Solid Oxide Fuel Cell Technology: Principles, Performance and Operations

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i

Solid oxide fuel cell technology

ii

Related titles: Materials for fuel cells (ISBN 978-1-84569-330-5) This authoritative reference work provides a comprehensive review of the materials used in hydrogen fuel cells, which are predicted to emerge as an important alternative energy option in transportation and domestic use over the next few years. The design and selection of the materials is critical to the correct and long-term functioning of fuel cells and must be tailored to the type of fuel cell. The book looks in detail at each type of fuel cell and the specific material requirements and challenges. Chapters cover material basics, modelling, performance and recyclability. Materials for energy conversion devices (ISBN 978-1-85573-932-1) The term electroceramic is used to describe ceramic materials that have been specially formulated with specific electrical, magnetic or optical properties. Electroceramics are of increasing importance in many key technologies including microelectronics, communications and energy conversion. This innovative book is the first comprehensive survey on major new developments in electroceramics for energy conversion devices. It presents current research from leading innovators in the field. Solid-state hydrogen storage: Materials and chemistry (ISBN 978-1-84569-270-4) The next several years will see an emergence of hydrogen fuel cells as an alternative energy option in both transportation and domestic use. A vital area of this technological breakthrough is hydrogen storage, as fuel cells will not be able to operate without a store of hydrogen. The book focuses on solid-state storage of hydrogen. Part I covers storage technologies, hydrogen containment materials, hydrogen futures and storage system design. Part II analyses porous storage materials, while Part III covers metal hydrides. Complex hydrides are examined in Part IV, and Part V covers chemical hydrides. Finally, Part VI is dedicated to analysing hydrogen interactions. Details of these and other Woodhead Publishing materials books can be obtained by: • visiting our web site at www.woodheadpublishing.com • contacting Customer Services (e-mail: [email protected]; fax: +44 (0) 1223 893694; tel.: +44 (0) 1223 891358 ext. 130; address: Woodhead Publishing Limited, Abington Hall, Granta Park, Great Abington, Cambridge CB21 6AH, UK) If you would like to receive information on forthcoming titles, please send your address details to: Francis Dodds (address, tel. and fax as above; e-mail: [email protected]). Please confirm which subject areas you are interested in.

iii

Solid oxide fuel cell technology Principles, performance and operations Kevin Huang and John B. Goodenough

CRC Press Boca Raton Boston New York Washington, DC

WOODHEAD

PUBLISHING LIMITED

Oxford

Cambridge

New Delhi

iv Published by Woodhead Publishing Limited, Abington Hall, Granta Park, Great Abington, Cambridge CB21 6AH, UK www.woodheadpublishing.com Woodhead Publishing India Private Limited, G-2, Vardaan House, 7/28 Ansari Road, Daryaganj, New Delhi – 110002, India Published in North America by CRC Press LLC, 6000 Broken Sound Parkway, NW, Suite 300, Boca Raton, FL 33487, USA First published 2009, Woodhead Publishing Limited and CRC Press LLC © 2009, Woodhead Publishing Limited The authors have asserted their moral rights. This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. Reasonable efforts have been made to publish reliable data and information, but the authors and the publishers cannot assume responsibility for the validity of all materials. Neither the authors nor the publishers, nor anyone else associated with this publication, shall be liable for any loss, damage or liability directly or indirectly caused or alleged to be caused by this book. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming and recording, or by any information storage or retrieval system, without permission in writing from Woodhead Publishing Limited. The consent of Woodhead Publishing Limited does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from Woodhead Publishing Limited for such copying. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. Library of Congress Cataloging in Publication Data A catalog record for this book is available from the Library of Congress. Woodhead Publishing ISBN 978-1-84569-628-3 (book) Woodhead Publishing ISBN 978-1-84569-651-1 (e-book) CRC Press ISBN 978-1-4398-1336-2 CRC Press order number: N10103 The publishers’ policy is to use permanent paper from mills that operate a sustainable forestry policy, and which has been manufactured from pulp which is processed using acid-free and elemental chlorine-free practices. Furthermore, the publishers ensure that the text paper and cover board used have met acceptable environmental accreditation standards. Typeset by Replika Press Pvt Ltd, India Printed by TJ International Limited, Padstow, Cornwall, UK

v

Contents

Author contact details Preface

ix xi

1 1.1 1.2 1.3 1.4 1.5

Introduction to solid oxide fuel cells (SOFCs) A brief history of the solid oxide fuel cell (SOFC) Advantages of the solid oxide fuel cell Applications of solid oxide fuel cells Solid oxide fuel cell components and functionality Solid oxide fuel cell designs

1 1 4 5 6 7

2 2.1

Thermodynamics of the solid oxide fuel cell (SOFC) Electromotive force (EMF) and Gibbs free energy change (∆G) Effect of concentration on electromotive force (EMF) Heat effects in a galvanic cell The temperature coefficient of electromotive force (EMF) The pressure coefficient of electromotive force (EMF) The thermal and chemical expansion coefficients

2.2 2.3 2.4 2.5 2.6 3 3.1 3.2 3.3 3.4 3.5 4 4.1 4.2

Electronic properties of solids for solid oxide fuel cells (SOFCs) General considerations Redox 4fn energies and polaronic conduction Ligand-field d-electron energies Localized versus itinerant d electrons Applications Transport of charged particles in a solid oxide fuel cell (SOFC) General bulk transport theory Effect of electronic conduction in electrolyte on electromotive force (EMF)

10 10 11 14 16 17 19

23 23 25 26 29 32

41 41 47

vi

Contents

4.3 4.4

Application to electrolyte: steady-state PO 2 distribution Application to electrolyte: electronic leakage current density Application to interconnect: steady-state PO 2 distribution Application to interconnect: ionic leakage current density Pressure effect on electronic leakage current density in the electrolyte

4.5 4.6 4.7

5 5.1 5.2 5.3 5.4 5.5 5.6 6 6.1 6.2 6.3 6.4 6.5 6.6 7 7.1 7.2 7.3 7.4 7.5 7.6 8 8.1 8.2 8.3 8.4

Oxide-ion electrolytes in solid oxide fuel cells (SOFCs) Quality criteria Phenomenology Random-walk theory Fluorites Perovskites Other oxides Current, gas flow, utilization, and energy balance in a solid oxide fuel cell (SOFC) Introduction Fuel flow, current, and fuel utilization Air flow, current, and oxygen utilization Fuel consumption Calculating stack fuel composition of reformed natural gas Energy balance in a solid oxide fuel cell system Voltage losses in a solid oxide fuel cell (SOFC) Ohmic polarization Activation polarization Concentration polarization A combined activation and concentration polarization of the cathode Distributions of electromotive force and current density Effect of leakage flux on the voltage–current curve Direct current (DC) electrical efficiency and power of a solid oxide fuel cell (SOFC) Direct current electrical efficiency Efficiency, fuel utilization, and electrical power The maximum direct current electrical efficiency Effect of the system pressure on direct current electrical efficiency

50 55 57 62 63

67 67 69 73 74 79 82

85 85 86 87 88 90 92 98 98 101 110 125 127 136

141 141 148 150 154

Contents

9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10

Performance characterization techniques for a solid oxide fuel cell (SOFC) and its components Electrochemical impedance spectroscopy Galvanic current interruption Voltage–current curve characterizations ‘Helium–oxygen shift’ Fuel sensitivity Determination of fuel loss Electrical conductivity and sheet resistance measurements Determination of porosity and density of a porous ceramic body Diffusivity and diffusion conductance measurements

vii

156 156 163 165 166 170 173 177 179 180

10.4 10.5

Steam methane reforming and carbon formation in solid oxide fuel cells (SOFCs) Introduction Thermodynamics of steam methane reforming Partial pressure of oxygen ( PO 2) and Nernst potential (E) of the steam-reformed methane Kinetics of steam methane reforming Carbon formation: thermodynamics versus kinetics

11 11.1 11.2 11.3 11.4

Poisoning of solid oxide fuel cell (SOFC) electrodes Sulfur poisoning of the anode Silica poisoning of the anode Phosphorus poisoning of the anode Chromium poisoning of the cathode

197 197 202 207 213

12 12.1 12.2 12.3 12.4 12.5 12.6

Materials for solid oxide fuel cells (SOFCs) ZrO2-based solid oxide fuel cells CeO2-based solid oxide fuel cells LaGaO3-based solid oxide fuel cells Interconnects and cell-to-cell connectors Fabrication techniques Conclusions

220 220 233 242 248 257 265

References

269

Appendix 1: Thermodynamic data of selected chemical reactions and substances

278

Appendix 2: The standard heat of combustion of common fuels

280

10.1 10.2 10.3

183 183 184 189 191 193

viii

Contents

Appendix 3: Gas diffusion properties of a cylindrical tubular cathode substrate

289

Appendix 4: Molecular diffusivities of selected binary gas systems

298

Commonly used physical constants

299

Nomenclature Greek symbols Roman symbols

300 300 304

Index

319

ix

Author contact details

Dr Kevin Huang Stationary Fuel Cells Siemens Energy, Inc. 1310 Beulah Road Pittsburgh PA 15235 USA E-mail: [email protected]

Dr John Bannister Goodenough The University of Texas at Austin Department of Mechanical Engineering ETC Building 204 East Dean Keeton St. Austin TX 78705 USA E-mail: [email protected]

x

xi

Preface

Conventional fossil fuels will continue to be the primary energy resource to power human society for at least the next 50 years. How to convert fossil fuels into useful forms of energy in the most efficient manner with a minimal environmental impact will be the theme of future energy development and policy making. High-temperature solid oxide fuel cells (SOFCs) emerge as the leader in terms of conversion efficiency, fuel flexibility, and environmental impact among all types of power generation systems. They are in an excellent position to offer clean and efficient power generation. The compactness, modularity, and durability of high-temperature SOFCs find particular applications in stationary, distributed power generation, a niche market in which conventional heat engines find it difficult to compete. It is these unique advantages of SOFCs that have attracted lasting interest in research and commercialization worldwide over the last few decades. Despite the promise of SOFCs, product cost and system reliability are the two foremost obstacles presently prohibiting the modern SOFC technology from being commercialized. Both hurdles are essentially rooted in the high operating temperature, a unique characteristic of SOFCs that has both positive and negative impacts. The high operating temperature activates the fuel reforming and oxidation reactions and avoids the CO poisoning of the electrodes commonly encountered in low-temperature fuel cells. It also permits the excessive heat in the exhaust to be further recovered for primary heat utilization or for secondary electricity production. Unfortunately, a high operating temperature unequivocally requires exceptional properties of materials and modules. Chemical and electrical incompatibilities between ceramics and metals, as well as delicate management of different rates of thermal expansion among different cell components, are just a few examples of the pressing issues that need to be addressed in the product development of SOFC technology. Although the SOFC technology has been intensely pursued worldwide over the last few decades, a systematic narrative of the technology with an in-depth fundamental and engineering analysis is still lacking. The present book is intended to fill the gap and provide an important and useful scientific and engineering tool for researchers and engineers in the field of SOFCs.

xii

Preface

The book is the fruition of R&D and commercialization efforts on SOFC technology in which the authors have been heavily involved. Chapter 1 gives an overview of SOFC technology including its history, advantages, potential markets, basic functional components, and leading stack designs. Chapters 2, 3, 4, and 5 cover the theoretical background for the SOFC technology. The electromotive force (EMF) of a concentration cell operating on the same chemical reaction as that of the SOFC is connected to such thermodynamic quantities of the reaction as the changes of the Gibbs free energy, the enthalpy, and the entropy. Principles of solid state physics and chemistry are used to interpret the physical and chemical factors that influence the electronic and ionic transport properties of the several component materials used and to provide a guide for the design of new materials, including the catalytic behavior of the anode and cathode. Irreversible thermodynamics is applied to the transport equations for mixed ionic/electronic conduction in the electrolyte and interconnects in order to reveal how mixed conduction effects the profiles of the partial pressure of oxygen and the relationships among leakage, ionic, and external-load current densities in order to establish criteria for selecting an optimum electrolyte material. Chapters 6, 10, and 11 focus on the issues associated with SOFC operations, such as how to set the electrical current to achieve desirable stack compositions with the required fuel and oxygen utilizations, how to estimate the exhaust temperature based on the energy balance principle, how to avoid carbon formation during steam reforming of methane, and how to understand the mechanisms of electrode poisoning. Chapters 7, 8, and 9 analyze the performance of SOFCs in great detail and the factors that determine this performance. The losses in electrical performance and electrical efficiency by various mechanisms are particularly discussed. Finally, a thorough technical review on materials in SOFC is given in Chapter 12. It is the authors’ sincere hope that this book can be of great help to the development and commercialization of the SOFC technology. Kevin Huang Siemens Energy

John B. Goodenough The University of Texas at Austin

1 Introduction to solid oxide fuel cells (SOFCs)

Abstract: Technology advancement to address the world’s growing demand for clean and affordable energy will require simultaneous advances in materials science and technology in order to meet the performance demands of new power-generating systems. Fuel cells emerge as highly efficient, fuel flexible, and environmentally friendly electricity producing devices. These unique characteristics advantageously differentiate fuel cells from conventional heat engines for power generation and therefore have attracted worldwide attention – from research and development activities in institutes to commercialization efforts in industries – for the last few decades. In this chapter, the history, advantages, applications, and designs of solid oxide fuel cells (SOFCs) are briefly reviewed. Key words: solid oxide fuel cell, advantages, applications, functionality, stack design.

1.1

A brief history of the solid oxide fuel cell (SOFC)1, 2

Fuel cells have been known to science for more than 150 years. As early as 1839, the Swiss scientist Christian Friedrich Schoenbein first asserted the possibility of a fuel cell that combined hydrogen with oxygen.3 One month later, the English scientist William Robert Grove published the experimental observation of voltage in a concentration cell (called a ‘gas cell’ at the time) when combining hydrogen with oxygen in the presence of platinum.4 A few years later, in 1845, he published the paper ‘On the gas voltaic battery – voltaic action of phosphoros, sulphur and hydrocarbons’,5 which formally confirmed the technical feasibility of a fuel cell as a power-generating device. However, it was not until the end of the nineteenth century with the discovery by the German scientist Walther Nernst of the so-called ‘Nernst mass’6 of a ceramic material consisting of 85 mol% ZrO2 and 15 mol% Y2O3 that the key solid electrolyte material for modern SOFCs was identified. Since then, many mixtures of ZrO2 with the rare-earth and alkaline-earth oxides have been systematically studied, revealing a range of compositions with high oxide-ion conductivity. After electrochemistry was connected with thermodynamics, the basic principle that establishes the relationship between the chemical energy of a fuel and the voltage of a fuel cell was explained by H. von Helmholtz in 1882.7 In 1894, W. Ostwald correctly pointed out that 1

2

Solid oxide fuel cell technology

a fuel cell could produce electricity in a more efficient way than a conventional steam engine.8 Such a realization undoubtedly became a stimulant for pursuing fuel cells as potential highly efficient power-generating devices in the twentieth century. If the nineteenth century was considered as an era of curiosity in fuel cells, the twentieth century was certainly the epoch for fuel cells to become the subject of intense research and development (R&D) and commercialization efforts. The conceptual SOFC was probably first demonstrated in 1937 by the Swiss scientists Emil Bauer and Hans Preis using zirconia ceramics as the electrolyte, Fe3O4 as the cathode, and C as the anode.9 Clearly, the problems of stability of the electrode materials and gas-phase diffusion were not recognized at the time. However, more concentrated and systematic studies on SOFCs started after the pioneering 1943 work by the German scientist Carl Wagner, who first recognized the existence of oxygen vacancies in mixed oxides such as doped ZrO2, and attributed the observed electrical conductivity at high temperatures to the movement of these oxygen vacancies under a gradient of oxygen partial pressure.10 In 1957, Kiukkola and Wagner published another landmark work describing thermodynamic investigations with concentration cells based on the solid electrolyte Zr0.85Ca0.15O1.85.11 It was this work that laid the theoretical foundation for the modern solid-state electrochemistry of the SOFC. A few years later, two scientists, Joseph Weissbart and Roswell Ruka, from the Westinghouse Electric Corporation, reported in 1961 the first solid-electrolyte-based device for measuring the oxygen concentration of a gas phase with a concentration cell,12 which later led to their patent ‘A solid electrolyte fuel cell’ issued in 1962.13 Based on these initial efforts, a group of Westinghouse engineers developed and successfully tested the first tubular ‘bell-and-spigot’ SOFC stack from 1962 to 1963. This development eventually became the foundation of today’s cathode-supported, tubular seal-less SOFCs developed by Westinghouse/Siemens. During the same period, advances in electrode materials for SOFCs have also taken place. The most noticeable progress was in the evolution of the cathode material. It started with the noble metals such as platinum and transitioned to doped In2O314 and finally settled on today’s doped LaMnO3. The evolution of cathode materials was clearly driven by the performance requirement, viz. the capability to activate effectively the oxygen-reduction process. The unique electrical and catalytic properties possessed by rareearth, transition-metal perovskite oxides best satisfy the cathode requirement. However, the requirement for a thermal-expansion match between cathode and electrolyte has narrowed the practical cathode material to the doped LaMnO3 for ZrO2-electrolyte-based SOFCs. Another important material, developed by Meadowcroft in 1969, was the doped LaCrO3 perovskite that is stable in both oxidizing and reducing atmospheres;15 it immediately found use as an interconnect in SOFCs. A patent filed by Spacil in 1964 described

Introduction to solid oxide fuel cells (SOFCs)

3

a composite anode consisting of Ni metal with a ZrO2-based electrolyte that has remained the standard choice of anode for SOFCs.16 Historically speaking, the period from the 1970s to the 1990s marks an important era in the technical development of SOFCs. In the 1970s, the electrochemical vapor deposition (EVD) process was invented in Westinghouse by an engineer of genius, Arnold Isenberg, who demonstrated the making of a perfectly dense ZrO2 electrolyte thin film on the substrate of a porous, tubular substrate at relatively low temperatures. Based on this important invention, Westinghouse successfully manufactured and tested a series of SOFC generator systems in the range of 5–250 kWe from the 1970s to 1990s and clearly positioned itself as the world leader in modern SOFC technology. It was also during this period that various SOFC stack designs flourished, from tubular to planar in geometry, and alternative materials for the cathode, the anode, and the interconnect were also explored as the substrate. A real advancement of anode-supported planar SOFCs took place after the pioneering work of de Souza et al. of Berkeley National Laboratory, published in 1997;17 they essentially demonstrated that an electrolyte on a porous anode substrate can be co-fired at high temperatures into a dense thin film without invoking chemical reactions. The cathode was applied afterwards and sintered at much lower temperatures to minimize chemical reactions. As a result, the single-cell performance has been significantly improved, which in turn has allowed an anode-supported SOFC to operate at lower temperatures where commercially available oxidation-resistant alloys such as thermalexpansion-compatible ferritic steels can be utilized as interconnect materials for SOFC stacks. A majority of today’s SOFC designs adopt the anodesupported planar geometry based on considerations of cost and performance. However, the reliability and stability appear to be the leading issues for commercialization at the present time. Looking through the history of SOFCs, it is not difficult to find that the ZrO2-based materials have remained the mainstream electrolytes since the discovery by Nernst over 100 years ago. As early as 1990, Goodenough et al.18 had pointed out that high oxide-ion conduction can exist in the perovskites and hence in other structures than the classical fluorite structure, giving hopes for finding a new family of oxide-ion conductors in other crystal structures. This prediction was favorably vindicated by the noteworthy discovery of the high oxide-ion conductivity perovskite Sr- and Mg-doped LaGaO3 (LSGM) by Ishihara et al.19 in 1994, immediately confirmed by Feng and Goodenough20 in the same year, followed by a systematic characterization of the system by Huang et al.21, 22 The high oxide-ion conductivity and the crystallographic compatibility with cathode materials make LSGM even more attractive for low-temperature SOFCs. Mitsubishi Materials has recently demonstrated an excellent stack performance of an SOFC based on LSGM electrolyte operating at 800 °C.23

4

Solid oxide fuel cell technology

In summary, the major driver for sustaining the development of SOFC technology is the intrinsically high electrical efficiency compared with a conventional heat engine. After a century of scientific research and commercial engineering, development in the areas of materials, designs, and system integration has advanced dramatically. A thorough review of SOFC materials and fabrication techniques is given in Chapter 12. The feeling is that commercialization of the technology is on the horizon.

1.2

Advantages of the solid oxide fuel cell

The fuel cell is a device that directly converts the chemical energy in fossil fuels into electrical power in an electrochemical manner. Unlimited by the Carnot cycle, a fuel cell has an inherently higher electrical efficiency than conventional heat engines, particularly for the less than 1 MW class. Higher electrical efficiency infers a reduced CO2 emission per unit electricity produced if hydrocarbons are used as fuels; this influence has become increasingly important as we endeavor to minimize the emission of greenhouse gases in future power generation. Another conceivable benefit is the minimal environmental impact from a fuel-cell generator compared with a conventional heat engine. Owing to its relatively low operating temperature, the formation and therefore the emission of nitrogen oxides (collectively known as NOx) are negligible. Use of a desulfurizer subsystem in a fuel-cell generator ensures almost zero emission of sulfur oxides (collectively known as SOx). In addition, fuel-cell power generators are much quieter and exhibit less vibration than a conventional engine during operation; they therefore represent a competitive alternative in distributed, stationary power generation. For SOFCs operating at higher temperatures, there are added advantages. High-temperature operation, typically in the range of 600–1000 °C, not only provides high-quality waste heat, but also effectively activates the processes of reforming and electrochemical oxidation of hydrocarbon fuels in the presence of catalysts. This realization is technically important for several reasons. First, it opens the opportunity for SOFCs to use most hydrocarbon fuels, either in the gaseous or liquid state, provided that they are properly cleaned and reformed into simple fuels such as H2 and CO. This is in contrast to lowtemperature fuel cells such as proton exchange membrane (PEM) fuel cells where CO poisons the anode. The second is that excessive heat produced from the electrochemical oxidation of fuels can be utilized by the highly endothermic steam reforming reaction simultaneously occurring, which makes internal on-cell reformation possible. Such integration further increases the overall system efficiency. Co-production of heat and power, often known as combined heat and power (CHP), is the third added advantage of hightemperature SOFCs. The recovery of waste heat along with the production of electricity enables the total energy efficiency of such a system to be in the

Introduction to solid oxide fuel cells (SOFCs)

5

range of 85–90%. Another way of recovering waste heat is to combine a micro gas turbine with an SOFC stack to form a hybrid system. In order to maximize the electrical efficiency, the SOFC/micro gas turbine hybrid is often operated under pressurization that would boost both the performance of the SOFC stack and the effectiveness of the micro gas turbine. In order to be even more efficient, a bottom cycle steam turbine can be added into the above hybrid system. This is particularly preferable for generators over 100 MWe. The hybrid SOFC generator system has been reported by Siemens/ Westinghouse to achieve a net AC electrical efficiency of 53% for a 220 kWe class. When compared with another type of high-temperature fuel cell, the molten carbonate fuel cell (MCFC), the all-solid components used in an SOFC system avoid the corrosion issues caused by the liquid electrolyte of the MCFC system, which prolongs the lifetime of an SOFC. In fact, with over 35 000 operating hours at an acceptable degradation rate, a Siemens/ Westinghouse 100 kWe unit is the longest-running SOFC generator.

1.3

Applications of solid oxide fuel cells

Determined largely by the unique advantages presented above, the best application of SOFC systems is distributed, stationary power generation. Depending on the size of the SOFC generator, stationary power generation can be further grouped into the following markets. •

•

•

•

Residential. The SOFC is targeted for powering a home with a power rating of 1–10 kWe. Hot water, house heating, and chilling can also be provided as a by-product. Pipeline natural gas or coal gas is the fuel of choice. The net AC efficiency is expected to be >35%. Industrial. The SOFC is targeted for powering a small industrial unit such as a credit-card data processing center or a hospital that cannot tolerate a power outage. The power rating typically ranges from 100 to 1000 kWe. Quality heat can also be provided as a by-product. Pipeline natural gas can be used as the fuel. The net AC efficiency is expected to be >45%. Dispersed. An extension of industrial SOFC generators can be targeted for powering a larger industrial unit or a small community with a power rating of 2–10 MWe. Natural gas or coal-derived fuel is the fuel of choice. The net AC efficiency is expected to be >48%. Central. The largest SOFC generator system would have a power rating of 100 MWe. In such a system, producing electricity by the most efficient way is the ultimate goal. Therefore, a hybrid SOFC system is the design choice. Natural gas and coal-derived gas can be used as fuels. The net AC efficiency is expected to be >60%.

6

1.4

Solid oxide fuel cell technology

Solid oxide fuel cell components and functionality

There are four basic functional elements in an SOFC: electrolyte, cathode, anode, and interconnect. For tubular SOFCs, there is an added element, the cell-to-cell connector. The materials for these components are either ceramics or metals. The functionality of an electrolyte is to transport oxygen continuously and solely in the form of O2– from cathode to anode under a gradient of chemical potentials of oxygen. In order to enable O2– ion migration across the electrolyte, the cathode, which is supplied with oxygen, has to convert O2 into O2–, a process commonly known as oxygen reduction. Similarly, the anode of the fuel supply accepts O2– delivered by the electrolyte and converts it into H2O, CO2, and electrons by reacting with hydrogen or hydrocarbon fuels. The electrons required for the cathode reaction are released by the anode and arrive at the cathode via an external load, by which the production of electricity is realized. Figure 1.1 illustrates the working principle of an SOFC in tubular geometry. The overall driving force for an SOFC is the gradient of chemical potentials of oxygen existing between a cathode of high partial pressure of oxygen and an anode of low partial pressure of oxygen. The maximum cell voltage of a typical single SOFC with air as an oxidant typically reaches up to 1.2 V, depending on temperature, system pressure, and fuel composition. This voltage is obviously inadequate for any type of practical application. In order to build up a sufficiently high voltage and Air Cathode Electrolytes Anode Load

H2 + O2– = H2O + 2e′ e′

O2 + 2e′ = O2–

+

–

e′ Cell-to-cell connector

Current collector

Current collector

Interconnect

Fuel

1.1 A schematic of the working principle of SOFCs. Tubular SOFCs are used for illustrative purposes.

Introduction to solid oxide fuel cells (SOFCs)

7

power, multiple single cells have to be connected in series and/or parallel with the aid of interconnects and/or cell-to-cell connectors; the latter should be pure electronic conductors and oxide-ion insulators. Like a battery, each SOFC component also exhibits an internal resistance to either electronic or ionic current flow, often expressed as voltage loss. The terminal cell voltage is, therefore, the open-circuit voltage (or electromotive force, EMF, if no fuel loss is caused by any means; EMF is discussed further in Chapter 2) reduced by the individual voltage loss of each cell component. One radical requirement for an operational SOFC is to realize oxygen transport across the electrolyte in the form of O2–, but not in the form of O2. In order to achieve this mandate, dense barriers between air and fuel have to be established. For the one-end-closed tubular SOFC design, such barriers are easily achieved by the electrolyte and interconnect in a unique geometry, only allowing air and fuel to meet at the open-end where combustion occurs after the fuel is mostly utilized over the entire cylindrical surface. No physical sealing material is needed. However, for the planar SOFC design, sealing materials have to be applied along the perimeters of interconnect/electrodes and electrolyte/ electrodes in order to prevent air from mixing with fuels, which often presents a challenge to the reliability and stability of the planar SOFCs.

1.5

Solid oxide fuel cell designs

Owing to the preference for high performance and low operating temperature, modern SOFCs typically consist of a thin electrolyte film and a supporting substrate (porous or channeled dense body for gas transport). Substrates have been made of electrodes (cathode or anode), interconnect (metal or ceramic), or inactive insulator. From a geometric standpoint, the substrate can be made into either a tubular (cylindrical or flattened and ribbed) or a planar shape. The use of tubular geometry with the one-end-closed feature enables the seal-less design. This is the strongest advantage over planar geometry where gas seals are needed along the perimeters of the stack. However, a particular substrate may be advantageous in combination with one specific type of geometry. For example, a cathode substrate with a tubular geometry is an excellent marriage simply because it allows cell-to-cell connections in a stack to take place in a reducing atmosphere where inexpensive transition metals such as Ni and Cu can be used. Figure 1.2 shows a schematic of cellto-cell connections in a cathode-supported tubular SOFC stack. If cathode substrates are not used, noble metals are needed for connecting anode-supported cells into stacks in an oxidizing atmosphere. On the other hand, an anode substrate is a good choice for planar stacks. High-power-density, anode-supported single cells permit a reduced operating temperature at which economic and commercially available oxidation-resistant

8

Solid oxide fuel cell technology

Cathode bus

Fuel Air

Air

Air Interconnection Fuel electrode Electrolyte

Nickel felt

Air

Air

Air

Air electrode

Anode bus

1.2 A schematic of cell-to-cell connections in a cathode-supported tubular SOFC stack. Upper interconnect Cathode Electrolyte Anode Fuel Air

Lower interconnect

1.3 A schematic of cell configuration in an anode-supported planar SOFC stack.

alloys can be utilized to connect single cells into a stack. In such a design, the oxidation-resistant alloys essentially provide the mechanical support for the stack and function as interconnects and current collectors simultaneously. Figure 1.3 shows a schematic of an anode-supported planar SOFC configuration with metal interconnect. Gas channels for air and fuel deliveries on a dense metal interconnect are particularly illustrated. Porous metal substrates have attracted considerable interest in recent years for application in SOFCs. The conceivable advantages include robustness and cost effectiveness of thus-made cells and stacks. Challenges include

Introduction to solid oxide fuel cells (SOFCs) Cathode

Electrolyte

9

Ceramic interconnect

Air

Anode Porous insulating support Fuel channel

Fuel

1.4 A schematic of the ‘segmented-in-series’ design adopted by Rolls-Royce.

how to fabricate dense electrolyte and/or interconnect layers on the substrate at such a low temperature that significant oxidation and chemical reactions between underlying layers are not encountered. Chromium vaporization resulting from the chromium-containing metal interconnects during operation could also degrade the cathode performance in the presence of air and moisture. Multiple cells have also been deposited in series on an electrochemically inactive and electrically insulating substrate. This design, termed ‘segmentedin-series’, has unique advantages such as low fabrication costs. More importantly, such an SOFC stack operates at higher voltage and lower current for a fixed power rating. This feature could help reduce the power losses on current connections, which is particularly important for large-class SOFC generators. Figure 1.4 shows a schematic of cell-component arrangements in the ‘segmented-in-series’ design adopted by Rolls-Royce. Gas manifolding and current collection within the stack appear to be challenging.

2 Thermodynamics of the solid oxide fuel cell (SOFC)

Abstract: Thermodynamics is the theoretical foundation for any type of electrochemical concentration (or galvanic) cell. It depicts the fundamental relationship between thermodynamic quantities and electrical quantities; this relationship not only allows determination of thermodynamic properties of materials by accurate electrochemical methods, but also defines the maximum cell voltage of a specific chemical reaction and its dependence on concentration, temperature, and pressure. In addition, the volume changes that occur with variations in temperature, oxygen stoichiometry, and partial pressure of oxygen can also be understood by the principles of thermodynamics. In this chapter, the laws of thermodynamics are applied to elucidate the electrochemical and mechanical behaviors of components of SOFCs. Key words: electromotive force, Gibbs free energy change, temperature coefficient, pressure coefficient, thermal expansion coefficient, chemical expansion coefficient.

2.1

Electromotive force (EMF) and Gibbs free energy change (∆G)

When a system undergoes a reversible process under isothermal and isobaric conditions, the decrease in the Gibbs free energy, G, of the system equals w max ′ , the work (other than work of expansion) done by the system. The increment of such a process is represented by24 –dG = δ w max ′

[2.1]

For a galvanic cell system such as an SOFC, the work δ w max ′ is performed by transporting an electric charge across a voltage difference, i.e. from one electric potential to another as the result of the occurrence of a chemical reaction. The electrical work δ w max ′ done by the galvanic cell is the product of the charge transported and the electric potential difference, ∆φ (volts), and the units are joules. If dn moles of ions of valence z are transported through a voltage difference ∆φ maintained between the electrodes of a cell, then

δ w max ′ = zF × ∆φ × dn

[2.2]

where F is Faraday’s constant, 96 485 C/mol. If the transport is conducted reversibly, the electric potential difference between the electrodes of the cell 10

Thermodynamics of the solid oxide fuel cell (SOFC)

11

is called the electromotive force (EMF), E, of the cell, and

δ w max ′ = zF × E × dn = –dG

[2.3]

For transporting 1 mole of ions, equation [2.3] becomes ∆G

= –zFE

[2.4]

Equation [2.4] is known as the Nernst equation and is the basis for calculating the E value of a specific chemical reaction. Under a standard state,* equation [2.4] is rewritten as ∆G

o

= –zFE o

[2.4a]

o

where E refers to the EMF under the standard state. As the standard state of a component is simply a reference state to which the component in any other state is compared, it follows that any state can be chosen as the standard state, and the choice is normally made purely on the basis of convenience. For gaseous systems, the standard state is usually taken as the pure substance at ideal gas conditions, 1 bar pressure, and the temperature of the system. For a condensed matter system, the standard state is taken as the pure substance at the temperature of the system and 1 bar pressure.

2.2

Effect of concentration on electromotive force (EMF)

For a general chemical reaction aA + bB = cC + dD

[2.5]

in which the reactants and products do not occur in their standard states, a Cc a Dd [2.6] a Aa a Bb and, from equation [2.4], the EMF of the cell in which the above reaction is occurring electrochemically is ∆G

= ∆G o + RT ln

ac ad E = E o + RT ln Ca Db [2.7] zF aAaB where Eo = –∆Go/zF. To give an example, consider the individual electrode and overall electrochemical oxidation reactions of H2† Cathode: 0.5O2(g) + 2e′ = O2– *The standard state of a component of a system is usually chosen as being the pure component in its stable state of existence at the temperature of the system. This is called the Raoultian standard state. † In the following context, electrochemical oxidation is referred to as electro-oxidation.

12

Solid oxide fuel cell technology

Anode: H2(g) + O2– = H2O(g) + 2e′ Overall reaction: H2(g) (in fuel) + 0.5O2(g) (in oxidant) = H2O(g) (in fuel)

[2.8]

Equation [2.7] for this reaction with z = 2 leads to aH 2O PH 2 O E = E o – RT ln = E o – RT ln 1/2 2F 2 F aH 2 a O2 PH 2 PO1/22

[2.9]

for the system pressure Pt = 1 atm. It is worth noting that the ∆Go of a chemical reaction is generally available in the thermodynamic database; therefore Eo of the cell is known. Table A1.1 in Appendix 1 lists the temperaturedependent ∆Go of oxidation reactions of common fossil fuels. The variations of Eo with temperature resulting from an electro-oxidation by a galvanic cell are shown in Fig. 2.1. As evident in the figure, H2 and CO oxidations produce an Eo more sensitive to temperature whereas the Eo from CH4 and C oxidations are almost temperature independent. This observation implies that CH4 and C may be more suitable as fuels for low-temperature SOFCs. The standard Gibbs free energy change, ∆Go, for most of the fuel oxidation reactions listed in Table A1.1 of Appendix 1 were taken from reference 25. Under a non-standard state, the E of a specific fuel oxidation reaction can

1.100 CH4(g) + 2O2(g) = 2H2O(g) + CO2(g)

E o (V)

1.050

1.000

C(s) + O2(g) = CO2(g)

H2(g) + 0.5O2(g) = H2O(g) 0.950 CO(g) + 0.5O2(g) = CO2(g) 0.900

0.850 500

600

700

800 t (°C)

900

1000

2.1 Eo of various electro-oxidations of fuels as a function of temperature.

1100

Thermodynamics of the solid oxide fuel cell (SOFC)

13

1200

E

1000

E (mV)

800 1000 °C 900 °C 800 °C

600 Ni oxidation EMF (vs air)

400

200

0 0.1

1

10

100

1000

PH 2 O / PH 2 (atm)

2.2 Variations of E of H2O–H2 fuel with temperature and H2O/H2 ratio.

also be calculated with a known condition. Figure 2.2 illustrates an example of how the E of the H2 electro-oxidation reaction changes with T and logarithm of PH 2 O / PH 2, with air as oxidant PO 2 = 0.2 atm. The E equation is developed from the thermodynamic data as a function of T and PH 2 O / PH 2

 PH O  E = 1275 – 0.3171T – 0.0430 T ln  2   PH 2 

(mV)

[2.10]

As expected, E decreases with PH 2 O / PH 2 and T. Also plotted in Fig. 2.2 is the ENi (Ni oxidation EMF (vs air)) representing the Ni–NiO equilibrium below which the oxidation of Ni takes place. According to the figure, for example, it would require a mixture of 99.9% H2O–0.1% H2 to oxidize Ni metal at 1000 °C. Since Ni and NiO are both solids, the degree of freedom or the number of independent variables is unity under isobaric conditions according to the Gibbs phase rule. Therefore, the ENi is only dependent on T for a fixed oxidant such as PO 2 = 0.21 atm, but independent of PH 2 O / PH 2. It follows in an SOFC that the cell voltage becomes invariant at the ENi if oxidation of Ni occurs. Similarly for CO fuel, the E equation is written from the thermodynamic data as a function of T and PCO 2 / PCO  PCO 2  [2.11] E = 1462 – 0.4830 T – 0.0430 T ln  (mV)   PCO  Figure 2.3 plots E as a function of PCO 2 / PCO at three representative temperatures along with the Ni oxidation potentials. As evident in the figure, oxidizing Ni

14

Solid oxide fuel cell technology 1200

1000

E

E (mV)

800 1000 °C 900 °C 800 °C

600 Ni oxidation EMF (vs air)

400

200

0 0.1

1

10

100

1000

PCO2 /PCO (atm)

2.3 Variations of E of CO2–CO fuel with temperature and CO2/CO ratio.

requires lower PCO 2 / PCO than PH 2 O / PH 2 at a given temperature, suggesting that a CO2–CO mixture has a greater oxidizing ability than an H2–H2O mixture.

