2 (b) (b) True

Chapter 6 6.3 6.4 6.11 6.14 6.15 6.16

29.6 mV (a) 1.19V, (b) ac = 0.196V, bc = 0.0284, (c) 0.1 Ω ⋅ cm2 , (d) 0.022 cm2 /s, (e) 1.33 A/cm2 , (g) 0.77 W/cm2 at 1.1 A/cm2 , (h) 0.418 (41.8%) (c) jL,c = 14.3 A/cm2 , jL,a = 18 A/cm2 (a) 1.0 V, (b) 4 A/cm2 , (d) 1.54 W/cm2 at 2.67 A/cm2 (a) 0.020 cm2 /s, (b) 3.32 A/cm2 , (c) 0.0035 cm2 /s, (d) 0.58 A/cm2 , (e) 0.016 V vs. 0.198 V, (f) 12 times (a) 1.18V, (b) 0.175 V, (c) 8.88 A/cm2 , (d) 0.012 V, (e) 1.20 V, (f) 0.117 V, (g) 88.8 A/cm2 , (h) 0.00113 V

APPENDIX H: ANSWERS TO SELECTED CHAPTER EXERCISES

Chapter 7 7.4 7.8 7.9 7.10 7.11

(a) Higher scan rate will show higher “apparent” fuel cell performance 760 True False 0.22 V

Chapter 8 8.5 20.4 kW 8.6 (a) 65.2%, (b) 72.6% 8.7 (b)

Chapter 9 9.7 9.9 9.12 9.13

5.7 years 𝛿 = nFDeff c0R ∕(2j) (a) 𝜂 ohmic = 5 mV, 𝜂 act = 360 mV, (b) 𝜂 ohmic = 7 mV, 𝜂 act = 240 mV 4 nm

Chapter 10 10.8 10.12 10.13 10.14

(a) 1128 W, (b) 45.1 W 60% (a) x = (BP/AV)1/2 , (b) x = 0.146 t = 5.6 h

Chapter 11 11.13 11.15 11.17 11.18 11.20 11.21

28% (without water gas shift), 42% (with complete water gas shift) (b) 89.8%, (c) 85.9% S/C = 1.5, 9.6 mol H2 /mol fuel (1) 3.23, (2) 3, (3) 3.23 32% 2%

Chapter 12 12.24 Area able to be heated = 148 m2 . Can divide into rooms assuming various room sizes/shapes as desired

553

554

APPENDIX H: ANSWERS TO SELECTED CHAPTER EXERCISES

12.27 47∘ C 12.29 (a) 11.9 kW 12.30 (a) 4.8 kW

Chapter 13 13.4 13.5 13.6 13.9

2.9 times as long 100 A/cm3 53.7 W 𝜆O2 = 9.21, PFC = 64.6 W

Chapter 14 14.18 14.19 14.25 14.26

∼4–7×108 metric tons/yr 46%, 4.6 ×1012 CO2 equiv/yr, $1.2–3.1 billion/yr H2 release may almost quadruple (0.6 MT H2 /yr vs. 0.16 MT H2 /yr) Part 1: coal 104.5, gasoline 71.9, ethanol 71.1, methanol 68.6, natural gas 56.8, methane 55, hydrogen 0; Part 2: coal 46.9, gasoline 23.8, ethanol 14.7, methanol 11.0, natural gas 11.4 methane 11.0, hydrogen 0

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INDEX

a, see Activity A, see Area Ac (catalyst area coefficient), xxiii Acceleration due to gravity (g), xxiv Accrual rates, greenhouse gas, 495–496 AC power, 364, 366, 368 Activated state, 83 Activation energy, 81–85 Activation energy barrier (ΔGact , ΔG‡ ), xxiv and exchange current density, 95–96 and potential, 89 and reaction rate, 81–82, 84, 86 voltage gradient modification to, 157 Activation kinetics, 97–100 Activation losses: and concentration losses, 179–180 on current–voltage curves, 246, 247 defined, 20 and fuel cell performance, 204 on Nyquist plots, 249, 250 in 1D SOFC models, 217 Activation overvoltage (𝜂act ), xxvi approximations based on, 97–98 in Butler–Volmer equation, 92, 93 and concentration, 177–178 at equilibrium, 112 Active catalyst area, 265 Active cooling, 353–356 Activity (a), xxiii, 50 catalytic, 107 in concentration cells, 54–55 and Gibbs free energy, 51, 52 mass, 315 specific, 315–317

water vapor, 136, 138 Activity coefficient (𝛾), xxv AD (anaerobic digestion), 408–409 ADG (anaerobic digester gas), 408–409 Adhesion strength, 107–108 Adiabatic conditions, 459, 475 Adsorption beds, PSA, 413 Adsorption charge (Qh ), xxiv Adsorption charge on smooth catalyst surface (Qm ), xxiv AFCs, see Alkaline fuel cells Afterburners, 424, 428 Air blowers, 354–355, 471, 472, 474, 475 Air operation, 54 Air pollution, 502–507 Air supply, for portable SOFC systems, 463 Aliovalent doping, 318, 320 Alkali-based soft glass, 336, 340 Alkaline-based direct methanol fuel cells, 285, 286 Alkaline fuel cells (AFCs), 278–280 advantages of, 279 catalysts for, 317 described, 13 disadvantages of, 280 other fuel cells vs., 298–300 reaction kinetics in, 102 All-vanadium redox flow batteries, 296 𝛼, see Charge transfer coefficient 𝛼* (channel aspect ratio), xxv 𝛼 (CO2 equivalent coefficient), xxv 𝛼 (ratio of water flux to charge flux), 210, 220 Ambient pollution, 503, 505 Ammonium borohydride (NH4 BH4 ), 287 Ampere, 78 Anaerobic digester gas (ADG), 408–409

565

566

INDEX Anaerobic digestion (AD), 408–409 Angular frequency (𝜔), xxvi Angular momentum quantum number, 534 Anodes: alkaline fuel cell, 278 catalysis at, 104–106 defined, 15–16 degradation of, 339 direct borohydride fuel cell, 287 direct formic acid fuel cell, 287 in fuel cell structure, 14 limiting current densities at, 175 molten carbonate fuel cell, 280 in 1D models, 212 PEMFC, 275, 313–314 phosphoric acid fuel cell, 274 of redox flow batteries, 296 SOFC, 282, 329–333, 339 of zinc–air cells, 291 Anode catalysts, 313–314 Anode funnels, 290 Anode-supported MEA design, 213, 327–329 Apollo missions, 278, 279 Aqueous electrolytes, 131–134 AR, see Autothermal reforming Area (A), xxiii, xxv, 124–126 Area-specific resistance (ASR), xxiii, 124–126, 146 Arhenius conductivity equation, 458 Aromatic hydrocarbon membranes, 305–306 Assumptions: modeling, 177–179, 231, 462 thermodynamic, 26 Atomic orbitals, 533–534 Autothermal reforming (AR), 396, 397, 402–407 Avogadro’s number (NA ), xxiv, 44, 517 Back diffusion, 141 Balance of plant (BOP) components, 462, 467 Ballard (company), 199 Banded electrolyte design, 350–351 Batteries: fuel cells vs., 3, 8–11, 386–387 redox flow, 296 salt water, 55 Beale, S. B., 180 𝛽 (CO2 equivalent coefficient), xxv BET (Brunauer–Emmett–Teller) surface area measurement, 240, 266–267 BIMEVOX family of materials, 322–323 Binary diffusion model, 214–215 Binary diffusivity, 214–215 Biogas, 365 Biological fuel cells, 288 Bipolar plate stacks, 338, 349, 350, 354 Bismuth oxides, 322–323 Black carbon, 491 Blocking electrodes, 254 Blowers, see Air blowers Bohr, Niels, 531, 533 Boltzmann’s constant (k), xxiv, 517 Bonds, 5, 27 Bond enthalpy calculations, 35 Boost regulators, 379 BOP (balance of plant) components, 462, 467