2.3

Heat effects in a galvanic cell

The change in the enthalpy of a system equals the heat q entering or leaving the system during a constant-pressure process only if the work of the volume change is the sole form of work performed on or by the system. If an electrochemical reaction is conducted in a galvanic cell, as a result of which electrical work w is performed, ∆H ≠ q. For a change of state at constant temperature and pressure ∆G

= q – w + p ∆V – T ∆S = q – w′ – T ∆S

[2.12]

If w′ = 0, then q = ∆G + T ∆S = ∆H. But, if the process involves the performance of work w′ and –w′ = – w max ′ = ∆G and if the process is conducted reversibly as in the case of reversible concentration cells, then from equation [2.3] q = T ∆S

[2.13]

The temperature-dependent ∆S(T) of a chemical reaction can be obtained from the following expression ∆S( T )

= ∆S298 +

∫

T

298

∆C p

T

dT

[2.14]

Thermodynamics of the solid oxide fuel cell (SOFC)

15

where ∆S298 is the entropy change of the reaction at 298 K; ∆Cp is the change of constant-pressure molar heat capacity of the reaction. For each component involved in the reaction, Cp typically follows the form of Cp = a + b × T + c × T 2 + d × T 3 + e × T –2

[2.15]

Substitution of equation [2.15] into equation [2.14] yields ∆S( T )

T  = ∆S298 + ∆a × ln   298 

+ ∆b × ( T – 298) + 1 ∆c × ( T 2 – 298 2 ) 2 1  + 1 ∆d × ( T 3 – 298 3 ) – 1 ∆ e ×  12 – 3 2 T 298 2 

[2.16]

o Under the standard state, the coefficients a, b, c, d, and e, and S 298 values for each constituent of a fuel oxidation reaction can be found in Tables A1.2 and A1.3 of Appendix 1, respectively. With equation [2.16], the variations of ∆ So with T for the fuel electro-oxidation reactions of interest are plotted in Fig. 2.4. The area between the T axis and each line represents the heat associated with the electro-oxidation reaction. Evidently, H2 and CO electro-oxidations are both exothermic (T ∆So < 0) with CO oxidation releasing more heat than H2 oxidation. The heat quantities associated with the direct CH4 and C electro-oxidations are less significant, as is evidenced by the very small areas. The difference is that the former reaction is weakly exothermic and the latter one is weakly endothermic. More heat released from the electro-

20 C(s) + O2(g) = CO2(g) 0

–20

–40 H2(g) + 0.5O2(g) = H2O(g)

∆S

o

(J/(K mol))

CH4(g) + 2O2(g) = 2H2O(g) + CO2(g)

–60

CO(g) + 0.5O2(g) = CO2(g)

–80

–100 850

950

1050

1150

1250

T (K)

2.4 Entropy changes of fuel oxidation reactions as a function of temperature.

1350

16

Solid oxide fuel cell technology

oxidation implies less chemical energy being converted into electrical energy and therefore lower electrical efficiency for H2 and CO fuels. In other words, C and CH4 could potentially be better fuels for achieving a higher electrical efficiency by a direct electro-oxidation. The factors that influence electrical efficiency will be further discussed in Chapter 8. A slightly endothermic effect for the C electro-oxidation reaction shown in Fig. 2.4 seems to suggest that to retain a direct electro-oxidation of C may require additional energy input. With the available T ∆So in Fig. 2.4, the heat production rate HPR (J/s) through the electro-oxidation of fuels can be estimated by

HPR =

Qfo × ( T ∆S o ) 22.4 × 60

[2.17]

where Qfo is the mass flow rate in slpm (standard liters per minute) of the fuel for 100% electrochemical utilization (see Chapter 6). For a partial utilization of a fuel in flow rate Qf by a galvanic cell to a degree of Uf, the HPR can be rewritten by HPR =

Qf × U f × ( T ∆S o ) 22.4 × 60

[2.17a]

The remaining unutilized fuel Qf × (1 – Uf) will be combusted and the heat associated with the combustion should be determined by the heat value of the fuel, as will be discussed in Chapter 8.

2.4

The temperature coefficient of electromotive force (EMF)

Differentiation of ∆G in equation [2.4] with respect to temperature at a constant pressure gives  ∂∆G   ∂E    = – zF   = – ∆S  ∂T  P  ∂T  P

[2.18]

Thus, for the cell reaction a ∆S  ∂E    = zF  ∂T  P

[2.18a]

makes the change of enthalpy of the reaction ∆H

 ∂E  = – zFE + zFT    ∂T  P

[2.19]

Application of ∆So of the electro-oxidation reactions shown in Fig. 2.4 to equation [2.18a] yields Fig. 2.5, where the (∂Eo/∂T)P is plotted against

Thermodynamics of the solid oxide fuel cell (SOFC)

17

0.10 C(g) + O2(g) = CO2(g)

(dEo/dT )P (mV/K)

0.00

CH4(g) + 2O2(g) = 2H2O(g) + CO2(g)

–0.10

–0.20 H2(g) + 0.5O2(g) = H2O(g) –0.30

–0.40

CO(g) + 0.5O2(g) = CO2(g)

–0.50 850

950

1050

1150

1250

1350

T (K)

2.5 Temperature coefficients of Eo of various electro-oxidation reactions of fuels as a function of temperature.

temperature. It is evident that the Eo values of the H2 and CO electro-oxidations have negative (∂Eo/∂T)P whereas those for CH4 and C are almost temperature independent or have a slight positive (∂Eo/∂T)P. Although the (∂Eo/∂T)P of all fuels appear to be insensitive to the temperature change, the magnitude of the (∂Eo/∂T)P is clearly different for each fuel. Higher absolute values of (∂Eo/∂T)P for CO and H2 infer more heat being released and less chemical energy being converted into electrical power. On the other hand, the Eo for the C electro-oxidation reaction has a slightly positive (∂Eo/∂T)P. Such a behavior can also be used to judge whether the formation of carbon occurs for a specific hydrocarbon fuel as the operating condition varies in an SOFC. This behavior will be further discussed in Chapter 10.

2.5

The pressure coefficient of electromotive force (EMF)

Differentiation of equation [2.4] with respect to pressure under a constant temperature gives

 ∂∆G   ∂E    = – zF   = ∆V  ∂P  T  ∂P  T Rearrangement of equation [2.20] yields

[2.20]

18

Solid oxide fuel cell technology ∆V  ∂E    = – zF  ∂P  T

[2.20a]

which implies that the E of a cell increases where the volume decreases in the reaction. In reaction [2.8], for example, the E of the H2 oxidation reaction increases with pressure as a result of the volume-decreasing reaction. Assuming ideal gas behavior for all gases involved, the partial molar volume VA of component A (for example A = H2, H2O, and O2) in the mixture is given by VA =

VA

Σ ni

=

X A RT PA

[2.21]

and the overall change of molar volume of the reaction [2.8] is ∆V

= VH 2 O – VH 2 – 1 VO 2 2 =

X H 2 O RT X O 2 RT X H 2 RT – – 1 PH 2 O PH 2 2 PO 2

= RT – RT – 1 RT = – 1 RT Pt Pt 2 Pt 2 Pt

[2.22]

Integration of equation [2.20a] from atmosphere Po to an elevated pressure Pt (z = 2) gives the increase ∆E in E for the H2 oxidation reaction ∆E

P = E ( P , T ) – E ( P o , T ) = RT ln to 4F P

[2.23]

Clearly, higher T and Pt values favor a greater enhancement in E. The complete equation of E as a function of T, concentration, and P for reaction [2.8] can then be written from equations [2.10] and [2.22] as E = 1275 – 0.3171T – 0.0430 T ln

PH 2 O P + 0.0215 T ln to PH 2 P

(mV) [2.24]

For CO fuel, the enhancement term of E is the same as equation [2.23], and therefore the complete equation of E with T, PCO 2 / PCO , and Pt as variables is written from equation [2.11] as  PCO 2  E = 1462 – 0.4830 T – 0.0430 T ln    PCO 

Pt (mV) [2.25] Po For direct electrochemical oxidations of C and CH4 fuels, the system pressure Pt should have no effect on the E value as the volumes of the reactants and + 0.0215 T ln

Thermodynamics of the solid oxide fuel cell (SOFC)

19

products involved in the reactions remain unchanged. However, if the primary fuels are reformed into simple fuels such as H2 and CO, the Pt effect on the simple fuels should be considered instead. Therefore, expressions [2.24] and [2.25] are the two basic equations for calculating E values of hydrocarbon fuels under various circumstances.

2.6

The thermal and chemical expansion coefficients

The dimensional stability of SOFC components is an important criterion to be considered in developing a reliable and robust SOFC product. Thermal expansion and chemical expansion coefficients, which are both material properties, are the two important factors effecting dimensional stability. The former is used to describe the linear volume change of a material with temperature whereas the latter defines the volume change of a material due to changes in chemical composition caused by surrounding atmospheres. In this section, we discuss from the perspective of thermodynamics the definitions of thermal and chemical expansion coefficients. For a multi-component oxide system in which the molar volume V of one component is a function of temperature T, system pressure P, and oxygen non-stoichiometry δ V = V(T, P, δ)

[2.26]

General differentiation of equation [2.26] gives  ∂V   ∂V   ∂V  dV =   dT +   dP +   dδ  ∂T  P,δ  ∂P  T ,δ  ∂δ  T ,P

[2.27]

Rearrangement of equation [2.27] by dividing by V yields

 ∂V   ∂V   ∂V  d ln V = 1  dT + 1  dP + 1  dδ [2.28] V  ∂T  P,δ V  ∂P  T, δ V  ∂δ  T, P where αt = V–1(∂V/∂T)P,δ is defined as the thermal expansion coefficient, γ = V –1(∂V/∂P)T,δ is the compressibility, and αc = V–1(∂V/∂δ)T,P is the chemical expansion coefficient. For an all-solid SOFC system, γ is negligible for most system pressure ranges of interest. Expansion in each of three spatial directions can be resolved by considering the relationship of molar volume V to the strain tensor ε. For a differential change in expansion, tr(dε) = d lnV is expected. Therefore, in the absence of mechanical stress (including constant system pressure), the uniaxial strain for an isotropic solid is given from equation [2.28]26

dε = 1 tr(dε ) = 1 α t dT + 1 α c dδ 3 3 3

[2.29]

20

Solid oxide fuel cell technology

Consider the total derivative of the strain dε with respect to temperature T at constant PO 2

 ∂ε   ∂ lnV    =1  = 1 α t + 1 α c  ∂δ  = TEC [2.30]   3  ∂ T  P 3 3 ∂T  P  ∂ T  PO  O O 2

2

2

Equation [2.30] represents a true definition of the thermal expansion coefficient (TEC). It shows that under a constant PO 2, the slope of the ε–T plot is composed of two terms: (a) ‘ordinary’ thermal expansion (1/3α) and (b) thermally induced chemical expansion 1/3α c ( ∂δ / ∂T ) PO 2. The second term could become significant, depending upon the thermal properties of materials at high temperatures, where oxygen non-stoichiometry δ becomes increasingly prevalent. As a result, thermal expansion curves of perovskite oxides that have a strong dependence of oxygen non-stoichiometry on temperature show curvature (deviation from linearity) at higher temperatures. One immediate exemplary material is La1–xSrxCoO3–δ (δ > 0) in which δ varies strongly with temperature. On the other hand, the total derivative of the expansion with respect to PO 2 at constant temperature T,

 ∂ε   ∂δ  1  ∂P  = 3 α c  ∂P   O2  T  O2  T

[2.31]

is often known as the isothermal chemical expansion. As evident from the equation, the volumetric change of the material, which can be either shrinkage or expansion, relies partially upon how the oxygen non-stoichiometry δ varies with PO 2. In order to illustrate graphically the relationship described in equation [2.31], Fig. 2.6 shows the thermal expansion curves measured for the perovskite La0.6Sr0.4Co0.2Fe0.8O3–δ (LSCF) at various PO 2 values. The material has been shown from a separate study (reference 115 in Chapter 12) to have ( ∂| δ |/ ∂PO 2 ) < 0 (δ < 0) from air towards low PO 2 at T > 600 °C, which implies that a volumetric expansion occurs according to equation [2.31]. As clearly shown in Fig. 2.6, the volume of the sample indeed expands as PO 2 decreases. On the other hand, a spin-state transition of a Co3+ ion could also contribute to the change in expansion, as will be discussed in Chapter 3. For the doped LaMnO3 system, many reports in the literature have shown from weight-change studies that the isothermal δ undergoes a change from δ > 0 to δ = 0 and to δ < 0 as PO 2 decreases.‡ According to equation [2.31], doped LaMnO3 should experience shrinkage ( ∂| δ |/ ∂PO 2 > 0, δ > 0) from ‡

Refer to Fig. 12.7 for an example of measured 3 + δ as a function of PO 2 in the doped

LaMnO3 system.

Thermodynamics of the solid oxide fuel cell (SOFC) 0.1 bar

13

0.01 bar

12

Chemical expansion

10–4 bar

11

Uniaxial expansion ε (103 ppm)

21

Return to 1.0 bar

10–3 bar

10

Hold at 792 °C

9

Linear ramp 25–792 °C PO 2 = 1.0 bar

8 7

Thermal expansion

6 5 4 3 2 1 0 0

1000

2000 3000 Time (min)

4000

5000

2.6 Thermal expansion and chemical expansion curves of LSCF.

pure O2 to air, and expansion at low PO 2 ( ∂| δ |/ ∂PO 2 < 0, δ < 0). This prediction is partially supported by the isothermal dilatometry measurement on the material. Figure 2.7(a) shows the volumetric shrinkage at 1000 °C as the atmosphere is changed from O2 to air. At low PO 2 , loss of oxygen in the lattice reduces the Mn, generally leading to an increase in lattice expansion. The overall chemical expansion and oxygen stoichiometry as a function of temperature are depicted schematically in Fig. 2.7(b). In summary, the commonly used thermal expansion coefficient of oxides includes two terms, one of which is caused by the change in temperature and another of which results from the change in oxygen non-stoichiometry (δ) induced by temperature, PO 2 , and/or in the case of cobalt, a change in the spin state of the ion. For δ-sensitive materials, the measured dimensional change as a function of temperature deviates from linearity as temperature increases. Under isothermal conditions, the volumetric change of material upon change in PO 2 is proportional to ∂δ/∂ PO 2 . An increase in δ with

Solid oxide fuel cell technology

Specimen expansion (%)

22

1.08

O2 atmosphere

1.067%

Air atmosphere 1.054%

1.06

1.044%

1.042% 1.04

1.02

1.00 5

10

15 20 Elapsed time (hours) (a)

25

30

LaMnO3-based

+ Volume

δ 0

–

0

PO 2 (b)

2.7 (a) Isothermal expansion curve of doped LaMnO3 measured at 1000 °C. (b) Schematic of isothermal chemical expansion and oxygen stoichiometry varying with P O 2 .

either decreasing or increasing PO 2 implies a volumetric expansion. Understanding of the isothermal change in δ with PO 2 in cathode materials is critical for cathode-supported SOFCs since the cathode/electrolyte interfacial PO 2 decreases considerably with loading current. If such a change is not properly managed, large stresses could be developed during loading, and the integrity of the cell could be compromised.

3 Electronic properties of solids for solid oxide fuel cells (SOFCs)

Abstract: The performance of an oxide used as an electrode or as an interconnect between cells of an SOFC depends critically on the electronic properties of the oxide. However, there are very few reports in the literature that discuss how the electronic properties of an oxide effect the electronically conducting electrodes and interconnect in an SOFC. In this chapter, the electronic properties of oxides are discussed from the perspective of solid-state physics, and the results are applied particularly to electrode and interconnect materials commonly used in SOFCs. Key words: polaronic conduction, redox 4fn energies; ligand-field d-electron energies, localized electrons, itinerant electrons.

3.1

General considerations

The spin-paired core electrons of an atom are not active in bonding; they do, however, screen the outer electrons from the nuclear charge, and differential screening of electrons of different orbital angular momentum is greater the heavier the atom. Of particular interest are the outer electrons active in bonding. The character of the outer electrons in a solid depends on the relative strengths of the interatomic versus intra-atomic interactions. The interatomic interactions between neighboring atoms depend on the electron energy transfer integral bij ≡ (Ψi, H′Ψj) = εij(Ψi, Ψj)

[3.1]

where H′ is the perturbation of the free-atom potential energy by neighboring atoms, εij is a one-electron energy, and (Ψi, Ψj) is the overlap integral of the interacting wavefunctions. With the exception of the lone-pair 5s2 or 6s2 electrons stabilized relative to the 5p or 6p states by core-electron screening, the atomic outer s and p orbitals on neighboring atoms overlap strongly to form energy bands of one-electron itinerant states occupying an energy band of width W. Each itinerant electron belongs equally to all like atoms on an array of energetically equivalent sites. In an oxide, the broad s and p bands are split by an energy gap into a lower band of bonding states and an upper band of antibonding states by a translational symmetry that distinguishes the metal and oxygen atoms. The magnitude of the energy gap depends on two terms: (a) the difference between an electrostatic Madelung energy EM and the energy EI to transfer an electron from the 23

24

Solid oxide fuel cell technology

cation to the anion at infinite separation, which provides the ionic component of the bonding, and (b) the covalent bonding between distinguishable atoms. The electrostatic model is illustrated schematically in Fig. 3.1 for MnO. Transfer of a second 4s electron from Mn: 3d54s2 to create Mn2+ and O2– ions separated at a great distance costs an energy EI = Ei – EA, where Ei is the ionization energy of the Mn+ ion and EA < 0 is the electron affinity of the O– ion. On assembling the ions into the periodic array of the rock-salt structure, the electric field between the atomic arrays lowers the O2– level and raises the Mn2+ energies by the Madelung energy EM to stabilize the O2– ions relative to the Mn2+ ions by the energy EM – EI > 0. The 3d5 manifold of the Mn: 3d54s2 atom is also raised by EM above the O2– energy in MnO. This electrostatic model needs to be modified by consideration of back electron transfer from covalent bonding. A strong overlap of the s and p orbitals of the Mn2+ and O2– ions induces a covalent mixing of Mn and O states into both the cation antibonding states and the anion bonding states. Although the covalent bonding lowers the ionic charges and therefore EM, it pushes apart the bonding and antibonding states to compensate for the lower EM, which leaves a large energy gap Eg between the bottom of the Mn-4s band and the top of the O-2p bands. The covalent interaction also transforms the s and p electronic states into itinerant-electron states within respective energy bands of width W. The 3d electrons of the 3d5 manifold have a smaller radial extension and therefore a smaller overlap integral, and the intra-atomic interactions become competitive with the interatomic interactions to leave a localized 3d5 manifold as is discussed below. Because of the large ionic component of the bonding in an oxide, it is customary to refer to the lower, bonding bands as the O-2p or O2–: 2p6 bands and the upper, antibonding bands as the cation-s bands. In PbO, differential screening of the 6s and 6p electrons by the core electrons leaves a 6s2 lonepair core with an energy in the gap between the Pb-6p and O-2p bands. In the rare-earth oxides, stronger covalent bonding with the 6s orbitals compared ε

O–/O2–

E

EA

Vac

Mn2+ : 4s0

Mn2+ : 4s0

εF 3d5

3d5

Ei

EM – EI

EI O2– : 2p6

O2– : 2p6

4s 3d5

Mn2+/Mn+

3.1 Schematic model of formation of electronic states in MnO.

Electronic properties of solids for solid oxide fuel cells (SOFCs)

25

with the 5d orbitals leaves the bottom of the conduction band mostly 5d in character. Where the cation dn or fn manifolds are raised into the energy gap between cation s and 5d and the anion p bands, the M–O covalent bonding creates ligand-field orbitals of d or f symmetry. Of particular interest are not only the location of the dn or fn manifolds within the energy gap, but also the relative strengths of M–M or M–O–M interactions and the intra-atomic electron– electron interactions of the ligand-field electrons. With one or more electrons or holes in an atomic dn or f n manifold, successive ionization energies are separated by a discrete energy difference U as a result of the intra-atomic electron–electron coulomb repulsions. Since the Pauli exclusion principle prevents occupation of the same d or f orbital by two electrons with the same spin, the intra-atomic electron–electron interactions are reduced if the electrons have the same spin; this stabilization is the intra-atomic exchange energy, ∆ex, that is responsible for Hund’s highest multiplicity rule for a free atom. The more covalent bonding extends the ligand-field orbitals in an oxide, the weaker are the intra-atomic energies ∆ex and U and the stronger are the interatomic interactions.

3.2

Redox 4fn energies and polaronic conduction

The small radial extension of the 4f orbitals on rare-earth (R) ions makes the intra-atomic interactions of the 4f n manifolds everywhere stronger than the interatomic interactions between 4f orbitals. Therefore, the 4fn configurations are always localized, i.e. are like those on the free atom, so long as the Fermi energy does not intersect both the 4fn energy and an itinerant-electron band. Since the energy U separating the successive 4f n manifolds is as large or larger than the energy gap between the R-5d and O-2p bands of the rareearth oxides, only one valence state is accessible for many rare-earth cations, viz. the R3+ ion; but where a 4f n energy lies in the energy gap between filled and empty bands, the Fermi energy may lie above or below the 4fn energy to give the R: 4fn or R: 4f n–1 valence states. If the Fermi energy lies at a 4fn energy within the gap between itinerant-electron bands, the system is then in a mixed-valent state. Since an energy U separates successive 4fn manifolds, a single-valent rare-earth array cannot contribute to the electronic conductivity; it is necessary to have a mixed-valent state with the Fermi energy at the 4f n energy. Moreover, since only two valence states are accessible at the 4f n energy, the 4fn configurations correspond to redox couples and their energies to redox energies*. The Ce: 4f1, Pr: 4f2, Eu: 4f7, Tb: 4f8, and Yb: 4f13 energies lie in the energy gap between R-5d and O-2p bands of a rare-earth oxide, which makes the following redox couples possible: Ce4+/Ce3+, Pr4+/ Pr3+, Eu3+/Eu2+, Tb4+/Tb3+, and Yb3+/Yb2+. *Charges signify formed valence states, not the actual charge on the cation.

26

Solid oxide fuel cell technology

ε ∆Ht

+ ∆Hp

∆Hp

3.2 The 4fo/4fl energies of CeO2–δ: —•— , 4fl; ——, 4fo · ∆Ht is the energy of trapping at an oxygen vacancy V o⋅⋅.

Disordered, mobile 4f charge carriers on a mixed-valent array do not change the translational symmetry, but electron–lattice interactions that trap charge carriers at single ionic sites prevent the electrons from becoming itinerant and permit the redox-couple description, e.g. Ce4+/Ce3+ in CeO2–δ. The charge carriers are trapped by local breathing-mode displacements of the oxygen atoms that reduce the M–O bond lengths at the ions of higher valence state relative to those at the ions of lower valence state as a result of the smaller ionic radius of the cation of higher valence, which contains fewer antibonding 4f electrons. Since the 4f electrons are antibonding, the oxygen displacements, which increase the R-4f and O-2p overlap at the cation of higher valence, raise the energy of the empty states relative to the occupied states of the 4fn couple by an energy ∆Hp, as illustrated in Fig. 3.2. The condition for a charge carrier to be trapped in a local site deformation is that the time τh for a charge carrier to hop to a neighboring cation should be longer than the period ω o–1 of the local breathing-mode vibration that traps it. A τh ≈ h /W follows from the uncertainty principle, and a τh > ω o–1 leads to the condition W < h ωo

[3.2]

where, in the tight-binding approximation, the bandwidth is W ≈ 2zb

[3.3]

The energy parameter b is that of equation [3.1] for the z nearest-neighborlike cations. Condition [3.2] is fulfilled for 4f charge carriers, and mobile charge carriers that are dressed in a local lattice deformation are called small dielectric polarons. Since charge transfer is between states of equal energy, thermal energy must be supplied for a polaron to hop to a neighboring site where its local deformation reforms. This activated motion is diffusive with a motional enthalpy ∆Hm = ∆Hp/2 entering the charge-carrier mobility. The formalism for the electronic conductivity σe by polarons is the same as that for the oxide-ion conductivity σ VO˙˙ developed in Chapter 5.

3.3

Ligand-field d-electron energies

Since the electronically conducting d-block transition-metal oxides of interest for SOFC materials have perovskite-related structures with the active transition-

Electronic properties of solids for solid oxide fuel cells (SOFCs)

27

metal compounds in octahedral sites, discussion is restricted to octahedralsite ligand-field wavefunctions. In an octahedral site, the fivefold-degenerate d orbitals are split into threefold-degenerate, π-bonding t orbitals (xy, yz, zx) and twofold-degenerate, σ-bonding e orbitals (3z2–r2, x2–y2) to give, from second-order perturbation theory, the ligand-field wavefunctions Ψt = Nπ(ft – λπφπ)

[3.4]

Ψe = Nσ(fe – λσφσ – λsφs)

[3.5]

where Nπ and Nσ are normalization constants, and ft and fe are atomic t and e orbitals. The φπ, φσ, φs are symmetrized 2pπ, 2pσ, and 2s orbitals of the oxygen ligands; the λ ≡ bca/∆E are covalent-mixing parameters in which bca is the energy integral of equation [3.1] for overlapping M-d and O-2p, 2s orbitals, and ∆E is the energy difference between the O-2p and fivefolddegenerate M orbitals. Since the σ-bonding overlap integral is larger than the π-bonding overlap integral, the antibonding Ψe orbitals are at an energy ∆c above the Ψt orbitals ∆c

≈

∆M

+ ( λ σ2 – λ π2 ) ∆E

[3.6]

where ∆M is a small electrostatic component and, for simplicity, the smaller contribution from λs is omitted. The term ( λ σ2 – λ π2 ) ∆E follows directly from second-order perturbation theory, ∆H ≈ (bca)2/∆E, where ∆E is the energy difference between the O-2p and the empty 3dn+1 configuration. In addition to the cubic-field splitting ∆c, there is the splitting ∆ex of states of different spin. Figure 3.3 illustrates the high-spin (HS) configuration (∆c < ∆ex) for the d5 configuration t3e2 of the Mn2+ ion of MnO. A ∆c > ∆ex suppresses the Hund highest-multiplicity rule as is illustrated for low-spin (LS) Ni3+: t6e1 versus an HS t5e2 state for Co2+. A ∆c ≈ ∆ex for octahedral-site Co3+ stabilizes the LS t6e0 state in bulk LaCoO3 at lowest temperatures, but HS t4e2 configurations are excited at higher temperatures. At the surface of LaCoO3, the intermediate-spin (IS) t5e1 configuration is stabilized in the tetragonal symmetry of a square-pyramidal site.27 In a cubic perovskite with 180° M–O–M interactions between like atoms, only the O-2p components of the ligand-field wavefunctions on neighboring cations overlap one another. Since each cation on opposite sides of the O2– ion tends to bond with the same O-2p orbital, the 180° M–O–M electrontransfer energies of equation [3.1] become

bσcac ≈ ε σ λ σ2

and

bπcac ≈ ε π λ π2

[3.7]

With z = 6 in the perovskite structure, the tight-binding bandwidths (equation [3.3]) for the Ψe and Ψt orbitals become Wσ ≈ 12εσ λ σ2

and

Wπ ≈ 12επ λ π2

[3.8]

where λπ < λσ. (Note: for simplicity of notation, the Ψt and Ψe orbitals are referred to as t and e orbitals in the following text.)

28

Solid oxide fuel cell technology e t ∆ex

e ∆c

t

(a) HS Mn2+: 3d5 e e

∆c

t ∆ex

t (b) LS Ni3+: 3d7

e t e t (c) HS Co2+: 3d7

e

t

e

t (d) IS Co3+: 3d6 (tetragonal c/a > 1)

3.3 High-spin (HS) versus low-spin (LS) and intermediate-spin (IS) states.

Because of the cubic-field splitting ∆c and the intra-atomic exchange splitting the effective intra-atomic energy that splits successive dn manifolds is

∆ex,

Ueff = U + ∆U

[3.9]

where ∆U = 0 for additions within an occupied tn manifold or en manifold of the same spin, ∆U = ∆c if the added electron requires an additional energy ∆c,

Electronic properties of solids for solid oxide fuel cells (SOFCs)

29

and ∆U = ∆ex if a first minority-spin electron is added. The condition for localized-electron behavior of a single-valent dn manifold is W < Ueff

[3.10]

σ It follows from equation [3.8] that a Wπ < Wσ makes possible a Wσ > U eff in σ the presence of a Wπ < U eff since the smaller λπ also makes Uπ > Uσ.

3.4

Localized versus itinerant d electrons

The 5d and 4d electrons have a larger radial extension than the 3d electrons and therefore a larger bca, which increases the λπ and λσ covalent-mixing parameters. The result is a larger ∆c and a smaller ∆ex. Consequently the 4dn and 5dn configurations with n > 3 are all low spin. Moreover, in AMO3 π , so even the singleperovskites these configurations all have a Wπ > U eff valent compounds form itinerant-electron states with a Fermi energy in a partially filled band except for n = 6, which gives a filled π* band below an empty σ* band. In a partially filled band, a large density of itinerant-electron states at the Fermi energy leads to metallic conductivity. However, with n > π to give ferromagnetism and 3, these configurations may have Wπ ≈ U eff metallic conduction in a single-valent compound, as is illustrated by the metallic perovskite SrRuO3. The smaller radial extension of the 3d electrons makes the competition between localized and itinerant electron behavior, i.e. between intra-atomic and interatomic interactions, much more interesting.28 For example, with 0 < x < 0.9 in the Sr1–xLaxTiO3 perovskite system, the condition Wπ > h ωo leads to a partially filled band of itinerant-electron states and, therefore, metallic rather than polaronic conductivity in the mixed-valent state. However, a Wπ < Uπ makes LaTiO3 an antiferromagnetic insulator, and the transition from metal to insulator in the interval 0.90 < x < 0.95 is first-order. This situation is illustrated in Fig. 3.4. The covalent-mixing parameters λ ≡ bcac/∆E depend on ∆E as well as on the orbital overlap, but λ nevertheless tends to decrease with increasing atomic number for a given valence state until a manifold is half-filled. Consequently, we find a Wπ > Uπ in the metallic system Sr1–xCaxVO3 but Wπ ≈ Uπ in Sr1–xCaxCrO3. It should be noted that the A-site cations of a perovskite σ-bond with the O-2p orbitals that π-bond with the octahedral-site cations. The more acidic Ca2+ ions compete more strongly than the Sr2+ ions for the O-2pπ orbitals, so the π* band of t-orbital parentage has a width Wπ in CaVO3 and CaCrO3 that is narrower than that in SrVO3 and SrCrO3. For a fixed atomic number, ∆E increases with decreasing cation valence. Consequently, LaVO3 is an antiferromagnetic insulator with a localized t2e0 configuration. Moreover, the larger the spin of the localized-electron configuration, the larger is the intra-atomic-exchange energy ∆ex.

30

Solid oxide fuel cell technology 250

(Pbnm)

(Ibmm)

(Pm3m)

248

Cell volume (Å3)

246 244 (Cell volume) × 4

242

240 238

0.0

0.2

0.4 0.6 x for La1–xSrxTiO3

0.8

1.0

(a) 160

1.6

TN (K)

140

M(10 K) (emu/cm3)

1.2

100

1.0

80

0.8

60

0.6

40

0.4

20

0.2

0 0.0

0.2

0.4 0.6 x for La1–xSrxTiO3

0.8

M(10 K) (emu/cm3)

TN (K)

120

1.4

0.0 1.0

(b)

3.4 (a) Unit-cell volume and (b) Néel temperature, TN (K), and magnetization at 10 K, M(10 K) versus x for La1–xSrxTiO3, after reference 29.

The t3e0 configurations of Cr3+ and Mn4+ are localized because the large Ueff = U + ∆c makes Wπ < Ueff. Similarly, the t3e2 configurations of HS Mn2+ and Fe3+ have a large Ueff = U + ∆ex to keep both the t3 and the e2 manifolds localized. That is why the 3d5 configuration in Fig. 3.1 is kept localized at the Mn2+ ion of MnO.

Electronic properties of solids for solid oxide fuel cells (SOFCs)

31

Where there is an orbital degeneracy of the localized-electron configuration on a cation, neither ∆c nor ∆ex enters Ueff, but the energy of the localizedelectron configuration is lowered by an electron–lattice interaction that distorts the octahedral site from cubic to lower symmetry. The Jahn–Teller stabilization30, 31 is resisted by the crystal elastic energy; therefore, the stabilization is stronger where the site distortions are cooperative.32 The HS Mn3+: t3e1 configuration has an e-orbital degeneracy, which makes it a strong Jahn–Teller ion. At low concentrations of Mn3+, the Jahn–Teller stabilization enhances the polaronic energy ∆Hp; at higher concentrations, cooperative Jahn–Teller distortions may stabilize localized versus itinerant electronic behavior. For example, LaMnO3 has a Wσ ≈ Uσ , and a cooperative Jahn– Teller distortion at TJT = 750 K changes it from a good conductor to an insulator33 that becomes antiferromagnetic below TN = 100 K. In the mixedvalent system La1–xSrxMnO3, the cooperative distortions are removed in the interval 0.2 ≤ x < 0.5, and the system becomes metallic; the bulk e electrons occupy a partially filled, itinerant-electron σ* band of e-orbital parentage having a Wσ > h ωo. However, the t3 configuration remains localized with a Wπ < Uπ + ∆ex, and a strong intra-atomic ∆ex aligns the σ*-electron spins parallel to the t3 spin S = 3/2 to give a ferromagnetic, metallic state below a Curié temperature Tc > 300 K.28 Nevertheless, localized Mn3+ ions are stabilized at square-pyramidal sites at the surface by the local tetragonal symmetry. The transition from localized to itinerant behavior of the e electrons as x increases in the interval 0 < x < 0.17 in the La1–xSrxMnO3 system is first order, which leads to a segregation of hole-rich and hole-poor regions by locally cooperative oxygen displacements at lower temperatures. For example, holes may become trapped in a two-atom Mn3.5+–O–Mn3.5+ Zener polaron. Zener34 first postulated the possibility of the two-atom polaron having a ∆Hm ≈ 0 in an attempt to account for the metallic ferromagnetism found in La0.7Sr0.3MnO3 with his spin–spin double-exchange mechanism. At x = 0.5, either small-polaron or Zener-polaron ordering is found in La0.5Sr0.5MnO3 and La0.5Ca0.5MnO3.28 In addition, the Co3+ and Ni3+ ions in perovskites are of particular interest. These ions have redox energies pinned at the top of the O-2p bands, which means that the Co4+/Co3+ and Ni4+/Ni3+ couples have a large O-2p component and a large λσ. Consequently, LaCoO3 has a ∆c ≈ ∆ex and LaNiO3 is metallic with a ∆c > ∆ex and a Wσ > Uσ at LS Ni3+: t6σ*1 ions. The mixed-valent La0.6Sr0.4CoO3 compound is a metallic ferromagnet with an IS configuration t5σ*0.6. The splitting between the t5 and t6 configurations signals a Wπ < Uπ in the presence of a Wσ > h ωo, see Fig. 3.5.35 Nevertheless, the tetragonal symmetry of a surface square-pyramidal site also stabilizes here a localized IS Co3+: t5e1 and LS Ni3+: t6e1, where the e1 electron occupies the 3z2–r2 orbital oriented toward the oxygen vacancy.

32

Solid oxide fuel cell technology σ*0.6

ε t6

εF

t5

Uπ

3.5 The IS configuration t5σ*0.6 of the CoO3 array in the metallic ferromagnet La0.6Sr0.4CoO3.

3.5

Applications

3.5.1

Anode

The function of the anode of an SOFC is to oxidize the gaseous fuel with O2– ions from the cathode that are resupplied via the electrolyte. The conventional SOFC operates on pure H2 as fuel and uses an electrolyte/NiO composite as the anode; the NiO is reduced to elemental nickel in the anode atmosphere to form a porous anode morphology with a percolating string of nickel beads on the pore surface; the nickel wets the oxide poorly, but percolating strings of nickel beads carry electron to the current collector. The H2 is chemisorbed on the nickel, and the H+ ions spill over to the electrolyte where they react with O2– ions to form water that is desorbed; the electrons donated to the nickel by the chemisorption are transferred via a current collector to the external circuit. This process involves proton spillover as well as fast transfer of O2– ions in the electrolyte to the triple phase boundry (TPB). This anode can not be used with a logistic fuel such as natural gas since hydrocarbons form coke deposits on a nickel catalyst, and sulfur impurities in the fuel react with nickel to form Ni sulfides. Therefore, there is a strong incentive to find a catalytically active mixed ionic/electronic conductor (MIEC) that is stable in the reducing atmosphere at the anode; the oxide-ion conductivity in the MIEC is needed to bring O2– ions to the site of chemisorption of the fuel since chemisorbed hydrocarbons are not mobile on an oxide surface. Given the function of an MIEC anode, the following specifications challenge the designer of a viable anode for operation on natural gas: • •

chemical stability in the electrode atmosphere and at anode/electrolyte and anode/interconnect interfaces; mechanical compatibility, i.e. a similar thermal expansion, with the electrolyte and the interconnect with which it makes contacts;

Electronic properties of solids for solid oxide fuel cells (SOFCs)

• • • •

33

good electronic conductivity with little contact resistance at the anode/ current collector interface; an oxide-ion conductivity comparable with or better than that of the electrolyte with no additional resistance to oxide-ion transport across the anode/electrolyte interface; fast dissociative chemisorption of the fuel at the anode surface; fast desorption of the chemisorbed fuel as an oxidized species at the operating temperature Top of the SOFC.

In Chapter 5, it will be shown that oxygen-deficient perovskites and fluorites are stable and allow isotropic oxide-ion conduction via oxygen vacancies. Moreover, these two oxide families can have comparable thermal expansion coefficients (TECs), which is needed for retention of a bonded electrode/ electrolyte interface on thermal cycling. Furthermore, we can expect good oxide-ion transfer across an interface between two materials having isotropic oxide-ion conductivity provided any impurity or cation transfer across the interface does not create unwanted interface phases. Therefore, exploration of MIEC anodes has concentrated on transition-metal or rare-earth oxides with the fluorite or the perovskite structure. Good electronic and ionic conductivity are compatible because disordered oxide-ion vacancies introduce mixed valence on a redox couple. However, in order to retain a mixed-valent cation array in an oxide in the anode atmosphere, it is necessary to have an operative redox energy that is not too stable. At the same time, the partially occupied redox couple must be at a low enough energy to accept electrons from the fuel in the chemisorption step. For example, loss of the energy ∆ex for the minority-spin electron of the Fe3+/Fe2+ couple raises the energy of this couple in FeO1–δ relative to the Mn3+/Mn2+ couple in MnO even though Fe is heavier than Mn, so Fe1–δ O is not reduced all the way to FeO. This condition is also fulfilled by the Ce: 4f′ level in CeO2; in H2 atmosphere, Sm-doped CeO2 (SDC) is only partially reduced, and the first MIEC to be tested as an anode was SDC.36 Partial reduction of Ce4+ to Ce3+ introduces polaronic conduction, which does not give a high enough electronic conductivity and it also increases the volume significantly, which causes mechanical mismatch with YSZ. Nevertheless, porous SDC/Cu composite anodes have been made and tested. CuO was impregnated into porous SDC and reduced to metallic Cu to reduce the path length from the site of the fuel oxidation reaction (FOR) to a metallic conductor. Although this construction is analogous to that of the nickel electrode, the chemisorption does not take place on the copper, but on the SDC surface. Completion of the FOR is accomplished on the MIEC by oxide-ion delivery to the site vacated by desorption of the oxidized product and electron hopping from the reaction site to a copper ribbon for transfer to the external circuit. Although dissociative chemisorption on a clean SDC surface appears to be fast, completion of the FOR is not fast enough to provide a competitive power output with this

34

Solid oxide fuel cell technology

anode. Nevertheless, this anode showed no poisoning by sulfur, no coke build-up, and a sufficient FOR activity to demonstrate the feasibility of an MIEC anode. The double perovskites Sr2MMoO6–δ, M = M2+, have given a more promising anode performance in H2 and methane.37 Although the M and Mo cations are ordered into alternate octahedral sites of the perovskite structure, the Mo-4d electrons have a large enough radial extension that the Mo–O–Mo interactions give adequate electronic conductivity of the Mo6+/Mo5+ mixed valence found in the anode atmosphere at a Top ≈ 800 °C. Moreover, the energy of this redox couple is low enough to receive electrons from the fuel, yet high enough to resist reduction to all Mo5+ in the anode atmosphere. The FOR steps for CH4 oxidation on Sr2MMoO6–δ are probably the following. First, the CH4 is dissociatively chemisorbed on two neighboring surface oxide ions at sixfold-coordinated M and Mo cations to form a surface O–CH2–O species with 2H+ and 4e– transferred, respectively, away to neighboring O2– ions and to the Mo6+/Mo5+ redox couple. The two H+ ions combine on a single O2– ion to form H2O that is desorbed and the four electrons are transferred to the external circuit. Next, the second 2H+ and 4e– are, respectively, similarly desorbed as H2O and delivered to the external circuit. Provided the M atom is stable in fourfold as well as sixfold oxygen coordination and the Mo is stable in fivefold as well as sixfold coordination, removal of the CO2 from the surface is facilitated. The formation of Mo==O species with both Mo6+ and Mo5+ ions allows the Mo to be stable in squarepyramidal coordination, and the best performance is found with the most basic of the ions, Mg2+, that are stable in both fourfold and sixfold oxygen coordination. The third and final step of the catalytic oxidation reaction CH4 + 4O2– = CO2 + 2H2O + 8e′

[3.11]

is replenishment of the surface O2– ions by oxide-ion conduction from the cathode oxygen-reduction reaction (ORR) through the electrolyte to the reaction site on the surface of the anode. The stability of both the Mo6+/Mo5+ and the M2+ ion in less than six-fold oxygen coordination also facilitates the mobility of oxygen vacancies. Another promising MIEC anode for operation of an SOFC on a logistic fuel has been reported (J. Haag, B. D. Madsen. S. A. Barnett, and K. Poeppelnicer, personal communication); it is a composite of the perovskite LaSr2Fe2CrO9–δ and Gd0.1Ce0.9O2–δ (GDC). The Cr3+ was added to Sr2Fe2O5–δ to prevent reduction of the iron to Fe0 in the anode atmosphere. However, the addition of Cr3+ also reduces the mobility of the oxygen vacancies as well as the electronic conductivity in the Fe3+/Fe2+ redox couple. The GDC was added to introduce a TPB that would decrease the distance O2– ions would need to travel to reach an FOR site in the perovskite. In this composite MIEC, the Fe3+/Fe2+ energy is near that of the Mo6+/Mo5+ couple; the two

Electronic properties of solids for solid oxide fuel cells (SOFCs)

35

overlap in the double perovskite Sr2FeMoO6. Removal of O2– ions from a surface Fe2+ ion creates tetrahedral-site Fe3+ having its Fe3+/Fe2+ redox energy above that of the octahedral-site cations as is found in Fe1–δO, which facilitates CO2 desorption from the surface. These investigations of MIEC anodes for an SOFC operating on a logistic fuel are preliminary. However, they illustrate well the several features that must be considered in the design of a superior anode material.

3.5.2

Cathode

The function of the cathode is to transfer electrons from the external circuit to adsorbed dioxygen molecules to reduce the O2 to 2O2– and to transfer the O2– ions generated to the electrolyte. Platinum is commonly used to catalyze the breaking of the double bond of the O2 molecule. Since noble metals are expensive, the cathode of choice for the SOFC is an oxide that is stable as a metallic conductor in air at the operating temperature Top and that also catalyzes the reduction of O2 to 2O2–. Since oxygen is mobile on the surface of an oxide, the cathode need not also be an oxide-ion conductor. However, an MIEC in air at Top that is a good catalyst for oxygen reduction allows utilization of bulk as well as surface O2–-ion conduction. The first step in the ORR is chemisorption of the O2 molecule on a metal atom. Whereas a fuel molecule chemisorbs on a surface O2– ion, the O2 molecule chemisorbs on an oxide surface at a surface oxide-ion vacancy. Breaking of the strong double bond of the O2 molecule in the chemisorption process requires that the cation attacked by the O2 molecule has an electron at a high enough energy that it can donate electrons to the antibonding orbitals of the O2 molecule. The conventional cathode of an SOFC is a metallic La1–xSrxMnO3 (LSM) perovskite with x ≈ 0.2. Its TEC is compatible with that of a YSZ electrolyte and, as discussed above, the (180° – φ) Mn–O–Mn e-orbital interactions are just strong enough to transform the σ-bonding electrons to itinerant electrons in a narrow σ* band of e-orbital parentage with Wσ > h ωo. The π-bonding t3 configuration remains localized with a spin S = 3/2, but the σ* electrons provide the desired metallic conductivity. However, the mixed-valent Mn4+/ Mn3+ redox energy is too high for this perovskite to retain sufficient oxideion vacancies in air at Top in the presence of the strong octahedral-site preference of the cations, so this perovskite is not an MIEC. Nevertheless, the surface provides the necessary bare-cation attack site with electrons capable of reducing the O2 molecule to the peroxide ion (O2)2–. One of the oxygen atoms of the peroxide ion is mobile on the surface and can travel to another attack site to create 2O2– ions, and the surface O2– ions can be transferred to a TPB to be absorbed by the oxide-ion electrolyte. Since LSM is not an MIEC, the LSM cathode must be made as a porous LSM/electrolyte composite to provide an

36

Solid oxide fuel cell technology

extensive TPB network. Unfortunately the catalytic activity of LSM for the ORR is inadequate at Top < 800 °C. An alternative approach is to use an oxide that is an MIEC in air at Top and is also catalytically active for the ORR. The specifications for an acceptable MIEC cathode include: • •

• • •

fast ORR kinetics at the surface at a desired Top; an O2–-ion conductivity σ V .. that replenishes oxygen vacancies at the O attack site as fast as oxide ions are leaving the anode surfaces without requiring a large overpotential; an electronic conductivity σe >> σ V .. ; O

O2–-ion transport across the cathode/electrolyte interface that is faster than can be supplied by the ORR; chemical and mechanical compatibility with both the electrolyte and the cell interconnect.

Once an optimum oxide has been chosen, the problem of fabrication of the optimum electrode morphology remains. Oxygen-deficient AMO3–δ perovskites are the most widely investigated MIECs. In these oxides, oxide-ion conduction is by oxygen vacancies. However, retention of oxygen vacancies in the air atmosphere at the cathode requires operation on a redox couple that is near or pinned at the top of the O2–: 2p6 bands. This condition also provides a large enough admixture of M-3d and O-2p orbitals in the redox couple that the (180° – φ) M–O–M interactions are strong enough to give a metallic σe >> σ V .. , and the electronic conductivity O

lowers the activation energy Ea for the O2–-ion transport to make σ V .. of the O cathode comparable with that of the electrolyte. The AMO3–δ perovskites containing Fe4+/Fe3+, Co4+/Co3+, and/or Ni4+/Ni3+ formal valences have redox couples near or pinned at the top of the O2–: 2p6 bands and are MIECs. Overpotentials at several non-porous perovskite cathodes at 800 °C are compared in Fig. 3.6. The large overpotential for a non-porous La0.85Sr0.15MnO3 film in this figure is due to its very small σ V .. because it contains few O

oxygen vacancies; the overpotential is greatly reduced where there is access to a TPB. Of particular interest is the superior performance of the electrodes containing cobalt, particularly those with Co0.8Ni0.2 and Co0.8Fe0.2 in contrast to the cathode with Fe0.8Ni0.2. This difference reflects a faster rate of the ORR with cobalt than with iron or nickel. The La0.6Sr0.4CoO3–δ perovskite also shows a relatively low overpotential, which confirms that the Co4+/Co3+ couple, like the Mn4+/Mn3+ couple, provides a surface site that is readily attacked by the O2 molecule. This observation provides a clue as to what makes a surface site active for the ORR. The surface of an oxide contains neutral oxygen vacancies VO× . At room temperature, these vacancies are occupied by bound water; a proton of the

Electronic properties of solids for solid oxide fuel cells (SOFCs)

37

0.6 Measured at 800 °C

LSM

0.5

La2NiO4

ηc (V)

0.4 LSFN

0.3

LSCo

0.2

LSCN SCF

0.1 0.0 0

200

400

600 800 J (mA/cm2)

1000

1200

3.6 Overpotentials (ηc) of several non-porous cathode films at 800 °C, after reference 38. LSFN, La(Sr)Fe(Ni)O3; LSCo, La(Sr)CoO3; LSCN, La(Sr)Co(Ni)O3; SCF, SrCo(Fe)O3.

bound water is transferred to other surface oxide ions to introduce surface OH– ions. This process allows all the surface metal atoms to complete their oxygen coordination. However, at the Top of an SOFC, the bound water is lost, which recreates the surface VO× . Perovskites having a partially filled M4+/M3+ redox couple will stabilize the lower valence M3+ at a surface VO× . What distinguishes the surface Mn3+ and Co3+ ions is a σ-bonding e1 orbital degeneracy: HS Mn3+: t3e1 and IS Co3+: t5e1 ions with an e1 configuration are stable in a square-pyramidal site in which the e1 electron is ordered into a 3z2–r2 orbital projected toward the oxygen vacancy. The Fe3+ ion, on the other hand, is stable in fourfold oxygen coordination; it therefore attracts two VO× to create a stable tetrahedral site at the surface where it is shielded from attack by an O2 molecule unless the O2 molecule can convert the site to an octahedral-site Fe(V) species. It is energetically more favorable to attack the VO× of a square-pyramidal site with a capture of the 3z2–r2 electron to stabilize M4+–( O 2– )s, especially if a second electron can be captured from a partially filled σ* band. These considerations lead to the following hypothesis for good ORR kinetics at an oxide surface containing HS Mn3+: t3e1, IS Co3+: t5e1, and/or LS Ni3+: t6e1. The ORR would proceed by an O2 -molecule attack at the VO× of a square-pyramidal surface site (M 3+ –VO× ) + e – + O 2 = M 4+ (O 2) s2–

[3.12]

where one electron has come from the M3+–(3z2–r2) electron and one electron from the external circuit via the σ* band of the bulk. The terminal oxygen

38

Solid oxide fuel cell technology

atom of the resulting (O 2 ) s2– species can move over the surface oxygen to another VO× where it accepts one electron from the M3+–(3z2 – r2) orbital and one electron from the external circuit via the bulk σ* band M3+–( VO× )s + (M4+–(O2)2–)s + e– = 2(M4+–O2–)s

[3.13]

In these reactions, the M4+ ions become coordinated by six O2– ions, and the stronger O-2p, M-3d hybridization raises the energy of the 3z2–r2 orbital to where it donates electrons to the O2 molecule. Trapping of electrons from the external circuit creates an electric field that drives the surface (O2–)s ions to .. a bulk oxygen vacancy VO either within the electrode, from where it is .. transported to the electrolyte, or directly to an electrolyte VO at a TPB by surface oxide-ion conductivity to reconstitute the surface ..

2(M4+–O2–)s + 2 VO = 2(M3+– VO× )s + 2 O O×

[3.14]

for a total ORR ..

4e′ + O2 + 2 VO = 2 O O×

[3.15] ..

where O O× is a bulk O2– ion neutralizing the lattice 2e′ charge on a bulk VO . In this discussion, formal valences and Kröger–Vink notation are used. In each case, the O-2p, M-3d hybridization is strong enough to transform σ-bonding e electrons into itinerant electrons of an antibonding σ* band in the bulk; only at the surface does a VO× capture a localized e1 electron in a (3z2–r2) orbital. In SrCo0.8Fe0.2O3–δ, the Fe4+/Fe3+ couple is at a higher energy than the Co4+/Co3+ couple, so the iron is present as HS Fe4+: t3e1, like the Mn3+ ions. However, the iron cations at the surface of La0.7Sr0.3Fe0.8Ni0.2O3–δ are Fe3+ ions in a surface tetrahedral site. Attack of this site by an O2 molecule requires more thermal energy, which slows the ORR. Unfortunately, cathodes with a significant amount of Co and Sr exhibit too large a TEC relative to that of the YSZ or LSGM electrolyte owing to the excitation of higher spin states in the bulk cobalt. The introduction of Fe or Ni for Co and La for Sr in SrCo0.8Fe0.2O3–δ and La0.8Sr0.2Co0.8Ni0.2O3–δ was designed to mitigate the TEC mismatch. An alternative approach explored by Lee and Manthiram39 is to optimize the trade-off between TEC and cathode performance by investigating these cation substitutions in oxygen-deficient Ruddlesden–Popper phases An+1MnO3n+1, which consist of an intergrowth of (AO)2 rock-salt layers and An–1MnO3n–1 perovskite layers (Fig. 3.7).

3.5.3

Interconnect

An SOFC consists of several individual single cells connected in series to provide a desired total output voltage and power. The individual cells are

Electronic properties of solids for solid oxide fuel cells (SOFCs)

n=1 (a)

39

n=2 (b)

3.7 Ruddlesden–Popper An+1MnO3n+1 structures.

connected electrically by an interconnect material. At the high temperatures 800 °C < T < 1000 °C of a conventional SOFC, the interconnect is an oxide that must be fabricated as a dense separator of the anode and cathode gases. This material must not only be stable, but must also be a good electronic conductor in both the cathode and anode atmospheres; it must also provide good electronic contacts to both electrodes, contacts that are not broken by thermal cycling. The Cr3+ ion has a strong octahedral-site preference, and the large Ueff = U + ∆c puts the Cr3+/Cr2+ redox energy out of reach in an oxide. Therefore, the insulator LaCrO3 is not reduced at low PO 2 . Substitution of Sr and/or Ca for La introduces a mixed Cr4+/Cr3+ valence with a Wπ ≈ h ωo; these oxides are good electronic conductors with a TEC matched to that of the other oxide .. components of the SOFC. Since the lattice charge on an oxygen vacancy VO attracts Cr3+ ions that are more stable in sixfold oxygen coordination, the reduction in the anode atmosphere of the Cr4+/Cr3+ couple to all Cr3+ is resisted. However, a Cr4+ ion can be introduced by substituting Sr2+ for La3+, and Cr4+ can be stabilized in square-pyramidal coordination to allow some loss of oxygen in the fuel atmosphere. Loss of oxygen on the fuel side not only reduces σe across the anode/interconnect interface; it also creates a change in volume between the air and fuel side of the interconnect. Moreover, the high melting point and potential structural changes at high temperatures

40

Solid oxide fuel cell technology

require optimizing the choice of dopants and the use of soft-chemistry techniques to fabricate dense ceramics below 1450 °C. Substitution of other cations for Cr to reduce the loss of oxygen in the anode atmosphere runs the risk of cation interdiffusion at the electrode/interconnect interface. Nevertheless, perovskites with a mixed Cr4+/Cr3+ valence are used successfully as interconnects in a conventional high-temperature SOFC.

4 Transport of charged particles in a solid oxide fuel cell (SOFC)

Abstract: One of the important aspects of SOFC components is the transport behavior of charged particles; each type of charged particle contributes a partial electrical current density, either ionic or electronic in nature. In this chapter, the law of irreversible thermodynamics is primarily applied to the transport phenomena of oxide-ions, electron holes, and excess electrons in the electrolyte and interconnect materials. The steady-state PO 2 profiles and ionic and leakage (electronic or ionic) current densities across the two dense electrolyte and ceramic interconnection membranes are particularly solved from the phenomenological transport equations. Key words: flux density, charge neutrality, tracer (self) diffusivity, chemical diffusivity, ambipolar diffusivity, PO 2 profile, leakage current density.

4.1

General bulk transport theory

The bulk transport theory is generally described by the principles of irreversible thermodynamics. Interested readers are recommended to look into topical monographs such as those given in references 40 and 41. To be precise, the flux equations are only considered to be valid in the following context given the following assumptions: (a) the diffusivity is independent of diffusion direction, i.e. it is isotropic; (b) the electrical field ∇φ is much smaller than RT so that a linear relationship exists between flux density j (or current density i) and ∇φ; (c) coupling between the simultaneous movement of charged particles is ignored, i.e. the Onsager cross-coefficients are equal to zero; (d) chemical equilibrium between the solid and gas phases is always locally established; and (e) all fluxes, forces, and gradients are along one given diffusion direction, the x-direction. Therefore, the concentrations and fluxes have the same value at any position on a given plane normal to this direction.

4.1.1

Flux density and current density

The driving force for the transport of a charged particle (or charge carrier) k in a solid is the gradient of electrochemical potential ∇ηk, which is defined to combine the effects of chemical potential µk and electrostatic potential φ

ηk = µk + zkFφ

[4.1] 41

42

Solid oxide fuel cell technology

where zk is the number of electrical charges of charge carrier k. By thermodynamic definition, the µk of charge carrier k can be further expressed by

µ k = µ ko + RT ln a k

[4.2]

where µ ko is the standard-state chemical potential, which equals µk at ak = 1. From irreversible thermodynamics, the flux density jk of the charge carrier k is jk = –

σk ∇η k z k2 F 2

[4.3]

where σk is the electrical conductivity of charge carrier k. The current density ik that is related to the flux density jk is, therefore, given by Faraday’s law ik = z k Fj k = –

σk ∇η k zk F

[4.4]

Equations [4.3] and [4.4] serve as the basis for calculating the distributions of various transport properties of charge carrier k in electrolyte and interconnect given in the following sections.

4.1.2

Charge neutrality

For a solid with multiple charge carriers, the net current density of all charge carriers must equal zero at steady state and in absence of external current since otherwise local charge neutrality could not be maintained everywhere42

Σ ik = 0 k

or

Σ zk jk = 0 k

[4.5]

Equation [4.5] can also be extended to include the ‘quasi-neutrality’ condition, which applies where the particles are forced to migrate together by an attractive electrostatic force that is formed by a slight mis-adjustment of the charge balance of the moving particles so that they have a common diffusivity. The concentration mis-adjustment is relatively so small that the concentrations may be taken as equal. The ‘quasi-neutrality’ condition holds well for sample dimensions larger than the Debye length. This means that the higher the charge-carrier concentrations, the more accurate is the approximation. The ‘quasi-neutrality’ condition is mathematically expressed by

Σ z k ∇c k = 0 k

[4.5a]

where ∇ck is the concentration gradient of species k. From the transport number tk of the charge carrier k tk =

σk σ

[4.6]

Transport of charged particles in a solid oxide fuel cell (SOFC)

43

where

σ = Σ σk k

is the total electrical conductivity, and can be obtained by combining equation [4.4] with equation [4.5]

t

Σ zk ∇η k = 0 k k

[4.7]

In order to illustrate the utility of equation [4.7], consider a system with two charge carriers. From equation [4.7], it follows that

t1 t ∇η1 + 2 ∇η 2 = 0 z1 z2

[4.8]

Since t1 + t2 = 1, equation [4.8] can be further simplified into t1  1 ∇η1 – 1 ∇η 2  = – 1 ∇η 2 z2 z2  z1 

[4.9]

The flux density j2 of the charge carrier 2 can then be obtained from equation [4.3] j 2 = – t1 ×

σ2 ×  1 ∇η 2 – 1 ∇η1  z1  z2  z2 F 2

[4.10]

and the current density i2 from equation [4.4] i2 = – t1 ×

σ2  1 × ∇η 2 – 1 ∇η1  F z1  z2 

[4.11]

No restrictive assumptions have been made in the above deduction, making equations [4.10] and [4.11] valid for all types of solids with two types of charge carriers.

4.1.3

Ohm’s law and the Nernst–Einstein equation

In a crystal, the electronic and ionic conductivities are generally tensor quantities relating the current density i to the applied electric field ∇φ. In an isotropic medium such as a cubic crystal or a polycrystalline ceramic, the conductivity is a scalar. In this case, Ohm’s law for the charge carrier k has the form i k = – σ k × ∇ φ = – c k × q × vk

[4.12]

where ck is the concentration of mobile charge carrier k; q = zkF (on a onemole basis) is the lattice charge of charge carrier k moving with velocity vk.*

*q = zke on a particle basis; RT should also be replaced by kT.

44

Solid oxide fuel cell technology

From the definition of charge-carrier drift mobility uk ≡ vk /∇φ, the conductivity of species k is

σk = c k × q × uk

[4.13]

The Nernst–Einstein equation describes the basic relationship of the particle diffusivity Dk and the drift mobility uk of charge carrier k in the following form

Dk = RT

uk zk F

[4.14]

Substitution of equation [4.14] into equation [4.13] leads to c k Dk σ = 2 k2 RT zk F

[4.15]

If σk is measurable and ck is known, equation [4.15] can be used to determine Dk. The electrical conductivity σk is typically measured by the DC or AC technique. Therefore, Dk is often referred to as conductivity-related diffusivity.

4.1.4

Correlations between different diffusivities

In this section, we will discuss the definitions and correlations among particle diffusivity, tracer (or self) diffusivity, chemical diffusivity, and ambipolar diffusivity. These terms are commonly used in the field of solid-state electrochemistry. If any correlation effect is ignored, viz. the atom jump probabilities do not depend on the directions of previous jumps, the tracer diffusivity D* of the total concentration of species k in an ionic solid,† Nk, can be substituted into equation [4.15] as ckDk = NkD* N k D* σ = 2 k2 RT zk F

[4.16]

D* describes the diffusive motion macroscopically, i.e. as if all ions (Nk) took part in the process and not only the corresponding lattice defects with concentration ck. If the correlation effect can not be ignored, Dk is correlated with the tracer diffusivity D* by the Haven ratio HR D* = HRDk

[4.17]

HR correlates charge conduction to the ionic motion in successful jumps and is of the order of unity for most cases unless different jumping distances and †

It includes both mobile and immobile k charged particles.

Transport of charged particles in a solid oxide fuel cell (SOFC)

45

frequencies can not be ignored. For a vacancy diffusion mechanism, the Haven ratio HR becomes the tracer correlation factor f, HR = f. From a diffusion standpoint, the correlation factor f is defined by the equation * * Dactual = f Drandom

[4.18]

* Dactual

is the tracer diffusivity under actual conditions, where the atom Here * is the tracer diffusivity one would follows a correlated walk, and Drandom obtain if the atom made the same number of jumps per unit time but successive atom jumps were independent of one another. For a self-diffusion by a vacancy mechanism, f ranges from 0.78 for a face-centered cubic structure to 0.50 for a diamond structure. For impurity diffusion by a vacancy mechanism, f becomes almost zero for a fast diffusing impurity or almost unity for a slow diffusing impurity. An f value near zero corresponds to a large correlation effect, whereas one near unity indicates that there are only weak correlations between successive jumps. For processes involving changes of stoichiometry in solids, it is necessary for ions (such as positively charged particles k) and electrons (such as negatively charged e′) to diffuse simultaneously in order to maintain the charge neutrality. The diffusivity that describes such a process is termed chemical diffusivity D˜ . The flux of neutral species k* relative to the virtually immobile lattice frame is given by Fick’s first diffusion law j k* = – D˜ ∇c k*

[4.19]

where the asterisk represents the neutral species. For the charged particles k and electrons e′, the fluxes can be expressed by equation [4.3] and the following equation, respectively

σ e′ ∇η e ′ [4.20] F2 According to the charge-neutrality law, equation [4.5], the fluxes of jk*, jk, and je′ hold the following relationship je′ = –

j k = j k* =

j e′ zk

[4.21]

If the internal defect reactions are fast enough to attain local chemical equilibrium so that the concentrations of involved charge carriers anywhere in the solid are fixed by the local value of the chemical potential µk*, i.e. z k * = k k + + zke′ and ∇µk + zk∇µe′ = ∇µk* j k = j k* = –

1 z k2 F 2

×

σ kσ e′ ∇µ k* (σ k + σ e ′ )

[4.22]

Use of the definition of chemical potential of equations [4.2] and [4.15] simplifies equation [4.22] into

46

Solid oxide fuel cell technology j k* = – ( Dk t e ′ )

∇ ln a k* ∇c k* ∇ ln c k*

[4.23]

where te′ = σe′/(σk + σe′). Comparing equation [4.23] with equation [4.19] yields ∇ ln a k* D˜ = ( Dk t e ′ ) ∇ ln c k*

[4.24]

which leads to the definition of the ‘thermodynamic factor’ or ‘enhancement factor’

ξ=

∇ ln a k* ∇ ln c k*

[4.25]

It is obvious that the factor becomes unity for ideal behavior of a solid. However, ξ will deviate from unity for non-ideal solids and can only be determined directly from measuring the stoichiometry of neutral species k* as a function of the partial pressure of k*. D˜ is a measurable property. In the literature, D˜ has been reported being determined as a parameter from a diffusion-related process such as the conductivity relaxation. Another interesting diffusion coefficient that is less restrictive in assumptions is the ambipolar diffusivity, a form of D˜ that describes a simultaneous diffusion process of ions and electrons. It is commonly used in the fields of astronomy physics, plasma physics, semiconductors, and electrolytic solutions. For a simultaneous conduction of ions k with a smaller concentration of electrons e′ in a solid, use of the ‘quasi-neutrality’ assumption equation [4.5] and equation [4.10] with equation [4.2] leads to the following expression   j k = j k* = – z k2 c k Dk t e ′  21 + 1  ∇c k n  zk ck

[4.26]

where n = ce′ and te′ are the concentration and the transport number of free electrons, respectively. The common diffusivity Da then becomes   Da = z k2 c k t e ′ Dk  21 + 1  n  zk ck 

[4.27]

or can be further modified by using equations [4.15] and [4.6] into Da = te′Dk + tkDe′

[4.28]

where tk is the ionic transference number and De′ is the electron diffusivity. Since De′ (or mobility) is much greater than Dk, Da is often simplified into Da ≈ tkDe′

[4.29]

It is not difficult to find that Da is greater than Dk and is usually very large.

Transport of charged particles in a solid oxide fuel cell (SOFC)

4.2

47

Effect of electronic conduction in electrolyte on electromotive force (EMF)

One of the basic requirements for a good solid electrolyte is that charge transport in the electrolyte is completely ionic in nature. In reality, this condition is often difficult to satisfy. Depending upon the electrolyte material, operating temperature, and PO 2 , the electrical conduction in the electrolyte could be principally carried by oxide ions but accompanied by a minor contribution from electrons, either in the form of electron holes or excess electrons. The production of electronic conduction can be described by the following defect equilibria established between gas and solid At low PO 2 , .. O O× = 1 O 2 + VO + 2e ′ 2

[4.30]

At high PO 2 ,

1 O + V .. = O × +2h• O O 2 2

[4.31]

where e′ and h• represent excess (free) electrons and electron holes, respectively. Kröger–Vink notation is used. Assuming that the defect concentrations are low and the interaction between defects is negligible, application of the mass-action law to equations [4.30] and [4.31] yields the concentrations of excess electrons n = [e′] and electron holes p = [h•] n=

K1[O O× ] –1/4 P = K1′ PO–1/4 .. 2 [VO ] O 2

[4.32]

..

p=

K 2 [VO ] ×] [O O

PO1/4 = K 2′ PO1/4 2 2

[4.33]

where K1 and K2 are the chemical equilibrium constants of reactions [4.30] .. and [4.31], respectively. [VO ] = c V .. and [O O× ] = c O × are the concentrations O

..

O

of defect VO and lattice O O× , respectively, which are considered as constants in this case. The electrical conductivity σk of charged particles k (k = .. × , h . , e ) can be linked to its concentration c by the Nernst–Einstein VO , O O ′ k relationship as shown in equations [4.13] and [4.15]

σ k = z k F × ck × uk =

Dk c k × z k2 F 2 RT

[4.34]

Dk has the form Dk = Dko exp(–∆Hk/(RT)), which will be discussed further in Chapter 5. Substitution of equations [4.32] to [4.34] into the total electrical conductivity σ = σ V .. + σ e ′ + σ h yields •

O

48

Solid oxide fuel cell technology

σ ( T , PO 2 ) = =

4 F 2 DV .. O

RT

σ Vo .. O

T

× c V .. + O

F 2 De ′ F 2 Dh ×n+ ×p RT RT •

σo ∆H e ′  ∆H m  exp – + e ′ PO–1/4 exp  – 2  RT  T  RT 

σ ho. 1/4 ∆H h  P exp  – +  RT  T O2 •

[4.35]

where σ Vo .. , σ eo′ , and σ ho. are the pre-exponential factors of Arrhenius O relationships for the electrical conductions of oxide ion, excess electron, and electron holes, respectively. ∆Hm (in J/mol) is the motional enthalpy for oxide-ion conduction, which will be discussed in detail in Chapter 5. ∆He′ and ∆H h (in J/mol), are the motional enthalpies for excess electron and electron hole conductions, respectively. Equation [4.35] suggests that the conduction of excess electrons or electron holes preferentially occurs at low PO 2 and high PO 2 extremes, respectively, whereas the pure oxide-ion conduction dominates in the moderate PO 2 range. The boundary of each domain is determined by the relative magnitude of electronic conductivity to oxide-ion conductivity, often expressed by the ionic transport number t V .. = σ V .. /σ •

O

–1/4 1/4   PO   PO   t V .. = 1 +  2  + 2  O  Ph    Pe ′  

O

–1

[4.36]

•

where two important parameters, Pe′ and Ph , are introduced. The definitions of Pe′ and Ph . are the partial pressures of oxygen at which σ VO.. = σe′ and σ V .. = σ h , respectively, and are therefore often known as the characteristic O PO 2 of the electrolyte for excess electron and electron hole conductions, respectively. The relationship of Pe′ and Ph with temperature can be established by equating the first term of the right-hand side in equation [4.35] with the second and third terms, respectively •

•

•

4

 σo Pe ′ =  oe ′  σ ..  VO

  4( ∆ H e ′ – ∆ H m )   exp  –  RT   

 σ Vo .. O Ph =  o  σh 

  4( ∆ H m – ∆ H h )   exp  –  RT   

•

•

[4.37]

4

•

[4.38]

Equations [4.37] and [4.38] suggest that a logarithmic plot of Pe′ and Ph as a function of reciprocal temperature follows a straight line; from its slope the •

Transport of charged particles in a solid oxide fuel cell (SOFC)

49

activation energy containing the motional enthalpy of defects can be determined. The values of Pe′ and Ph can be experimentally measured by the techniques of coulometric titration and electrolyte breakdown voltage. Some Pe′ values are listed in Table 4.1 for ZrO2-based,43 CeO2-based,44 and LaGaO3-based electrolyte systems.45 It is clear that ZrO2-based electrolytes have the lowest electronic conduction, followed by LaGaO3-based and CeO2-based electrolytes. The Ph values for these electrolyte systems are well above one atmosphere before electron hole conduction becomes appreciable. The only material system that exhibits appreciable electron hole conduction under atmospheric conditions is the ThO2-based electrolyte. The presence of electronic conduction in a solid electrolyte of an electrochemical device, such an SOFC, leads to a lowered open-circuit voltage by an electronic leakage current, and therefore reduced electrical efficiency. The EMF after considering the electronic conduction is given by41 •

•

( PO′′2 )1/4 + ( Ph )1/4   ( PO′ 2 )1/4 + ( Pe ′ )1/4 E = RT  ln + ln  1/4 1/4 F  ( PO′′2 ) + ( Pe ′ ) ( PO′ 2 )1/4 + ( Ph )1/4  •

•

[4.39]

where PO′ 2 and PO′′2 are partial pressures of oxygen at cathode/electrolyte and anode/electrolyte interfaces, respectively. For most electrolytes, except for ThO2-based, electron hole conduction is negligible and equation [4.39] can be simplified into Table 4.1 Characteristic partial pressures of oxygen as a function of reciprocal temperature for excess (Pe′) and hole (Ph ) electronic conductions in solid electrolyte systems •

Solid electrolytes ZrO2– 8 mol% Y2O3 ZrO2– 6.4 mol% CaO ZrO2– 6.9 mol% MgO CeO2– 10 mol% Gd2O3

LaGaO3– 10 mol% SrO– 20 mol% MgO

Expression

Temperature range (°C)

log10P e ′ (atm) = – 56 500 + 19.3 T

500–1200

log10P e ′ (atm) = – 51 800 + 17.9 T

700–1600

log10P e ′ (atm) = – 74 370 + 29.42 T

1300–1600

log10P e ′ (atm) = – 36 970 + 18.00 T

>400

 2.475 eV  –1/ 4 σ e ′T = 3.456 × 10 9 exp –  PO 2 (S/cm) kT  

 14.0 ± 0.5 eV  P e ′ (atm) = (1.8 ± 0.5) × 10 24 exp  –  800–1000 kT   Ph (atm) = (6 ± 2) exp  – 1.9 ± 0.6 eV    kT •

50

Solid oxide fuel cell technology

 ( PO′ 2 )1/4 + ( Pe ′ )1/4  E = RT  ln F  ( PO′′2 )1/4 + ( Pe ′ )1/4 

[4.40]

Figure 4.1 shows a graphical representation of equation [4.40] calculated for T = 700 °C. As can be seen from this figure, appreciable deviation begins to occur as PO 2 approaches the characteristic Pe′. Therefore, use of the Pe′ value as a criterion for selecting a quality solid electrolyte or selecting an operating window of a given electrolyte is commonly practiced in the SOFC community. In Section 8.3, the effect of electronic leakage current on the electrical efficiency and power density of SOFCs will be discussed in detail.

4.3

Application to electrolyte: steady-state P O 2 distribution

One of the classical examples of using the aforementioned transport equations is to calculate the PO 2 profile across the electrolyte membrane under a steady-state condition with the assumption that the positively charged oxygen .. vacancy, VO , is the major charge carrier and the negatively charged excess electron, e′, is the minor charge carrier in electrolytes such as ZrO2-based and CeO2-based materials. The current density (or flux density) for each charge carrier is written according to equation [4.4] as σ V .. O [4.41] iV .. = – ∇η V .. O O 2F σ ie ′ = e ′ ∇η e ′ [4.42] F 1.40 1.20

No electronic conduction

E (V)

1.00 0.80 0.60 0.40 0.20

t = 700 °C P O′ 2 = 0.21 atm Pe′ = 10–20 atm

0.00 1×10–28

1×10–24

1×10–20

1×10–16 1×10–12 PO′′2 (atm)

1×10–8

1×10–4

4.1 A graphical representation of equation [4.40] at 700 °C.

1×100

Transport of charged particles in a solid oxide fuel cell (SOFC)

51

If an external load (assuming free electron in nature) with electronic conductivity σL is also considered, the current density for the load, iL, can be written in analogy to equation [4.42] as iL =

σL ∇η e ′ F

[4.43]

The charge-neutrality requirement mandates that the following condition should be met iV .. + ie ′ + i L = 0

[4.44]

ie ′ + i L = iO 2–

[4.45]

O

or with the current density of oxide ions iO 2– replacing iV .. . O

Substitution of equations [4.41] to [4.43] into equation [4.44] leads to

σ V .. O ∇η e ′ = 1 × ∇η V .. O 2 σ e′ + σ L

[4.46]

As ∇η V .. is not directly measurable, it must be replaced with another O experimentally measurable quantity such as partial pressure of oxygen. Consider chemical equilibrium established between charged defects in the electrolyte and gas phase via reaction [4.31]. The gradient of chemical potential of each defect holds

1 ∇µ + ∇µ .. + 2 ∇µ = 0 O2 e′ VO 2

[4.47]

Combining the definition of η in equation [4.1] with equation [4.47] yields ∇η V .. = – 1 ∇µ O 2 – 2 ∇η e ′ O 2

[4.48]

Substitution of equation [4.48] into equation [4.46] leads to

σ V .. O ∇η e ′ = – 1 × ∇µ O 2 4 σ V .. + σ e ′ + σ L

[4.49]

O

With equation [4.49], the current density equations [4.41] to [4.43] can be rewritten as

σ V .. × (σ e ′ + σ L ) O iV .. = 1 × ∇µ O 2 O σ V .. + σ e ′ + σ L 4F

[4.50]

σ V .. × σ e ′ O ie ′ = – 1 × ∇µ O 2 4 F σ V .. + σ e ′ + σ L

[4.51]

O

O

52

Solid oxide fuel cell technology

σ V .. × σ L O iL = – 1 × ∇µ O 2 4 F σ V .. + σ e ′ + σ L

[4.52]

O

It is evident that iV .. = – ie ′ when σL = 0 or in open-circuit conditions. For O simplicity, only a one-dimensional profile is considered. With the definition of µ in equation [4.2], equation [4.50] can be rewritten as

σ V .. × (σ e ′ + σ L ) O iV .. dx = RT × d ln PO 2 O σ V .. + σ e ′ + σ L 4F

[4.53]

O

Integration of equation [4.53] from 0 to x on the left-hand side and from PO 2 (0) to PO 2 (x) on the right-hand side, as shown in Fig. 4.2, yields the accumulative current density iV .. ( x ) O

x × iV .. ( x ) = RT O 4F

∫

ln PO 2 ( x )

σ V .. × (σ e ′ + σ L ) O

σ V .. + σ e ′ + σ L

ln PO 2 (0)

d ln PO 2

[4.54]

O

Similarly, the accumulative current density over the entire thickness L of the electrolyte, iV .. ( L ) , is given by O

ZrO2- or CeO2-based solid electrolytes PO2 (0) Low PO2 side

..