Borax, 287 Bottleneck processes, 482, 483, 485–487, 508–509 Boundary conditions, 455–458 Brunauer–Emmett–Teller (BET) surface area measurement, 240, 266–267 Building heat loops, 428, 429 Bulk (flow channel) reactants, 170 Butler–Volmer equation, 177, 452 approximations for, 97–98 and Nernst equation, 108–112 potential and reaction rate in, 89–94 BZY (yttrium-doped barium zirconate), 326 C, see Capacitance c* (concentration at reaction surface), xxiii c (concentration), xxiii c (mass transport constant for loss), xxiii, 180 Calcium-doped LaCrO3 , 335 Capacitance (C), xxiii, 251, 266 Capacitors, 252, 257 Carbon: black, 491 doped carbon catalysts, 317 low carbon fuels, 495–497 oxidation of, 338 sooty carbon deposits, 331 steam-to-carbon ratio, xxv, 394 Carbon cloth, 311, 312 Carbon dioxide, 280, 412–413, 492, 501, 525 Carbon dioxide equivalent (CO2equivalent ), 497–499 Carbon monoxide: clean-up, 372, 411–414 combustion of, 400 external costs of, 505 health effects of, 504 from methanol oxidation reaction, 313–314 pressure swing adsorption for, 413 selective methanation of, 411–412 selective oxidation of, 412–413 thermodynamic data, 524 tolerance for, 274, 285 yield of, 410 Carbon monoxide poisoning, 101, 286, 314, 362 Carbon paper, 311, 312 Carnot cycle, 61, 62 Carriers: charge, 129–131, 147 concentration of, 129–131, 147 hydrogen, 357, 360–362 intrinsic vs. extrinsic, 148–149 mobility of, 129, 147 Carrier system effectiveness, 360, 549 Catalysis, 94–97, 104–107 Catalysts: anode, 313–314 cathode, 314–317 on cyclic voltammogram, 265 deactivation of, 410 degradation of, 337–338 and electrode design, 103–104 governing equations for, 216 in 1D models, 212 PEMFC, 308–317 selecting, 96, 107–108, 540

INDEX SOFC, 326–329, 337–338 Catalyst area coefficient (Ac ), xxiii Catalyst layers, 104, 170, 309, 310 Catalytic activity, 107 Cathodes: alkaline fuel cell, 278 catalysis at, 106–107 defined, 15–16 degradation of, 339 direct borohydride fuel cell, 287 direct formic acid fuel cell, 287 in fuel cell structure, 14 limiting current densities at, 175 molten carbonate fuel cell, 280 PEMFC, 275, 314–317 phosphoric acid fuel cell, 274 of redox flow batteries, 296 SOFC, 282, 333–334 of zinc–air cells, 291 Cathode catalysts, 314–317 Cathode-supported MEA design, 327, 328 CCHP (combined cooling, heating, and electrical power) systems, 382–383 Cdl , see Double-layer capacitance CellTech, 293 Central control units, 368 Ceramic: electrolytes of, 13, 146–151 extrinsic defect concentrations in, 150–151 flow structures of, 197 interconnects of, 335 SOFC cathodes of, 334 Ceramic glasses, 336, 340 Ceria-based anodes, 331–332 Ceria-based electrolytes, 320–322, 339 Cermet(s), 283, 284, 326, 329–332 CFD, see Computational fluid dynamics Change in enthalpy per unit time, 429 Change in quantity (Δ), xxv, 32 Change in reaction (rxn), xxvi Channel aspect ratio (𝛼*), xxv Characterization techniques, 237–269 about, 239–240 electrochemical, 240–265 equations for, 549 ex situ, 265–268 importance of, 20–21 properties examined by, 238–239 Charge (Q), xxiv adsorption, xxiv carried by charged species, 118 in electrode kinetics, 79–80 on electrons, 451–452, 517 forces and movement of, 117–120 fundamental, xxiv Charge conservation equations, 449, 451–452 Charged particles, electrochemical potential of, 51 Charge flux ( j), 118, 160–161 Charge transfer coefficient (𝛼), xxv, 89, 96, 456–458 Charge transfer reactions, 82–84 Charge transport, 117–164 and conductivity, 128–131, 153–154, 156–160

and diffusivity, 153–160 electrical driving forces in, 160–161 and electrolyte classes, 132–153 equations for, 548 forces and charge movement, 117–120 and ion conduction in oxide electrolytes, 161–163 resistance, 124–128 voltage loss with, 121–123 Chemical bonds, 5, 27 Chemical determinations, 240, 267 Chemical driving forces, 119 Chemical potential (𝜇), xxvi, 50, 51 Chemical reactions, 78 CHP (combined heat and power) systems, 369–383, 425 Chromium-based metallic interconnects, 335–336, 339–340 Chromium poisoning, 336 Chromium volatilization, 339–340 Circular flow channels, friction factors for, 189–190 Climate change, 490, 495–497 Closed systems, 9, 10 CO2equivalent (carbon dioxide equivalent), 497–499 CO2 equivalent coefficient (𝛼, 𝛽), xxv Coal, 407–408, 486, 496 Coefficient of performance (COP), 382–383 Cogeneration, 371 Coking, 101 Cold streams, 424, 426–432 Column buckling analogy, 537 Combined cooling, heating, and electrical power (CCHP) systems, 382–383 Combined heat and power, 371 Combined heat and power (CHP) systems, 369–383, 425 Combustion, 3, 6, 35, 400–401, 486 Combustion engines, 3–5, 8–9, 11, 28, 482 Complete combustion, 401 Compressed hydrogen, 358, 359, 364, 365 Compression force, 246 Computational fluid dynamics (CFD), 447–462 assumptions in, 462 boundary conditions in, 455–458 building fuel cell models, 453–455 flow structure analysis with, 183 governing equations for, 448–453 and modeling, 227–230 results analysis, 460, 462 solution process in, 459–461 volume conditions in, 455, 459 COMSOL Multiphysics, 448 Concentration (c), xxiii. See also Reactant concentration carrier, 129–131 and chemical potential, 51 and exchange current density, 95 extrinsic defect, 150–151 and Nernst voltage, 176–177 and reaction rate, 92, 177–178 and reversible cell voltage, 50–54 time dependence of, 171–172 vacancy, 148, 151 Concentration at reaction surface (c*), xxiii Concentration cells, 54–59 Concentration gradients, 160–161, 169 Concentration losses, 167–168 on current–voltage curves, 178–180

567

568

INDEX Concentration losses, (continued) defined, 20, 171 in diffusive transport, 180–183 and fuel cell performance, 204 in 1D SOFC models, 217 Concentration overvoltage (𝜂 conc ), xxvi, 175 Condensers, 373, 424, 427, 428 Conduction, 17–18, 120, 306, 307, 415. See also Ionic conduction Conductivity (𝜎), xxvi atomistic origins of, 153–154, 156–160 and diffusivity, 156–160 electrical, 329–330, 332 of electrode materials, 311 of electrolyte materials, 304, 306, 318–320 electron, 130–131 electronic, 130–131, 318, 321, 322, 334, 456–458 ionic, 131, 320–322, 324, 456–458 in 1D models, 215, 223 partial electronic (hole), 325 physical meaning of, 128–131 proton, 325–326 thermal, 451, 456–458 and transport processes, 120 Conductors: area of, 125–126 electronic vs. ionic, 129–130 with hopping mechanisms, 153–160 mixed ionic–electronic, 152–153, 283–284, 334 thickness of, 126–127 Configurations, system, 28 Conservation laws, 210 Constant-flow-rate condition, 63, 64 Constant-phase elements, 257 Constant-stoichiometry condition, 63–65 Consumption rate, 67, 468 Contact resistance, 127 Continuity equations, see Mass conservation equations Control systems, 368, 369 Control volume analysis, 483, 502 Convection, 120, 168, 415 Convective mass flux ( JC ), xxiv Convective transport, 183–199 diffusive vs., 168–170 in flow channels, 188–192 flow structure design for optimal, 196–199 fluid mechanics of, 183–188 gas depletion in, 192–196 in 1D fuel cell models, 212 Conversion factors, 517–518 Cooling, 353–356 Cooling cells, 354 Copper concentration cell, 57–59 Corrosion, of bipolar plates, 338 Counter-flow heat exchangers, 426 Coupling coefficient of flow and flux (Mik ), xxiv, 119 cp , see Heat capacity Cryogenic hydrogen, 358, 365 Crystalline ceramic, 150–151 Crystalline solid electrolytes, 131 Current (i), xxiv calculating predicted, 460 in capacitors, 252

in characterization techniques, 240 and consumption of reactants, 65 in electrode kinetics, 78–79 fuel leakage, xxiv, 205–206 and fuel utilization efficiency, 63, 64 as fundamental electrochemical variable, 241–242 response of, to voltage perturbation, 248 steady-state value of, 241 and voltage efficiency, 63 Current density ( j), xxiv. See also Exchange current density; j–V curves and Butler–Volmer equation, 90, 92–94 and CFD, 460, 462 and diffusive transport, 172, 175–176 in electrode kinetics, 80 and flux balance, 208 and fuel cell efficiency, 65 and overvoltage, 98 temperature effects on, 99–100 Current interrupt measurement, 239, 242, 261–264 Current–voltage (i–V) curves, 18–19, 259, 263–264. See also j–V curves Cyclic voltammetry (CV), 240, 242, 264–265 D, see Diffusivity Darcy’s law, 450 Davisson, C. J., 531 DC–AC inverters, 366, 368 DC–DC converters, 366, 367, 464, 472, 473 DC power, converting AC to, 364 Deactivation effect, 315 Dead zones, 228 de Broglie, Louis, 531 Decay rate, 84–85 Degradation, materials, 330, 337–340 ΔG‡ , see Activation energy barrier ΔGact , see Activation energy barrier Δ (change in quantity), xxv, 32 𝛿 (diffusion layer thickness), xxv 𝛿 phase, bismuth oxide, 322, 323 Density (𝜌), xxvi, 194–196, 456–458 Density functional theory (DFT), 105, 533, 540–541 Dependent variables, 30 Devolatilization, 407 DFMCs, see Direct methanol fuel cells DFT, see Density functional theory Dielectric breakdown 127 Diffusion (dif f ), xxvi back, 141 binary model, 214–215 convection vs., 168 and diffusivity, 154–155 Maxwell–Stefan model, 214, 215 reactions driving, 170–174 transport via, 120 Diffusion flux, 174 Diffusion layer, 170 Diffusion layer thickness (HE , 𝛿), xxiv, xxv Diffusive transport, 170–183 concentration and Nernst voltage in, 176–177 concentration and reaction rate, 177–178 concentration loss and j–V curve, 178–180 convective vs., 168–170 electrochemical reactions in, 170–174

INDEX fuel cell concentration loss, 180–183 limiting current density in, 175–176 in 1D fuel cell models, 212 Diffusivity (D), xxiii, 120 atomistic origins of, 153–160 binary, 214–215 and conductivity, 131, 147–148, 156–160 effective, 173–174, 456–458 ion, 541 nominal, 173 as volume condition, 456–458 water, 143–146, 221 Digestion, anaerobic, 408–409 Dilute solutions, activity of, 50 Direct alcohol fuel cells, 313–314 Direct borohydride fuel cells, 287 Direct electro-oxidation, 361, 362 Direct flame SOFCs, 292–293 Direct formic acid fuel cells, 287 Direct internal reforming, see Internal reforming Direct liquid-fueled fuel cells, 285–288 Direct methanol fuel cells (DMFCs), 276, 285, 286, 313–314 Distance conversion factors, 517 Dopants, 148, 149, 318, 320, 321, 325 Doped carbon catalysts, 317 Doped ceria, 320–322 Doped lanthanum chromite, 332 Doped perovskites, 318 Doping, aliovalent, 318, 320 Double-layer capacitance (Cdl ), xxiii, 251, 266 dTmin (pinch point temperature), 437–440 dTmin, set , see Minimum temperature difference Dual-layer approach to MEAs, 309–311, 326–329 Durability: of PEMFC materials, 337–338 of SOFC materials, 338–340 Dynamic characterization techniques, 242 Dynamic equilibrium, 87, 154 Dynamic potentiostatic techniques, 241 E (electric field), xxiii, 133 (electrical subscript), xxvi E (thermodynamic ideal voltage), xxiii. See also Reversible cell voltage EBOP (Electrical Balance of Plant), 281 Edge tabs, 350 ef f (effective property), xxvi Effective diffusivity, 173–174, 456–458 Effective porosity, 266 Effective property (ef f ), xxvi Effective thermal conductivity, 451, 456, 457 Efficiency (𝜀), xxv of DC–DC converters, 366 defined, 60 fuel cell, 60–65, 68–69 in fuel cell system design, 466 of fuel processing subsystems, 414–416 fuel processor, xxv, 398 fuel reformer, xxv, 373, 375, 398 fuel utilization, 63–64 gasification, 408 gross electrical, 376, 378 heat recovery, xxv hydrogen storage, 359 E

mass storage, 359 overall, xxv, 371, 381, 510 reversible thermodynamic, 63, 68, 71 voltage, 63 EIS, see Electrochemical impedance spectroscopy ELAT (Electrode Los-Alamos Type), 312 (electrical subscript), xxvi elec Electrical Balance of Plant (EBOP), 281 Electrical conductivity, 329–330, 332 Electrical driving forces, 119, 120, 160–161 Electrical efficiency (𝜀R ), xxv, 376, 378, 381–383 Electrical potential (𝜙), xxvi, 43 Electrical subscript (E, e,elec ), xxvi Electrical work, 39–42 Electric field (E), xxiii, 133 Electricity generation plants, 486, 496 Electricity production, in fuel cells, 16–18 Electric load, changes in, 369 Electric power generation systems, 487, 507–510 Electric power plants, 501–503, 505 Electric wall conditions, 459, 460 Electrocatalysis, 94–97 Electrochemical characterization techniques, see In situ electrochemical characterization Electrochemical equilibrium, 57–59 Electrochemical half reactions, 6 Electrochemical impedance spectroscopy (EIS), 246–261 basics of, 246–249 as dynamic technique, 242 and equivalent circuit modeling, 250–261 and fuel cells, 249–250 as in situ electrochemical characterization technique, 239–240 Electrochemical potential (𝜇), xxvi, 51, 56, 159–160 Electrochemical processes, 78, 353, 368 Electrochemical reactions: for CFD, 449, 452–453 chemical reactions vs., 78 in diffusive transport, 170–174 in equivalent circuit modeling, 251–255 in fuel cells, 17 half-reactions in, 45–46 potential in, 80–81 Electrochemical waste heat, 353 Electrodes: attachment of, 309 blocking, 254 catalysts and design of, 103–104 convective mass transport to, 191–192 degradation of, 338 governing equations for, 214–215 mass transport in, 168–183 PEMFC, 308–313, 338 SOFC, 326–336 Electrode kinetics, 77–82 Electrode Los-Alamos Type (ELAT), 312 Electrode potentials, 44–46, 80–81, 529 Electrolysis, 68–69, 297 Electrolytes, 273–301. See also Polymer electrolytes about, 273 alkaline fuel cell, 278–280 aqueous, 131–134 biological fuel cell, 288

569

570

INDEX Electrolytes, 273–301. See also Polymer electrolytes (continued) ceramic, 13, 146–151 ceria-based, 320–322, 339 classes of, 132–153 comparison of, 298–300 crystalline solid, 131 defined, 6 degradation of, 338–339 direct liquid-fueled fuel cell, 285–288 in fuel cell structure, 14 governing equations for, 215–216 membraneless fuel cell, 289–290 in metal–air cells, 290–291 and mixed ionic–electronic conductors, 152–153 molten carbonate fuel cell, 280–282 nonstandard fuel cell, 284–298 oxide, 161–163 PEMFC, 304–308 phosphoric acid fuel cell, 274–275 protonic ceramic fuel cell, 294–295 and reaction kinetics, 102 in redox flow batteries, 296 requirements of, 304 in reversible fuel-cell electrolyzers, 297–298 SOFC, 282–284, 291–294, 317–326 solid-acid fuel cell, 295–296 types of, 12–14 Electrolyte resistance, 128 Electrolyte-supported MEA design, 327–328 Electrolytic cells, 16, 297 Electrolyzer mode, 67–70, 297–298 Electrons: activity of, in metals, 50 charge on, 451–452, 517 number transferred in reaction (n), xxiv, 44–45, 118 potential and energy of, 80–81 transport of ions vs., 117 Electron density distribution, 105, 541 Electronic conduction, 17–18 Electronic conductivity (𝜎 elec ), 130–131, 318, 321, 322, 334, 456–458 Electronic conductors, 129–130 Electron mass, 517 Electro-osmotic drag, 140–141 Elements, periodic table of, 543 Emissions, 485–486, 490–507 End indicators, 240 Endothermic reactors, 353 Energy: activation, 81–85 and bonds, 5 conversion factors for, 518 defined, 7 of electrons, 80–81 and entropy, 29 free, 26 heat and work as transfer of, 28 input rate, 65 internal, xxv, 26–27, 29–32, 34–35 kinetic, 535 negative changes in, 32 potential, 535, 538

specific, 8 Energy buffers, 369 Energy conservation, 27, 449, 451, 483 Energy density, 8, 11, 12, 357, 385–386, 549 Energy flows, 483, 488–489, 509 Enthalpy (H, h), xxv change in, 429 fluid, 451 and Gibbs free energy, 37–39 intuition about, 26 of reactions, 34–37 temperature and, 432–437 as thermodynamic potential, 31, 32 Entropy (S, s), xxiv, 27–32, 35, 37–39, 48 Environmental impact, 21–22, 481–511 and air pollution, 502–507 of electric power production, 507–510 of emissions, 490–507 equations for, 550 and global warming, 490–502 life cycle assessment of, 481–490 quantifying, 497–507 EPA (U.S. Environmental Protection Agency), 488, 500 𝜀 (strain rate), xxv, 184 𝜀, see Efficiency; Porosity 𝜀FP (fuel processor efficiency), xxv, 398 𝜀H (heat recovery efficiency), xxv, 380–383 𝜀O , see Overall efficiency Equilibrium, 34, 57–59, 86–89, 112, 154 Equivalent circuit modeling, 250–261 Equivalent weight, 142–143 𝜀R , see Electrical efficiency ET (temperature-dependent thermodynamic voltage), xxiii 𝜂, see Overvoltage 𝜂 act , see Activation overvoltage 𝜂 conc (concentration overvoltage), xxvi, 175 𝜂 ohmic , see Ohmic overvoltage Ethanol, carbon content of, 496 Ethanol oxidation reaction, 314 Ethermo (thermodynamic ideal voltage), xxiii Euler buckling load, 533–534, 537 Evaporation, 399 Exchange current density ( j0 ), xxiv, 86–87, 92, 94–97, 456–458 Exchange current density at reference concentration ( j00 ), xxiv, 92 Exothermic reactors, 353 Ex situ characterization techniques, 239, 240, 265–268 External costs, 499–501, 505–507 External heating, 355 External heat transfer, 426–427 External reforming, 361–363 Extrinsic carriers, 148–149 Extrinsic defect concentrations, 150–151 Extrinsic quantities, 32 F, see Faraday constant; Helmholtz free energy f (friction factor), xxiii, 189–190 (quantity of formation subscript), xxvi f f (reaction rate constant), xxiii Fans, cooling by, 354–355 Faradaic resistance (Rf ), xxiv, 251 Faraday constant (F), xxiii, 44–45, 78, 517 Feedback loops, 368