VO

PO2 (L ) High PO2 side

iL

e′

Load

L

0

x

(a) i V ..

O

iL

iL + i e ′ + i V .. = 0 O

ie′ (b)

4.2 (a) Schematic of transport of charged species in the electrolyte; (b) charge-neutrality diagram.

Transport of charged particles in a solid oxide fuel cell (SOFC)

L × iV .. ( L ) = RT O 4F

∫

ln PO 2 ( L )

σ V .. × (σ e ′ + σ L )

ln PO 2 (0)

O

σ V .. + σ e ′ + σ L

d ln PO 2

53

[4.55]

O

At steady state, iV .. ( x ) = iV .. ( L ) and division of equation [4.54] by equation O O [4.55] leads to

x = L

∫

ln PO 2 ( x )

∫

ln PO 2 ( L )

ln PO 2 (0)

ln PO 2 (0)

 σ VO.. × (σ e ′ + σ L )    d ln PO 2  σ VO.. + σ e ′ + σ L   σ VO.. × (σ e ′ + σ L )    d ln PO 2  σ VO.. + σ e ′ + σ L 

[4.56]

For electrolytes, σ V .. is independent of PO 2 as the concentration of oxygen O vacancies is fixed by the level of dopants, whereas σe′ varies with PO 2 , particularly in the low PO 2 range. The dependence given in equation [4.35] . Substitution of this relation into typically has the form of σ e ′ = σ eo′ × PO–1/4 2 equation [4.56] gives

x = L

 σ V .. + σ eo′ × PO 2 ( x ) –1/4 + σ L ln  O  σ .. + σ o × PO (0) –1/4 + σ L 2 e′  VO  σ V .. + σ eo′ × PO 2 ( L ) –1/4 + σ L ln  O  σ .. + σ o × PO (0) –1/4 + σ L 2 e′  VO

 σ o  + ln  e ′  σo   e ′  σ o  + ln  e ′  σo   e ′

+ (σ V .. + σ L ) × PO 2 ( x ) 1/4  O  + (σ V .. + σ L ) × PO 2 (0)1/ 4   O + (σ V .. + σ L ) × PO 2 ( L ) 1/4  O  + (σ V .. + σ L ) × PO 2 (0)1/4   O

[4.57] For known values of σ V .. , σ e ′ , and σ L , PO 2 as a function of x/L across the O electrolyte can be calculated. As an example, the CeO2-based electrolyte system is used as a model to demonstrate the PO 2 profile. In the calculation, the electrical properties compiled by B. C. H. Steele44 for Rhodia-made lowSiO2 (<30 ppm) Ce0.9Gd0.1O1.95 (GDC10) were adopted: 0.64 eV  σ V .. T = 1.09 × 10 5 exp – O kT  

(S/cm)

(T > 400 °C) [4.58a]

0.77 eV  σ V .. T = 1.00 × 10 6 exp – O kT  

(S/cm)

(T < 400 °C) [4.58b]

log 10 Pe ′ = –

36 970 + 18.00 T

(atm)

[4.59]

54

Solid oxide fuel cell technology

σ e ′ T = 3.456 × 10 9 exp  – 

2.475 eV  –1/4 P kT  O2

(S K/cm)

[4.60]

Figure 4.3 shows the calculated PO 2 profiles at four different temperatures and a current density of 1.0 A/cm2. As is evident in the figure, the overall PO 2 profile is mainly dominated by the low PO 2 , indicating the significant influence from electronic conduction. This profile has been favorably confirmed by the experimental results.46 The profile under open-circuit voltage (OCV) conditions shows essentially no difference from that under load, suggesting that the loading condition has minimal impact on the PO 2 profile across the electrolyte layer. With the PO 2 profile available, the profile of oxide-ion transference number, t V .. , can then easily be calculated by O

–1/4    PO  t V .. = 1 +  2   O P  e′   

–1

[4.61]

The results are shown in Fig. 4.4. Evidently, the effective thickness for a good electrolyte can be considerably reduced by elevating the temperature. Therefore, CeO2-based material is widely considered to be a candidate electrolyte for low-temperature SOFC applications.

1 × 100 1 × 10–3 1 × 10–6

10 mol% Gd2O3-doped CeO2 Loading current density: 1.0 A/cm2 PO2 (0) ~ determined by PH2O /PH2 = 1 PO2 (L ) = 0.209 atm

PO2 (atm)

1 × 10–9

1200 °C

1 × 10–12

1000 °C

1 × 10–15 800 °C 1 × 10–18 1 × 10–21 600 °C 1 × 10–24 0.0

0.2

0.4

0.6

0.8

x /L

4.3 PO2 profiles across CeO2-based electrolytes under load and at different temperatures.

1.0

Transport of charged particles in a solid oxide fuel cell (SOFC)

55

1.00

0.80

600 °C

tion

0.60

0.40 800 °C 0.20

10 mol% Gd2O3-doped CeO2 Loading current density: 1.0 A/cm2 PO2 (0) ~ determined by PH2O /PH2 = 1 PO2 (L ) = 0.209 atm

1000 °C 1200 °C

0.00 0.0

0.2

0.4

0.6

0.8

1.0

x /L

4.4 Profiles of ionic transference number of CeO2-based electrolytes at different temperatures.

4.4

Application to electrolyte: electronic leakage current density

The basic equation for calculating the electronic leakage current density is given by integrating equation [4.51] over the thickness L of the electrolyte with the electronic conductivity dependence on PO 2 as σ e ′ = σ eo′ × PO–1/4 2 ie ′ =

RTσ V ..

O

FL

 σ V .. + σ eo′ × PO 2 (L) –1/4 + σ L O ln   σ .. + σ eo′ × PO 2 (0) –1/4 + σ L  VO

  

[4.62]

Similarly, the ionic current density iVO.. can be obtained by integrating equation [4.50] iV .. = – O

–

RTσ V ..

O

FL

 σ V .. + σ eo′ × PO 2 (L) –1/4 + σ L O ln   σ .. + σ eo′ × PO 2 (0) –1/4 + σ L  VO

  

RTσ V .. σ L

 σ eo′ + (σ V .. + σ L ) × PO 2 (L)1/4  O [4.63] ln + σ L )  σ eo′ + (σ .. + σ L ) × PO 2 (0)1/4  VO  

O

FL (σ V ..

O

One must be cautious when using equation [4.62] to calculate ie′ as a function of load current density iL because the interfacial PO 2 (0) and PO 2 ( L ) vary

56

Solid oxide fuel cell technology

with load current iL as a result of current-dependent electrode polarizations. Therefore, the interfacial PO 2 (0) and PO 2 ( L ) have to be first solved with iL, by which ie′ can be further calculated from equation [4.62]. In order to do so, the boundary conditions of ionic current density at steady state are considered in the electrolyte, cathode, and anode layers (refer to Fig. 4.11 for the meanings of symbols)

iV .. (in electrolyte) = iV .. (in cathode) = iV .. (in anode) O O O

[4.64]

With linear polarizations in cathode and anode, the following relations hold

 PO (air)  iV .. (in cathode) = RT × 1 × ln  2  O 4 F Rc  PO 2 ( L ) 

[4.65]

 PO 2 (0)  iV .. (in anode) = RT × 1 × ln   O 4 F Ra  PO 2 (fuel) 

[4.66]

By equating equation [4.64] with equations [4.65] and [4.66], respectively, PO 2 (0) and PO 2 ( L ) can be determined as a function of load current iL. Figure 4.5 shows the solved interfacial PO 2 as a function of load current density. As clearly illustrated, the difference between PO 2 (0) and PO 2 ( L ) becomes smaller as iL increases. With interfacial PO 2 available, the variation of ie′ with iL is further calculated for three temperatures and is shown in 1 × 100

1 × 100 700 °C

1 × 10–2

1 × 10–5

1 × 10

1 × 10–10

500 °C 500 °C

600 °C

700 °C

1 × 10–15

1 × 10–6 1 × 10–20 1 × 10–8

Electrolyte thickness: 50 µm PO2 (b ) = 0.209 atm P O 2 (f ) ~ PH 2 O /PH 2 = 1

1 × 10–10 0.0

0.5

1.0

1.5 2.0 iL (A/cm2)

2.5

3.0

PO2 (0) (atm)

PO2 (L ) (atm)

600 °C –4

1 × 10–25 1 × 10–30 3.5

4.5 Interfacial partial pressures of oxygen as a function of load current density at different temperatures. Solid lines, PO2 (L); dashed lines, PO2 (0); anode resistance index, Ra (Ω cm2) = 1.3 × 10–5 exp(9261/T); cathode resistance index, Rc (Ω cm2) = 2.5 × 10–8 exp(13 952/T). After reference 47.

Transport of charged particles in a solid oxide fuel cell (SOFC)

57

Fig. 4.6. It is evident that ie′ decreases with iL but increases with T, implying that the open-circuit condition gives rise to the highest electronic leakage current density.

4.5

Application to interconnect: steady-state P O 2 distribution

The state-of-the-art ceramic interconnect for SOFCs is the doped LaCrO3 perovskite. In this section, we take Ca-doped LaCrO3 as an example to discuss the defect model and to calculate the PO 2 profile across the interconnect.48 • In Ca-doped LaCrO3, the predominant defect species are Ca ′La , h , and .. VO in accordance with Kröger–Vink notation. As pointed out in Chapter 3, the electrical conduction is dominated by small-polaron hopping where the • charge carriers are localized at the Cr sites. Therefore, the notation of h has • . . the same meaning as CrCr (Cr4+). In what follows, h and CrCr are used interchangeably. The chemical equilibrium between the defect species and surrounding atmosphere is established via

1 O + V .. + 2Cr × = 2Cr . + O × O Cr Cr O 2 2

[4.67]

or

1 O + V .. = O × + 2h• O O 2 2

3.50 3.00

ie′ (A/cm2)

2.50 2.00 1.50 700 °C

1.00 0.50

600 °C 500 °C

0.00 0.00

0.10

0.20

0.30 iL (A/cm2)

0.40

0.50

4.6 Electronic leakage current density as a function of load current density of a CeO2-based electrolyte at different temperatures.

58

Solid oxide fuel cell technology •

[4.67a] e′ + h = 0 Application of the mass-action law and only considering concentrations yields the chemical equilibrium constant

K=

. 2 [CrCr ] × [O O× ] .. × 2 [CrCr ] × [VO ] × PO1/22

[4.68]

Application of the ‘quasi-neutrality’ condition, equation [4.5a], leads to ..

. [CrCr ]+ 2[VO ] = [Ca ′La ]

[4.69]

and the site requirement mandates

[O O× ] + [VO ] = 3

NA Vm

[4.70]

. × [CrCr ] + [CrCr ]=

NA Vm

[4.71]

..

where NA is Avogadro’s number and Vm is the molar volume. The drift mobility u h of a small polaron is given by the small-polaron hopping theory •

u h = (1 – p ) × •

q × a 2 × ν ho E × exp  – h  = (1 – p ) × u ho  kT  kT •

•

•

[4.72] where p is the fraction of sites occupied by the polarons, q = e is the charge of polarons, a is the hopping distance per jump, ν ho is the phonon frequency, and E h is the hopping enthalpy; u ho is the drift mobility with zero occupancy of the polarons. According to equation [4.13], the hole conductivity σ h is given by •

•

•

•

. σ h = [CrCr ] × q × u h = p × (1 – p ) × e × u ho × •

•

•

NA Vm

[4.73]

. where p = [CrCr ] NA/Vm. From equation [4.73], p is solved by

1–

1–

4σ h × Vm e × u ho × N A 2 •

•

p=

[4.74]

Combining equation [4.68] with equations [4.70] and [4.71], and using y = [Ca ′La ] NA/Vm, yields

K=

p 2 × (6 – y + p ) (1 – p ) 2 × ( y – p ) × PO1/22

[4.75]

Transport of charged particles in a solid oxide fuel cell (SOFC)

59

Substitution of equation [4.74] into equation [4.75] leads to a relationship between σ h and PO 2 . On the other hand, a well-accepted empirical form of σ h as a function of PO 2 has also been shown experimentally to be •

•

σh = •

1 k1 + k 2 POk23

[4.76]

where k1, k2, and k3 are composition- and temperature-dependent constants. From the σ h values measured at different PO 2 , the parameters such as K and u ho can be obtained from non-linear least-squares fitting. One set of such data obtained for La0.7Ca0.3CrO3–δ, which will be used in the following calculations, is shown in Table 4.2 for different temperatures. The transport of charged species in LaCrO3-based interconnects is •

•

Table 4.2 Constants used for calculating the P O 2 profile and leakage-current density in La0.7Ca0.3CrO3–δ o

T (°C)

k1

k2

k3

K

uh

900 950 1000 1050

1.67 ×10–2 1.57 ×10–2 1.53 ×10–2 1.56 ×10–2

5.42 ×10–6 1.55 ×10–5 2.36 ×10–5 2.91 ×10–5

–0.231 –0.217 –0.217 –0.224

5.75 ×107 1.83 ×107 7.00 ×106 2.44 ×106

0.0879 0.0914 0.0900 0.0923

•

LaCrO3-based interconnects PO2 (0) Low PO2 side

PO2 (L ) High PO2 side

..

VO

iL

h·

Load

L

0

x

(a)

i V ..

O

iL + i V .. = i h •

iL

O

ih •

(b)

4.7 (a) Schematic of transport of charged species in an LaCrO3-based interconnect; (b) charge-neutrality diagram.

60

Solid oxide fuel cell technology

schematically illustrated in Fig. 4.7, where oxygen vacancies and electron holes are considered. From the basic current density equation [4.4], the current density for electron holes is

σh [4.77] ∇η h F .. Considering the local chemical equilibrium between VO and h• as shown in equation [4.67], the following relationships hold •

ih = – •

•

1 ∇µ + ∇µ .. = 2 ∇µ O2 h VO 2

•

∇µ e ′ + ∇µ h = 0

[4.78]

•

∇η e ′ + ∇η h = 0 •

Combining equations [4.1], [4.2], and [4.50] with equations [4.77] and [4.78] leads to the one-dimensional distributions of ionic current density iV .. and O electron hole current density i h •

σ V .. × (σ h – σ L ) d ln PO O 2 iV .. = RT × × O σ h – σ V .. – σ L 4F dx

[4.79]

σ V .. × σ h d ln PO 2 O i h = RT × × 4 F σ h – σ V .. – σ L dx

[4.80]

•

•

O

•

•

•

O

Evidently, iV .. = i h when σL = 0 or the OCV condition is operated. Note that O both σ V .. and σ h in LaCrO3-based materials depend on PO 2 . This dependence O is in contrast to the situation demonstrated for oxide-ion conductors where σ V .. is considered to be independent of PO 2 . According to equation [4.15], O σ V .. can be further expressed by O •

•

..

σ V .. = O

4 F 2 × [VO ] × DV ..

O

RT

[4.81] ..

where DV .. denotes the particle diffusivity of oxygen vacancies. Let [VO ] = O δ and consider the ‘quasi-neutrality’ condition of equation [4.69], p + 2δ = y; the correlation between δ and PO 2 , and therefore σ V .. and PO 2 , can then O be obtained from K=

p 2 × (6 – y + p ) (3 – δ ) × ( y – 2δ ) 2 = δ × (1– y + 2δ ) 2 × PO1/22 (1 – p ) 2 × ( y – p ) × PO1/2 2

[4.82]

According to Fig. 4.7, the charge-neutrality condition holds in the form

Transport of charged particles in a solid oxide fuel cell (SOFC)

61

i h = i L + iV ..

[4.83]

i L = i h + iO 2–

[4.83a]

•

O

or •

Combining equations [4.79] to [4.83] leads to the PO 2 profile across the thickness L of the interconnect

x = L

∫

ln PO 2 ( x )

∫

ln PO 2 ( L)

ln PO 2 (0)

ln PO 2 (0)

 σ VO.. × (σ h – σ L )    d ln PO 2  σ h – σ VO.. – σ L   σ VO.. × (σ h – σ L )  d ln PO 2  σ – σ V .. – σ L  O  h  •

•

[4.84]

•

•

Figure 4.8 shows the calculated PO 2 profiles across the interconnect at four different temperatures with the constants listed in Table 4.2 and a fuel mixture of CH4 and H2O (CH4/H2O = 2.5).48 It is evident that the PO 2 profile and 0

log PO2 (atm)

–5

–10 1000 °C 1050 °C –15 950 °C

900 °C –20 –0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

x /L

4.8 Calculated PO2 profiles across the thickness of a 30 mol% Cadoped LaCrO3 interconnect at different temperatures, from equation [4.84] and parameters in Table 4.2. The fuel is assumed to be a mixture of CH4 and H2O in a ratio of 2.5. Reproduced by permission of The Electrochemical Society.

62

Solid oxide fuel cell technology

therefore electrical conductivity are dominated primarily by the low PO 2 as evidenced by the sharp decrease near the air surface. This phenomenon resembles that of Fig. 4.3.

4.6

Application to interconnect: ionic leakage current density

In contrast to the situation discussed in Section 4.3, the boundary PO 2 for PO 2 distribution in the interconnect is very close to the PO 2 in bulk air and the fuel stream. The load current density has a marginal effect on the boundary PO 2 . As a result, equation [4.79] can be used to derive directly the ionic leakage current density iV .. . The calculated iV .. × x as a function of PO 2 is O O shown in Fig. 4.9 for four different temperatures. Under steady state, iV .. is O constant at any location in the interconnect so that PO 2 (x) is essentially proportional to x. It is seen that PO 2 changes almost linearly with x up to about 90% of the total thickness, followed by a sharp increase just before reaching the air–side surface. The iV .. × x at which PO 2 sharply changes O increases as the temperature rises, indicating that the ionic leakage current density increases with an increase in temperature.

log PO2 (atm)

1050 °C

1000 °C

950 °C

–5

900 °C

0

–10

–15

–20 0.00

0.02

0.04

0.06

0.08

0.10

0.12

i V .. × x (A/cm) O

4.9 Calculated PO2 and ionic current density i VO˙˙ from equation [4.79] and parameters in Table 4.2 at different temperatures. The PO2 at the fuel side is assumed to be 10–18 atm. Reproduced by permission of The Electrochemical Society.

Transport of charged particles in a solid oxide fuel cell (SOFC)

63

450 1050 °C

400

i V .. (mA/cm2) O

350

1000 °C

300 250 950 °C

200 150

900 °C

100 50 –2500

–2000 –1500

–1000 –500 iL (mA/cm2)

0

500

1000

4.10 Ionic current density as a function of load current density of a 30 mol% Ca-doped LaCrO3 interconnection with a thickness of 3 mm at different temperatures. Reproduced by permission of The Electrochemical Society.

The variation of iV .. with load current iL is shown in Fig. 4.10. Negative O current density means that the current flows through the interconnect from the air side to the fuel side, which corresponds to the operation in an SOFC. The ionic leakage current density of an SOFC mode decreases linearly with increasing load current. This result is in agreement with data for the case of oxide-ion conductors discussed in Section 4.4. This result can also be understood by the fact that the gradient of electrochemical potential of oxide ions increases with the decrease of the load current density because the gradients of chemical potential of oxygen and electrostatic potential are established in opposite directions.

4.7

Pressure effect on electronic leakage current density in the electrolyte

As pointed out in Section 4.4, the key to calculate the electronic leakage current ie′ in the electrolyte is to determine the interfacial PO 2 . In order to do so, a linear polarization is assumed for both cathode and anode, by which a steady-state oxygen-ion flux is established across the interfacial boundaries. As the total system pressure Pt is elevated, PO 2(L) at the cathode/electrolyte interface is considerably increased as a result of an increased PO 2(b) and decreased activation polarization suggested by equation [7.22] in Chapter 7, see Fig. 4.11. However, the PO 2(f) of the bulk fuel is only dependent on

64

Solid oxide fuel cell technology Anode

Electrolyte

Cathode

Pt > 1 atm

PO2 (b)

PO2 (L )

Pt > 1 atm

PO2 (0) PO2 (f)

L

4.11 Schematic of PO2 profiles across SOFC layers under two different system pressures. 1 × 10–24

Pe′ (atm)

1 × 10–25

1 × 10–26

Pt = 1 atm 1 × 10–27 ScSZ electrolyte made by plasma spray 1 × 10–28 0.76

0.78

0.80

0.82

0.84 0.86 1000/T (K–1)

0.88

0.90

0.92

0.94

4.12 Specific partial pressure of oxygen of plasma-sprayed ScSZ electrolyte as a function of reciprocal temperature.

PH 2 O / PH 2 , not Pt , at a fixed temperature. The PO 2(0) at the anode/electrolyte interface can only be reduced by an elevated Pt as a result of reduced activation polarization as implicated by equation [7.30] in Chapter 7. Therefore, both the increased PO 2(L) and decreased PO 2(0) enlarge the PO 2 gradient across the electrolyte membrane, leading to an increased electronic leakage current density. In the following discussion, a case study is given for a cathodesupported cylindrical SOFC with Sc2O3-doped ZrO2 as electrolyte to illustrate the true effect of Pt on ie′. The governing equation for the calculation is equation [4.62], and the key is to solve PO 2(0) and PO 2(L) with the presumption of linear cathode and anode polarizations. Since σ VO.. is independent of PO 2, σ eo′ in equation

Transport of charged particles in a solid oxide fuel cell (SOFC)

65

[4.62] is affected by Pt via

σ eo′ = σ V .. ( X e ′ Pt )1/4

[4.85]

O

where Xe′ is the Pe′ at Pt = 1 atm. Figure 4.12 shows the variation of Xe′ (Pe′) as a function of temperature measured via the OCV method described in Section 9.6 of Chapter 9. With Xe′ available, the interfacial PO 2 (0) and PO 2 (L) at different iL are calculated out from equation [4.62] for 900 °C, as is shown in Fig. 4.13. The parameters used in the calculation can be found 2.50 × 10–16

2.00 1.80 Dashed line: PO2 (0)

Pt = 10 atm 1.50 × 10–16

1.20 1.00

1.00 × 10–16

0.80 0.60

Pt = 1 atm

0.40

5.00 × 10–17

Pt = 1 atm

0.20 0.00 0.00

0.10

0.20 0.30 iL (A/cm2)

0.40

0.00 0.50

4.13 Variations of interfacial PO2 with load current iL of an ScSZbased SOFC at 900 °C and two system pressures. 0.018 0.016

Pt = 10 atm

0.014

ie′ (A/cm2)

0.012 0.010 0.008

Pt = 1 atm

0.006 0.004 0.002 0.000 0.000

0.100

0.200 iL (A/cm2)

0.300

0.400

4.14 Electronic leakage current ie′ as a function of load current iL of an ScSZ-based SOFC at 900 °C and two system pressures.

PO2 (0) (atm)

Pt = 10 atm

1.40

PO2 (L ) (atm)

2.00 × 10–16

Solid line: P O 2 (L)

1.60

66

Solid oxide fuel cell technology

in Table 4.1. The gradient of interfacial PO 2 across the electrolyte is evidently increased by an elevated Pt, which infers an increased driving force for electronic leakage current. With the PO 2 profile available, the electronic leakage current ie′ can then be calculated, as is shown in Fig. 4.14. As expected, ie′ at higher Pt is higher than ie′ at lower Pt. However, the trend that ie′ decreases with iL remains the same regardless of Pt. The fact that electronic leakage current density increases with the system pressure is technically important for a pressurized hybrid SOFC system, which is widely believed to be the most efficient system for electrical power generation. Higher leakage current means more fuel losses and therefore lower electrical efficiency, which is counterproductive to the highly efficient pressurized SOFC system. Therefore, minimizing leakage currents in SOFC components is highly desirable for high-temperature and high-pressure hybrid SOFC systems.

5 Oxide-ion electrolytes in solid oxide fuel cells (SOFCs)

Abstract: An electrolyte is an ionic conductor and an electronic insulator. The oxide-ion electrolyte is the core functional component of an SOFC. The level and the stability of oxide-ion conductivity in both oxidizing and reducing atmospheres largely determine whether a material is suitable as an oxide-ion electrolyte. Higher oxide-ion conductivity means less voltage loss for a given temperature or a lower operating temperature for a given layer thickness. On the other hand, the magnitude of any electronic conductivity in an oxide-ion conductor has a great impact on the maximum voltage and electrical efficiency achievable by the SOFC. Therefore, understanding the conducting behaviors of ceramic oxide-ion conductors is critically important to the design of new high-performance solid electrolyte materials and to the performance of an SOFC. In this chapter, the criteria for a good candidate oxide-ion conductor, a phenomenological description of conducting behavior, and the effect of crystal structure will be specifically discussed. Key words: oxide-ion conduction, phenomenology, random walk theory, fluorites, perovskites.

5.1

Quality criteria

Ideally, an oxide-ion electrolyte conducts only O2– ions (or oxygen vacancies .. VO ) and remains an electronic insulator under operating conditions. The total conductivity σ = σ V .. + σe + σi contains contributions from the oxideO ion conductivity, σ V .. , and any electronic conductivity (electronic holes or O excess electrons), σe. Normally, there are no mobile cations at the operating temperature Top of an SOFC to give an additional conductivity component, σi, to the total conductivity; but at lower temperature, absorbed water may give a protonic conduction. A quality criterion σ V .. >> σe of an oxide-ion O electrolyte is commonly expressed by the transport number t V ⋅⋅ ≡

σ V ..

O [5.1] ≈1 σ In an ideal electrochemical power cell, the ionic current through the electrolyte inside the cell matches an electronic current through an external load. Because ionic conductivities are much smaller than the electronic conductivities of the external circuit, the solid oxide-ion electrolyte of an SOFC is in the form of a membrane of small thickness L and large area A that separates the two electrodes of the cell electronically. The internal resistance of a cell is O

67

68

Solid oxide fuel cell technology

R=

L +R +R +R c a ohm σ V .. A

[5.2]

O

where Rohm is the total ohmic resistance resulting from the electrodes; Rc and Ra represent, respectively, the resistances associated with the kinetics of the surface reactions and ionic/electronic transport in the cathode and anode, respectively, including any resistance to O2–-ion transfer across the electrode/ electrolyte interface or at a triple phase boundary (TPB) consisting of electrode, electrolyte, and air or fuel. Rc and Ra can be current-dependent. For a current I through a cell, the overpotential η = ∫ R d I is a voltage drop to be minimized to yield a higher power P P = I × (VOC – η)

[5.3]

where VOC is the open-circuit voltage, which is equivalent to electromotive force (EMF) or E discussed in Chapter 2 if no leaks (chemical and physical) are involved in the electrochemical cell. Figure 5.1 illustrates a theoretical performance (polarization) curve for an electrochemical power cell regardless of actual fuel utilization. The voltage drop η at low current, region (i), includes the contribution from the kinetics of the chemical reactions at the electrode/gas interfaces, whereas the linear drop in V = VOC – η with increasing current in region (ii) is due to the resistance to ionic conduction in the electrolyte and electronic current in the electrodes. The final voltage drop at high current densities in region (iii) is due to a depletion of acceptor sites or mobile ions at an interface of the cell; it is known as the diffusion-limited region. A higher σ VO.. on the low- σ VO.. side of the interface displaces region (iii) to higher current densities and increases the power density.

V

VOC η

(i)

0

(ii)

(iii)

I

5.1 Typical polarization curve for an electrochemical power cell.

Oxide-ion electrolytes in solid oxide fuel cells (SOFCs)

69

These simple considerations lead to the following general criteria for the quality of a solid-electrolyte material to be used in an SOFC: (a) ease of fabrication into a mechanically strong membrane of arbitrary shape with small L and large A; (b) even with a small L/A ratio, an oxide-ion conductivity σ VO.. > 10–2 S/cm at Top is required; for polycrystalline and cubic materials, a scalar σ V .. O is used; cubic electrolytes are preferred as a σ⊥ ≠ σ|| in a one- or twodimensional oxide-ion conductor can lower σ V .. as well as introduce O anisotropic thermal expansions; (c) chemical and mechanical compatibility of the electrode/electrolyte interfaces as well as fast oxide-ion transfer across these interfaces or, where the surface chemical reaction occurs at a TPB, to/from the reaction site; (d) t V .. ≡ σ V .. /σ ≈ 1 ; O O (e) chemical stability in the working environments; (f) mechanical stability against thermal cycling between ambient temperature and Top; ceramic strength is improved where cell design retains the ceramic membrane under a compressive stress. In order to meet these criteria for the electrolyte of an SOFC, it is necessary to understand how the criteria limit the choice of materials.

5.2

Phenomenology

Oxides containing filled arrays of crystallographically equivalent, and therefore energetically equivalent, oxygen sites are not oxide-ion conductors. Oxideion conduction requires either the introduction of interstitial oxide ions, O 2– i , in an array of empty interstitial sites at some higher energy, and/or mobile oxygen vacancies in the array of occupied sites. Oxide-ion electrolytes are oxides that contain mobile oxygen vacancies. Oxides with main-group cations have a large energy gap Eg > 5 eV between the filled O-2p valence bands and the empty cation conduction bands. In these oxides, substitution of a main-group cation of lower valence introduces oxygen vacancies that trap two positive charges rather than holes in the O2p bands. In Kröger–Vink notation, the doubly charged oxygen vacancies .. are represented as VO . The situation is similar with transition-metal cations having empty d or f states. However, in a conventional SOFC where fuel oxidation occurs at a TPB, the electrolyte must not be reduced by the fuel gas. If a transition-metal oxide is used, the empty d and/or f states must be at too high an energy to accept electrons from the fuel if the requirement t V .. ≈ 1 is to be met. This constraint limits the use of transition-metal cations O to Sc3+, Y3+, Zr4+, La3+, Hf4+, and Ta5+. Rare-earth cations that, like Gd3+, have no 4fn configuration in the energy gap Eg, may also be used.

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Solid oxide fuel cell technology

Application of equation [4.13] to oxygen-vacancy conduction leads to the conductivity

σ V .. = c V .. × q × u V .. O

O

O

[5.4]

where c V .. = N × c m is the concentration of mobile oxygen vacancies; N is O the number of oxygen sites per unit volume on which the vacancy can move and cm is the fraction of available oxygen sites containing mobile oxygen vacancies; q = 2e is the lattice charge of an oxygen vacancy; u V .. ≡ v V .. / ∇φ O O is the drift mobility of the oxygen vacancy. Since oxide-ion conductivity is diffusive with a motional Gibbs free energy ∆Gm = ∆Hm – T ∆Sm, we use the Nernst–Einstein relationship (equation [4.14]) between the ionic diffusivity DV .. = D o exp (–∆Gm/kT) and the drift mobility O u V .. to obtain a conductivity O

E σ V .. = A exp  – a  O  kT  T

[5.5]

If c is the total fraction of oxygen vacancies and cm = c is temperatureindependent E a = ∆Hm A=

Nq 2 ∆S × c × D o × exp  m   k  k

[5.6] [5.7]

A disordered, mobile oxygen vacancy moves as an oxide ion hops into it from a neighboring site through a potential-energy saddle-point position having a shortest distance Rb from the center of the saddle-point to the nearest cation site. Where Rb is less than the sum of the radii of the oxide ion and the peripheral cation, thermal energy must be supplied to open up the saddle-point site. Even where Rb is roughly equal to the sum of the ionic radii, some thermal energy is needed to excite the mobile ion to the maximum in the potential-energy pathway at the saddle-point position. The motional enthalpy ∆Hm

= ∆Hb + ∆Hr

[5.8]

has two components: a barrier energy ∆Hb for the oxide ion to hop when the acceptor vacancy and donor site have the same potential energy, and a relaxation energy ∆Hr to make the two potential energies equal. ∆Hr is present because the time it takes for a mobile ion to hop across the barrier is longer than the time it takes for the host matrix to relax the bond lengths at the vacancy and occupied sites to their equilibrium values; the bond-length relaxations raise the potential at the vacancy site relative to that at an occupied site. The minimum value of ∆Hm is found where Rb is only a little smaller than the sum of the ionic radii and where the peripheral cations are easily polarized.

Oxide-ion electrolytes in solid oxide fuel cells (SOFCs)

71

Where the introduction of vacancies is done by doping with a cation of lower valence, the smaller charge on the dopant traps the charged oxygen vacancy it introduces by an energy ∆Ht. Although the creation of oxygen vacancies by aliovalent doping is analogous to the introduction of holes into the valence band of silicon by doping with boron, the trapping energy ∆Ht of an oxygen vacancy at a dopant cation is greater than the trapping energy of a hole at an acceptor site in silicon. Holes in the valence band of silicon carry only a single electronic charge and they are itinerant, which means they can be described by a hydrogenic model in which the dielectric screening parameter κ > 10 enters as κ –2. The diffusive motion of an oxygen vacancy reduces the dielectric screening of the coulomb component ∆Hc of ∆Ht to κ –1 so that 0.1 < ∆Hc < 1 eV. In addition, an ionic dopant of different size creates a local strain that may attract the oxygen vacancy by a lattice relaxation energy ∆Hre to give a total trapping energy ∆Ht

= ∆Hc + ∆Hre

[5.9]

In order for the trapped oxygen vacancy to contribute to a DC conductivity, it must free itself from the dopant, which makes ∆H c m = c × exp  – t   2kT 

[5.10]

and the activation energy of equation [5.6] becomes

Ea = ∆H m + 1 ∆H t 2

[5.11]

Alternatively, the vacancy must find a tortuous percolation pathway through the solid via dopant near-neighbors. However, a large enough concentration of dopant for percolation makes the oxide metastable with respect to vacancy ordering as a result of the coulomb repulsions between the oxide-ion vacancies. A few structures contain intrinsic oxygen vacancies; these vacancies are not introduced by aliovalent doping, so there is no ∆Ht. However, in these cases the fraction of vacant oxygen sites is large, and electrostatic interactions between the charged oxygen vacancies introduce long-range ordering below a temperature Tt that raises the potential energy of a vacancy site above that of the occupied sites by a gap energy ∆Hg, as illustrated schematically in Fig. 5.2. Ordered vacancies are not mobile; they are trapped by the energy ∆Hg, which makes the fraction of mobile vacancies vary as  ∆H g  c m = c × exp  –   2kT 

[5.12]

and the activation energy of equation [5.6] becomes

Ea = ∆ H m + 1 ∆ H g 2

[5.13]

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Solid oxide fuel cell technology

∆Hg(T ) ∆Hm

Order

Disorder (a)

∆Hg(T )

∆Hg(T )

∆Hg(0)

1.0

∆Hg(0)

1.0

T1

T

T1

T

(b) ln(σT )

ln(σT )

1/T

1/T (c)

5.2 (a) O2–-ion energies (filled circles) and oxygen-vacancy energies for ordered and disordered partial occupation of a set of sites that are crystallographically equivalent if the ions are disordered. (b) Order parameter and (c) Arrhenius plots for smooth versus firstorder transitions.

Since ∆Hg = ∆Hg(T) decreases as cm increases, the order parameter for the order–disorder transition at Tt can be defined as ∆Hg(T)/∆Hg(0). At temperatures T > Tt, all the vacancies may become mobile, in which case only the motional enthalpy ∆Hm enters the activation energy Ea of equation [5.6]. However, short-range order persists above Tt, and the electrostatic interactions between the vacancies introduce strong correlations of the vacancy motion that modify the diffusion coefficient calculated by random-walk theory and/or trap some vacancies in ordered clusters.

Oxide-ion electrolytes in solid oxide fuel cells (SOFCs)

5.3

73

Random-walk theory

For small values of c, the vacancies move independently of one another and therefore randomly; the ionic motions are not correlated. In this case, the theory of random walk can be used to obtain a general expression for Do in equation [5.7]. Because two atoms cannot occupy the same site, the transition probability for a vacancy to move to a neighboring site is w(1 – c)fν(E), where (1 – c)w is the probability of finding an O2– ion on the w nearneighbor sites and f is the correlation factor of order unity that depends on the jump path. The jump frequency ∆H ν ( E ) = ν O exp  – m   kT 

[5.14]

contains the free energy ∆Gm = ∆Hm – T ∆Sm for the oxide ion to reach a oxide-ion saddle-point and contains the attempt frequency νO of the mobile r optical-mode vibrations (1012–1013 Hz). In an electric field ∇ φ, the enthalpy ∆Hm – (q × ∇φ × l/2) for jumps in the direction of the field is lower than the ∆Hm + (q × ∇φ × l/2) for jumps in the opposite direction; l/2 is the vector distance to the saddle-point between sites. The net drift velocity vx in the direction lx is the product of (lx/2) and the difference in the jump probabilities in the forward and back directions. Because q × ∇φ × l/2 << kT holds at temperatures where the ions are mobile, it follows from expansion of the exponentials that vx ≈

q × ∇x φ kT

×

l x2 2

× w × (1 – c ) × f × ν

[5.15]

where ν = ν (E = 0). For an isotropic medium, l x2 = l y2 = l z2 allows replacing l x2 /2 by l2/6 where l is the distance between oxygen sites. Since from equation [4.14] we have ux = vx /∇xφ = qD/kT, it follows from equations [5.14] and [5.15] that 2 D o = l × w × (1 – c ) × f × ν O 6

[5.16]

in equation [5.7] for three-dimensional conduction. From these simple phenomenological considerations, it follows that the design of a solid electrolyte for the SOFC requires 0 < c(1 – c) < 1/4 and a structure that minimizes Ea of equation [5.6]. Although there are several candidate oxide-ion electrolytes, none is completely satisfactory for an SOFC operating at a Top < 800 °C. The different problems encountered with these candidates are illustrative of the challenge facing the designer of an oxideion electrolyte.