INDEX Fermi level, 80 Feynman, Richard, 26 Fick’s diffusion equation, 450 Fick’s law, 193, 214 Figures of merit, 7 First law of thermodynamics, 26, 27, 431 Fitting constants, 204 Fixed charge sites, 135 Fixed-flow-rate condition, 246 Fixed parasitic power loads, 376 Fixed-stoichiometry condition, 246 Fk (generalized force), xxiii Flip flop configuration, 351 Flooding, 18, 212, 312 Flow channels, 183, 188–196 Flow channel reactants, 170 Flow field plates, 17 Flow rates, 67, 245–246, 468 Flow structures, 168–170, 183–199, 453–454 Fluids, 183–184 Fluid enthalpy, 451 Fluid mechanics, 183–188 Fluorite crystal structure, 318, 319 Flux ( J), 117–119 charge, 118, 160–161 diffusion, 174 and diffusivity, 154–155, 158, 159 mass, xxiv, 191–194 molar, xxiv, 141 in 1D fuel cell models, 215–216 Flux balance, 206, 208–210, 448 Force(s), xxiii, 117–120, 133, 246 Formation enthalpy, 35 Formic acid, 287 Forward activation barrier, 87–90 Forward current density, 86, 94 Free electrons, 535 Free energy, xxiii, 26, 31, 32. See also Gibbs free energy (G, g) Free-energy curves, 84, 95–96 Free-energy maximum, 83 Free radicals, 337 Free volume, 135 Frequency, xxvi, 107–108, 248, 254–255 Frictional drag force, 133 Friction factor ( f ), xxiii, 189–190 Fuel(s): availability and storage of, 11, 12 crossover of, 127 for electric power generation, 487 liquid, 285–288 low carbon, 495–497 for LTA-SOFCs, 293 natural gas, 371–372 and reaction kinetics, 101 storage effectiveness of, 357–358 Fuel cell(s), 3–23 advantages of, 8–11 basic operation of, 14–18 batteries vs., 3, 8–11, 386–387 combustion engines vs., 3–5, 8–9, 11, 28 disadvantages of, 11–12 efficiency of, 60–65

and electrochemical impedance spectroscopy, 249–250 electrolysis cells vs., 297 life cycle assessments of, 484–489 performance of, 18–20 properties for characterization, 238–240 simple, 6–8 sizing of, for portable systems, 383–385 technologies using, 21 thermodynamics and boundaries of, 25 types of, 12–14, 273–301 Fuel cell design: boundary conditions in, 455–458 building models in, 453–455 governing equations for, 448–453 results analysis, 460, 462 solution process in, 459–461 via computational fluid dynamics, 447–462 volume conditions in, 455, 459 Fuel cell efficiency, 60–65, 68–69 Fuel cell mass transport, see Mass transport, fuel cell Fuel cell mode, reversible fuel cells in, 67–70 Fuel cell performance, 18–20, 25, 180–181, 204–205, 239, 353. See also j–V curves Fuel cell subsystem, 348–352, 372, 376–378 Fuel leakage current ( jleak ), xxiv, 205–206 Fuel processing, as bottleneck process, 485 Fuel processing subsystem, 357–365, 372–375, 393–418 Fuel processors, xxv, 398, 414–417 Fuel reformers, xxv, 373, 375, 398, 414–417 Fuel reforming, see Reforming Fuel reservoirs, 383–385 Fuel supply systems, 463 Fuel utilization efficiency, 63–64 Fundamental charge (q), xxiv G, see Gibbs free energy g (acceleration due to gravity), xxiv Gadolinia-doped ceria (GDC), 318, 320–322, 324–325, 332 Galvanic cells, 16 Galvani potentials, 87–91 Galvanostatic techniques, 241, 245 𝛾 (activity coefficient), xxv Gases: active cooling with, 354–355 activity of, 50 anaerobic digester, 408–409 as fluids, 184 number of moles of, xxiv one-dimensional electron, 536–537 viscosity of mixtures of, 186 Gas channel thickness (HC ), xxiv Gas depletion, 192–196, 224–228, 230 Gas diffusion layer (GDL), 104, 310–313 Gasification, 407–408, 486 Gasoline, 365, 496 Gas permeability, 240, 266, 267 Gas-phase transport, see Mass transport, fuel cell Gaussian, xxvi, 105, 533 GDC, see Gadolinia-doped ceria Generalized force (Fk ), xxiii Germer. L. H., 531 Gibbs free energy (G, g), xxiv, 37–46 and activity, 51, 52 calculating, 37–39

571

572

INDEX Gibbs free energy (G, g), (continued) change in, 33, 37–39, 51, 412 and chemical potential, 50 defined, 37 and electrical work, 39–42 and lower heating values, 61 and reversible cell voltage, 47, 48 and spontaneity, 42–43 and standard electrode potentials, 44–46 as thermodynamic potential, 30–32 and voltage, 43–44 Global warming, 490–502 Global warming potential (GWP), 497 Gottesfeld, S., 309 Governing equations, 210, 213–216, 448–453 Gradient vector, 535 Graphite, 196 Gravimetric energy density (specific energy), 8, 357, 385–386, 549 Gravimetric power density (specific power), 7 Greek symbols, xxv–xxvi Greenhouse effect, natural, 490–491 Greenhouse gases, 491–493, 495–496 Grid generation, for modeling fuel cells, 454–455 Gross current produced at electrodes, 206 Gross electrical efficiency, 376, 378 Grove, William, 7 GT-based materials, 333 GWP (global warming potential), 497 H, see Enthalpy; Heat h (Planck’s constant), xxiv, 517 Haile, S., 295 Hamiltonian, 535, 540 HC (gas channel thickness), xxiv HE , (diffusion layer thickness), xxiv Health effects, of air pollution, 503–505 Heat (H, Q), xxiv. See also Stationary combined heat and power (CHP) systems combined heat and power, 371 consumption of, 68–70 dissipation of, by electrochemical processes, 353 and efficiency of reversible fuel cells, 68 and enthalpy, 34–35 and first law of thermodynamics, 27 and thermal balance, 66 transfer of energy associated with, 28 unrecovered, 415 Heat capacity (cp ), xxiii, 36–37, 469 Heat capacity flow rate (mcp ), xxiv, 429 Heat exchangers, 423, 426, 432–434, 437–440, 470–472 Heat/expansion engines, 61–62 Heat generation rate, 468–469 Heat loops, building, 428, 429 Heat management, 464 Heat of combustion, 35 Heat recovery, xxv, 353, 355–356, 380–383, 426–427 Heat-to-power ratio, 371 Heat transfer rate, 470 Height (z), xxv Helmholtz free energy (F), xxiii, 31, 32 Heterogeneous processes, 78 Heteropolyacids (HPAs), 307, 317 Higher heating value (HHV), xxvi, 61, 62

High-surface area carbon materials, 310, 317 hm , see Mass transfer convection coefficient Hohenberg, P.C., 541 Home Energy System, xxvi Honda FCX, xxvi Honda Home Energy Station, 363 Hopping mechanisms, 17, 129, 153–160 Hopping rate (𝑣), xxv, 156–158 HOR, see Hydrogen oxidation reaction Hot reformate stream, 424, 427 Hot spots, 127 Hot streams, 424, 426–432 HPAs (heteropolyacids), 307, 317 Humidifiers, 351 Hydraulic diameter, 189 Hydrogen: and air pollution, 502–503 combustion of, 3, 6, 400 compression of, 485 as fuel, 11, 12, 365, 496, 497 and global warming, 492–495 liquid, 358, 359 palladium–silver membrane separation of, 414 for portable SOFC systems, 463 storage of, 358–360 thermodynamic data, 520 yield of, 394 Hydrogen atom models, 531, 538–539 Hydrogen carriers, 357, 360–362 Hydrogen concentration cells, 55–56 Hydrogen economy, 21–22 Hydrogen fuel cells, 313 Hydrogen generators, 486 Hydrogen oxidation reaction (HOR), 15, 78, 100, 101, 313 Hydrogen pump mode cyclic voltammogram, 264 Hydrogen storage efficiency, 359 Hydrogen supply rate, 467 Hydronium formation, 105–106 Hydroperoxy radicals, 337 Hydrophobic treatment, for GDL materials, 312 Hydroxy radicals, 337 Hyundai ix35 fuel cell vehicle, 275 i, see Current i (species subscript), xxvi ICE (internal combustion engine), 482 Ideal gas constant (R), xxiv Ideal gases, 50 Ideal solutions, 50 Impedance (Z), xxv, 247–249, 251, 261. See also Electrochemical impedance spectroscopy (EIS) Incomplete combustion, 401 Incomplete conversion, 415–416 Independent variables, 30 Inductors, 257 Infinite Warburg, 255, 257 Ink, in MEA fabrication, 309 Inlet conditions, 459 In-plane conductivity, 311 In situ electrochemical characterization techniques, 240–265 current interrupt measurement, 261–264 current–voltage measurements, 244–246 cyclic voltammetry, 264–265 defined, 239

INDEX electrochemical impedance spectroscopy, 246–261 end indicators of, 240 fundamental variables, 241–242 methods in, 239–240 test station requirements, 242–244 Interconnects, 335–336, 339–340 Interdigitated flow, 198, 199 Interfaces, 212, 309 Interfacial potentials, 88–89 Internal combustion engine (ICE), 482 Internal energy (U), xxv, 26–27, 29–32, 34–35 Internal heating, 355 Internal heat transfer, 427 Internal reforming, 361, 362, 393–394 International System of Units (SI), 7 Interstitials, 129 Intrinsic carriers, 148–149 Intrinsic quantities, 32 Intrinsic vacancy concentration, 148 Ions, 117, 136, 451–452 Ion diffusivity, 541 Ionic conduction: in aqueous electrolytes and ionic liquids, 131–134 in ceramic electrolytes, 146–151 in fuel cells, 17–18 in oxide electrolytes, 161–163 in polymer electrolytes, 135–146 in SOFCs, 317–318 Ionic conductivity (𝜎 ion ), 131, 320–322, 324, 456–458 Ionic conductors, 129–130 Ionic contamination, 337 Ionic liquids, 131–134 Ionic (electrolyte) resistance, 128 iR-free curves (iR-corrected curves), 263–264 Iron-doped lanthanum cobaltites, 334 Isobaric conditions, 40–42 Isothermal conditions, 38, 40–42, 459 Iterative solution processes, 459, 460 i–V curves, see Current–voltage curves j, see Current density J (joule), 7 Jˆ (mass flux), xxiv, 191–194 J (molar flux), xxiv, 141 j0 , see Exchange current density j 00 (exchange current density at reference concentration), xxiv, 92 JC (convective mass flux), xxiv JL (limiting current density), xxiv, 175–176 jleak (fuel leakage current), xxiv, 205–206 Joule (J), 7 j–V curves, 93–94 from CFD analysis, 228, 229 comparisons of, 238 concentration losses and, 178–180 and DC–DC converters, 367 in electrochemical characterization, 239 fuel cell system design based on, 475 interpreting, 246 for modeling, 204 of 1D PEMFC models, 219–224 of 1D SOFC models, 216–219, 227 and in situ characterization techniques, 239 steady-state, 241, 244–245

for system design, 466 test conditions for, 245–246 k (Boltzmann’s constant), xxiv Kilowatt-hours (kWh), 7 Kinematic viscosity, 184 Kinetics, see Reaction kinetics Kinetic energy, 535 Kinetic Monte Carlo (KMC) techniques, 162, 163 KOH (potassium hydroxide): in alkaline fuel cells, 132, 134, 278–279 conductivity of aqueous, 134 in direct liquid-fueled fuel cells, 287 in metal–air cells, 290 Kohn, W., 533, 541 kWh (kilowatt-hours), 7 L (length), xxiv Lacorre, P., 323 Laguerre polynomial, 539 𝜆, see Water content 𝜆 (stoichiometric coefficient), xxvi, 52 𝜆 (stoichiometry factor), 64, 65 Laminar flow, 186–187, 289 LAMOX series, 323 Lanthanum chromites, 332, 335, 339 Lanthanum gallate, 324–325 Lanthanum–strontium cobaltite ferrite (LSCF) cathodes, 334 Latent heat of vaporization, 61 Lawrence Livermore National Laboratories (LLNL), 358 LCAs, see Life cycle assessments Leaching, 316 Leakage, 414, 493 Le Chatelier’s principle, 49, 398, 410, 411 Legendre polynomial, 539 Legendre transforms, 30 Length (L), xxiv LHV, see Lower heating value Life cycle assessments (LCAs), 481–490, 507–510 Limiting current density ( JL ), xxiv, 175–176 Linear momentum, 535 Linear systems, 249 Liquids, active cooling with, 355 Liquid-fueled reformer + fuel cell systems, 287–288 Liquid hydrogen, 358, 359 Liquid-tin anode solid-oxide fuel cells (LTA-SOFCs), 293–294 LLNL (Lawrence Livermore National Laboratories), 358 Lone-pair substitution (LPS), 323 Loss(es). See also Activation losses; Concentration losses; Ohmic losses efficiency, 414–416 Nernstian, 170, 178–179 reaction, 171 voltage, 121–123 Low carbon fuels/fuel cells, 495–497 Lower heating value (LHV), xxvi, 61, 497 LPS (lone-pair substitution), 323 LSCF (lanthanum–strontium cobaltite ferrite) cathodes, 334 LSCV–YSZ anodes, 332 LSGM series, 324–325, 339 LSM (strontium-doped lanthanum manganite), 152–153 LSM–YSZ cathodes, 334, 339 LTA-SOFCs (liquid-tin anode solid-oxide fuel cells), 293–294

573

574

INDEX m (mass), xxiv M (mass flow rate), xxiv, 431 M (molar mass), xxiv Marginal emissions, 502 Mass (m), xxiv Mass activity, 315 Mass balance, 67, 467–468 Mass conservation equations, 449, 483 Mass flow rate (M), xxiv, 431 Mass flows, in LCAs, 483, 488–489, 509 ˆ xxiv, 191–194 Mass flux (J), Mass storage efficiency, 359 Mass transfer convection coefficient (hm ), xxiv, 191, 196 Mass transport, fuel cell, 167–200 convective, 183–199 defined, 167–168 diffusive, 170–183 in electrodes vs. flow structures, 168–170 equations for, 548 in equivalent circuit modeling, 255–257, 259 Mass transport constant for loss (c), xxiii, 180 Mass transport resistance, 328 Matrix material, electrolyte, 132 Maximum quantity of heat recoverable, 431 Maxwell–Stefan model, 214, 215 MBOP (Mechanical Balance of Plant), 281 MCFCs, see Molten carbonate fuel cells mcp (heat capacity flow rate), xxiv, 429 MEAs, see Membrane electrode assemblies Mean flow velocity (¯u), xxv Mean free time (𝜏), xxvi Mechanical Balance of Plant (MBOP), 281 Mechanical driving forces, 119 Mechanical integrity, membrane, 126, 337 Mechanical work, 27 Mediator approach, 288 Mediator-free approach, 288 Membranes, 275–276, 304–307, 337, 414 Membrane electrode assemblies (MEAs), 276, 308–309, 327–329 Membraneless fuel cells, 289–290 Mercury porosimetry, 266 Metals, 50, 130–131, 335–336 Metal–air cells, 290–291 Metal–ceria cermets, 332 Metal hydride, 358–360, 364, 365, 473 Metal macrocycles, 316–317 Metal plates, in flow structures, 197 Methanation, selective, 411–412 Methane, 331, 365, 400, 411–412, 492, 496, 526 Methanol, 313–314, 360–361, 365, 496, 528 Microporous layers, 312 Microstates, 28–29 MIECs, see Mixed ionic–electronic conductors Migration energy barriers, 161–162 Mik (coupling coefficient of flow and flux), xxiv, 119 Minimum temperature difference (dT min,set ), 431–440 Mixed ionic–electronic conductors (MIECs), 152–153, 283–284, 334 Mobility (u), xxv, 129–131, 133, 147 Models, fuel cell, 203–231 basic structure, 203–206