74

5.4

Solid oxide fuel cell technology

Fluorites

The discovery by Faraday of fast F–-ion conduction in PbF2 signaled that the fluorite structure of Fig. 5.3 is capable of conducting anions. However, PbF2 is remarkable because all the F– sites are occupied so that conduction requires excitation across a ∆Hg to the interstitial octahedral sites through a shared site face. Fast F–-ion conduction means that F– ions become disordered over the octahedral and tetrahedral sites with a ∆Hm for hopping through a shared site face that is made small by the polarizability of the 6s2 core electrons on the Pb2+ ions. No AO2 oxide with the fluorite structure contains an A4+ cation comparable to Pb2+; there is only the high-temperature δ-Bi2O3 phase having an oxygen-deficient fluorite structure. In this case, disordering of the oxideion vacancies over the tetrahedral sites requires a jump through the octahedral sites via two saddle-point shared faces. Here also the ∆Hm to hop through a shared site face is made small by the 6s2 lone pair electrons on the Bi3+ ion; however, the difference between the octahedral-site and tetrahedral-site energies, ∆Hg, must also be small. Therefore, early attempts to obtain a fast oxide-ion conductor concentrated on aliovalent doping on the A-sites of AO2 oxides so as to introduce oxygen vacancies, or on substitutions for Bi in Bi2O3 so as to retain the δ-Bi2O3 structure to room temperature. These studies were extended to the fluorite-related A2B2O6O′ pyrochlore structure containing an ordered array of A3+ and B4+ cations.

5.4.1

Stabilized zirconia

High-temperature zirconia (ZrO2) has the cubic fluorite structure of Fig. 5.3; on cooling from its melting point (2680 °C), it becomes tetragonal below 2370 °C and then monoclinic below 1170 °C. The cubic phase can be stabilized

Ca or Zr

5.3 Cubic fluorite structure.

O or F

Oxide-ion electrolytes in solid oxide fuel cells (SOFCs)

75

to room temperature by substitution of Zr4+ with larger cations of lower valence, which also introduces oxygen vacancies and therefore oxide-ion conductivity. The amount of dopant required to fully stabilize the cubic structure is about 12–13 mol% for CaO, 8–9 mol% for Y2O3 and Sc2O3, and 8–12 mol% for other rare-earth oxides R2O3.49 The trapping energy ∆Ht is greater for Zr1–xCaxO2–x than for a trivalent dopant. Zr1–xScxO2–0.5x has the smallest Ea because the size of the Sc3+ ion is most closely matched to that of Zr4+ and has only a single electron difference in charge, which minimizes both ∆Hre and ∆Hc of equation [5.9]. Nevertheless, Zr1–xYxO2–0.5x (YSZ) with 8–9 mol% Y2O3 is the cubic phase of choice. The tetragonal ZrO2 phase is stabilized with a low dopant content, i.e. 2– 2.5 mol% Y2O3.50 The cubic phase has low strength, toughness, and thermalshock resistance whereas the tetragonal phase has extremely high strength and toughness due to a stress-induced transformation to monoclinic zirconia. The tetragonal phase can be fabricated as a thin (20-µm thick), flexible membrane capable of supporting the two electrodes whereas a thin cubic phase of 20 µm thickness must be supported on a porous electrode, normally chosen as the anode that is fabricated to be under a compressive stress. With a larger value of c, the cubic phase has the highest σ V .. . O Compositions containing 3–7 mol% Y2O3 or R2O3 segregate into the cubic and tetragonal phases with time at Top even if quenched from the sintering temperature, to form a t′-phase with tweed-like microstructure. Segregation of the vacancies into a non-percolating, vacancy-rich phase reduces σ V .. ; a O segregation over time of vacancies into a vacancy-ordered phase or a nonpercolating vacancy-rich phase reduces the fraction cm of mobile oxygen vacancies. This phenomenon is known as ‘aging’. A change in the slope of the Arrhenius plots of ln (σ V .. T ) versus 1/T is O consistently observed near 800 °C in zirconia-based systems having no impurity phases formed in grain boundaries; a higher activation energy Ea is observed at lower temperatures, and the low-temperature value of Ea increases with dopant concentration. This behavior is a characteristic of dopant ions acting as nucleating centers for the formation of ordered-vacancy clusters. The simplest model for such a condensation consists of a critical temperature T* below which the oxygen vacancies are progressively trapped out into ordered clusters with decreasing temperature; above T* the vacancies are dissolved into the matrix of oxygen sites. If the chemical potentials of the condensate and the solid solution with a random distribution of vacancies are µ v* (cond) and µ v* (sol), respectively, then at equilibrium

µ v* (cond) = µ v* (sol) = µ v* (sol) + kT ln( c m / c )

[5.17]

At temperatures T < T*, the free energy for trapping vacancies into clusters is ∆Gt(T) = µ v* (cond) – µ v* (sol) . Since ∆Gt(T*) = 0 at equilibrium and the standard states of the disordered and ordered phases are the same, it follows that

76

Solid oxide fuel cell technology ∆Ht c ln  m  = – ×  1 – T*    c  kT T 

[5.18]

provided that ∆H and ∆S for the condensate are independent of temperature and ∆Ht = T ∆St. Combining equation [5.18] with equations [5.5], [5.6], and [5.7] gives

ln(σ V .. T ) = ln A – O

Ea kT

[5.19]

 Nq 2  2 O A=γ×   ×c×l ×ν  k 

[5.20]

∆S γ = f × w exp  m  6  k 

[5.21]

where for T < T*

Ea = ∆ H m + ∆ H t  1 – T*   T 

[5.22]

and for T ≥ T* E a = ∆Hm

[5.23]

It follows that an Arrhenius plot would give two straight lines intersecting at T* with activation energies E1 = ∆Hm and E2 = ∆Hm + ∆Ht with pre-exponential terms, A1 = A and A2 = A exp (∆Ht/kT*), respectively, and ∆Ht E – E1 A ln  2  = = 2  A1  kT * kT *

[5.24]

In summary, yttrium-stabilized cubic zirconia has an acceptable oxide-ion conductivity σ V .. at temperatures T > T* ≈ 800 °C provided thin (thickness O ≤ 20 µm), dense ceramic membranes can be fabricated free of grain-boundary impurities and the dopant concentration is kept near 9 mol% Y2O3 at the edge of the cubic-phase domain. Smaller dopant concentrations are plagued by aging; higher dopant concentrations encounter segregation of orderedvacancy phases. Although the sintering temperatures of zirconia are high (near 2000 K), the zirconias are chemically inert to the gaseous reactants of an SOFC and to most of the preferred electrode materials. However, electrodes containing La, if fabricated above 1200 °C, react with zirconias to form La2Zr2O7 pyrochlore interface phases that block O2–-ion transport.

5.4.2

Doped ceria

Cerium, praseodymium, and terbium can be stabilized as R4+ ions, and RO2 has the cubic fluorite structure. However, PrO2 and TbO2 are easily reduced

Oxide-ion electrolytes in solid oxide fuel cells (SOFCs)

77

as the Pr3+: 4f2 and Tb3+: 4f8 redox energies lie deep in the energy gap Eg between the R-5d conduction band and the O-2p valence band. The Ce3+: 4f1 level also falls within Eg, but at a higher energy. Nevertheless, CeO2 is also partially reduced to CeO2–x in the reducing atmosphere at the anode of an SOFC. .. The formal lattice charge at an oxygen vacancy VO is 2+, and the nearest.. neighbor cations tetrahedrally coordinated to a VO move away from the vacancy while the neighboring O2– ions move towards it. These atomic displacements raise the potential energy at the vacancy site by a relaxation enthalpy ∆Hr. In CeO2–x, the vacancies also trap the two electrons, each introduced as localized 4f1 configurations at neighboring Ce3+ ions; once freed from a trap site, these electrons move diffusively as small polarons to give an electronic component σe′ to the total conductivity σ = σ V .. + σe′. O In order to introduce oxygen vacancies into ceria without reducing the cerium to Ce3+, a rare-earth R3+ ion having no 4fn energy within Eg is substituted for Ce in Ce1–xRxO2–0.5x. Although this strategy gives a transport number t V .. ≈ 1 in air or an inert atmosphere such as argon, additional oxygen vacancies O and therefore Ce3+ ions are introduced in a reducing atmosphere to give a measurable electronic component σe′ to σ = σ V .. + σe′. The electronic O component is a polaronic charge transfer 4f1 to 4f0 on the mixed-valent Ce4+/ Ce3+ array. In a polycrystalline ceramic, the oxide-ion resistivity ρV .. = ρb + O ρgb has two components: an intragrain (bulk) resistivity ρb and a grainboundary resistivity ρgb that can be distinguished in an AC impedance spectrum, but not in a DC measurement. The Arrhenius plot of the bulk σ V .. versus O 1/T for Ce0.9Gd0.1O1.95 in air gave a T* = 583 ± 45 °C, ∆Hm = 0.63 ± 0.01 eV, and ∆Ht = 0.19 ± 0.01 eV = 2.57 kT*. A measured ∆Ht > kT* reflects the presence of short-range order among the disordered vacancies. The bulk electronic conductivity σe′ = σ − σ V .. can be obtained by measuring O

σ and σ V .. bulk conductivities in a reducing atmosphere and in air, respectively, O since the change in cm is small in a reducing atmosphere. The Arrhenius plots51 for σe′ gave a polaronic activation enthalpy Ea = ∆Hp = 0.40 ± 0.01 eV for T > T* and Ea = ∆Hp + ∆Ht ≈ 0.51 ± 0.01 eV for T < T*. Comparison of the activation energies of σe′ and σ V .. shows that the bulk polaronic conduction O above T* ≈ 583 °C is comparable with that for oxide-ion conduction. In view of the oxidizing atmosphere at the cathode, the effective transport number t V .. for oxide-ion conduction across a doped ceria electrolyte in an SOFC is O not prohibitively low. Moreover, a T* < 600 °C compared with a T* ≈ 800 °C in yttria-stabilized zirconia (YSZ) makes doped ceria competitive with YSZ for an SOFC operating below 600 °C. In addition, doped ceria is effectively chemically inert with respect to the electrode; interdiffusion of lanthanide ions of the ceria with those of a perovskite cathode does not create new phases at the interface that block oxide-ion transport, which may simplify co-firing of the oxides of the cell.

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Solid oxide fuel cell technology

The Sm3+ and Gd3+ ionic radii have the best match to that of the Ce4+ ion, and Sm0.2Ce0.8O1.9–δ (SDC) gives the highest oxide-ion conductivity of the rare-earth-doped cerias.52 Moreover, reduction of the Ce4+ at the surface of the SDC catalyzes the oxidation of H2. Therefore, SDC is considered a promising candidate electrolyte and anode component for a low-temperature SOFC.

5.4.3

Stabilized δ-Bi2O3

High-temperature δ-Bi2O3 has the fluorite structure with one-quarter of the oxygen sites randomly occupied by vacancies. It has a σ V .. ≈ 2.3 S/cm at O 800 °C, but it is only stable between its melting point at 804 °C and a firstorder order–disorder transition at Tt = 730 °C; the vacancy-ordered α-Bi2O3 phase is monoclinic. In the δ-Bi2O3 phase, the oxide ions neighboring an oxygen vacancy are displaced from the center of their tetrahedral site toward the common face with the empty octahedral site neighboring it and the vacancy.53 As on the Pb2+ ion of PbF2, the 6s2 lone pair at the Bi3+ ion is highly polarizable, which lowers ∆Hm for an O2–-ion jump across the empty octahedral site to the neighboring oxygen vacancy. The transition at Tt corresponds to a melting of the anion sublattice, which results in an abrupt collapse of ∆Hg and an increase in the oxide-ion conductivity by nearly three orders of magnitude, as is illustrated in Fig. 5.4. A large volume change at Tt degrades the mechanical properties of a ceramic membrane on thermal cycling through Tt. The stability range of the cubic phase can be extended to room temperature by substitution of one-quarter of the Bi3+ by an R3+ ion.54, 57 However, substitution of R 3+ ions lowers σ V .. progressively with the R 3+-ion O concentration. Figure 5.4 compares the Arrhenius plots of the conductivity of Bi2O3 with those of Bi0.75Y0.25O1.5 and several other oxide-ion electrolytes. The curve for Bi0.8Er0.2O1.5 is similar, but slightly higher, than that shown for Bi0.75Y0.25O1.5. Although the first-order transition between the α and δ phases of Bi2O3 is suppressed by a rare-earth substitution, a knee at a T* ≈ 873 K, i.e. 600 °C, reveals a condensation of vacancies into ordered clusters below T*. Values of σ V .. ≈ 0.11 and 0.23 S/cm at 650 °C for Bi0.75Y0.25O1.5 and O Bi0.8Er0.2O1.5, respectively, are quite acceptable for a low-temperature fuel cell; however, these stabilized phases have been shown to age at 600 °C owing to transformation to a vacancy-ordered rhombohedral phase having a significantly lower oxide-ion conductivity. This aging phenomenon can be suppressed by the addition of ZrO2, ThO2, or CeO2.58 Unfortunately, Bi2O3-based electrolytes are reduced in the atmosphere at the anode of an SOFC. Attempts to coat the anode side with a thin, dense layer of another electrolyte, e.g. Sm-doped CeO2 (SDC), have so far not been reported to be successful.

Oxide-ion electrolytes in solid oxide fuel cells (SOFCs)

79

1 6 0

log σ (S/cm)

–1

1

2 3 4

5

–2

–3

–4

–5

0.8

1.0

1.2 1.4 1000/T (K–1)

1.6

1.8

5.4 Arrhenius plots of total conductivity in air of several oxide-ion electrolytes: curve 1, Bi0.75Y0.25O1.5;54 curve 2, the perovskite La0.9Sr0.1Ga0.8Mg0.2O2.85;21 curve 3, Ce0.8Gd0.2O1.9;55 curve 4, Zr0.91Y0.09O1.955;56 curve 5, the brownmillerite Ba2In2O5;18 curve 6, Bi2O3.54

5.4.4

Pyrochlores

The cubic A2B2O6O′ pyrochlore structure is derived from an oxygen-deficient fluorite structure with an ordering of both the cations and the anion vacancies. If the ratio rA/rB of the ionic radii of the A3+ and B4+ ions is small enough, the cations may become disordered above a transition temperature Tt to give a cubic fluorite structure. Although such a transition has been achieved,59 the presence of two aliovalent cations makes the σ V .. of the disordered phase O non-competitive.

5.5

Perovskites

The ideal ABO3 perovskite structure of Fig. 5.5(a) consists of a cubic array of corner-shared BO6/2 octahedra with a large A cation at the body-center position. The A2B2O5 brownmillerite structure of Fig. 5.5(b) is derived from the perovskite structure by an ordering of the oxygen vacancies into alternate BO2 sheets to give layers of corner-shared BO6/2 octahedra alternating with layers of corner-shared BO4/2 tetrahedra; the tetrahedra and octahedra share

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Solid oxide fuel cell technology

B

O

A (a)

(b)

5.5 The structure of (a) cubic perovskite and (b) brownmillerite.

Oxide-ion electrolytes in solid oxide fuel cells (SOFCs)

81

corners along the c-axis. This structure is formed where the B cations are stable in both octahedral and tetrahedral symmetry, as is the case for the main-group ions Mg2+, Al3+, Ga3+, and In3+. The O2– ions of the tetrahedral sites are displaced toward tetrahedral sites defined by two A and two B cations, sites that would be the saddle-points for an oxygen jump in an oxygen-deficient perovskite. In order to test whether these displacements signal a small ∆Hm for oxide-ion motion in an oxygen-deficient perovskite, the σ V .. of the brownmillerite Ba2In2O5 was measured18 through an order– O disorder transition at Tt ≈ 930 °C. Since there are no aliovalent dopants to introduce the vacancies, the activation energy in a disordered phase should be Ea = ∆Hm. The Arrhenius plot for Ba2In2O5 in Fig. 5.4 shows an abrupt drop in ∆Hg at Tt just as in Bi2O3; the slope change near 650 °C is due to the onset of a temperature-dependent ∆Hg(T). Although long-range vacancy ordering is destroyed at Tt, considerable short-range order persists above Tt; the entropy change at Tt is only 4% of that calculated for a total order– disorder transition.60 Nevertheless, the data reveal an activation energy Ea that approaches ∆Hm comparable with that found in oxides with the fluorite structure. At room temperature, Ba2In2O5 takes in water from the atmosphere to become cubic BaInO2OH, which makes it a good proton conductor below 400°C.61 Problems with water insertion and the first-order order–disorder transition are overcome by substitution of La3+ for Ba2+. Figure 5.6 shows that Ba0.6La0.4InO2.7 has an oxide-ion conductivity σ V .. similar to that of O YSZ above 800 °C; it is a little higher below T* ≈ 800°C. These data show that oxygen-deficient perovskites containing B-site cations stable in octahedral, tetrahedral, or square-pyramidal oxygen coordination are candidate oxideion electrolytes competitive with oxygen-deficient oxides with the fluorite structure. The oxygen-deficient oxide-ion electrolyte with the highest σ V .. thus O far identified is La 0.8Sr0.2Ga 0.83Mg0.17O 2.815 (LSGM); 21 the Arrhenius plot for La0.9Sr0.1Ga0.8Mg0.2O2.85 is also shown in Fig. 5.6. The bulk conductivity of LSGM is almost independent of oxygen partial pressure from 10–20 ≤ PO 2 ≤ 1 atm, which indicates a transport number t V .. ≈ 1. It O exhibits a T* ≈ 600 °C with a σ V .. > 10–2 S/cm at 600 °C. In the system O La0.8Sr0.2Ga1–y MgyO3–0.5(0.2 + y), first identified by Ishihara et al.,19 two deleterious impurity phases have been identified.22 In the interval 0.05 ≤ y ≤ 0.10, LaSrGa3O7 forms at the grain boundaries to block O2–-ion transport; in the interval 0.25 ≤ y ≤ 0.30, the impurity phase LaSrGaO4 eliminates the grain-boundary contribution to the impedance. In order to use LSGM as the O2–-ion electrolyte of an SOFC, it is necessary to have chemical compatibility with the electrodes. Unless a thin barrier layer with equal La3+-ion activity is placed between the electrode and the electrolyte, interdiffusion of La3+ ions can create unwanted interfacial phases such as LaNiO3 or LaSrGa3O7 that block O2–-ion transport. The fluorite

82

Solid oxide fuel cell technology 0

LSGM 40% La YSZ

log σ (S/cm)

–1

–2

–3

–4

–5 0.8

1

1.2 1.4 1000/T (K–1)

1.6

1.8

5.6 Conductivity σ VO˙˙ (T) of La0.9Sr0.1Ga0.8Mg0.2O2.85 (LSGM), La0.4Ba0.6InO2.7 (40% La), and (ZrO2)0.9(Y2O3)0.1 (YSZ).

phase Ce0.6La0.4O1.8 has proven to make a good barrier layer at the anode/ electrolyte interface.62

5.6

Other oxides

5.6.1

Bimevox

Aurivillius phases have an intergrowth of (Bi2O2)2+ sheets and perovskite blocks (An–1BnO3n+1)2– that contain a layer of octahedral B sites. The (Bi2O2)2+ layer consists of a planar square array of oxygen atoms with Bi3+ ions above and below alternate squares, each situated above an A-site position of the adjoining perovskite layer. Below 604 °C, the oxidation catalyst Bi2MoO6 forms a tetragonal γ phase with the Aurivillius n = 1 structure of Fig. 5.7. Bi2VO5.5 has an oxygen-deficient Aurivillius n = 1 structure. On heating, it undergoes two transitions: from a monoclinic α to an orthorhombic β phase at 450 °C and from the β to a tetragonal γ phase at 570 °C. The vacancydisordered γ phase can be stabilized to room temperature by substituting

Oxide-ion electrolytes in solid oxide fuel cells (SOFCs)

83

Bi

O(1) O(2)

O(3) V

5.7 The ideal structure of Bi2MoO6.

other elements for vanadium; these stabilized tetragonal phases are referred to as BIMEVOX, where ME represents the substituted metal atom.63, 64 The highest oxide-ion conductivities first obtained were with BICUVOX-10, Bi2V0.9Cu0.1O5.35. This composition contains the lowest concentration of copper at which the room-temperature structure is tetragonal. Although BICUVOX has a transport number t V .. ≈ 1 in air, it is decomposed in the O reducing atmosphere at the anode of an SOFC.

5.6.2

Oxides with isolated tetrahedral polyanions

Structures containing isolated (MO4)n– polyanions can undergo an order– disorder transition temperature Tt above which the anions rotate freely. Introduction of oxygen vacancies by aliovalent doping may create cornershared (M2O7) units or, at room temperature in a wet atmosphere, MO2(OH)2 units as in the ferroelectric KH2PO4. With a smaller concentration of oxygen vacancies in wet atmosphere, the proton-transfer reaction

84

Solid oxide fuel cell technology

MO2(OH)2 + MO4 = 2MO3(OH)

[5.25]

in wet atmosphere may not require too high an activation energy for proton transfer to give an H+-ion electrolyte. At higher temperatures, the water is lost and free rotations of the polyanions with an asymmetric M–O…M bond in the (M2O7) unit allows transfer of the oxygen vacancy from one polyanion to another in a mechanism analogous to the Grotthus mechanism of proton transfer in an aqueous medium. These processes are now under active investigation, but a competitive oxide-ion electrolyte with isolated tetrahedral polyanions for the SOFC has yet to emerge.65

6 Current, gas flow, utilization, and energy balance in a solid oxide fuel cell (SOFC)

Abstract: This chapter describes the basic relationships among current, gas flows, and utilizations of fuel and air, two reactants of an SOFC. Examples are given to calculate the stack compositions and the exhaust temperature from the energy balance in an SOFC system. Key words: fuel flow, air flow, load and ionic currents, fuel and oxygen utilizations, fuel consumption, energy balance.

6.1

Introduction

The fact that an SOFC converts chemical energy in fuel directly into electrical power in an electrochemical manner infers a direct link between the electrical current (a measure of electrical power) and fuel flow (a measure of chemical energy). The classical Faraday’s law lays the foundation for such a connection. For an electro-oxidation reaction of a simple fuel A taking place at the anode A + z O 2– = AO z /2 + ze ′ 2

[6.1]

a total of z electrons is transferred. For a full utilization of fuel A with flux Jo (mol/s), the total current I (A) generated via the reaction is governed by Faraday’s law I = z × F × Jo

[6.2]

For a partial utilization of fuel flux J at current I to a degree of Uf < 1, equation [6.2] can be modified into I = z × F × Uf × J

[6.3]

where Uf is commonly termed the fuel utilization. It is evident that J is greater than Jo for a fixed I. The division of equation [6.2] by equation [6.3] outlines the true definition of Uf at a given current I o Uf = J J

[6.4]

The physical meaning of U f is understood to be how much fuel is electrochemically oxidized via reaction [6.1] relative to the total fuel input. Note that the total current I as shown in equations [6.2] and [6.3] should be 85

86

Solid oxide fuel cell technology

considered as the ionic current Iion (or I V .. ) in the electrolyte, as mentioned O in Chapter 4, not the external load current IL. Iion differs from IL by the amount Ie′ when electronic conduction is present in the electrolyte. This phenomenon has been systematically discussed in Chapter 4 and will be explored further in Chapter 9.

6.2

Fuel flow, current, and fuel utilization

For H2 and CO fuels, the electrochemical oxidation reaction taking place at the anode is expressed by H2(g) (or CO(g)) + O2– = H2O(g) (or CO2(g)) + 2e′

[6.5]

Rewriting equation [6.3] with the substitution z = 2 and F = 96 485 C/mol yields the following mass flow rate of fuel Qf in slpm (standard liters per minute)

Qf =

0.006 96 × I ion 22.4 × 60 × I ion = Uf 2 × 96 485 × U f

(slpm H2 or CO) [6.6]

This equation suggests that for a 100% utilization of fuel H2 or CO, 1 ampere is equivalent to 0.00697 slpm H2 or CO. The subscript ‘ion’ has the same physical meaning as ‘oxygen vacancy’ discussed in Chapter 4. For a generic hydrocarbon fuel CnH2n+2, where n represents the number of carbons, the anodic electro-oxidation reaction is expressed by CnH2n+2(g,1) + (3n + 1)O2– = nCO2(g) + (n + 1)H2O(g) + (6n + 2)e′ [6.7] with z = 2 × (3n + 1). Following equation [6.3], the mass flow rate of fuel Qf is then written by

Qf =

0.006 96 × I ion 22.4 × 60 × I ion = (3 n + 1) × U f 2 × (3 n + 1) × 96 485 × U f

(slpm CnH2n+2)

[6.8]

If there are N cells in an SOFC stack, the total fuel flow is determined by

Qf =

0.006 96 × I ion × N (3 n + 1) × U f

(slpm CnH2n+2)

[6.9]

Industrial natural gas usually contains a mixture of CnH2n+2. If the molar fraction for CnH2n+2 is yn, the total number of electrons transferred during the electro-oxidation reaction is 2 × Σ nm=0 (3 n + 1) y n , from which the total natural gas flow is given by

Current, gas flow, utilization, and energy balance

Qf =

0.006 96 × I ion × N  m  (3 n + 1) y n  × U f  nΣ  =0 

(slpm natural gas)

87

[6.10]

Two examples of how to calculate fuel flows are illustrated. For pure CH4, n = 1, and z = 8, the CH4 flow given by equation [6.9] is

Qf =

0.006 96 × I ion × N 4 × Uf

(slpm CH4)

[6.11]

Another example is given for an industrial natural gas 13AO, with compositions as listed in Table 6.1. The average number of electrons transferred via the anodic oxidation reaction is 2 × 4.60 for an average number of carbons n = 1.2. By equation [6.10] the fuel flow rate set for 13AO natural gas is given by

Qf =

6.3

0.006 96 × I ion × N 4.60 × U f

(slpm 13AO natural gas)

[6.12]

Air flow, current, and oxygen utilization

For the oxygen reduction process, every mole of O2– transferred consumes 0.5 mole of O2 as indicated by the following reaction 1 O + 2e ′ = O 2– 2 2(g)

[6.13]

On the other hand, reaction [6.7] suggests that every mole of CnH2n+2 consumes Σ nm=0 (3 n + 1) y n mole O2–. Therefore, the O2 flow QOo 2 at 100% utilization of O2 can be rewritten from equation [6.10] as m

QOo 2 = 1 × Σ (3 n + 1) y n × Qf × U f = 0.00348 × I ion × N 2 n =0 (slpm O2) [6.14] Table 6.1 Composition of an industrial natural gas 13AO n

yn

(3n + 1)yn

1 2 3 4

0.88 0.06 0.04 0.02

3.52 0.42 0.40 0.26

m

Σ (3n + 1) y n = n =0

4.60

88

Solid oxide fuel cell technology

Let Ua be the oxygen utilization, then Ua =

QOo 2 = 1 × 0.003 48 × I ion × N QO 2 QO 2

[6.15]

Rearrangement of equation [6.15] with S = 1/Ua, which is termed ‘oxygen stoich’, yields QO 2 = 1 × 0.003 48 × I ion × N = 0.003 48 × S × I ion × N [6.16] Ua In most cases, air is used as a practical oxidant and the flow rate of air Qair is then given by Qair =

0.003 48 × S × I ion × N = 0.016 65 × S × I ion × N 0.209

[6.17]

It is to be noted that the air flow rate is only a function of oxygen stoich, cell current, and number of cells. In SOFC generator operation, air is commonly used as a vehicle to deliver external heat to the SOFC stack during start-up and to reject heat generated from the cell under normal operation. Therefore, a lower S or higher Qair is often preferable.

6.4

Fuel consumption

Equation [6.4] gives the definition of Uf, which depicts the consumption of fuel by the electrochemical reaction that generates useful external current. However, there are some other sources that consume fuels without producing external current. These sources include physical leaks, fuel bypass away from the reacting surface (anode), and – most importantly – the leakage current induced by electronic conduction in the electrolyte and ionic conduction in the ceramic interconnect. The latter, which has been discussed in detail in Chapter 4, constitutes the majority of fuel losses. Therefore, it is essential to coin a general term to describe the total consumption of the fuel within an SOFC. From a practical viewpoint, it is the load current IL, not Iion, that measures the performance of an SOFC generator. In order to continue to use the form of equation [6.10] to set fuel flows, but with a consideration of total leakage flux QL (in slpm) and the rate of fuel bypass B, the fuel consumption Cf is introduced by Cf =

(0.006 96 × I L + QL ) × N   (3 n + 1) y n  × Qf × (1 – B )  nΣ  =0  m

[6.18]

Note that equation [6.18] reflects an average way of correcting for fuel losses. The total leakage flux QL = QLchem + QLphy includes the chemical

Current, gas flow, utilization, and energy balance

89

contribution from the electronic leakage current in the electrolyte and ionic leakage current in the interconnect and physical contribution from the pores and microcracks in the electrolyte and interconnect layers. The former is a function of the load current IL whereas the latter should be a constant independent of IL. According to the analysis presented in Chapter 4, increasing IL generally decreases Ie′ in the electrolyte and I V .. in the ceramic interconnect; O in both cases QLchem is decreased. Therefore, Cf should decrease with IL and become closer to Uf. The rate of fuel bypass represents the percentage of fuel that is not electrochemically oxidized by the cell and typically is in the range of 2–3%, largely dependent on the stack design and assembly. In Chapter 9, the experimental technique to measure Cf directly will be discussed. Comparing equations [6.10] and [6.18] gives the ratio of Uf to Cf

0.006 96 × I L + QL Cf = Uf 0.006 96 × I ion + (1 – B )

[6.19]

Experimentally, Cf can be determined by analyzing the exit gas composition via the following equation

QH + QCO Cf = Total fuel consumed = 1 – o 2 o Total fuel input QH 2 + QCO

[6.20]

where Qo and Q represent inlet and exit mass flow rates of fuel components, respectively. Let ro and r represent the oxide-to-fuel ratio of inlet and exit fuels, respectively ro =

r=

o QHo 2 O + QCO 2 o QHo 2 + QCO

QH 2 O + QCO 2 QH 2 + QCO

[6.21]

[6.22]

Considering H and C mass balances

QHo 2 + QHo 2 O = QH 2 + QH 2 O o o QCO + QCO = QCO + QCO 2 2

[6.23]

substitution of equations [6.21] to [6.23] into equation [6.20] yields o Cf = 1 – 1 + r [6.24] 1+r The validity of equation [6.24] can be easily verified with a known condition. For example, consuming 85% of an inlet composition of 89% H2–11% H2O by an SOFC would produce a mixture of 13% H2–87% H2O at the exit. If we take r = 87/13 and ro = 11/89, Cf is equal to 85%. The key to obtaining Cf is how to determine the ratio r at the fuel exit. The oxide-to-fuel ratio r can be obtained experimentally by measuring PO 2 with

90

Solid oxide fuel cell technology

the oxygen sensor technique. For simple H2–H2O and CO–CO2 fuels, equations [2.10] and [2.11] can be used to calculate r. In Chapter 9, more details will be given on how to conduct such an experiment.

6.5

Calculating stack fuel composition of reformed natural gas

In order to demonstrate how to calculate the fuel compositions in an SOFC stack, Fig. 6.1 presents a schematic illustration of the fuel flow in a stack, including steam methane reforming, electrochemical oxidation, and recycling of depleted fuel. For an SOFC generator fueled by commercially available natural gas that is composed mainly of CH4 , but with minor amounts of higher hydrocarbons and a trace of impurities, the steam reforming reaction can be expressed by CnH2n+2(g,1) + nH2O(g) = (2n + 1)H2(g) + nCO(g)

[6.25]

The inlet molar fraction of H2, CO, and H2O at the stack entrance (point 1 in Fig. 6.1) can be calculated by m

[H 2 ] o = Σ (2 n + 1) y n n =0 m

[CO] o = Σ ny n

[6.26]

n =0 m

[H 2 O] o = – Σ ny n n =0

Exhaust

Fresh natural gas

Recycled anode gas

Exit 2

3

SOFC stack

Inlet

Pre-reformer

1

6.1 Flow diagram of fuel in a natural gas-fueled SOFC generator.

Current, gas flow, utilization, and energy balance

91

If the incoming, fully reformed natural gas is led into the SOFC stack and oxidized by a degree Cf (including both fuel cell and other leaks), the composition of the spent fuel at the exit (point 2 in Fig. 6.1) is then calculated by m

[H 2 ] = (1 – Cf ) Σ (2 n + 1) y n n =0 m

[CO] = (1 – Cf ) Σ ny n n =0

m

m

n =0 m

n =0

[6.27]

[H 2 O] = Cf Σ (2 n + 1) y n – Σ ny n [CO 2 ] = Cf Σ ny n n =0

The percentage of unspent fuel ([H2] + [CO]) in the gas stream at the exit, which is directly related to Cf of the SOFC, is therefore given by m

[H 2 ] + [CO] = (1 – Cf ) [H 2 ] + [CO] + [H 2 O] + [CO 2 ]

Σ (3n + 1) y n n =0 m

[6.28]

Σ (2 n + 1) y n

n =0

For the left side of equation [6.28], the actual composition of the spent fuel at point 2 in Fig. 6.1, excluding the H2O content, can in practice be measured by gas chromatography (GC). To work out the H2O content in the spent fuel, a mass balance must be used. In this case, the H-to-C ratio, HTC, is chosen and held as a constant throughout the reaction; it is given by m

HTC =

Σ (2 n + 2) y n n =0

[6.29]

m

Σ ny n n =0

If the measured molar fractions for H2, CO, and CO2 are represented by a, b, and c, respectively, together with the derived H2O content d, the Cf can be calculated via equation [6.28] as shown by m

Cf = 1 –

Σ (2 n + 1) y n n =0 m

Σ (3n + 1) y n

×

a+b a+b+c+d

[6.30]

n =0

Note that the composition of the mixture of depleted fuel and fresh natural gas at point 3 in Fig. 6.1 can also be calculated out with a known recycling factor. In order to validate the above equations and demonstrate how to use them, practical examples are given in the following.

92

Solid oxide fuel cell technology

Example 1 Table 6.2 shows the measured composition of an industrial natural gas and the calculated reformed natural gas compositions. With knowledge of the incoming natural gas composition, the HTC is calculated from equation [6.29] to be 3.826. The gas composition measured at the exit (point 2 in Fig. 6.1) by GC was: H2, 27.44%; CO, 19.94%; CO2, 39.32%. Combining this composition with the known HTC value, the H2O content is calculated to be 85.92%. By substituting the above spent-fuel and incoming natural gas compositions into equation [6.30], the Cf is calculated to be 0.796. Example 2 Another example is given for a simplified case. Where n = 1, a pure CH4 is fed into an SOFC stack. Equation [6.30] is simplified into

a+b Cf = 1 – 3 × 4 a+b+c+d

[6.31]

For a Cf = 0.85 (relative to the incoming CH4), the unspent fuel (H2 + CO) at the exit (point 2) amounts to 0.20.