CFD for, 227–230, 453–455 equations for, 548 importance of, 20–21 limitations of simple, 447–448 1D, 206–227 Mo-doped GT, 333 Molar flow rate (𝑣), xxv Molar flux ( J), xxiv, 141 Molar mass (M), xxiv Molar quantities, 32–33 Molar volume, 151 Moles, number of (N), xxiv Molecular orbitals, 534 Mole fraction (x), xxv Molten carbonate fuel cells (MCFCs), 13, 280–282, 298–300, 355 Momentum, 449, 450, 535 𝜇, see Electrochemical potential 𝜇, see Chemical potential; Viscosity Multicomponent diffusion model, 214, 215 Multielectron systems, 540 n (electrons transferred in reaction), xxiv, 44–45 N (number of moles), xxiv NA , see Avogadro’s number Nafion, xxvi, 136–141, 143–146, 221, 304–307 NASA, 385 National Emission Inventory (NEI), 500 Natural gas, 371–372, 486, 496 Natural greenhouse effect, 490–491 Neat hydrogen, 365 NEI (National Emission Inventory), 500 Nernst equation, 53–54, 56–59, 108–112, 176 Nernstian losses, 170, 178–179 Nernst voltage, 176–177 Net efficiency, 376, 378, 381–383, 474 Net electrical power, 376, 379 Net energy flows, 509–510 Net power output, 473 Net reaction rate, 85–86 Neutral system water balance, 373 Neutral water balance, 375 Newtonian fluids, 184n2 ng (number of moles of gas), xxiv NH4 BH4 (ammonium borohydride), 287 Nickel–YSZ (Ni–YSZ) cermet anodes, 284, 329–332, 339 Ni-GDC cermet, 332 Nitrates, 492 Nitrogen gas, 467, 527 Nitrogen oxides, 504, 505 Nitrous oxide, 492 Ni–YSZ anodes, see Nickel–YSZ cermet anodes Nominal diffusivity, 173 Nonideal gases, 50 Nonideal solutions, 50 Nonspontaneous processes, 29, 42 Nonstandard fuel cells, 284–298 Normalizability, of wave function, 535 No-slip condition, 186 Nusselt number, see Sherwood number (Sh) Nyquist plots, 248–258

INDEX Ohmic losses, 122–123 from current interrupt measurements, 263 defined, 20, 117 and fuel cell performance, 204 on Nyquist plots, 249, 250 in 1D models, 212, 215, 217 in PEMFCs, 246 Ohmic overpotential, 220 Ohmic overvoltage (𝜂 ohmic ), xxvi, 146, 223–224 Ohmic resistance, 250–251 𝜔 (angular frequency), xxvi 1D fuel cell models, 206–227 considerations with, 227 examples of, 216–217 flux balance in, 208–210 gas depletion effects in, 224–227 governing equations for, 213–216 simplifying assumptions for, 210–213 1D PEMFC models: governing equations for, 214–216 j–V curve predictions from, 219–224 simplifying assumptions for, 211–213 SOFC models vs., 207, 209, 210 1D SOFC models: of anode-supported structures, 213 gas depletion effects, 224–227 governing equations for, 215 j–V curve predictions from, 216–219 PEMFC models vs., 207, 209, 210 simplifying assumptions for, 211–213 One-dimensional electron gases, 536–537 Open systems, 9, 10 Operating fuel rich (term), 401 Operating temperature, 466, 469 Operating voltage, 63, 68–69 Orbitals, 533–534 ORR, see Oxygen reduction reaction Outlet conditions, 459 Overall efficiency (𝜀O ), xxv, 371, 381, 510 Overpotential, 220, 452–453 Overvoltage (𝜂), xxvi. See also Activation overvoltage (𝜂 act ) concentration, xxvi, 175 and current density, 98 ohmic, xxvi, 146, 223–224 in 1D fuel cell modeling, 216, 218–219 in 1D PEMFC models, 223–224 in 1D SOFC models, 226, 227 Oxidation: carbon, 338 defined, 15 on doped ceria, 331 ethanol, 314 hydrogen, 15, 78, 100, 101, 313 methane, 331 methanol, 313–314 partial, 400–401 selective, 412–413 and standard electrode potentials, 45–46 Oxide electrolytes, 161–163 Oxygen, 333, 401, 467, 521 Oxygen-ion-conducting perovskite oxides, 323–325 Oxygen reduction reaction (ORR), 15, 100, 314–317 Ozone, 490n.2, 503

p, see Pressure P, see Power (parasitic subscripts), xxvi P P (product subscripts), xxvi Pacific Northwest National Laboratory microfuel processor, 363 PAFCs, see Phosphoric acid fuel cells Palladium membrane separation, 414 Parallel flow, 197, 198 Parallel impedance elements, 253–255, 257 Parallel–serpentine flow, 198, 199 Parasitic power, 355, 376 Parasitic power load (X), xxv, 376 Parasitic subscripts (P ), xxvi Partial combustion (partial oxidation), 400–401 Partial electronic (hole) conductivity, 325 Partial oxidation reforming, 396, 397, 400–402 Particulate matter, 504, 505 Passive cooling, 353, 354 Pauli principle, 535 PBI (phosphoric acid doped polybenzimidazole), 306 PCFCs (protonic ceramic fuel cells), 294–295 PEEK (polyetheretherketone) materials, 305–306 PEM electrolysis cells, 297–298 PEMFCs, see Polymer electrolyte membrane fuel cells Percolation theory, 329, 330 Perfluorinated polymers, 304–305 Performance, 94–97, 107, 303. See also Fuel cell performance Periodic table of elements, 543 Permeability, 240, 266, 267, 450, 456–458 Perovskite oxides, 318, 323–326, 332–334 Phase factor (𝜙), xxvi 𝜙 (electrical potential), xxvi, 43 Phosphates, 307 Phosphoric acid doped polybenzimidazole (PBI), 306 Phosphoric acid fuel cells (PAFCs), 13, 274–275, 295, 298–300 Physical constants, 517 Physical domains, 454 Pinch point analysis, 424–440 Pinch point temperature (dT min ), 437–440 Planar interconnection configurations, 349–350 Planck’s constant (h), xxiv, 517 Plates: bipolar plate stacks, 338, 349, 350, 354 flow between, 185–186 flow field, 17 metal, in flow structures, 197 Platinum alloys, 313–316 Platinum catalysts, 313, 315 Platinum dissolution, 337–338 Platinum-free catalysts, PEMFC, 316–317 Poisoning: at anodes, 274, 331 carbon monoxide, 101, 286, 314, 362 of catalysts, 308, 314, 316, 410 at cathodes, 336 chromium, 336 from external reforming, 362 and fuel processing subsystems, 393 sulfur, 101, 274, 331 Polarization curves, 465

575

576

INDEX Pollution, air, see Air pollution Polyetheretherketone (PEEK) materials, 305–306 Polymer electrolytes, 13, 135–146, 304–308 Polymer electrolyte membrane fuel cells (PEMFCs), 275–277, 303–317 advantages of, 276 catalysts for, 103–105, 107, 308–317 CFD modeling of, 228–230 cooling for, 354 current density and flux balance in, 208 described, 13 disadvantages of, 276 durability and lifetime of, 337–338 electrode materials, 308–313 electrolyte materials, 304–308 external humidifiers for, 351 fuel cell stacking in, 349 ion conduction in, 141 modeling basic, 206, 207 ohmic losses in, 246 1D models, 185–190, 207, 209 other fuel cells vs., 298–300 SOFCs vs., 13–14 solid-acid fuel cells and, 295 test stations for, 242–243 volume conditions for, 457 Polymer–inorganic composite membranes, 307 Polymorphism, 322 Polytetrafluoroethylene (PTFE, Teflon), 304, 312 Porosity (𝜀), xxv defined, 266 effective, 266 and effective diffusivity, 174 in ex situ characterization, 240, 266 and mass conservation, 449 as volume condition, 456, 457 Porous bounded Warburg model, 255–257 Porous transport layer, see Gas diffusion layer (GDL) Portable fuel cell systems, 347, 348, 383–387, 463–475 Postulates, quantum mechanical, 532, 534–535 Potassium hydroxide, see KOH Potential(s): chemical, xxvi, 50, 51 electrical, xxvi, 43 electrochemical, xxvi, 51, 56, 159–160 electrode, 44–46, 80–81, 529 Galvani, 87–91 interfacial, 88–89 of reaction at equilibrium, 87–89 and reaction rate, 89–94 thermodynamic, 29–32 work, 37, 39 Potential energy, 535, 538 Potentiostatic techniques, 241 Potientiostats, 242 Power (P), xxiv. See also Stationary combined heat and power (CHP) systems combined heat and, 371 consumption of, 473 conversion factors for, 518 defined, 7 from fuel cells, 19 net electrical, 376, 379

parasitic, 355, 376 specific, 7 Power conditioning, 364 Power conditioning devices, 378, 379 Power density (P), xxiv, 7, 11, 19–20, 385–386, 466 Power electronics subsystem, 364, 366–369, 372, 378–379 Power inversion, 364, 366, 368 Power regulation, 364, 366, 367 Power supply management, 368, 369 Preleached platinum alloy catalysts, 316 Pressure (p), xxiv conversion factors for, 518 for current–voltage measurements, 245 and Gibbs free energy, 40 and mass transport in flow channels, 188–191 and Nernst equation, 53–54 operation of thermodynamic engine at constant, 40–42 and palladium membrane separation, 414 and reversible cell voltage, 48–50 and thermodynamic potential, 30–32 and viscosity, 186 Pressure resistance, 475 Pressure swing adsorption (PSA), 413 Principal quantum number, 534 Process chain analyses, see Life cycle assessments (LCAs) Products, xxvi, 18, 84–85 Protons, movement of, 140–141 Proton-conducting perovskite oxides, 325–326 Proton conduction, 306, 307 Proton conductivity, 325–326 Protonic ceramic fuel cells (PCFCs), 294–295 PSA (pressure swing adsorption), 413 Pt/C catalyst approach, 313 PTFE (polytetrafluoroethylene, Teflon), 304, 312 Pulse-width modulation, 366, 368 Pumps, cooling by, 354–355 PureCell, xxvi, 275 Pure components, 50 Purging, 414, 494 Pyrochlore-type oxides, 333 Q, see Charge; Heat q (fundamental charge), xxiv Qh (adsorption charge), xxiv Qm (adsorption charge, smooth catalyst surface), xxiv Quantity, change in (Δ), xxv Quantity of formation subscript (f ), xxvi Quantum mechanics, 104–107, 531–541 Quantum number of angular momentum z component, 534 R, see Resistance R (ideal gas constant), xxiv (reactant subscripts), xxvi R Radial frequency, 248 Radiative heat transfer, 415 Ragone plots, 384–386 Rate of reaction, see Reaction rate(s) Raw materials, 482, 484 RC circuits, 252–255 Re, see Reynolds number Reactants, 17, 65, 67, 170 Reactant concentration, 95, 174, 176–178, 181 Reactant crossover, 132 Reactant subscripts (R ), xxvi

INDEX Reaction(s): change in (rxn), xxvi electrons transferred in, xxiv exchange current density and sites of, 96–97 variations in reaction kinetics and, 100–103 Reaction enthalpies, 34–37 Reaction kinetics: activation energy and, 82–85 Butler–Volmer and Nernst equations, 108–112 and catalyst-electrode design, 103–104 and catalyst selection, 107–108 charge transfer reactions, 82–84 defined, 77 electrode, 77–82 equations for, 548 exchange currents and electrocatalysis, 94–97 net rate of reaction, 85–86 in 1D fuel cell models, 212 potential and rate of reaction, 89–94 potential at equilibrium, 87–89 and quantum mechanical framework for catalysis, 104–107 rate of reaction at equilibrium, 86–87 simplified activation kinetics, 97–100 and spontaneity, 42–43 variations in reactions and, 100–103 Reaction losses, 171 Reaction rate(s), 81–82, 84–87, 89–94, 177–178 Reaction rate constant (f ), xxiii Reaction rate per unit area (V), xxv Reactors, 353, 409–411, 416–417, 428 Real (practical) efficiency of fuel cells, 62–65 Rectangular flow channels, 189–190 Redox flow batteries, 296 Reduction, 15, 45–46, 100, 314–317, 321 Reference state (0 ), xxvi Reformate stream, 373 Reforming, 394, 396–409 anaerobic digestion, 408–409 autothermal, 396, 397, 402–407 external, 361–363 gasification, 407–408 of hydrogen carriers, 361–363 internal, 361, 362, 393–394 partial oxidation, 396, 397, 400–402 steam, 396–400, 486 Relaxation parameters, CFD, 460 Renewable fuels, 409 Residence time, 416 Resistance (R), xxiv, 124–128 additive nature of, 127–128 and area, 124–126 area-specific, xxiii, 124–126, 146 contact, 127 defined, 246 electrolyte, 128 Faradaic, xxiv, 251 ionic (electrolytic), 128 mass transport, 328 ohmic, 250–251 in 1D PEMFC models, 223 pressure, 475 and thickness, 126–127 voltage loss due to, 121

Resistivity (𝜌), xxvi Resistors, 251, 257 Reversibility, in thermodynamics, 34 Reversible cell voltage, 34, 43, 47–60 Reversible efficiency of fuel cells, 60–63, 68–69 Reversible fuel cells, 68–71, 297–298 Reversible thermodynamic efficiency, 63, 68, 71 Reynolds number (Re), xxv, 184, 189, 190 Rf (Faradaic resistance), xxiv, 251 𝜌, see Density 𝜌 (resistivity), xxvi Ruthenium, 314 (change in reaction), xxvi rxn S, see Entropy Sabatier principle, 107–108 Saddle points, 541 SAFCell (company), 295–296 SAFCs (solid-acid fuel cells), 295–296 Salt bridges, 59 Salt water batteries, 55 Samaria-doped ceria (SDC), 320 S∕C (steam-to-carbon ratio), xxv, 394 Scenario analysis, 437–440 Schrödinger, E., 531–532 Schrödinger equation, 532, 535–540 SDC (samaria-doped ceria), 320 Sealants, degradation of, 338, 340 Sealing, SOFC, 336, 349–351 Second law of thermodynamics, 27–29, 431 Selective methanation, 411–412 Selective oxidation, 412–413, 428 Self-heating, 354 Series impedance elements, 252–253, 257 Serpentine flow, 198, 199, 228–230 Sets, fuel cell, 347 Sh, see Sherwood number Shear stress (𝜏), xxvi, 184 Sherwood number (Sh), xxv, 191, 192 Shorting, 127, 152 SI (International System of Units), 7 Siemens-Westinghouse, 283, 351, 352 Sievert’s law, 414 𝜎, see Conductivity 𝜎 (Warburg coefficient), xxvi, 255 Single-chamber flow structures, 183 Single-chamber SOFCs, 291–292 Single-phase AC power, 366, 368 Sintering, 410 (stack subscripts), xxvi SK Slip boundary conditions, 186 Slow-scan j–V curves, 245 Small-signal voltage perturbations, 249 Sodium borohydride, 287 SOFCs, see Solid-oxide fuel cells Software, CFD, 448 Solar cells, 9–11 Solid-acid fuel cells (SAFCs), 295–296 Solid-acid membranes, 307–308 Solid-oxide fuel cells (SOFCs), 282–284, 291–294, 317–336 advantages of, 284 catalyst materials, 326–329 cooling for, 355 described, 13

577

578

INDEX Solid-oxide fuel cells (SOFCs), (continued) direct flame, 292–293 disadvantages of, 284 durability and lifetime of, 338–340 electrode materials, 326–336 electrolyte materials, 317–326 fuel cell stacking in, 351, 352 interconnect materials, 335–336 ionic conduction in, 146–151 liquid-tin anode, 293–294 materials for, 303 mixed ionic–electronic conductors in, 152–153 modeling basic, 206, 207 1D models, 207, 209, 216–219, 224–227 other fuel cells vs., 13–14, 298–300 protonic ceramic fuel cells and, 294 reaction kinetics in, 101, 102 sealing materials, 336 single-chamber, 291–292 solutions from model of, 460, 461 test stations for, 242–244 volume conditions for, 456 Solid-oxide fuel cell (SOFC) systems, 462–475 Solutions, activity of, 50 Solution process, CFD, 459–461 Sooty carbon deposits, 331 Space velocity (SV), 416–417 Species conservation equations, 449–451 Species source (species sink), 450 Specific activity, 315–317 Specifications, system, 447 Specific energy, 8 Specific power, 7 Specific surface area, 315 Spontaneity, 42–43, 46 Spontaneous processes, 29, 42 SR, see Steam reforming Stacks, fuel cell: in fuel cell subsystems, 348–352 in fuel cell system design, 475 hot streams related to, 424, 427, 428 for portable SOFC systems, 463, 465–466 Stack subscripts (SK ), xxvi Standard electrode potentials, 44–46, 529 Standard state (0 ), xxvi, 33, 35 Standard temperature and pressure (STP), 33 Starvation, fuel cell, 17 Stationary combined heat and power (CHP) systems, 369–383 Stationary waves, 532 Steady state, 241, 242, 244–245 Steam reforming (SR), 396–400, 486 Steam-to-carbon ratio (S∕C), xxv, 394 Step-down converters, 366, 367 Step-up converters, 366, 367 Stoichiometric amount, 401 Stoichiometric coefficient (𝜆), xxvi, 52 Stoichiometric number, 225, 226 Stoichiometry factor (𝜆), 64, 65 Storage density, volume, 359, 549 Storage effectiveness, 357–358 Storage efficiency, 359 STP (standard temperature and pressure), 33