6.6

Energy balance in a solid oxide fuel cell system

Understanding the energy balance in an SOFC system is critically important for estimating the distribution of energy efficiency. The basic principle of energy balance for an SOFC system is the energy conservation law as shown by ∑ Energy rates in = ∑ Energy rates out

[6.32]

Figure 6.2 shows a schematic of energy input and output up to the level of stack DC electrical power. The net AC power can be easily integrated into Table 6.2 Chemical compositions of incoming and reformed natural gas (NG) Incoming NG

Reformed NG

n

yn

H2

CO

H2 O

1 2 3 4 5 6

0.923 02 0.037 68 0.014 71 0.008 90 0 0

2.7691 0.1884 0.1030 0.0801 0 0

0.923 02 0.075 36 0.044 13 0.035 60 0 0

–0.923 02 –0.075 36 –0.044 13 –0.035 60 0 0

Total

0.9831

3.1406

1.078 11

–1.078 11

Current, gas flow, utilization, and energy balance Energy in fuel Qf, Tf, ∆cHf

93

DC electrical power

Pexh PDC

Pf SOFC generator

Pair-PL Energy in cooling air for power leads Qair-PL, TPL, ∆HPL

Ploss Heat loss

Pair Energy in process air Qair, Tair, ∆Hair

6.2 Energy distribution diagram in an SOFC system. For notation see text.

Fig. 6.2 by adding a PCS (power conditioning system) subsystem. The rate of extraction of chemical energy stored in fuels Pf is a function of fuel flow Qf, fuel inlet temperature Tf, and enthalpy change ∆cHf of the fuel oxidation reaction. Another source of rate of energy input comes from air. As mentioned previously, air is commonly used to heat up the SOFC stack. The rate of energy input from air Pair consists of the air flow Qair, air inlet temperature Tair, and enthalpy change ∆Hair due to the temperature change. Pair-PL is the energy rate used to cool the power leads when the generator is running at high power. The rate of energy consumption includes the produced DC electrical power PDC, heat loss Ploss, and heat in the exhaust Pexh. The DC electrical power PDC is the product of stack current and stack voltage. Heat loss refers to the heat conducted through insulation and any other exposed surfaces. The heat left in the exhaust can be expressed as a function of exhaust flow Qexh, exhaust temperature Texh, and enthalpy change ∆Hexh due to temperature changes. Based on equation [6.32] and Fig. 6.2, the balance of energy rate follows Pf + Pair + Pair-PL = Pexh + PDC + Ploss

6.6.1

[6.33]

Energy rate of fuels, Pf

In order to calculate Pf, an SOFC generator fueled by natural gas is used as an example. Since the industrial natural gas is composed mainly of CH4 with minor amounts of higher hydrocarbons, the generic reaction between fuel and oxygen is expressed by

C n H 2 n+2(g) +

(3 n + 1) O 2(g) = n CO 2(g) + ( n + 1) H 2 O (g) 2

[6.34]

with yn being the molar fraction of each CnH2n+2. Note H2O(g) is used in the reaction, which means a lower heating value (LHV) of the fuel considered in

94

Solid oxide fuel cell technology

the calculation. The standard enthalpy change oxidation at temperature T follows LHV =

∆ c H To , n

=

o ∆ c H 298, n

+

∫

∆ c H To , n

or LHV for CnH2n+2

T

∆ C p,n d T

[6.35]

298

o where ∆ c H 298, n is the standard enthalpy change of reaction [6.34] at 298 K, or termed standard heat of combustion. ∆Cp,n is the change in constantpressure molar heat capacity of the reaction, which is equal to

∆ C p,n

= ( n + 1) C p,H 2 O + nC p,CO 2 – 3 n + 1 C p,O 2 – C p,C n H 2 n+2 2 [6.36]

For individual species, Cp typically follows the form of equation [2.15]. o Both ∆ c H 298, n and Cp values of some fuels can be found in Table A2.1 of Appendix 2. Substitution of equations [6.36] and [2.15] into equation [6.35] yields ∆ c H To , n

1 o 2 2 = ∆ c H 298, n + ∆ a n ( T – 298) + 2 ∆ bn ( T – 298 )

+ 1 ∆ c n ( T 3 – 298 3 ) + 1 ∆ d n ( T 4 – 298 4 ) – ∆ e n (1/ T – 1/298) 3 4

[6.37] where ∆an, ∆bn, ∆cn, ∆dn, and ∆en are coefficients dependent on n, a, b, c, d, and e of the reactants and products of reaction [6.34]. However, in many cases, the assumption that Cp is a temperature-independent constant is a good estimation. Equation [6.37] can then be simplified into o LHV = ∆ c H To , n = ∆ c H 298, n + ∆ a n ( T – 298)

[6.38]

o For all gas-phase reactions, ∆an is typically marginal. Therefore, ∆ c H 298, n is in practice a good estimate for LHV at all temperatures of interest in SOFC o applications. The temperature-dependent ∆ c H 298, n values of some common fuels can also be found in Table A2.1. For a mixture of hydrocarbons such as industrial natural gas, the total enthalpy change or the total chemical energy is given by m

LHV =

∆ c H To ,NG

= Σ y n ∆ c H To , n n =0

[6.39]

where the individual LHV is given by equations [6.37] or [6.38]. As an example, Table 6.3 lists the calculated LHV of an industrial natural gas, 13AO, at 1273 K. For a natural gas fuel at a flow rate of Qf (in slpm), the total energy rate is finally given by

Current, gas flow, utilization, and energy balance

95

Table 6.3 Lower heating value of an industrial natural gas, 13AO, at 1273 K n

yn

o ∆ c H 1273, n (kJ/mol)

o y n × ∆ c H 1273, n (kJ/mol)

1 2 3 4

0.88 0.06 0.04 0.02

–803.39 –1430.43 –2228.13 –2920.31

–706.98 –85.83 –89.13 –58.41

o

∆ c H 1273,NG

Pf (kW) = –

= –

Qf (slpm) × 22.4 × 60 ∆ c H To ,NG

22.4 × 60

×

(kJ/mol) =

–940.35

∆ c H To ,NG (kJ/ mol)

0.006 96 × I L × N + QL × N   (3 n + 1) y n  × Cf × (1 – B )  nΣ =0   m

[6.40]

6.6.2

Energy rate of process air, Pair

For a closed system of fixed composition undergoing a change in temperature from T1 to T2 at the constant pressure P, the change in enthalpy, ∆H, is governed by ∆H

= H ( T2 , P ) – H ( T1, P ) =

∫

T2

C p dT

[6.41]

T1

For the process air in an SOFC generator, the enthalpy change due to composition is negligible. Substituting equation [2.15] into equation [6.41] gives the standard enthalpy change ∆ H To ,dry-air of dry air as a function of temperature as ∆ H To ,dry-air

= a air ( T2 – T1 ) + 1 bair ( T22 – T12 ) + 1 c air ( T23 – T13 ) 2 3 + 1 d air ( T24 – T14 ) – eair (1/ T2 – 1/ T1 ) 4

[6.42]

where the constants a, b, c, d, and e can be found in Table A1.2 in Appendix 1. If the air is humid, the thermal properties of water vapor should also be considered. Let xS be the specific humidity at saturation and pWS and pA be the partial pressures of water at saturation and atmospheric pressure, respectively. The following equation is then established x s = 0.621 98 ×

p Ws p A – p Ws

[6.43]

96

Solid oxide fuel cell technology

The enthalpy change of humid air with regard to temperature can then be corrected by ∆ H To ,hum-air

= ∆ H To ,dry-air + x S ∆ H To ,water

[6.44]

In equation [6.44], ∆ H To ,water should have a very similar form to that of equation [6.42]. The corresponding data for steam can also be found in Tables A1.2 and A1.3 of Appendix 1. The energy rate of air is, finally, given by Pair (kW) =

Qair (slpm) × ∆ H To ,air (kJ/ mol) 22.4 × 60

= 1.24 × 10 –5 × ∆ H To ,air × S × I × N

[6.45]

where ∆ H To ,air can be given by either equation [6.42] or equation [6.44], depending on the humidity of the air.

6.6.3

Direct current (DC) electrical power, PDC

The DC electrical power PDC produced directly from the SOFC stack is given by N

PDC (kW) =

I L × VS = 1000

I L × Σ Vc ( k ) k =1

1000

[6.46]

where VS and Vc(k) are the stack and individual cell voltages, respectively; it is assumed that N cells are connected in series. Note that the DC current used must be the load current IL.

6.6.4

Energy rate of exhaust, Pexh

Similar to air, the enthalpy change of depleted fuel at a fixed composition resulting from changing one temperature to another can be treated with equation [6.41]. Since the depleted fuel contains mainly N2, O2, CO2, and H2O(g), the total enthalpy change is the composition-weighted sum of the enthalpy changes of the individual components in the same form as equation [6.39]. A typical fully combusted SOFC exhaust contains roughly 8.9 mol% H2O, 2.4 mol% CO2, 14.9 mol% O2, and 73.8 mol% N2. The enthalpy change of individual species ∆ H To , n is given by a similar form of equation [6.37] o o with a replacement of ∆ c H 298, n by ∆ f H 298, n , the standard enthalpy of formation of species n at 298 K. The total enthalpy change of the exhaust ∆ H To ,exh is, therefore, given by

Current, gas flow, utilization, and energy balance

97

m

∆ H To ,exh

= Σ y n ∆ H To , n n =1

[6.47]

o Substituting Cp and ∆ f H 298, n listed in Tables A1.2 and A1.3 of Appendix 1, as well as the exhaust composition yn, into the above equation gives ∆ H To ,exh as a function of exhaust temperature Texh as

∆ H To ,exh (kJ/ mol)

2 = –39.37 + 0.0270 Texh + 4.46 × 10 –6 Texh

3 4 –1 – 6.11 × 10 –10 Texh + 2.43 × 10 –14 Texh – 4.09Texh

[6.48] Texh is the absolute temperature (in K). The energy rate of the exhaust, Pexh, is given by Pexh (kW) =

Qexh (slpm) × ∆ H To ,exh (kJ/ mol) 22.4 × 60

= Pf + Pair + Pair-PL – Ploss – PDC

[6.49]

From equations [6.48] and [6.49], the exhaust temperature Texh can be estimated if Pair-PL and Ploss are known.

7 Voltage losses in a solid oxide fuel cell (SOFC)

Abstract: The maximum voltage achievable by a single SOFC is governed by its electromotive force (EMF) under the open-circuit voltage (OCV) condition. Upon delivering electrical current, the components of an SOFC exhibit resistance, resulting in voltage losses. The cell voltage useful for power generation is, therefore, the cell EMF subtracted by the voltage losses of the individual components. The knowledge of these voltage losses as well as their distributions in an SOFC is critically important for maximizing the power output, an ultimate goal of any power-generating device. Based on the nature of the resistance, the voltage losses can generally be classified into three categories: ohmic, activation, and concentration polarizations. In this chapter, the voltage loss from each type of polarization will be discussed extensively. Key words: ohmic loss, activation polarization, concentration polarization, distribution of electromotive force, distribution of current density.

7.1

Ohmic polarization

The ohmic polarization is the voltage loss across the components due to an electric current that follows Ohm’s law in the continuum form of equation [4.12]. The ohmic behavior in an SOFC is present in almost every functional component, but the contribution of each component varies greatly among different designs. For example, the planar SOFC design produces a very small portion of ohmic voltage loss (IR) from the electrodes and electrolyte, but a very large IR loss from the contact resistances related to interconnect and current collection. In contrast, the tubular design, particularly the cathodesupported design, yields a larger ohmic IR loss resulting from the electrodes, but a smaller contribution from the interconnect and the current collector. The most commonly used form of Ohm’s equation is based on a macroscopically averaged-out version with the assumption of a pathindependent ∇φ over a length L

Ik [7.1] = σ k ∆V A L where Ik and A are, respectively, the electrical current of species k and the cross-sectional area through which the current passes. The voltage drop ∆V across L is equal to –∇φ × L as shown in equation [4.12]. Rearrangement of equation [7.1] gives 1 × L =I ×R [7.2] ∆V = I k × k k σk A 98 ik =

Voltage losses in a solid oxide fuel cell (SOFC)

99

where Rk is the ohmic resistance of the species k and its relationship with the geometry is therefore Rk = 1 × L σk A

[7.3]

If the conductivity of species k of a material is known, the resistance Rk of a regular geometry can be easily calculated out, from which the ohmic IR loss, ∆V, can be obtained at a known current Ik. It has to be noted, however, that the definitions of geometric parameters such as L and A depend on the direction of a current path, which may lead to a very different expression of Rk. This situation is explained in Fig. 7.1 where two probable current pathways commonly encountered in an SOFC are illustrated. Case (a) represents a current passing through a large cross-section with a small length whereas case (b) represents that of a smaller cross-section with a greater length. The latter case is widely encountered in the semiconductor industry where lateral (or in-plane) electrical conduction in thin films is considered. A useful and convenient term for describing Ohm’s behavior of thin films is the ‘sheet resistance’ (also termed ‘in-plane’ resistance in some applications). The sheet resistance, Rs, is a measure of resistance of a two-dimensional entity such as a thin film that has a uniform thickness t. The utility of Rs, as opposed to resistivity, is that it can be directly measured with a four-probe configuration, as is discussed in detail in Chapter 9. Rs is measured in Ω/square. It is equivalent to resistivity as used in three-dimensional systems. When the term Rs is used, the current must flow along the plane of a sheet, not perpendicular to it. For a regular three-dimensional conductor shown in Fig. 7.1(a), the resistance can be expressed by equation [7.3]. Note that L is always the length along which a current flows. In the case of Fig. 7.1(a), L is the thickness of the conductor. For a thin-film, two-dimensional conductor as shown in Fig. 7.1(b), however, t is the thickness of the conductor while L is kept as the current path length. By splitting the cross-sectional area A into the width W and the thickness t, equation [7.3] can be rewritten for a two-dimensional, thin-film conductor as I L W

A

A=W×L

I

t L (a)

(b)

7.1 Two likely current pathways encountered in SOFC components.

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Solid oxide fuel cell technology

Rk = 1 × L = Rs × L σk t×W W

ρ Rs = 1 = k σ kt t

[7.4]

Because the resistivity ρk in Rs is divided by a dimensional quantity, the units of Rs are ohms. The term Ω/square is usually used because it gives the resistance in ohms of current passing from one side of a square conductor to the opposite side regardless of the size of the square. For a square, L = W. Therefore, R = Rs is valid for any size of square. In order to estimate the ohmic polarization loss, a knowledge of areaspecific resistance (ASR) in Ω cm2 of the component is necessary. For a current path perpendicular to the plane of a film as shown in Fig. 7.1(a), the ASR is simply equal to Lρk according to equation [7.3]. For a current path parallel to the plane of a thin film as shown in Fig. 7.1(b), Rs can be conveniently measured by a four-probe technique (see Chapter 9). However, owing to the difference in Rs and ASR, a correlation has to be established between the two terms. For the case of an electrical current in the electrodes of a tubular cell shown in Fig. 7.2, the current path is parallel to the film and enters or exits the electrolyte layer sequentially as the current travels along the electrode films. Locally, the ASR(x) is given by [7.5] ASR(x) = R(x) × W × x = Rs(x) × x × W × x = Rs × x2 W Statistically, the average ASR over the entire current path length L, which is close to half of the circumference of the tube, is given by

∫ ASR =

L

ASR( x ) dx

0

L

ρ = 1 × Rs × L2 = 1 × k × L2 t 3 3

W

I L t

I

7.2 Current pathways in electrodes of a tubular SOFC.

[7.6]

Voltage losses in a solid oxide fuel cell (SOFC)

101

If ρk values are known for cathode and anode, the ASR values can be calculated out. Note that equation [7.6] is also applicable to any planar geometry although cylindrical tubular geometry is used here as an example to derive the relationship.

7.2

Activation polarization

In classic chemistry, activation of molecules is required for a chemical reaction to occur. A level of energy, often known as the activation energy, must be acquired to reach the transition state before the reactants can successfully react with each other to produce the products. Applying the same principle to an SOFC, energy is needed to convert molecular O2 into O2– at the cathode and to convert O2– into oxygen-containing products at the anode. In order to understand such an activation process, elucidation of the elementary steps involved in the kinetics of the chemical reactions at the cathode and anode is needed.

7.2.1

Cathode

A general mechanism accepted by the SOFC community for the oxygenreduction process taking place at the cathode/electrolyte interface can be generalized into the following three elementary steps (some variations are commonly found in the literature, but the three basic steps are generally accepted): 1 2

3

Adsorption of O2 on the surface of a cathode, dissociation of O species into adsorbed Oad and/or O ′ad (negatively charged). Surface diffusion of Oad and/or O ′ad to the triple phase boundaries (TPBs) where electrolyte (e.g. YSZ), cathode (e.g. LSM), and gas (e.g. air) meet. Conversion of Oad and/or O ′ad to O O× (O2–) of an electrolyte by electron transfer from the cathode to the electrolyte material at the TPB.

Figure 7.3 shows the schematic of each individual step. A shortcut that bypasses step 2 is step 1, which takes place directly at the TPBs. Determination of the rate-limiting step is critical to arrive at an analytical solution to the activation polarization loss because oxygen flux through the activation process is dictated by the rate of the limiting step. The rate-limiting step is often found to be dependent on materials. One classical example is that step 1 has been experimentally shown as the rate-limiting step for mixed electronic and O2–-ion-conducting cathode materials such as doped LaCoO3, whereas step 3 has been determined to be the rate-limiting step for only electronic conducting materials such as doped LaMnO3. Such a variation makes the generalization of an activation polarization model very difficult. In what follows, the discussion

102

Solid oxide fuel cell technology 1 O 2 2

1 e′

2

Oad /Oad′

1 O 2 2 1

2e′ 3

LSM

1 TPB

O2– (OO× )

1 O 2 2

2e′

YSZ

3 O2– (OO× )

7.3 A schematic of elementary steps involved in the cathode activation polarization process.

is mainly focused on a composite cathode composed of an electronically conducting cathode and O2–-ion electrolyte materials. Step 1: adsorption/dissociation of oxygen species Consider the adsorption/dissociation process of oxygen species as the following reaction k

ad,f →  O 2 + 2(s) ← 2O ad (s)  k ad,b

[7.7]

where (s) is a surface site, or k

ad,f →  O 2 + 2(s) + 2e ′ ←  2O ′ad (s)

k ad,b

[7.8]

The rate of the reaction (or flux), rad, is therefore written by rad = k ad,f PO 2 θ s2 – k ad,bθ o2

[7.9]

where θo and θs are the fractions of sites occupied and unoccupied by the oxygen species on the surface of a composite cathode, respectively, and kad,f and kad,b are the rate constants of forward and backward reactions (equation [7.7]), respectively. At equilibrium, the equilibrium constant Kad is given by 2

θ K ad =  o  1  θ s  PO 2

[7.10]

Since θo + θs = 1, the site fractions can be further written as

θo =

θs =

PO 21/2 1/2 + PO 21/2 K ad

1/2 K ad

1/2 K ad + PO 21/2

[7.11]

[7.12]

Voltage losses in a solid oxide fuel cell (SOFC)

103

It is evident that equations [7.11] and [7.12] are the derivative form of the Langmuir isotherm, which relates the coverage or adsorption of molecules to gas pressure of a medium above the surface of a solid at a fixed temperature. Step 2: surface diffusion of oxygen species The one-dimensional flux density jO(x) of oxygen species diffusing along the surface, either in neutral or charged form, can be described by the Nernst– Planck equation under an electric field

j O ( x ) = – DO

dφ dc O ( x ) z O F – DO c O ( x ) RT dx dx

[7.13]

where DO, cO, and zO are the diffusivity, concentration, and charge of the oxygen species, respectively; φ is the electrostatic potential present during diffusion. Clearly, for neutral oxygen species, the second term of the righthand side of equation [7.13] can be ignored, which leads to a classical form of Fick’s first law. Step 3: conversion of Oad ′ and/or Oad ′ to O O× of an electrolyte by electron transfer from the cathode electronic conductor at a triple phase boundary – charge-transfer process The charge transfer and the subsequent incorporation process at the TPB can be expressed by the following reaction in Kröger–Vink notation ..

O ad (s) + VO (el) + 2e ′ = O O× (el) + (s)

[7.14]

or ..

O ′ad (s) + VO (el) + e ′ = O O× (el) + (s)

where (el) represents the electrolyte. Since electrons are involved, an electrical field should have an effect on the kinetic rate of the process. In practice, this behavior can be used to determine whether a process is related to the chargetransfer reaction based on its response to an external electrical field. The classic Bulter–Volmer (B–V) equation is widely used to describe quantitatively the relationship between the electrode overpotential ηact of a charge-transfer process and the applied (or resulting) current density i in an electrochemical cell. For a general electrode reaction R i O + ze′, the B–V equation is written by   α Fη act   α Fη act   i = iex  exp  a – exp  – c [7.15]  RT    RT    where iex is referred to as the exchange current density; αc and αa are the

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Solid oxide fuel cell technology

apparent transfer coefficients for cathodic and anodic reactions, respectively. The exchange current density, iex, which is analogous to the rate constant in chemical kinetics, is one of the most important parameters in the B–V equation. It is often viewed as a kind of ‘idle current’ for charge exchange across the interface. By use of the symmetry factor β,* which represents the fraction of applied potential η that promotes the cathodic reaction, equation [7.15] can be re-written as   (1 – β ) zFη act   β zFη act   [7.16] i = iex  exp  – exp  –  RT RT       Clearly, αa = (1 – β)z and αc = βz, where z is the number of electrons transferred. Equation [7.16] is the most popular form of the B–V equation that is used to describe the η–i relationship of an electrode reaction. It implies that for a large cathodic (negative) overpotential, the anodic component is negligible and vice versa. Depending upon the magnitude of the applied electrical field, equation [7.16] is often simplified into the following forms.

Low-field approximation (i/iex < 1): linear characteristic at small ηact At small ηact, equation (7.16) is reduced into the following form based on the exponential approximation ex ≈1 + x

η act =

RT × i = RT × i zF iex (α a + α c ) F iex

[7.17]

Equation [7.17] suggests that the net current is linearly related to the overpotential in a narrow potential range near equilibrium potential. The ratio η/i has units of resistance and is often called the charge-transfer resistance, Rct Rct = RT × i [7.18] zF iex High-field approximation (i/iex > 4): Tafel equation For large values of ηact (either negative or positive), one of the bracketed terms in equation [7.16] becomes negligible, leading to the following Tafel forms  (1 – β ) zF act  η  i = iex exp  RT    β zF act  η  i = – iex exp  –  RT 

(1 – β ) zF act   η RT   for RT   β zF    for RT η act – RT   

[7.19]

* Similarly, 1 – β is the fraction of the applied potential that promotes the anodic reaction.

Voltage losses in a solid oxide fuel cell (SOFC)

105

The Tafel form can be expected to hold whenever the back reaction (i.e. the anodic process, when a net reduction is considered, and vice versa) contributes less than 1% of the net current. Inverse hyperbolic sine (IHS) approximation (αa = αc = 0.5z): a general form For a broad range of current, neither the low-field nor the high-field approximation is applicable. In this case, equation [7.16] can be converted into the following expressions for β = 0.5 (β is frequently assumed to be 0.5) act i = sinh  0.5 zFη    RT 2 iex  

[7.20]

or RT arcsinh  i   2 iex  0.5 zF 2     = RT  ln  i +  i  + 1    2 iex  0.5 zF   2 iex    

η act =

[7.20a]

Note that equation [7.20] or [7.20a] is valid only for one electrode reaction, either the cathodic or the anodic reaction. A single electron transfer (z = 1) is often assumed in SOFC modeling. In order to demonstrate the applicability of low-field, high-field, and IHS approximations, Fig. 7.4 plots the charge-transfer overpotential as a function of current density under T = 900 °C and iex = 0.30 A/cm2. As expected, the low-field approximation has a good fit in the low-current density range and deviates from the IHS approximation at a higher current density. Similarly, the high-field approximation shows a good fit at high current density, but fails to produce a reasonable fit at low current density. Only the IHS form gives the best approximation over the entire current density range. Therefore, the IHS form of the B–V model is widely used for electrochemical modeling of SOFCs. In order to demonstrate the usefulness of equation [7.20], Fig. 7.5 shows the cathode activation overpotentials of an ScSZ-based SOFC measured at 900 °C as a function of current density. The iex is obtained as a constant by fitting the experimental data with equation [7.20]. Microstructural effect on the charge-transfer resistance The charge-transfer resistance Rct can be directly measured with experimental methods such as electrochemical impedance spectroscopy (EIS), to be discussed

106

Solid oxide fuel cell technology 0.45 IHS approximation

0.40 0.35

Low-field approximation

hact (V)

0.30 0.25 0.20

T = 900 °C iex = 0.30 A/cm2

0.15 0.10 High-field approximation

0.05 0.00 0.00

0.50

1.00

1.50

2.00

2.50

i (A/cm2)

7.4 Plot of activation overpotential as a function of current density under different approximations.

0.050 0.045 0.040 0.035

hact (V)

0.030 0.025 0.020 0.015

T = 900°C iex = 0.443 A/cm2

0.010 0.005 0.000 0.00

0.05

0.10

0.15

0.20

0.25

i (A/cm2)

7.5 Plot of activation polarization overpotentials of a cathode against current density at 900 °C. Diamonds are experimental data, and line is the fitted curve using the B–V form of equation [7.20a] with z = 1.

in Chapter 9. On the other hand, theoretical analysis can also connect Rct with the physical/chemical properties of a cathode such as, for instance, the TPB density at the electrolyte/cathode interface. Tanner et al.66 correlate the microstructural parameters with the effective charge-transfer resistance Rct(eff) in the low-current-density limit by the following equation

Voltage losses in a solid oxide fuel cell (SOFC)

Rct(eff) ≈

Rct dρi (1 – ε c )

107

[7.21]

where Rct is considered as the intrinsic charge-transfer resistance; d and ρi are the grain size and ionic resistivity of the oxide-ion conducting phase (e.g. YSZ), respectively; εc is the porosity of the reactive layer near the interface. Equation [7.21] suggests that smaller grain size and higher conductivity of the oxide-ion conducting phase would decrease the effective charge-transfer resistance. Therefore, a mixture of doped LaMnO3 with fine-grained Scdoped ZrO2 should give and has been shown to give a better performance than that with YSZ.67 Equation [7.21] also implies that lower porosity εc would decrease the Rct(eff). However, lowering εc could also increase the concentration polarization (see Section 7.3). Therefore, a balanced εc is necessary in making the optimal composite cathode microstructure. This balanced εc is typically found in the range of 30–35%. Pressure effect on charge-transfer process The effect of system pressure Pt on the activation polarization of a cathode is essentially that on the exchange current density iex. The relationship between iex and PO 2 (= X O 2 × Pt ) has the following form68 * iex = iex

( PO 2 / PO*2 )1/2 1 + ( PO 2 / PO*2 )1/2

[7.22]

* and PO*2 are the material- and temperature-dependent characteristic where iex * have been reported in an constants. For an LSM/YSZ interface, PO*2 and iex 68 Arrhenius form

PO*2 = 4.9 × 10 8 exp  –  * iex = 5.8 × 10 4 exp  – 

200 000 (J/ mol)  RT 

110 000 (J/ mol)  RT 

(atm)

(A)

[7.23]

[7.24]

Applying equation [7.22] with conditions of T = 900 °C, X O 2 = 0.209, and PO 2 = X O 2 × Pt leads to the plot in Fig. 7.6, where iex is shown as a function of Pt. Clearly, iex increases with Pt, implying that the charge-transfer resistance decreases with Pt. Elevating the system pressure by a factor of 20 can increase the iex by a factor of 2. Recall from Chapter 2 that elevating the system pressure increases the electromotive force (EMF); decreasing the activation polarization is another benefit resulting from elevating the system pressure.

108

Solid oxide fuel cell technology 0.6 0.5

iex (A/cm2)

0.4 0.3 0.2

T = 900 °C XO = 0.209

0.1

2

0.0 0

5

10

15

20

25

Pt (atm)

7.6 Effect of system pressure Pt on iex of an LSM/YSZ cathode at 900 °C.

7.2.2

Anode

In order to develop an analytical expression for the anode activation overpotential in the B–V form similar to equation [7.16], it is helpful to begin by elaborating the elementary steps by which the H2 oxidation reaction is taking place (H2 fuel is used as an example here). Although considerable debate exists on the elementary pathways and rate-limiting steps, two fundamental elementary reactions in the Ni-YSZ anode are generally accepted by the SOFC community. 1

Adsorption/desorption (or spillover) of H2 on the Ni surface H2(g) + 2(s)Ni = 2H(s)Ni

2

[7.25]

Electron transfer from the electrolyte (el) to the nickel at the TPBs ..

2H(s) Ni + O O× (el) = 2(s) Ni + VO (el) + H 2 O (g) + 2e ′

[7.26]

The combination of reactions [7.25] and [7.26] leads to the global reaction of H2 electro-oxidation ..

H 2(g) + O O× (el) = VO (el) + H 2 O (g) + 2e ′

[7.27]

Similar to the oxygen adsorption/desorption process, the equilibrium coverage of H2 on the Ni surface as described by reaction [7.25] can also be expressed by the Langmuir isotherm

θo =

PH1/22 1/2 + PH1/22 K ad

[7.28]

Voltage losses in a solid oxide fuel cell (SOFC)

θs =

1/2 K ad 1/2 K ad + PH1/22

109

[7.29]

Similar to the cathode counterpart, the charge-transfer process (equation [7.26]) of an Ni–ZrO2 cermet anode can be written in a B–V form of equation [7.16]. The applicability of equation [7.16] to the anode charge-transfer reaction is evidently confirmed by Fig. 7.7, where the experimental values measured from an ScSZ + Ni anode of a cathode-supported SOFC agree well with simulated values. Pressure effect on charge-transfer process In order to study the effect of the system pressure on the charge-transfer process, one has to know the relationship between the system pressure and the exchange current density of the charge-transfer process, from which the activation overpotential can be determined based on equation [7.16]. Analogous to equation [7.22], such a relationship has been given as a function of partial pressures of H2 and H2O iex = iH* 2

( PH 2 / PH* 2 )1/4 ( PH 2 O ) 3/4 1 + ( PH 2 / PH* 2 )1/2

[7.30]

where iH* 2 and PH* 2 are two material- and temperature-dependent constants. The latter is determined from the balance between adsorption and desorption of hydrogen on the Ni as depicted by equation [7.25], whereas the former 0.016 0.014

hact a (V)

0.012 0.010 0.008 0.006 0.004

T = 900 °C iex = 1.005 A/cm2

0.002 0.000 0.00

0.05

0.10

0.15 i (A/cm2)

0.20

0.25

0.30

7.7 Plot of activation polarization overpotential of anode against current density at 900 °C. Diamonds are experimental data, and line is the fitted curve using the B–V form of equation [7.20a] with z = 1.

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Solid oxide fuel cell technology

can be derived from parameters associated with the charge-transfer reactions, although it is often used as an empirical constant owing to the complication of the anode kinetics. Figure 7.8 plots the variations of iex with Pt at 900 °C. The values of parameters iH* 2 and PH* 2 are taken arbitrarily for illustration. As the Pt changes, the partial pressures of H2 and H2O change correspondingly. As evident in the plot, iex increases, and therefore η aact decreases, with Pt for each PH 2 / PH 2 O plotted. Moreover, a higher H2O content appears to facilitate the anodic kinetics and enlarge the benefit from elevating the system pressure.

7.3

Concentration polarization

The electrochemical reactions taking place at the cathode and the anode of an SOFC consume oxygen and produce H2O (and CO2 if a hydrocarbon fuel is used), respectively. To sustain a steady-state rate of reaction, oxygen and fuel have to be constantly supplied to the cathode and the anode, respectively, and H2O (and CO2) have to be removed from the anode. The mass transport process of these gaseous species in a solid porous structure and the resulting electrochemical resistance are the focuses of this section.

7.3.1

Molecular and Knudsen diffusion

Mass transport in a porous solid is a complex phenomenon. Under a constant system pressure, the mass flux is diffusive in nature and may involve ordinary molecular diffusion, Knudsen diffusion, and surface migration. In what follows, 45.0

T = 900 °C * = 8.0 A/cm2 iex PH* = 5.0 atm

40.0 35.0

PH /PH

2

2

2O

= 50/50

iex (A/cm2)

30.0 25.0 20.0

PH /PH

15.0

2O

2

10.0

= 89/11

PH /PH 2

2O

= 97/3

5.0 0.0 0

5

10

15 Pt (atm)

20

25

30

7.8 Effect of system pressure on iex of an anode under different fuelto-oxide ratios.

Voltage losses in a solid oxide fuel cell (SOFC)

111

only molecular diffusion and Knudsen diffusion are considered, because the surface migration only occurs where the diffusing gases are adsorbed in a mobile layer and this is negligible in most studies of SOFCs. Molecular diffusion Ordinary gas diffusion results from movements of gas molecules. Modern kinetic theory of gases considers forces of attraction and repulsion between the molecules when gaseous molecules diffuse. Using the Lennard–Jones potential to evaluate the influence of the molecular forces, the equation for the diffusion coefficient for gas pairs of non-polar, non-reacting molecules A and B is given by69

0.001858T 3/2  1 + 1   MA MB  D AB = 2 Pt σ AB Ω D

1/2

[7.31]

where DAB is the mass diffusivity of A through B (in cm2/s); MA and MB are the molecular weights of A and B (in g/mol), respectively; σAB is the ‘collision diameter’, a Lennard–Jones parameter (in Å); and ΩD is the ‘collision integral’ for molecular diffusion, a dimensionless function of the temperature and of the intermolecular potential field for one molecule of A and one molecule of B. Based on equation [7.31], we can predict the diffusivity at any temperature and at any pressure below 25 atmospheres from a known experimental value obtained at T1 and Pt1 by P T 3/2 D ABT2, Pt2 = D ABT1, Pt1  t1   2   Pt2   T1 

Ω DT1 Ω DT2

[7.32]

Since an SOFC typically operates under a constant system pressure, equation [7.32] can be further simplified into

T D ABT2 = D ABT1  2   T1 

3/2

Ω DT1 Ω DT2

[7.33]

The values of ΩD at different temperatures are well documented and can be easily found in many gas handbooks.70, 71 Appendix 4 lists diffusivity data for some binary gases calculated by equations [7.31] and [7.33]. Knudsen diffusion Consider the diffusion of gas molecules through very small capillary pores. If the pore diameter is smaller than the mean free path length of the diffusing

112

Solid oxide fuel cell technology

gas molecules, and the density of the gas is low, the gas molecules will collide with the pore walls more frequently than with each other. This process is known as Knudsen diffusion. The gas flux is reduced by the wall collisions. The diffusivity of species A for Knudsen diffusion, DK, A (in cm2/s), is obtained from the self-diffusivity derived from the classical kinetic theory of gases70 DK,A = 1 d pore 3

8 RT = 4850d pore πM A

T MA

[7.34]

where dpore is the mean pore diameter (in cm). It is obvious that DK,A ∝ T1/2 and is a function of the pore size. Mass flux density including molecular diffusion and Knudsen diffusion A general expression of a one-dimensional diffusion process for a multicomponent gas system is described by the Stefan–Maxwell equation

X j – X m jn jm P dX m + Σ n m =– t DK,m n =1 Dmn RT dx

[7.35]

n≠m

where jm and jn (in mol/(s cm2)) are the molar flux densities of components m and n, respectively; DK,m and Dmn are the Knudsen diffusivity for component m and binary diffusivity of m through n, respectively; Xm and Xn are the molar fractions of components m and n, respectively. Application of equation [7.35] to special gas mixtures gives more meaningful insights. Taking air as a mixture of O2–N2, equation [7.35] is simplified into jO 2 X N 2 jO 2 – XO 2 j N 2 P dX O 2 + =– t DK,O 2 DO 2 N 2 RT dx

[7.36]

Compare equation [7.36] with the classical Fick’s first law for the O2 component jO 2 = –

Pt DO 2 dX O 2 RT dx

[7.37]

and with X O 2 + X N 2 = 1 and j N 2 = 0 under a steady-state SOFC operation, it is not difficult to find DO 2

1 – XO2   =  1 + DO 2 N 2   DK,O 2

–1

[7.38]

Generally speaking, the Knudsen process is significant only at low pressure and small pore diameter. However, there are instances where both Knudsen diffusion and molecular diffusion can be important. Equation [7.38] implies that Knudsen diffusion and molecular diffusion compete with one another by a ‘resistances in parallel’ model. For straight cylindrical pores, such a

Voltage losses in a solid oxide fuel cell (SOFC)

113

competition can also be conveniently evaluated by the Knudsen number Kn, which is defined by66 Kn =

λ d pore

[7.39]

λ=

kT 2 πσ A2 Pt

[7.40]

where λ is the mean free path length; σA is the Lennard–Jones diameter of the spherical molecule A. By rule, ordinary molecular diffusion predominates where Kn < 1 whereas Knudsen diffusion predominates where Kn > 10. Application of this rule to a porous cathode substrate with average pore size of 10 µm and σ O 2 ≈ 0.3433 nm, T = 1073 K, and Pt = 101300 Pa reveals Kn = 0.0028, suggesting that Knudsen diffusion in such a substrate is negligible in comparison with ordinary molecular diffusion. Note that the above relationships of diffusivity are based on diffusion within straight, cylindrical pores aligned in a parallel array. The interaction between gaseous species and the porous body is not considered. However, in most porous materials, pores with various diameters are twisted and interconnected with one another, and the paths for diffusion of gas molecules within the pores are tortuous. For these materials, if an average pore diameter is assumed, a reasonable approximation for the effective diffusivity of a binary gas AB in random pores is given by eff D AB = ε D AB τ

[7.41]

where ε and τ are porosity and tortuosity, respectively. In some cases, τ = 1/ε can be assumed, which leads to eff D AB = ε 2 D AB

[7.41a]

From equation [7.41a], the porosity can be estimated if the effective diffusivity is measurable. In Chapter 9, an experimental method to measure the porosity of ceramic bodies directly is described.

7.3.2

Cathode

In what follows, the O2 diffusion process is primarily discussed in cylindrical coordinates. The one-dimensional diffusion process under Cartesian coordinates can be easily described otherwise. Figure 7.9 shows a schematic of a tubular fuel cell in cylindrical coordinates. Use of the Stefan–Maxwell equation for the O2–N2 binary system while ignoring the Knudsen diffusion leads to† † For simplification, DOeff2 is used to represent DOeff2 N 2 for O2 diffusivity in air.

114

Solid oxide fuel cell technology

Porous cathode substrate

r2

r1

tion

nec

on terc

In 0

2θ

L

PO (2) 2

PO (1) 2

7.9 A schematic of tubular geometry for O2 diffusion in polar coordinates. The area of interconnection inactive to O2 diffusion is also marked.

XN2 JO2 – XO2 J N2 = –

DOeff2 × Pt dX O 2 ×A× RT dr

[7.42]

where the molar flux J (in mol/s) is used for convenience; DOeff2 is the effective diffusivity of oxygen molecules; A is the cross-sectional area for O2 diffusion, i.e. A = (2π – 2θ) × L × r. Rearrangement of equation [7.43] with X N 2 = 1 – X O 2 and J N 2 = 0 leads to the following equation for any given radius r eff dr = – DO 2 × (2 π – 2θ ) × L × Pt × dX O 2 r RT JO2 1 – XO2

[7.43]

Integration of equation [7.43] from r1 (inner radius) to r2 (outer radius) (in cm) on the left-hand side and the corresponding X O 2 (1) to X O 2 (2) on the right-hand side yields

DOeff2 P r  1 – X O 2 (2)  ln  2  = × (2 π – 2θ ) × L × t × ln    r1  RT JO2  1 – X O 2 (1)  [7.44] At steady state, the current I (or current density i) passing through the cathode/ electrolyte interface is governed by Faraday’s law and has the following relationship with the oxygen flux J O 2 I = (2π – 2θ) × L × r2 × i = 4 × F × J O 2

[7.45]

Voltage losses in a solid oxide fuel cell (SOFC)

115

Combining equations [7.44] and [7.45] gives the oxygen molar fraction X O 2 or partial pressure PO 2 (2) at the cathode/electrolyte interface‡   RTr2   r2   X O 2 (2) = 1 – (1 – X O 2 (1)) × exp   × i × ln  eff  r1      4 FDO 2 Pt  [7.46] or   RTr2   r2   PO 2 (2) = Pt – ( Pt – PO 2 (1)) × exp   × i × ln  eff  r1      4 FDO 2 Pt  [7.46a] If the system pressure is at atmospheric, equation [7.46a] is simplified into   RTr2  r  PO 2 (2) = 1 – (1 – PO 2 (1)) × exp   × i  × ln  2   eff  r1      4 FDO 2  [7.46b] The resulting voltage loss or concentration polarization loss, the cathode is then given by72   PO (1)  η cconc = RT ln  2  = RT ×  ln 4F  PO 2 (2)  4 F  

η cconc ,

across

   PO 2(1)) – ln  Pt – ( Pt – PO 2 (1)  

  RTr2 × exp   eff   4 FDO 2 Pt

   r2    ln ×    r1       [7.47]

η cconc

It is clear that is a function of temperature, bulk PO 2(1) in air, effective O2 diffusivity, inside and outside radii of the cathode tube, and of course the current density. The limiting current density, iLM (in A/cm2), is defined as the current density at which the interfacial oxygen partial pressure PO 2 (2) (or X O 2 (2)) becomes zero. It is a characteristic of the material and a measure of the material’s resistance to concentration polarization. A higher iLM implies a less polarized electrode. With equation [7.46] set to zero, the following equation can be obtained ‡ Strictly speaking, it should be the interface between the activation layer and the substrate, see Fig. 7.2.

116

Solid oxide fuel cell technology

i LM = –

ln(1 – X O 2 (1)) 4 FDOeff2 × × Pt RTr2 r2   ln  r1 

[7.48]

It is evident that iLM depends on the tube geometry, diffusivity, temperature, and the product of DOeff2 × Pt . With the dependence of DO 2 on temperature given by equation [7.33] and DO 2 ,293K as a reference, equation [7.48] can be further modified into

i LM = –

0.5 ln (1 – X O 2(1)) × 4 F × T 1.5 × ε × DO 2 , 293K × Pt τ R r 293 r2 × ln  2   r1 

[7.48a] Clearly, iLM is a weak function of temperature, but a strong function of effective oxygen diffusivity or porosity. Substituting equation [7.48] into equation [7.47], the interfacial PO 2 (2) and concentration polarization loss can be rewritten into the following simplified form PO 2 (2) = Pt – Pt (1 – X O 2 (1))1– ( i / iLM )

[7.49]

1– ( i / i LM )       RT   [7.50] = ×  ln X O 2 (1) – ln 1 –  1 – X O 2 (1) 4F       The derivative of equation [7.50] with respect to the current density i leads to the area-specific resistance Rcconc of the concentration polarization

η cconc

 ∂η conc  Rcconc =  c   ∂i  T, X O

2 (1)

ln (1 – X O 2 (1)) 1 = RT × × i LM 4F 1 – (1 – X O 2 (1)) ( i / iLM –1)

[7.51]

which implies that Rcconc increases with i exponentially. For a one-dimensional linear O2 diffusion in Cartesian coordinates, the limiting current density iLM can be simplified into

i LM = – ln (1 – X O 2 (1)) ×

4 FDOeff2 × Pt RTt c

[7.52]

where tc is the wall thickness of a cathode through which O2 diffuses. The cathode/electrolyte interfacial PO 2 (2) and concentration polarization loss have the same forms as equations [7.50] and [7.51].

Voltage losses in a solid oxide fuel cell (SOFC)

117

In the following, equations [7.49] to [7.52] are used as the basis to demonstrate the effects of substrate porosity, current density, bulk PO 2 in oxidant, wall thickness, and temperature on the concentration polarization loss of the cathode. In all calculations, r1 and r2 are taken as 0.9 and 1.1 cm, respectively. The effect of DOeff2 (or ε) is shown in Fig. 7.10, where the concentration polarization and interfacial PO 2 (2) are plotted as a function of DOeff2 at 1000 °C and bulk PO 2 (1) = 0.209 atm. The trend of voltage loss or PO 2 (2) decrease with DOeff2 and current density i are as expected. It appears that the effect of DOeff2 becomes significant for DOeff2 < 0.015 cm2/s, implying that concentration polarization begins to dominate the cathode performance. This behavior is best illustrated by Fig. 7.11, where the resistance of the concentration polarization is plotted against DOeff2. The effect of current density i is presented in Fig. 7.12 where variations of η cconc and PO 2 (2) with i are shown at different levels of DOeff2 under T = 1000 °C and PO 2 (1) = 0.209 atm. Consistent with Fig. 7.11, the η cconc –i relationship becomes more sensitive at lower DOeff2 values. This is the direct result of lowered limiting current density iLM, which is shown in Fig. 7.13 as a function of DOeff2 at 1000 °C and 900 °C, respectively. It is no surprise to see that iLM increases with DOeff2 . Temperature does not effect the iLM significantly. The effect of the oxygen partial pressure at the air/cathode surface, PO 2 (1) , on concentration polarization is noteworthy. Figure 7.14 illustrates variations of η cconc and PO 2 (2) with PO 2 (1) at 1000 °C; it shows that operating at high 0.20

60

0.18 50

0.16

0.12

2

i = 300 mA/cm2 i = 200 mA/cm2 i = 300 mA/cm2 i = 200 mA/cm2

20

10

2

0.10

30

PO (2) (atm)

ηcconc (mV)

0.14

T = 1000 °C PO (1) = 0.209 atm

40

0.08 0.06 0.04 0.02

0 0.000

0.010

0.020

0.030

0.040

0.00 0.050

eff

DO (293 K) (cm2/s) 2

7.10 Concentration polarization of the cathode as a function of D Oeff2 .

118

Solid oxide fuel cell technology 0.600

T = 1000 °C

R conc (Ω cm2) c

0.500

i = 300 mA/cm2

0.400 0.300 0.200 0.100

i = 200 mA/cm2 0.000 0.000

0.010

0.020

0.030

0.040

0.050

DOeff (293 K) (cm2/s) 2

7.11 Concentration polarization resistance as a function of D Oeff2. h at 0.021 cm2/s

h at 0.018 cm2/s

h at 0.015 cm2/s

PO at 0.021 cm2/s

PO at 0.018 cm2/s

PO at 0.015 cm2/s

2

2

2

70

0.24

60

0.20 0.16

40 0.12

2

30

PO (2) (atm)

hconc (mV) c

50

T = 1000 °C PO (1) = 0.209 atm

20

0.08

2

0.04

10 0 0

100

200

300

400

500

0.00 600

i (mA/cm2)

7.12 Concentration polarization loss as a function of current density.

oxygen utilization may be harmful to the cell performance. For example, with a 50% oxygen utilization the pore polarization could be increased by a factor of 5 (from 20 mV to 100 mV) at i = 300 mA/cm2. The effect of wall thickness on η cconc for a cylindrical tube relies on the absolute inner (r1) and outer (r2) radii because the sizes of r1 and r2 determine the wall thickness of the tube. Figure 7.15 shows how η cconc varies with r1

Voltage losses in a solid oxide fuel cell (SOFC)

119

1400

iLM (mA/cm2)

1200

1000 °C

PO (1) = 0.209 atm 2

1000

900 °C

800 600 400 200 0 0.000

0.010

0.020

0.030

0.040

0.050

eff

DO (293 K) (cm2/s) 2

7.13 iLM as a function of

at 1000 °C and 900 °C and Pt = 1 atm.

D Oeff2

0.30

140 eff

DO = 0.018 cm2/s 2 T = 1000 °C

120

0.25

0.20 80 0.15

2

60

PO (2) (atm)

hconc (mV) c

100

h at 300 mA/cm2 0.10

h at 200 mA/cm2

40

2

PO at 300 mA/cm 2

PO at 200 mA/cm2 2

20 0 0.00

0.10

0.20 PO (1) (atm)

0.30

0.05

0.00 0.40

2

7.14 Concentration polarization loss as a function of bulk PO2 at 1000 °C.

and r2 at 1000 °C. Surprisingly, it is found that the change in r1 has a greater impact on polarization than r2 at i = 300 mA/cm2. For instance, increasing the wall thickness from 0.20 cm to 0.35 cm by decreasing r1 from 0.90 cm to 0.75 cm at a fixed r2 = 1.10 cm, the concentration polarization was increased by 626% (from 19 to 138 mV). However, when wall thickness was increased by the same amount by increasing r2 from 1.10 cm to 1.25 cm at a fixed

120

Solid oxide fuel cell technology

r1 = 0.90 cm, the pore polarization was increased by only 384%. In either case, r1 and r2 do not have appreciable influence on η cconc at 200 mA/cm2 or lower. The results shown in Fig. 7.15 suggest that a larger r1 is beneficial for cell performance. (a)

h at 300 mA/cm2

PO at 300 mA/cm

h at 200 mA/cm2 2

2

PO at 200 mA/cm2 2

0.25

70 60

0.20

0.15

30

T = 1000 °C PO (1) = 0.209 atm

20

Deff O2

2

40

PO (2) (atm)

hconc (mV) c

50

0.10

2

2

= 0.018 cm /s r2 = 1.1 cm

0.05

10 0 0.70

0.80

(b)

0.90 r1 (cm)

0.00 1.10

h at 300 mA/cm2

h at 200 mA/cm2

PO at 300 mA/cm2

PO at 200 mA/cm2

2

2

60

0.25

50

0.20 0.15

30 20

T = 1000 °C PO (1) = 0.209 atm

0.10

Deff = 0.018 cm2/s O2 r1 = 0.9 cm

0.05

2

0.00

10 0 0.90

2

40

PO (2) (atm)

hconc (mV) c

1.00

–0.05

1.00

1.10

1.20

1.30

–0.10 1.40

r2 (cm)

7.15 Concentration polarization loss as a function of r1 and r2.

Voltage losses in a solid oxide fuel cell (SOFC)

7.3.3

121

Anode

The Stefan–Maxwell equation (equation [7.36]) can also be used to describe the gas diffusion process in a porous Ni–cermet anode. For simplicity, only the H2–H2O binary system is considered for illustration. Readers can do exercises with other systems using the same principle. The Stefan–Maxwell equation for flux density j of the H2–H2O system with ignoring Knudsen diffusion is XH2O jH2 – XH2 jH2O

DHeff2 –H 2 O dX H 2 =– × RT dx

[7.53]

The mass balance requires a net flux density of H2 and H2O to be zero, j H 2 + j H 2 O = 0 , and X H 2 + X H 2 O = 1 . At steady state, the net current density i passing through the cell is related to the net j H 2 arriving at the anode/ electrolyte interface and is given by i = 2 F × j O 2– = 2 F × j H 2

[7.54]

Integration of equation [7.53] from the surface of the anode to the anode/ electrolyte interface yields the interfacial partial pressures PH 2 (5) and PH 2 O (5) of H2 and H2O (refer to Fig. 7.20 for the meanings of various partial pressures)

t PH 2 (5) = PH 2 (6) – RT × eff a ×i 2F DH 2 –H 2 O × Pt

[7.55]

t PH 2 O (5) = PH 2 O (6) + RT × eff a ×i 2F DH 2 –H 2 O × Pt

[7.56]

where PH 2 (6) and PH 2 O (6) (in atm) are the partial pressures of H2 and H2O in the bulk of the fuel stream, respectively; ta is the thickness of the anode (in cm); DHeff2 –H 2 O is the effective diffusivity of H2 in H2O (in cm2/s), equal to DH 2 –H 2 O multiplied by microstructural factors such as porosity εa and tortuosity τa of the anode. If the Knudsen diffusion is also considered in the overall gas diffusion process, DH 2 –H 2 O should be replaced by DH 2 as follows

  1 DH 2 =  1 +  D D  K,H 2 H 2 –H 2 O 

–1

[7.57]

where DK,H 2 is the Knudsen diffusivity of H2. In this discussion, no Knudsen diffusion is taken into account owing to a large pore size present in the plasma-sprayed anode of a cathode-supported SOFC, which is used as an example. At limiting current density, i LM , PH 2 (5) = 0 , which leads to the iLM of an anode DHeff2 –H 2 O [7.58] i LM = 2 F × × PH 2 (6) RT ta

122

Solid oxide fuel cell technology

Rearrangement of equations [7.55], [7.56], and [7.58] leads to PH 2 (5) = PH 2 (6) ×  1 – i   i LM  PH 2 O (5) = PH 2 O (6) +

[7.58a]

i P (6) i LM H 2

[7.58b]

The concentration polarization η cconc of the anode is, therefore, written by

  η cconc = RT ×  ln  1 + i × 1o  – ln  1 – i      i LM i 2F   r LM  ro =

[7.59]

PH 2 O (6) PH 2 (6)

The derivative of equation [7.59] with regard to i gives the area-specific resistance Raconc of the polarization by  1 1 Raconc = RT ×  + i LM – 2 F  i + r o × i LM

 i 

[7.60]

The implication of equation [7.60] is important. It indicates that Raconc decreases with iLM, i, and inlet oxide-to-fuel ratio ro. In what follows, the effects of T, εa, ta, PH 2 (6) , and PH 2 O (6) on Raconc are shown in a series of plots against cell current density i for a cathode-supported SOFC. The molecular diffusivity of the H2–H2O binary system at different temperatures can be found in Appendix 4. Figure 7.16 demonstrates how Raconc varies with the cell current density at different temperatures. The relative magnitude of Raconc is much smaller (50–100 times lower) than that of the cathode, indicating the insignificance of pore polarization in the anode of a cathode-supported SOFC. Figure 7.17 presents the thickness effect of the anode. As expected, Raconc increases with the thickness almost linearly. Again, the relative magnitude of Raconc is small (<0.01 Ω cm2) even with a 300 µm-thick anode. The porosity effect of the anode is shown in Fig. 7.18 where εa is varied from 0.20 to 0.30 at ta = 100 µm. Again, Raconc increases as expected with εa. Although the relative change of magnitude is approximately a factor of 2.5 by varying εa from 0.20 to 0.30, the absolute value is still under 0.01 Ω cm2 with εa = 0.20. From Figs 7.17 and 7.18, it is also interesting to see how large the concentration polarization contribution is with the worstcase scenario. For example, with ta = 300 µm and εa = 0.20, the calculated Raconc is roughly 0.018 Ω cm2 at T = 900 °C and i = 300 mA/cm2 with PH 2 (6) = 0.89 atm and PH 2 O (6) = 0.11 atm. In comparison to the level of total cell resistance at this temperature, it is reasonable to assert that the anode concentration polarization is insignificant for a cathode-supported SOFC.

Voltage losses in a solid oxide fuel cell (SOFC) 0.0031

ea = 0.30

0.0031

1000 °C

0.0030

R aconc (Ω cm2)

123

ta = 0.01 cm PH (6) = 0.89 atm 2 PH O(6) = 0.11 atm 2

0.0030

900 °C

0.0029 0.0029

800 °C

0.0028 0.0028 0.0027 0

200

400

600 i (mA/cm2)

800

1000

1200

7.16 The dependence of R aconc on current density at different temperatures. 0.011

ta = 0.03 cm

R aconc (Ω cm2)

0.008

0.006

ta = 0.02 cm

0.004

ta = 0.01 cm 0.002 ea = 0.30, T = 900 °C PH (6) = 0.89 atm, PH O(6) = 0.11 atm 2

2

0.000 0

200

400

600 800 i (mA/cm2)

1000

1200

7.17 The dependence of R aconc on current density at different thicknesses.

As the ratio of H2O/H2 in the bulk of the fuel stream varies along the flow direction of a practical fuel cell owing to the progressive electro-oxidation, it is also valuable to examine the impact of this ratio on Raconc. Figure 7.19 shows Raconc as a function of i at various H2O/H2 ratios. A higher H2O/H2

124

Solid oxide fuel cell technology 0.008 0.007

ea = 0.20

R aconc (Ω cm2)

0.006 0.005 ea = 0.25 0.004 ea = 0.30

0.003 0.002

ta = 0.01 cm, T = 900 °C PH (6) = 0.89 atm, PH O(6) = 0.11 atm

0.001

2

2

0 0

200

400

600 i (mA/cm2)

800

1000

1200

7.18 The dependence of R aconc on current density at different porosities.

0.0035 H2/H2O = 89/11 0.0030

R aconc (Ω cm2)

0.0025

ta = 0.01 cm T = 900 °C ea = 0.30

0.0020

H2/H2O = 70/30

0.0015 0.0010 H2/H2O = 50/50 0.0005 0.0000 0

200

400

600

800

1000

1200

i (mA/cm2)

7.19 The dependence of R aconc on current density at different H2O/H2 ratios.

ratio (or higher H2O partial pressure) clearly favors a smaller Raconc. This observation suggests that use of a moistened H2 fuel is favorable for reducing concentration polarization loss of an anode. The same assertion is also true for the activation polarization of an anode, discussed in Section 7.2.2.

Voltage losses in a solid oxide fuel cell (SOFC)

7.3.4

125

Pressure effect

The critical factor in concentration polarization loss of either the cathode or the anode is the limiting current density iLM. Both equations [7.50] and [7.58] show that iLM is proportional to DAB × Pt, not Pt only. At low to moderate pressures, binary diffusivities vary inversely with pressure as suggested by equation [7.31], which implies that there is no Pt effect on iLM and therefore no effect on concentration polarization loss. However, at high pressures (typically >30 atm) the product of DAB × Pt is no longer a constant, but decreases with an increase in Pt. This trend infers that iLM can be decreased, and therefore the concentration polarization can be increased, by elevating the pressure above 30 atm.

7.4

A combined activation and concentration polarization of the cathode

For electrode-supported SOFCs, the activation and concentration polarization processes should not be treated separately since the exchange current density of the charge-transfer process is a function of the interfacial PO 2 , which is closely controlled by the gas mass transport–concentration polarization. The PO 2 -dependent iex can be found in equation [7.22] for the cathode and equation [7.30] for the anode. In what follows, an analytical equation that combines both activation and concentration polarizations is derived for a cathodesupported SOFC, but the solution is not limited to this case. The link between the activation polarization and the concentration counterpart of a cathode is the interfacial PO 2 (2) between the reactive layer and the substrate of the cathode as illustrated in Fig. 7.20. Moreover, for an

Anode

Electrolyte

PO (3)

PO (1)

Cathode

2

PO (2) 2

2

H2, H2O diffusion layer

PO (5) 2

PO (6)

PO (4) 2

2

O2 diffusion layer

H2 activation layer O

O2 activation layer

L

x

7.20 A schematic of PO2 profile across all functional layers of an SOFC.

126

Solid oxide fuel cell technology

activation polarization process with a single-electron, single-step reaction and β = 0.5, the polarization is described by equation [7.20a], where iex should be considered as a function of PO 2 as shown in equation [7.22]. Combining equations [7.49], [7.22], and [7.20a] yields the total cathode polarization η ctot  1 – (1 – X O 2 (1))1– ( i/iLM )  η ctot = RT arcsinh  i  – RT ln    2 iex  4 F X O 2 (1) 0.5 F   * iex = iex ×

[ Pt × (1 – X O 2 (1))1– ( i/iLM ) / PO*2 ]1/2 1 + [ Pt × (1 – X O 2 (1))1– ( i/iLM ) / PO*2 ]1/2

[7.61]

In most cases, PO 2 (2) PO*2 is satisfied and 1 + PO 2 (2)/ PO*2 ≈ 1 leads to the following simplified expression

η ctot

1/2    Pt  RT i = arcsinh  *  *  × [1 – (1 – X O 2 (1))1– ( i/iLM ) ] –1/2  0.5 F  2 iex  PO 2  

 1 – (1 – X O 2 (1))1– ( i/iLM )  – RT ln   4F  X O 2 (1) 

[7.61a]

Figure 7.21 shows the measured η ctot (diamonds) of a cylindrical tubular ScSZ-based SOFC as a function of current density at 950 °C. The cathode 0.16 0.14 0.12

htot (V) c

0.10

T = 950 °C * = 1.329 A/cm2 iex PO* = 0.386 atm 2 iLM = 0.663 A/cm2

0.08 Fitted line

0.06 0.04 0.02 0.00 0.00

0.10

0.20

0.30

0.40

0.50

0.60

i (A/cm2)

7.21 η ctot as a function of i measured at 950 °C from an ScSZ-based cylindrical tubular SOFC. Diamonds are experimental data points, and line is the fitted curve of equation [7.61a].

Voltage losses in a solid oxide fuel cell (SOFC)

127

interlayer is composed of a mixture of ScSZ and cathode. The line is produced from the non-linear least-squares fitting of equation [7.61a] with the best-fit * = 1.329 A/cm2, and PO*2 = 0.386 atm. parameters iLM = 0.663 A/cm2, iex In summary, the key for quantifying the activation and concentration polarization processes of a cathode by a unified equation is to solve the interfacial PO 2 (2) , which acts as a linking parameter between the two. The PO 2 (2) can be mathematically solved based on the theory of the O2 diffusion process.

7.5

Distributions of electromotive force and current density

For a large-scale SOFC, the EMF varies with spatial positions as a result of progressive electro-oxidation along the direction of fuel flow. Closely related to the cell EMF are the distributions of current density along and perpendicular to the direction of fuel flow, which are also dependent on local resistance. In this section, the one-dimensional EMF distribution along the direction of fuel flow and two-dimensional current density distribution are discussed on the basis of cylindrical tubular geometry. The same principle is also applicable to other geometries.

7.5.1

One-dimensional distribution of the electromotive force

The distribution of EMF(x) (or E(x)) along the fuel flow direction x is an important characteristic of an SOFC. It not only depicts the highest achievable cell voltage, but also influences the local current density i(x). The latter helps to identify areas where highly localized current densities are present and to understand whether a failure mechanism, if any, is associated with the high i(x). Therefore, a knowledge of the distributions of E(x) and i(x) is scientifically interesting and important practically for operating an SOFC generator. Fundamentally speaking, the one-dimensional distribution of the E(x) of an SOFC is the result of progressive electro-oxidation of fuel, which makes the local interfacial partial pressures of oxygen, PO 2 ( x) , vary correspondingly as the fuel travels from the entrance to the exit of the cell. At the entrance, E(x) is clearly the highest owing to the highest fuel-to-oxide ratio whereas E(x) is reduced to its lowest value at the exit after fuel is electro-oxidized over the entire active surface of the cell. However, the actual profile of E(x) from entrance to exit can be very different, depending upon operating parameters such as average current density,§ fuel utilization/consumption, and distribution of component resistance. Analytical solutions to such profiles § Here the average current density is equivalent to the normal current density discussed previously.

128

Solid oxide fuel cell technology

are often difficult to obtain owing to the sophistication of the progressive electro-oxidation process. However, the numerical method based on the principle of iterative optimization is probably the most convenient means of calculating the local E(x) and current density i(x). The equations used in the modeling for calculating local electrochemical quantities are based on the energy conservation law Vc = E(x) – Vohm(x) – ηc(x) – ηa(x)

[7.62] ||

where cell voltage Vc is held invariant along the fuel flow direction x. With the partial pressure of oxygen PO 2( x) available, E(x) is calculated by PO 2 ( x)   E ( x) = RT × ln   2 4F  ( PH 2 O ( x)/ PH 2 ( x)/ K ) 

[7.63]

where K is the chemical equilibrium constant of reaction H2(g) + 0.5O2(g) = H2O(g), which can be determined from thermodynamic data given in Appendix 1. The ohmic IR loss, Vohm(x), is given by Vohm(x) = i(x) × Rohm(x)

[7.64]

where Rohm(x) is the local ohmic area-specific resistance including all cell components as discussed in Section 7.1; it is equal to the total ohmic resistance divided by the number of selected finite elements. On the other hand, the total cathode ηc(x) and anode ηa(x) polarization losses can be expressed by

η c ( x) = η cact ( x) + η cconc ( x) η a ( x) = η aact ( x) + η aconc ( x)

[7.65]

The individual activation and concentration polarization losses of a cathode have been formulated in equations [7.20a] and [7.50], respectively, whereas those of an anode have been given in equations [7.20a] and [7.59], respectively. For solving the local interfacial PO 2( x) , equation [7.49] for the cathode and equations [7.55] and [7.56] for the anode are used, respectively. In order to calculate in practice the local electrochemical quantities, an SOFC is divided axially to consist of k finite elements, each in parallel electrically. The current density, i(k), for each element and therefore for each axial position, x, is associated with a position-independent cell voltage, Vc, and local partial pressures of reactants and products. Reactant flows are assumed to occur in series through the elements. Appropriate changes in ||

Operation of the fuel cell is also referred to as iso-potential operation, which is very true for stack or generator operation where the stack resistance is much higher than that of the current-collecting bus bars.

Voltage losses in a solid oxide fuel cell (SOFC)

129

partial pressures of the reactants are calculated, corresponding to fuel consumption and oxygen utilization by the preceding elements. A schematic of the cylindrical tubular cell segment representing the finite elements is shown in Fig. 7.22. In order to reduce the number of variables, temperature, bulk PO 2 (1) of air, and the fuel inlet PH 2 O (6)/ PH 2 (6) ratio are fixed in the modeling. Current density, fuel consumption, and ohmic resistance are varied to study the responses of the local electrochemical quantities. A total of 50 elements are selected to represent a 50 cm-long cylindrical tubular cell with an active area of 270 cm2. For each element, approximately 15–25 iterations are required to reach stable i(x) values that satisfy the condition [7.62] within an accuracy of 10–5. In what follows, several examples are given to illustrate how the average current density and fuel consumption influence the E(x) profile for a cathodesupported YSZ-electrolyte SOFC. The parameters used are listed in Table 7.1. The effect of current density on E(x) is computed with T = 1000 °C, Cf = 0.85, PH 2 (6)/ PH 2 O (6) = 89/11, S = 6, PO 2 (1) = 0.209 atm, and Rohm = 0.40 Ω cm2; the result is shown in Fig. 7.23. Although all profiles are confined within the boundary of Ein = 0.987 V (reflecting PH 2 / PH 2 O = 89/11) and Eex = 0.765 V (reflecting PH 2 / PH 2 O = 13/87), the actual profile under each current density is very different. At low current density, the E(x) profile features a sharp decrease at the entrance region and gradually flattens out towards the exit. With increasing current density, however, the profiles become more linear, with a shrinking difference between each profile. The graphical area under the curve from x = 0 to x = 50 cm represents the average E, i.e. mathematically

EMF(k), Vc(k), i(k), Uf(k), Uo(k), PO (1)(k), PO (2)(k) 2 2 PO (3)(k), PO (4)(k), hc(k), ha(k) 2

2

Element k k + 1

x (cm)

Fuel flow direction

0

7.22 Schematic of cell segments representing finite elements used in the modeling.

Solid oxide fuel cell technology Table 7.1 Parameters and equations used for modeling a cylindrical, tubular, cathode-supported YSZ-electrolyte SOFC Parameter

Value

M, number of elements Lcell (cm) Acell (cm2) r1 (cm) r2 (cm) ta (cm) εc εa T (°C) Pt (atm) PO2 (1) (atm) PH2 (6) (atm) PH2O(6) (atm) Rohm (Ω cm2) UO Cf Cathode iex (A/cm2) Anode iex (A/cm2) Cathode iLM (A/cm2) Anode iLM (A/cm2) iave (A/cm2)

50 50 270 0.9 1.1 0.01 0.30 0.25 1000 1 0.209 0.89 0.11 0.40 0.1667 0.70–0.90 0.481 1.007 0.848 112 0.05–0.50

Equation

80 000 × exp(–1.32 eV/kT) Equation [7.48] Equation [7.58]

1.000

T = 1000 °C, Cf = 0.85 0.950 0.900

E (x) (V)

130

0.850

0.20

0.30 0.40

0.10 0.800

iave = 0.05 A/cm2

0.750 0.700 0

10

20 30 Position x (cm)

40

50

7.23 E(x) profiles at various average current densities.

∫ E=

50

E( x)dx

0

∫

[7.66]

50

0

dx

Voltage losses in a solid oxide fuel cell (SOFC)

131

Numerical integration of E at different iave indicates that E increases with iave in a logarithmic form, see Fig. 7.24. At a high current density, for example iave ≥ 0.25 A/cm2, the profile can be approximately linearized, leading to the following simplified form of E

E ≈ 1 ( E in + E ex ) 2

[7.67]

The effect of fuel consumption Cf on E(x) is shown in Fig. 7.25, where Cf is varied from 0.70 to 0.90 under a fixed iave = 0.305 A/cm2. A direct result from varying Cf is the change in Eex, i.e. Eex decreases with Cf under a fixed current density. As the graphical area under each curve reduces with a higher Cf, E becomes smaller. However, the approximation of a linear profile remains valid for Cf at this current density. As the iave decreases to 0.10 A/cm2, the effect of Cf appears to be more pronounced, see Fig. 7.26. The onset of the axial position where the profile becomes flattened reduces with increasing Cf, implying a reduced average E .

7.5.2

Current density distribution

Axial current density distribution With the availability of the E(x) profile and component resistances, the local current density i(x) can be calculated out and is shown as a function of x at different iave in Fig. 7.27. Similar to the trend seen in Fig. 7.23, the highest current density is located at the entrance. For instance, i(x → 0) can be as 0.87

T = 1000 °C, cf = 0.85

0.86 0.85 0.84

E (V)

0.83

Or

E = 0.0343 ln(iave) + 0.8937 E = 0.0343 ln(lave) + 0.702

R2 = 0.9934

0.82 0.81 0.80 0.79 0.78 0.00

0.10

0.20 0.30 iave (A/cm2)

0.40

0.50

7.24 E as a function of current density at T = 1000 °C and Cf = 0.85.

132

Solid oxide fuel cell technology 1.000

T = 1000 ϒC, iave = 0.305 A/cm 0.950

E(x) (V)

0.900

0.850 2

0.70 0.80

0.800

Cf = 0.90 0.750

0.700 0

10

20 30 Position x (cm)

40

50

7.25 E(x) profiles with various fuel consumptions at iave = 0.305 A/cm2. 1.000

T = 1000 ϒC, iave = 0.100 A/cm2

0.950

E(x) (V)

0.900

0.850 0.70

0.750 0.800

0.80

0.750

Cf = 0.90

0

10

20 30 Position x (cm)

40

50

7.26 E(x) profiles with various fuel consumptions at iave = 0.100 A/cm2.

high as 0.60 A/cm2 with iave = 0.305 A/cm2. More importantly, the majority of i(x) is shifted to the entrance region as iave decreases. The i(x) profiles calculated under iave = 0.305 A/cm2 and different Cf are shown in Fig. 7.28. As Cf increases, more i(x) shift to the entrance region.

Voltage losses in a solid oxide fuel cell (SOFC)

133

0.700 0.600

i (x) (A/cm2)

0.500

Cf = 0.85 0.400 0.300

iave = 0.40 A/cm2

0.200

0.30 A/cm2 0.05 A/cm2

0.100

0.10 A/cm2

0.20 A/cm2

0.000 0

10

20 30 Position x (cm)

40

50

7.27 i(x) profiles at various average current densities, Cf = 0.85, and T = 1000 °C.

0.700

T = 1000 °C, iave = 0.305 A/cm2 0.600

Cf = 0.80 i (x) (A/cm2)

0.500 0.400 Cf = 0.70 0.300 0.200

Cf = 0.90

0.100 0.000 0

10

20 30 Position x (cm)

40

50

7.28 i(x) profiles at various Cf, T = 1000 °C, and iave = 0.305 A/cm2.

This trend becomes more pronounced at a lower iave and is better illustrated in Fig. 7.29 where the i(x) profile is computed at iave = 0.10 A/cm2. In summary, both E(x) and i(x) profiles shift to the entrance region as Cf increases and iave decreases. This shift calls for caution when operating under these conditions because a more concentrated current density could increase the risk of breaking the cell.

134

Solid oxide fuel cell technology 0.600

T = 1000 °C, iave = 0.100 A/cm2

i (x) (A/cm2)

0.500

0.400

Cf = 0.80

0.300

0.200 Cf = 0.70 0.100

Cf = 0.90 0.000 0

10

20

30

40

50

Position x (cm)

7.29 i(x) profiles at various Cf, I = 1000 °C, and iave = 0.100 A/cm2.

Contact

L

i

f1 (0) = u

i1 (x ) i3 (x ) i2 (x )

R 1 (Ω )

Anode

R3 (Ω cm2)

Electrolyte

R 2 (Ω )

Cathode

f 2 (L ) = 0

i

x=0

x=L

7.30 Schematic of lateral current pathway in the functional layers of an SOFC.

Lateral current density distribution Lateral current density distribution in this context refers to local current density variations in the direction perpendicular to the fuel flow x. An analytical solution shown below is applicable to any SOFC configuration. Figure 7.30 illustrates a simplified pathway of electrical current traveling in electrodes and electrolyte of an SOFC. Owing to the nature of the in-plane conduction in the electrodes, the concept of sheet resistance Rs, as discussed in Section 7.1, is utilized in deriving the equation of current density distribution. With the assumption that partial current densities in anode, electrolyte, and cathode

Voltage losses in a solid oxide fuel cell (SOFC)

135

are represented by i1, i3, and i2, respectively, and the corresponding resistances are R1 (Ω), R3 (Ω cm2), and R2 (Ω), respectively, the following relationships can be established by Ohm’s law under a one-dimensional Cartesian coordinate ∂φ1 i1 ( x ) = – 1 R1 ∂x

(A/cm)

∂φ 2 i2 ( x ) = – 1 R2 ∂x

(A/cm)

i3 ( x ) = – 1 (φ1 – φ 2 ) R3

(A/cm2) [7.68]

The boundary conditions are

φ2(L) = 0

[7.69]

φ1(0) = u

[7.70]

i1(0) = i2(L) = i

[7.71]

i1(L) = i2(0) = 0

[7.72]

and the continuum condition requires

φ1 – φ 2 ∂i1 ( x ) ∂i ( x ) = – 2 = – i3 ( x ) = – R3 ∂x ∂x

[7.73]

The second-order derivative of i1(x) with respect to x is then given by

∂i ( x ) ∂ 2 i1 ( x )    ∂φ ∂φ  =– 3 = – 1  1 – 2  = 1  i1 ( x ) R1 – i2 ( x ) R2  2 R R ∂ x ∂ x ∂ x   ∂x 3  3  [7.74] From equations [7.68], it is evident that i1(x) + i2(x) = i

[7.75]

where i is the position-independent total cell current. Substitution of equations [7.75] into equation [7.74] yields R ∂ 2 i1 ( x ) R1 + R2 = × i1 ( x ) – 2 i R3 R3 ∂x 2

[7.76]

Integration of equation [7.76] with the boundary conditions of equations [7.69] to [7.72] leads to tanh (γ x )  sinh (γ x )  i1 ( x ) R1 cosh (γ x )  R2  = 1– + 1– i R1 + R2  R1 + R2  tanh (γ L )  tanh (γ L )  [7.77]

136

Solid oxide fuel cell technology

where the constant γ is defined as

γ=

R1 + R2 R3

[7.78]

From equation [7.76], the lateral current density in the cathode, i2(x), can be derived i2 ( x ) R1 = i R1 + R2

+

tanh (γ x )     1 – cosh (γ x )  1– tanh (γ L )    

R2  1 – sinh (γ x )  R1 + R2  tanh (γ L ) 

[7.79]

Similarly, the lateral current density in the electrolyte, i3(x), can be derived from equation [7.73] i3 ( x ) i

=

λ cosh (γ x )  R1 R2  (1 – tanh (γ x ) + tanh (γ L )) + R1 + R2  tanh (γ L ) sinh (γ L )  [7.80]

Equations [7.77], [7.79], and [7.80] serve as general analytical expressions of the lateral current density distributions in the anode, cathode, and electrolyte layers, respectively. Application of equation [7.80] to the electrolyte layer of a cylindrical tubular SOFC is exemplified by Fig. 7.31 where the normalized current density i3(x)/i is plotted against x/L. The sheet resistances of anode and cathode, namely R1 and R2, are chosen as 0.015 and 0.060 Ω/SQ, respectively, for illustrative purposes. R3 is also used as a variable ranging from 0.40 to 0.80 Ω cm2 in the plot. It is evident that each curve features a minimum occurring between x/L = 0.20 and 0.30, and more current is collected towards the exit, i.e. the interconnect region. A higher R3 tends to spread current out more evenly than a lower R3. Owing to the nature of normalization, however, the graphical area under each curve and x axis should remain the same for each R3 case.

7.6

Effect of leakage flux on the voltage–current curve

In Chapter 4, it was demonstrated that the leakage current density resulting from mixed conduction in the electrolyte and interconnect layers prevails primarily under the condition of OCV and lower current density. In other words, the effect of leakage current density becomes less significant at higher

Voltage losses in a solid oxide fuel cell (SOFC)

137

0.50 0.48

L = 2.78 cm R1 = 0.015 Ω/SQ R2 = 0.060 Ω/SQ

0.46 0.44

i R3 = 0.40 Ω cm2

L

0.60

i3(x)/i

0.42

0.80

0.40 0.38

i

0.36 0.34 0.32 0.30 0

0.2

0.4

0.6

0.8

1

x /L

7.31 Normalized distribution of lateral current density in the electrolyte layer of a cathode-supported cylindrical tubular SOFC.

current density. In practice, this assertion is further confirmed by a lowcurrent-density bending-over phenomenon of a V–I curve measured at a constant Uf if loss of fuel is present. In this section, the way in which the leakage current density influences the bending-over behavior is discussed specifically. The basis of correlating leakage current density with cell voltage is the fact that the cell EMF varies with the leakage flux QL as a result of altered PH 2 (or QH 2 ) and PH 2 O (or QH 2 O ) of an H2–H2O fuel mixture. This is particularly accurate for a cathode-supported SOFC where anode polarization is generally marginal. Under a constant Uf condition and a fixed H2 leakage flux QL,¶ the leakage fluxes of H2 and H2O at the exit are given by

QH 2 O ( I ) = QHo 2 O ( I ) + QL

[7.81]

QH 2 ( I ) = QHo 2 ( I ) – QL

[7.82]

QHo 2

QHo 2 O

and are the inlet H2 and H2O mass flow rates, respectively, where which can be calculated by equation [6.5] with a known Uf and cell current I. Based on equation [2.10], the difference between a leakage-free and a leakage-present EMF, ∆E, is then given by ¶ For simplification, a constant H2 leakage flux QL is assumed although it was shown that QL decreases with load current density. This case will represent a conservative scenario.

138

Solid oxide fuel cell technology

∆ E ( QL , I ) = 0.0430

 QHo O ( I ) + QL QHo 2 ( I )  × T × ln  o2 × o   QH 2 ( I ) – QL QH 2 O( I ) 

(mV)

[7.83]

For a practical V–I curve, the cell voltage Vc can be corrected from a leakagefree Vco by  (6.97 × I )/ U f + QL / r o  Vc ( I ) =Vco ( I ) – 0.0430 × T × ln    (6.97 × I )/ U f – QL  (mV)

[7.84]

where r = I = i × Acell where Acell is the cell active area (in cm2); the leakage flux QL has a unit of sccm H2. Equation [7.84] clearly states that the leakage-dependent Vc varies with Vco, T, I, Uf, ro, and QL. A higher T and QL, and lower ro, would result in a lowered Vc. In the last section, the variation of EMF with axial position was discussed. It was shown that the EMF also varies with cell current density as a result of progressive electro-oxidation of the fuel. An empirical equation that correlates the average EMF of a leakage-free cell with the average current I is given by o

QHo 2 O

/ QHo 2 ;

E = 0.0343 × ln(I) + 0.702

(V)

[7.85]

where an inlet r = 11/89, Uf = 0.85, T = 1000 °C, Acell = 270 cm , and Rcell = 4 mΩ are utilized. Substitution of equation [7.85] into equation [7.84] leads to the QL-dependent V–I equation o

2

Vc(I) = 34.3 × ln I + 702 – I × Rcell – 0.0430 × T  (6.97 × I )/ U f + QL / r o  × ln    (6.97 × I )/ U f – QL 

(mV)

[7.86]

The maximum Vc at a critical current Ic can be obtained by differentiating Vc with respect to I at a constant T and Uf ∂Vc ( I ) ∂I

T ,U f

m (1 + 1/ r o ) QL = 34.3 – Rcell + 0.043 T × I ( mI + 1/ r o QL )( mI – QL )

[7.87] where the constant m = 6.97/Uf. By setting equation [7.87] to zero, Ic can be solved. In order to simulate the bending-over behavior of the V–I curve with equation [7.86], an SOFC with Rcell = 4 mΩ and an active surface area of 270 cm2 is virtually perceived to operate at 1000 °C with Uf = 85% of an inlet fuel containing 89%H2 and 11%H2O. Figure 7.32 shows the simulated

Voltage losses in a solid oxide fuel cell (SOFC)

139

800 2O

2

30

650

= 0.11 atm

Uf = 0.85 Ac = 270 cm2 Rcell = 4 mΩ

10

700

Vc (mV)

T = 1000 °C Inlet PH = 0.89 atm, PH

QL = 0 sccm H2

750

600 50

550 500 450 400 0

20

40

60

80

100

I (A)

7.32 Effect of leakage flux QL on the V–I characteristics of an SOFC. 35 30

Ic ( a )

25 20 15

r° = PH /PH 2

10

2O

= 89/11

Uf = 0.85, m = 8.2 Rcell = 4 mΩ T = 1000 °C Ac = 270 cm2

5 0 0

20

40

60 QL (sccm H2)

80

100

120

7.33 Calculated critical current Ic as a function of leakage flux QL.

V–I curves under various H2 leakage fluxes. It is clearly seen that the bendingover behavior occurs in the low-current region and becomes more pronounced at a higher leakage flux. The underlying mechanism for a pronounced bendingover of the V–I curve at low current is the relatively lower fuel-flow-toleakage-flux ratio under a constant Uf, by which the amount of fuel presented to the cell can be easily overwhelmed by the amount of leakage flux. The

140

Solid oxide fuel cell technology

critical current Ic as a function of QL is shown in Fig. 7.33. Ic increases with QL as predicted. The significance of bending-over behavior resulting from the fuel loss to the electrical efficiency is to be discussed in detail in Chapter 8.