Strain rate (𝜀), xxv, 184 Strangulation, fuel cell, 18 Strontium-doped lanthanum manganite (LSM), 152–153 Structure determinations, 240, 267 Structured grids, 454 Subscripts, xxvi Sulfates, 491–492 Sulfonated hydrocarbon polymers, 305–306 Sulfur oxides, 504, 505 Sulfur poisoning, 101, 274, 331 Superprotonic phase transitions, 295 Superscripts, xxvi Supply chains, 482–485, 507–508 Supply chain analyses, see Life cycle assessments (LCAs) Supply management devices, 379 Supply rates, 467 Supply temperature, 429 Surface area, 96–97, 240, 266–267, 315 SV (space velocity), 416–417 Symmetry conditions, 459 (system subscripts), xxvi SYS Systems, fuel cell, 347–389 CCHP, 382–383 CHP, 369–383, 425 equations for, 549 fuel cell subsystem, 348–352 fuel processing subsystem, 357–365 goals of, 347 portable, 383–387, 463–475 power electronics subsystem, 364, 366–369 thermal management subsystem, 353–357 System actuation, by control systems, 368 System design, 447–477 and goals, 347 portable fuel cell sizing, 383–387 solid-oxide fuel cell system, 462–475 stationary combined heat and power system, 369–383 via computational fluid dynamics, 447–462 System monitoring, 368 System subscripts (SYS ), xxvi T, see Temperature t (thickness), xxv Tafel equation, 97–99, 253 Tafel slope, 98 Target temperature, 429 𝜏 (mean free time), xxvi 𝜏 (shear stress), xxvi, 184 𝜏 (tortuosity), 174, 456–458 TEC, see Thermal expansion coefficient Teflon (polytetrafluoroethylene) (PTFE), 304, 312 TEM (transmission electron microscopy), 267 Temperature (T), xxv in activation kinetics, 99–100 and conductivity, 150 and current density, 99–100 for current–voltage measurements, 245 and entropy, 37 and exchange current density, 96 and Gibbs free energy, 38, 40 near-surface, 492, 493 and Nernst equation, 53–54 operating, 466, 469 and palladium membrane separation, 414

INDEX and reaction enthalpies, 36–37 and reversible cell voltage, 47–49 for SOFC operation and fabrication, 338 for SOFC testing, 244 and standard state conditions, 33 thermodynamic engine at constant, 40–42 and thermodynamic potential, 30–32 and viscosity, 184 Temperature-dependent thermodynamic voltage (ET ), xxiii Temperature difference between hot and cold streams (dT), 426, 427 Temperature–enthalpy (T –H) diagrams, 432–437 Temperature profiles, 460, 462 Test station requirements, 242–244 Thermal balance, 65–66, 69–71, 468–471 Thermal bottleneck, 28 Thermal compatibility, 327–329 Thermal conductivity (k), 451, 456–458 Thermal data, for pinch point analysis, 429–431 Thermal decomposition, 400 Thermal expansion coefficient (TEC), 329, 330, 336, 351 Thermal fuel cell modeling, 227 Thermal gradients, 353 Thermal management subsystem, 353–357, 373, 379–380, 423–441 Thermodynamics, 25–72 defined, 25, 26 equations for, 547 first law of, 26, 27 fuel cell efficiency, 60–65 Gibbs free energy, 37–46 and internal energy, 26–27 molar quantities, 32–33 reaction enthalpies, 34–37 reversibility in, 34 of reversible fuel cells, 67–71 reversible voltage variations, 47–60 second law of, 27–29, 431 standard state conditions and, 33 thermal and mass balances in fuel cells, 65–67 Thermodynamic data, 520–528 Thermodynamic engines, 40–42 Thermodynamic ideal voltage (E, Ethermo ), xxiii. See also Reversible cell voltage Thermodynamic plots, 432–437 Thermodynamic potentials, 29–32 Thermodynamic standard state, 33 Thermoneutral voltage, 66, 69 Thickness (t), xxv Three-phase power, 366 Time, 241–242, 416 Time-independent wave function, 532 Tortuosity (𝜏), 174, 456–458 Total annual CO2 emissions, 501–502 TPBs (triple phase boundaries), 103 Transfer coefficient, see Charge transfer coefficient (𝛼) Transmission electron microscopy (TEM), 267 Transport, charge, see Charge transport Transport losses, see Concentration losses Triple phase boundaries (TPBs), 103 Triple phase zone, 309 Tubular geometries, 351, 352, 399 Tungsten bronze oxides, 333

Turbulent flow, 186–187 Turnover frequency, 107–108 Two-phase flow models, 212 u, see Mobility U, see Internal energy u¯ (mean flow velocity), xxv U.S. Department of Energy, 362 U.S. Environmental Protection Agency (EPA), 488, 500 United Technologies Corporation (UTC), 279 Universal gas constant, 517 Unrecovered heat, 415 Unstructured grids, 454 Upfront size cost, 386–387 UTC (United Technologies Corporation), 279 Utility grid, 379 Utility-scale stationary power generation, 347, 348 V, see Voltage; Volume 𝑣 (hopping rate), xxv, 156–158 𝑣 (molar flow rate), xxv V (reaction rate per unit area), xxv 𝑣 (velocity), xxv Vacancies, 129, 131, 318, 320 Vacancy concentration, 148, 151 Vacancy fraction (xV ), xxv, 151 Valence, 142 van’t Hoff isotherm, 52 Vaporization, latent heat of, 61 Variable parasitic power loads, 376 Vehicles, 502, 503, 505 Vehicle mechanism, 135–136 Velocity (𝑣), xxv Vertical plate stacks, see Bipolar plate stacks Viscosity (𝜇), xxvi, 120, 184–186, 456–458 Volatile organic compounds (VOCs), 502, 504, 505 Volatilization, chromium, 339–340 Volcano plots, 107–108 Voltage (V), xxv. See also Current–voltage (i–V) curves; Reversible cell voltage in characterization techniques, 240 and charge transport, 121–123 in fuel cell models, 203–204 of fuel cells, 20 from fuel cell subsystems, 348–349 as fundamental electrochemical variable, 241–242 and Gibbs free energy, 43–44 Nernst, 176–177 in 1D PEMFC models, 224 in 1D SOFC models, 218, 226 operating, 63, 68–69 perturbations in, 248, 249 reversible variations in, 47–60 steady-state value of, 241 thermodynamic, xxiii thermoneutral, 66, 69 Voltage efficiency, 63 Voltage gradients, 157, 160–161 Voltage profile, fuel cell, 88 Voltage rebound, 263 Volume (V), xxv, 30–33, 135, 455, 459, 517 Volume storage density, 359, 549 Volumetric air flow rate, 468 Volumetric energy density, 8, 11, 12, 357

579

580

INDEX Volumetric power density, 7, 11 Vulcan XC-72, 310 W, see Work W (watt), 7 Wall conditions, 459 Warburg coefficient (𝜎), xxvi, 255 Warburg elements, 255–256 Waste heat, 496 Water: absorption of, 136–138 back diffusion in, 141 and efficiency of fuel cells, 61 electrolysis of, 68–69, 297 in flux balance, 208–210 in fuel cell modeling, 207 movement of protons and, 140–141 production rate, 468 thermodynamic data, 522 Water balance, 373, 375 Water content (𝜆), xxvi, 137–140, 144–145, 221–223 Water diffusivity, 143–146, 221 Water–gas shift (WGS) reaction, 39, 398, 399, 410–411 Water–gas shift reactors, 409–411, 428 Water management, 304–305 Water vapor, 136, 138–139, 207, 212, 523 Watt (W), 7 Wave functions, 532, 534–535 Weight, 142–143, 517 WGS reaction, see Water–gas shift reaction

Wheel-to-wheel analyses, see Life cycle assessments (LCAs) Wilson, M. S., 309 Window pane designs, 349 Work (W), xxv, 27, 28, 39–42 Work potential, 37, 39. See also Gibbs free energy (G, g) x (mole fraction), xxv X (parasitic power load), xxv, 376 xV (vacancy fraction), xxv X-ray diffraction (XRD), 267 X-ray photoelectron spectroscopy (XPS), 267–268 Yield, xxv, 394, 410 Yttria-stabilized zirconia (YSZ), 146–150, 161–163 anode compatibility with, 329, 332 in anodes, 284, 329–332, 339 electrolytes of, 318–320, 338–339 LSM–YSZ cathodes, 334, 339 perovskite oxides vs., 324, 325 in SOFCs, 282, 283, 318–320 Yttrium-doped barium zirconate (BZY), 326 Z, see Impedance z (height), xxv 0 (reference state, standard state), xxvi zi (charge carried by charged species), 118 Zero-cost cathode catalyst, volumetric catalytic activity, 316 Zinc–air cells, 290–291 Zirconia, yttria stabilized, see Yttria-stabilized zirconia (YSZ) Z-profile, calculating, 215

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