8 Direct current (DC) electrical efficiency and power of a solid oxide fuel cell (SOFC)

Abstract: One of the unique advantages of SOFCs is the intrinsically high efficiency of producing electrical power. Depending on the power rating and stack configuration, the net AC electrical efficiency of an SOFC generator ranging from a few to hundreds of kilowatts varies from 35 to 55%. When combined with the waste-heat recovery cycle, the overall energy conversion efficiency of an SOFC system falls into the range of 80–90%. The high electrical and system efficiencies of SOFCs are believed to be one of the major drivers for the enduring interest in SOFC development in the last few decades. In this chapter, the DC electrical efficiency and factors that effect it are extensively discussed. The overall electrical efficiency is not the topic of this chapter; it is simply a multiplication of the DC electrical efficiency with the conversion efficiencies of the individual power conditioning system (PCS) components. Key words: DC electrical efficiency, thermodynamic efficiency, cell efficiency, Faradaic efficiency, voltage efficiency, Carnot heat engine efficiency.

8.1

Direct current electrical efficiency

The DC electrical efficiency ε is defined in this book as the ratio of total DC electrical power output generated from the SOFC stack to the total thermal power input of the fuel. The latter is the product of the enthalpy change ∆ c H To (in J/mol) at temperature T of the fuel oxidation reaction and the fuel flow rate Qf (in slpm) as defined in equation [6.10]

ε=

Stack power output = Fuel thermal power input

I L × VS – ∆ c H To × Qf ×

( 22.41 × 601 ) [8.1]

where IL and VS are the load current and stack voltage of an SOFC generator, respectively. ∆ c H To can be readily calculated from the available thermodynamic data with equation [6.35], which are tabulated in Table A2.2 of Appendix 2 for some common fuels. For a given Uf, the correlation between the fuel flow Qf and the ionic current Iion (equivalent to I V .. as discussed in Chapter 4) O follows from equation [6.10]. Substituting equation [6.10] into equation [8.1] yields 141

142

Solid oxide fuel cell technology

 m  2 ×  Σ (3 n + 1) y n  × F  n =0  V I × Uf × L × S ε= – o I ion N ∆ c HT

[8.2]

If there are N cells connected in series, the single cell voltage Vc equals VS/ N. The DC electrical efficiency of equation [8.2] can then be rewritten as

 m  2 ×  Σ (3 n + 1) y n  × F × E o   n =0 ε= – o ∆ c HT

× Uf ×

V IL × co = ε o × ε I × ε V I ion E

[8.3]

where ε is divided into a product of three sub-terms, εo, εI, and εV  m  2 ×  Σ (3 n + 1) y n  × F × E o =0 n   εo = – ∆ c H To

εI = εV =

IL × Uf I ion Vc Eo

[8.4]

εo, εI, and εV are termed thermodynamic efficiency, current (or Faradaic) efficiency, and voltage efficiency, respectively. The physical meaning of εo is understood to be the maximum electrical efficiency for electrochemically converting chemical energy stored in a fuel into electrical power under the standard state; εI represents the fraction of ionic current Iion being converted into useful load current IL; εV manifests the degree of voltage loss owing to total polarizations from cell components. The product of εI and εV is often termed the cell efficiency εc, which is discussed further in the following section. Note that ε can be further separated into two categories in practical applications: lower heating value (LHV) and higher heating value (HHV); the former refers to a combustion product containing the gaseous H2O whereas the latter refers to the liquid H2O. Refer to Appendix 2 for more detail on the subject. Owing to the fact that the ∆ c H To of the fuel oxidation reaction involving liquid H2O has a higher value than the former one, ε (HHV) is typically lower than ε (LHV).

Direct current (DC) electrical efficiency and power

8.1.1

143

Solid oxide fuel cell thermodynamic efficiency εo and Carnot heat-engine efficiency h

With the aid of equation [2.4a] and the average number of electrons

(

)

transferred 2 × Σ nm=0 (3 n + 1) y n during the oxidation reaction of a mixture of CnH2n+2, the εo in equation [8.4] can be modified into

εo =

∆GTo ∆ c H To

T × ∆ STo   = 1 –  ∆ c H To  

[8.5]

where ∆GTo and ∆ STo are the standard Gibbs free energy change and entropy change of the fuel oxidation reaction [6.34], respectively. With the availability of ∆ c H To , ∆GTo, and ∆ STo listed for various fuels in Appendices 1 and 2, the εo value can be calculated for a specific fuel as a function of temperature, and is shown in Fig. 8.1. As evident from the plot, C and CH4 have the highest εo values, close to 100%, followed by H2, syngas, and CO. The underlying reason for the variation in efficiency with types of fuels lies primarily in the magnitude of the heat term T × ∆S as discussed in Section 2.3. Sufficient heat is necessary from the electro-oxidation reaction in order to sustain the stack temperature. Therefore, a direct utilization of C and CH4 may not be practical for high-temperature SOFCs owing to an insufficient heat liberation despite the high thermodynamic efficiency. As the operating temperature is lowered, the demand for the heat to sustain the stack temperature decreases, making direct utilization of C and CH4 plausible. On the other hand, lowering the operating temperature considerably increases the εo values of H2, CO, and 1.05

C

1.00

CH4 0.95

e°(LHV)

0.90 0.85 0.80 0.75

H2

0.70

Syngas CO

0.65 0.60 550

600

650

700

750

800 T (°C)

850

900

950

1000 1050

8.1 Thermodynamic efficiency calculated from equation [8.5] for various fuels.

144

Solid oxide fuel cell technology

syngas, a major benefit for SOFCs fuelled by these simple fuels. The real competitor to an SOFC generator is the conventional heat engine. The maximum efficiency of a heat engine is determined by the Carnot cycle, which is a particular thermodynamic cycle proposed by Nicolas Léonard Sadi Carnot in 1824 and expanded upon by Benoit Paul Émile Clapeyron in the 1830s and 1840s. The Carnot cycle is the most efficient cycle possible for converting a given amount of thermal energy into work or, conversely, for using a given amount of work for refrigeration purposes. Figure 8.2 shows the four basic steps for the Carnot cycle when acting as a heat engine. 1

2

3

Reversible isothermal expansion of the gas at the ‘hot’ temperature, TH (isothermal heat addition). During this step (A to B in Fig. 8.2) the expanding gas causes the piston to do work on the surroundings. The gas expansion is propelled by absorption of heat from the high-temperature reservoir. Isentropic (reversible adiabatic) expansion of the gas. For this step (B to C in Fig. 8.2) the piston and cylinder are assumed to be thermally insulated so that no heat is gained or lost. The gas continues to expand, doing work on the surroundings. The gas expansion causes it to cool to the ‘cold’ temperature, TC. Reversible isothermal compression of the gas at the ‘cold’ temperature, TC (isothermal heat rejection) (C to D in Fig. 8.2). The surroundings do

T

TH

A

B

W = Q H – QC

TC

D

C

QC

SA

SB

8.2 A Carnot cycle acting as a heat engine illustrated in a temperature–entropy diagram.

S

Direct current (DC) electrical efficiency and power

145

work on the gas, causing the heat to flow out of the gas to the lowtemperature reservoir. 4. Isentropic compression of the gas (D to A in Fig. 8.2). The piston and cylinder are assumed to be thermally insulated. During this step, the surroundings do work on the gas, compressing it and causing the temperature to rise to TH. At this point the gas is at the same state as at the start of step 1. Based on Fig. 8.2, the amount of energy transferred as work during the Carnot cycle is given by

W=

∫ P dV = ( T

H

– TC ) × ( SB – SA )

[8.6]

The total amount of thermal energy transferred between the hot reservoir and the system is QH = TH × (SB – SA)

[8.7]

where SB is the entropy at TC, and SA is the entropy at TH. The total amount of thermal energy transferred between the system and the cold reservoir is QC = TC × (SB – SA)

[8.8]

The efficiency η of the Carnot cycle is therefore given by

T η= W =1– C QH TH

[8.9]

Figure 8.1 may be compared to η = 64%, which is the Carnot efficiency for a good modern large-scale steam plant that might attain an actual overall efficiency of 40%. It is shown clearly that the potential advantage of SOFCs in thermodynamic efficiency εo over steam power plant will disappear with increasing temperature, not to mention the polarization and fuel losses, unless the rejected heat is properly used. This is a much more serious consideration in our day than in Ostwald’s,8 when a fuel cell would have been called on to compete with steam plants having an overall efficiency of only 10%. However, SOFC generators still have the undisputable higher electrical efficiency than steam power generation in the range of the < 1 MW class. Therefore, SOFCs are widely believed to be more suitable for small-scale distributed powergeneration applications.

8.1.2

Cell efficiency εc

The product of current and voltage efficiencies is defined as the cell efficiency εc in this book. It measures how efficiently an SOFC converts the energy ∆Go of a given fuel oxidation reaction to electricity. Since IL, Vc, and Iion in

146

Solid oxide fuel cell technology

equation [8.4] are all interrelated, a unique analytical solution to εc as a function of IL is difficult to derive. However, with the aid of transport equations solved in Chapter 4 for oxide ions and electrons in the electrolyte, the cell efficiency can be estimated with the assumption of current-independent electrode resistances by

ε c = Uf ×

E – I V .. Rcell V IL I O × co = U f × L × o I V .. I V .. E E O O

[8.10]

where Rcell (in ohms) is the sum of total ohmic resistance Rohm, cathode resistance Rc, and anode resistance Ra. The IL-dependent I V .. analytical O equation is given by equation [4.63]. Eo and E are the Nernst potentials of the cell reactions at standard and non-standard conditions. The former is only dependent on temperature and is shown in Fig. 2.1 whereas the latter varies with temperature, fuel composition, and system pressure in the form of equations [2.24] and [2.25] for H2 and CO fuels, respectively. The calculation of I V .. in a mixed-conducting electrolyte is very similar O to that of Ie' demonstrated in Fig. 4.6 for CeO2-based electrolytes. With the I V .. –IL relationship in hand, εc as a function of IL can be easily computed O from equation [8.6] with a known Rcell and fuel composition. Figure 8.3 shows the results of the calculated εc of a 50 µm-thick CeO2 electrolyte as a 1.00

L = 0.005 cm H2 /H2O = 1/1

0.90 0.80 0.70 0.60

700 °C

ec 0.50 0.40 600 °C

0.30 0.20 0.10 0.00 0.00

500 °C

0.50

1.00

1.50 2.00 iL (A/cm2)

2.50

3.00

8.3 Variations of εc with iL of a 50 µm-thick CeO2-based electrolyte with fuel composition of H2/H2O = 1/1. Parameters in reference 44 and Table 4.1 were adopted for the calculation. Dotted lines represent zero leakage current density.

3.50

Direct current (DC) electrical efficiency and power

147

function of iL (or IL) at three temperatures. The Uf effect was averaged into the fuel composition H2/H2O = 1/1 in the calculation. The appearance of a peak in the efficiency is in striking contrast to the case with zero electronic conduction shown by the dotted lines in the figure. The peak efficiency appears to be a unique characteristic of mixed-conducting electrolytes if used in an SOFC. It is a direct result of the I e′ decreasing with IL. It is also indirectly evidenced by the decrease in the peak efficiency with temperature since the electronic conductivity increases with temperature in CeO2-based electrolytes. Figure 8.3 strongly suggests that the SOFC with a mixedconducting electrolyte has to operate above a certain IL in order to prevent the penalty of low power with low efficiency in the low IL region. With an εc– IL curve available, the εc–P curve can be further calculated out with P = IL × Vc and is shown in Fig. 8.4. Similar to Fig. 8.3, a peak in the efficiency occurs at a certain power, breaking the rule that a lower efficiency is always associated with a higher power. However, the plots seem to suggest that as the temperature increases, a higher efficiency can be achieved over a broader power density, but with a lowered maximum efficiency. With a ceramic interconnect, the mixed conduction in the interconnect at the operating temperature of an SOFC could further lower the cell voltage by Vc′, which leads to the following more general expression for εc

ε c = Uf ×

E – I V .. Rcell – Vc′ V IL I O × co = U f × L × I V .. I .. E Eo VO O

[8.11]

where Vc′ represents the voltage drop across the ceramic interconnect. Either 0.80 500 °C 0.70

600 °C

700 °C

0.60

εc

0.50 0.40 0.30 0.20

L = 0.005 cm H2 /H2O = 1/1

0.10 0.00 0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

P (W/cm2)

8.4 Variations of εc with P of a 50 µm-thick CeO2-based electrolyte with fuel composition of H2/H2O = 1/1. Parameters in reference 44 and Table 4.1 were adopted for the calculation.

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Solid oxide fuel cell technology

higher oxide-ion conductivity or lower electron hole conductivity would increase Vc′, lowering εc. The Vc′ is primarily dictated by the oxide-ion conduction in the electron-dominated ceramic interconnect. In order to estimate Vc′, knowledge of PO 2 distribution across the interconnect is first required; from this the distributions of PO 2-dependent oxide-ion conductivity σ V .. O and electron hole conductivity σ h are known. The voltage drop across the interconnect, Vc′, can then be calculated by integrating the gradient of the electrochemical potential of electron holes, dη h /dx , over the thickness of the interconnect L •

•

Vc′ =

∫

L

0

dη h

•

dx = – RT 4F dx

∫

σ V ..

ln PO′ 2

ln PO′′ 2

O

σ h – σ V .. – σ L •

d ln PO 2

[8.12]

O

where PO′ 2 and PO′′2 are the partial pressures of oxygen at the air and fuel sides, respectively. Interested readers can use the parameters given in Table 4.2 for further calculations. In practice, a pair of voltage tabs can be applied across the interconnect to measure the voltage loss during the operation directly.

8.2

Efficiency, fuel utilization, and electrical power

The general form of ε shown in equation [8.3] can be simplified into more convenient and useful expressions for some simple fuels. With pure H2 as an example, n = 0, yn = 1, zero leakage current (Iion = IL and εI = Uf), and ∆ c H To = –249.259 kJ/mol at T = 1000 °C (see Table A2.2 in Appendix 2), equation [8.3] gives

ε = – 2 F o × U f × Vc = 0.774 × U f × Vc ∆ c HT

[8.13]

It is evident that ε is only a function of the type of fuel, fuel utilization, and cell voltage. It is independent of current for a leakage-free SOFC. Another example is given for pure CH4, where n = 1, yn = 1, and ∆ c H To = –803.394 kJ/mol at 1000 °C, and zero leakage current

ε = – 8 F o × U f × Vc = 0.961 × U f × Vc ∆ c HT

[8.14]

Figure 8.5 plots the calculated ratio χ = – zF/ ∆ c H To as a function of temperature for different fuels. Clearly, CH4 and C are advantageous over H2 and CO in that they achieve higher electrical efficiency for a given Uf and Vc. Moreover, C is less sensitive to the temperature owing to the fact that ∆ c H To is less temperature-dependent.

Direct current (DC) electrical efficiency and power

149

1.00 C, z = 4 0.95

CH4, z = 8

0.90

–zF/∆cH T°

0.85 0.80 H2, z = 2

0.75 0.70

CO, z = 2

0.65 0.60 0

200

400

600 T (°C)

800

1000

1200

8.5 Coefficient of –zF/∆c H To as a function of temperature for various fuels.

1.10 e = 0.50

Solid lines: H2

1.00

Dashed lines: CH4 e = 0.45

Vc (V)

0.90

e = 0.40

0.80 0.70 0.60 e = 0.50 e = 0.45

0.50

e = 0.40 0.40 0.55

0.65

0.75 Uf

0.85

0.95

8.6 Variation of Uf with Vc under a fixed electrical efficiency (LHV) at 1000 °C.

To further illustrate the relationship in equations [8.13] and [8.14], the variations of Uf with Vc under a fixed ε at 1000°C are plotted as an example in Fig. 8.6. It is no surprise that for a fixed operating voltage, CH4 requires a lower Uf to achieve the same efficiency as H2.

150

8.3

Solid oxide fuel cell technology

The maximum direct current electrical efficiency

The fact that an SOFC is intrinsically more efficient at producing electric power than traditional combustion engines is well-accepted. Although the Carnot cycle does not constrain SOFC power generation, an upper limit of the electrical efficiency of an SOFC still exists; it is the maximum electrical efficiency of an SOFC that will be discussed in the following subsections.

8.3.1

Theoretical basis

As discussed in Chapter 7, the EMF of an SOFC and the cell current I, under a constant Uf and T, vary with the axial position along the fuel flow direction x. When the cell current approaches zero, E becomes flattened out to the value reflecting the composition of depleted anode gas at the exit, i.e. the highest cell voltage at a specific Uf is the E of depleted anode gas at the exit lim Vc = Vmax = E ex I →0

[8.15]

In a calculation of the maximum efficiency, the leakage current density can be ignored as the leakage current density always lowers the efficiency. The key is to determine the composition of depleted fuel at the exit, from which Eex can be calculated by using either equation [2.24] for H2 fuel or equation [2.25] for CO fuel at any given T, fuel-to-oxide ratio, and Pt. In Chapter 6, an approach to calculating the stack composition of an SOFC fuelled by natural gas has also been demonstrated. Interested readers are encouraged to use this example for a further exercise. For more complicated fuel systems, the chemical equilibrium of multi-components can be easily calculated by commercial software such as NASA code, by which the equilibrium PO 2 and therefore Eex can be determined. With Eex in hand, the maximum electrical efficiency εmax for a mixture of fuel generically formulated as CnH2n+2 can be calculated by

ε max

 m  2 ×  Σ (3 n + 1) y n  × F  n =0  = – × U f × E ex o ∆ c HT

[8.16]

where zero leakage current is assumed.

8.3.2

Under atmospheric conditions

In the following, a simple H2–H2O fuel mixture is used as an example to illustrate the relationships of Vmax–Uf – εmax. The calculated variations of Vmax with Uf at three representative temperatures and two possible inlet

Direct current (DC) electrical efficiency and power

151

compositions of H2–H2O fuel are shown in Fig. 8.7. As expected, Vmax decreases with Uf at a given temperature as a result of increased oxide-to-fuel ratio at the exit. The plots also suggest that the effect of temperature on Vmax is much more pronounced than the effect of inlet anode gas composition. Lowering temperature clearly promises a higher Vmax and therefore a greater εmax. With the data from Fig. 8.7, εmax can be further calculated and is shown in Fig. 8.8 as a function of Uf. In contrast to Fig. 8.7, however, εmax increases with Uf at a given temperature and decreases with increasing temperature at a given Uf. The inlet anode gas composition has a marginal effect on εmax. A noticeable trend is that εmax tends to flatten out with Uf > 0.90. This finding suggests that a further increase in Uf to >90% may not be beneficial in boosting electrical efficiency. Another important implication from Fig. 8.8 is that in order to achieve a targeted electrical efficiency, the maximum tolerable energy loss by polarizations and fuel losses can be projected. For example, it is seen in Fig. 8.8 that εmax ≈ 0.55 at 900°C and Uf = 0.90 for the H2–H2O fuel. If the parasitic consumptions account for 0.05, then the εmax achievable by the SOFC stack is 0.50. This result implies that the total energy loss due to the leakage current and polarization cannot exceed 0.05 (= 0.50–0.45) if the desirable maximum efficiency is set to be 0.45. The maximum 5% energy loss would be the guideline in the cell development for lowering leakage current and polarization losses. The effect of temperature on εmax is further expanded to a wide temperature range in Fig. 8.9. It is very clear that lowering the operating temperature is an effective way of increasing εmax. Simply lowering temperature from 0.950 800 °C

0.900

900 °C 1000 °C

Vmax (V)

0.850 0.800 0.750

Fuel inlet: 89% H2-11% H2O

0.700

Fuel inlet: 97% H2-3% H2O 0.650 0.50

0.60

0.70

0.80

0.90

1.00

Uf

8.7 Variation of Vmax of H2 fuel with Uf at two different inlet fuel compositions.

152

Solid oxide fuel cell technology

0.65 Fuel inlet: 89% H2–11% H2O Fuel inlet: 97% H2–3% H2O

0.60

800 °C 900 °C

emax (LHV)

0.55

1000 °C 0.50

0.45

0.40

0.35 0.50

0.60

0.70

0.80

0.90

1.00

Uf

8.8 Variation of εmax of H2–H2O fuel with Uf at two different inlet fuel compositions. 0.70

emax (LHV)

0.65 0.60

Uf = 85%, H2 89% Uf = 90%, H2 89% Uf = 85%, H2 97% Uf = 90%, H2 97%

Uf = 0.90 Uf = 0.85

0.55 0.50 0.45 0.40 500

600

700

800 T (°C)

900

1000

1100

8.9 Effect of temperature on εmax of H2–H2O fuel with two different Uf values and two different inlet fuel compositions.

1000°C to 800°C at Uf = 0.90 can lead to ~6% increase in efficiency. Moreover, lowering temperature would also decrease leakage current, which can in turn help to increase the actual electrical efficiency. However, the polarization losses would in general increase as a result of lowering temperature. Therefore, a high-performance, low-temperature SOFC is ideal for achieving high electrical efficiency.

Direct current (DC) electrical efficiency and power

8.3.3

153

Under pressurized conditions

Equation [2.24] indicates that Vmax can be enhanced by 0.0215T ln(Pt/Po) (in mV) under an elevated system pressure Pt over atmospheric pressure Po. The variations of Vmax with Uf under pressurization at 1000 °C and with two inlet anode gas compositions are plotted in Fig. 8.10. Three pressure levels – 1, 10, and 20 atm – are chosen for the purposes of illustration. Initial pressurization up to 10 atm apparently has a greater effect on Vmax than further pressurizing to 20 atm. Similar to the findings under atmospheric conditions, the inlet anode gas composition has a marginal effect for all temperatures and pressures. The εmax–Uf plots are shown in Fig. 8.11 under the same conditions. As the inlet anode gas composition has a marginal effect, only one inlet anode gas composition is considered. Figure 8.11 shows a roughly 5% increase in εmax as a result of elevating the pressure from 1 to 10 atm, after which only ~1% enhancement can be achieved by elevating pressure to 20 atm. However, it has to be noted that elevating the system pressure from 10 to 20 atm could significantly improve the effectiveness of an integrated micro-turbine, here the total electrical efficiency of the hybrid system could still be higher. The appearance of a plateau after Uf = 0.90 is still true for all pressures and temperatures, further confirming that greater than 90% fuel utilization may not be beneficial in boosting efficiency further. In addition, the trend of the variation of efficiency with temperature remains unchanged at any pressure level.

0.950 0.900

Vmax (V)

0.850 0.800 0.750 0.700

20 atm 10 atm

T = 1000 °C Inlet fuel compositions:

1 atm

89% H2–11% H2O 97% H2–3% H2O

0.650 0.55

0.65

0.75

0.85

0.95

1.05

Uf

8.10 Effect of pressure on Vmax of H2–H2O fuel with two different inlet fuel compositions at a temperature of 1000 °C.

154

Solid oxide fuel cell technology

0.65

T = 1000 °C Inlet fuel composition: 89% H2–11% H2O

emax (LHV)

0.60

20 atm 10 atm

0.55 1 atm 0.50

0.45

0.40

0.35 0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

Uf

8.11 Effect of pressure on εmax of H2–H2O fuel at 1000 °C.

8.4

Effect of the system pressure on direct current electrical efficiency

In Section 8.1, the effect of leakage current on the cell efficiency was discussed with an example of CeO2-based electrolytes. In this section, case studies of ScSZ-based SOFCs are given to illustrate the effect of system pressure on electrical efficiency in the presence of leakage current. Parameters listed in Table 4.1 are used for the calculations. Figure 8.12 compares ε–iL curves for atmospheric and pressurized conditions, from which a general trend is observed; viz. there is a peak efficiency occurring in the lower current-density range. It appears that pressurization shifts the peak efficiency towards higher cell current density, implying that a higher stack electrical efficiency with a higher power is possible for a pressurized SOFC. However, the peak efficiency is apparently lowered by the pressurization. This reduction in efficiency is undoubtedly the result of increased leakage currents by the pressurization, as shown in Fig. 4.14. Only a slightly higher efficiency for a pressurized cell is observed at higher current density. On the other hand, the ε–P curves shown in Fig. 8.13 indicate a reasonably high efficiency over a wider current density range for pressurized systems, also suggesting that pressurization of an SOFC is capable of achieving a higher stack electrical efficiency at a higher power density.

Direct current (DC) electrical efficiency and power 0.60 0.50

e (LHV)

0.40

Pt = 10 atm Pt = 1 atm

0.30 0.20

T = 900 °C

0.10 0.00 0.000

0.100

0.200

0.300

0.400

iL (A/cm2)

8.12 DC electrical efficiency as a function of load current iL with H2–H2O mixture as a fuel under 1 and 10 atm system pressures.

0.60 0.50

Pt = 10 atm

e (LHV)

0.40

Pt = 1 atm

0.30 0.20

T = 900 °C

0.10 0.00 0.000

0.100

0.200

0.300

0.400

0.500

P (W/cm2)

8.13 DC electrical efficiency as a function of power with H2–H2O mixture as a fuel under 1 and 10 atm system pressures.

155

9 Performance characterization techniques for a solid oxide fuel cell (SOFC) and its components

Abstract: Advances in the development of SOFC technology require an improved understanding of the fundamental electrochemical and physical processes. Conventional electrochemical and physical methods are important assets for supporting such an understanding, by which characteristic responses of an SOFC to various signal variables can be accurately measured and effectively analyzed with advanced electrochemical models. This chapter describes several well-established experimental techniques that are widely used in the study of SOFCs. Key words: electrochemical impedance spectroscopy, galvanic current interruption, helium–oxygen shift, fuel sensitivity, fuel loss.

9.1

Electrochemical impedance spectroscopy

Electrochemical impedance spectroscopy (EIS) stands out among all other electrochemical techniques for SOFC applications since information regarding ohmic losses, electrochemical kinetics, and mass transfer processes can be individually characterized in a single experiment, in remarkable contrast to the traditional DC technique. The principle of this technique is to perturb the electrochemical cell with an AC signal of small magnitude and to observe the ways in which the system follows the perturbation at steady state. The advantages of this technique include: (a) the ability to make high-precision measurements because the response may not be perfectly steady and can therefore be averaged over a long term; (b) the ability to treat the response theoretically by linearizing current-potential characteristics; and (c) measurement taken over a wide frequency range (10–4 to 106 Hz), in which processes that possess various time constants can be studied. Modern EIS measurements can be performed in either a frequency domain with a frequency response analyzer (FRA) or in a time domain by using Fourier transformation of a spectrum analyzer. The basic principle of applying an FRA to the study of an electrochemical cell is shown in the block diagram of Fig. 9.1. The FRA generates a voltage signal e(t) = ∆E sin(ω t) to the potentiostat. This will add to any existing DC bias EDC and is applied to the cell in a sum. The resulting current I(t), more precisely a voltage signal proportional to the current, is fed back to the analyzer, mixed with the input signal, and integrated 156

Performance characterization techniques

Frequency generator

157

Frequency response analyzer

Multiplier

Integrator

×

∫

Multiplier

Integrator

×

∫

cos(wt)

Zim

π/2 Output

0

Zre

sin(wt)

I (t) Potentiostat

e (t) EDC CE

Ref.

WE

9.1 Diagram illustrating the working principle of the modern FRA for an EIS measurement. CE, counter electrode; Ref., reference electrode; WE, working electrode; Zim, the imaginary part of the impedance; Zre, the real part of the impedance.

over several signal periods to yield signals that are proportional to the real and imaginary parts of the impedance (or equivalently the magnitude and phase angle of the impedance). A plot of the real part against the imaginary part is commonly called the impedance spectrum.

9.1.1

Application to a symmetrical solid electrolyte system

The early EIS measurements were primarily performed on liquid electrochemical systems. The real application of the EIS to solid-electrolyte systems was first introduced in 1969 by J. E. Bauerle,73 who demonstrated

158

Solid oxide fuel cell technology

that the impedance spectrum on a complex plane of a typical symmetrical cell, measured with a solid electrolyte and two identical electrodes in a single atmosphere, consists of distinctive contributions from the bulk and the grain boundary of an electrolyte and the electrodes (double layer effect) as the frequency of the AC signal is swept from high end to low end. A representative impedance spectrum (also known as a Nyquist plot) obtained from such a symmetrical cell is shown in Fig. 9.2, where Rb and Rgb are the bulk and grain-boundary resistances of the electrolyte, respectively. Re is the electrode resistance (including concentration and activation polarizations). The above impedance spectrum is commonly interpreted in connection with an equivalent electrical circuit (also known as Randles equivalent circuit) shown in Fig. 9.3. In this circuit, the two semicircles of the electrolyte process at high frequencies are represented by two resistances, Rb and Rgb, paralleled by two capacitances, Cb and Cgb (typically Cgb > Cb), respectively. The electrode process is, however, much more complex than the electrolyte counterpart. It depends largely on the material and microstructure of the electrode. A general model for describing the kinetics at the interface of the electrolyte/noble metal electrodes is the constant-phase-element (CPE) impedance. The analytical form of the CPE impedance ZCPE is expressed by

Z″ w increasing 1

Bulk Grain boundary

Rb

0

45° Rb + Rgb

2

Electrode process

Rb + Rgb + Re

Z′

9.2 A typical impedance spectrum representing a thick-film solid electrolyte symmetrical cell. Electrolyte

Electrode

Cb

Cgb

CPE

Rb

Rgb

Re

9.3 Equivalent electrical circuit representing a symmetrical thick-film solid electrolyte cell.

Performance characterization techniques

Z CPE (jω ) =

λ (jω ) α

159

[9.1]

where λ and α are real constants. With de Moivre’s theorem, equation [9.1] can be expressed in a more convenient form

απ απ Z CPE (jω ) =  λα  cos  – j λα  sin  ω  ω   2   2 

[9.2]

Figure 9.4 gives a graphical illustration of equation [9.2], where the angle α represents the degree to which the semicircle is being suppressed. In some special cases, as for semi-infinite diffusion, α = 0.5, and the ZCPE is simplified into the Warburg impedance ZW

ZW =

RT z F c O 2ωDO 2

2

(1– j)

[9.3]

where z is the number of electrons transferred in the electrode reaction; cO is the surface concentration of oxygen species (which can be either charged or neutral); DO is the diffusivity of oxygen species at the surface; and ω is the angular frequency. Under such circumstances, the impedance spectrum is shown by a straight line, intersecting the real axis by 45°, as represented by 햲 in Fig. 9.2. For a finite-length oxygen diffusion, for instance surface diffusion, α = 0.25, ZCPE becomes

ZD =

tanh(δ jω / Do ) RT 2 z F cO jω / Do

[9.4]

2

where δ is the diffusion distance of the oxygen species along the electrolyte/ electrode interface. The impedance spectrum becomes curved towards the real axis as illustrated by 햳 in Fig. 9.2. Z″

wmax to = 1 a p /2

u 0

v

(1 – a)p /2

Z′

o

9.4 Graphical representation of CPE on the complex plane.

160

Solid oxide fuel cell technology

Furthermore, the CPE would represent a resistor (R) for α = 0, a capacitor (C) for α = 1, an inductor (L) for α = –1, an R–C combination for 0 ≤ α ≤ 1, an R–L combination for –1 ≤ α ≤ 0, and an R–C–L combination for –1 ≤ α ≤ 1. Therefore, CPE is the most versatile element used in equivalent circuits of EIS studies of SOFCs, particularly thin-film-based SOFCs where dielectric loss of the electrolyte becomes appreciable. For an electrode-supported thin-film electrolyte system, performing EIS measurements in a single atmosphere and interpreting the results require particular caution since the symmetry requirement of the impedance cell is difficult to satisfy in most circumstances. The interfacial condition of an electrolyte/working electrode and an electrolyte/counter electrode may vary greatly even though the electrode material remains the same. As a consequence, the measured impedance spectrum includes indistinguishable contributions from two different electrode/electrolyte interfaces, making the results ambiguous. However, non-symmetrical impedance cells find their utility in screening the property of materials. For instance, with a fixed counter electrode such as platinum, the impedance spectra can be measured on different working electrode materials (the substrate in this case), from which the substrate electrode materials with the lowest electrode resistance can be selected. Figure 9.5 shows an example of a cathode-supported thin-film electrolyte cell with platinum as the counter electrode. The overall impedance measured over a limited range of frequency (0.1–105 Hz) is dominated by the electrode process owing to an insignificant contribution from the thin electrolyte. The electrode process consists of two sub-processes: charge transfer (activation polarization) and oxygen diffusion (concentration polarization through the cathode substrate). A good electrical equivalent circuit to simulate the spectrum is given in Fig, 9.6 where two CPE elements are used in parallel with the resistances corresponding to each process. Another precaution needed for conducting EIS measurements is the application of a superimposed DC bias to the small AC signal. Applying a Z″

w increasing Electrode process 1 2

0

Rohm

Rct + Rohm

Rdiff + Rct + Rohm

Z′

9.5 A typical impedance spectrum representing a thin-film solidelectrolyte cell.

Performance characterization techniques CPE

161

CPE

Rohm Rct

Rdiff

9.6 Equivalent electrical circuit representing a thin-film solidelectrolyte cell.

RE1 Anode

CE

GPIB 1287 PST

Solid electrolyte Cathode

1252 FRA RE2

WE

9.7 Experimental set-up for an EIS measurement in the study of SOFCs. PST, potentiostat; GPIB, general-purpose interface bus.

DC bias to the symmetrical solid-electrolyte cell would force one electrode to undergo the oxygen reduction reaction whereas the other electrode would undergo the oxidation reaction. The resulting impedance spectrum would include the two simultaneous, opposing electrode reactions, making it difficult to discern whether the two concurrent electrode reactions exhibit different impedance behaviors. However, just as with the non-symmetrical cell discussed above, DC-biased EIS measurements can be used as a screening tool for material selections. The modern EIS measurements are performed with commercially available FRAs and potentiostats. One renowned brand name is the Solartron 1250 series FRA and 1280 series potentiostat. The model 1252 FRA is designed for low-frequency application, capable of sweeping the frequency from 6.5 µHz to 100 kHz with adjustable AC amplitude. The 1287 potentiostat can supply up to 12 V or 2 A DC bias. Of the four cables used, two are connected to counter and working electrodes (CE and WE, respectively) for carrying current and the other two are connected to the reference electrode 1 (RE1) and reference electrode 2 (RE2) for sensing the voltage. Figure 9.7 shows a diagram of the equipment set-up. Along with the equipment, the software is also a powerful tool for data acquisition and processing. ZPLOT is the application program that controls the measurement in the perspectives of frequency sweeping, signal mode (galvanostatic or potentiostatic), AC amplitude, DC bias level, integration time, and so on. The acquired data are displayed in situ in the ZVIEW program in the form of Complex (Z′–Z″) or

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Solid oxide fuel cell technology

Bode (|Z|–f and f–θ) modes. More importantly, ZVIEW also contains a variety of equivalent circuits that allow users to analyze and simulate the measured spectrum from which the desired electrochemical parameters – such as resistance, capacitance, and inductance – can be extracted. However, the model adopted has to be physically meaningful in order to obtain a unique solution of the simulation since many equivalent circuits can lead to a good fit of the data but have no meaning in physical applications.

9.1.2

Application to a single solid oxide fuel cell

One of the most important applications of the EIS technique is to measure the impedance spectrum of a complete SOFC under an operating condition, from which the ohmic, anode, and cathode resistances can be determined. The instrumentation set-up is no different from that for the symmetrical solid-electrolyte cell. In order to simulate a real SOFC operating condition, the potentiostat applies a galvanic DC current in the same direction as in the SOFC operation during frequency sweeping. Since the time constants for anode and cathode processes are very different, the measured spectrum generally features two discernible semicircles (or arcs) within the frequency range studied. Figure 9.8 gives an example of the impedance spectrum produced from a cathode-supported SOFC under a DC galvanic mode. As is clearly seen, the anode process occurs at a higher frequency range whereas the cathode process appears at a lower frequency range, indicating that the cathode

–0.50 Cathode-supported SOFC T = 900 °C H2O/H2 = 1/1 i = 270 mA/cm2

–0.40

Z ″ (Ω cm2)

–0.30

–0.20

–0.10

Anode 15.69 kHz 415 Hz

Cathode 0.03 Hz

1.44 Hz

0.00 23.90 kHz 0.10 0.40

0.50

0.60

0.70

0.80

0.90

1.00

Z ′ (Ω cm2)

9.8 A typical impedance spectrum of a cathode-supported SOFC.

Performance characterization techniques

163

polarization is a slower process. The high-frequency intercept with the Z′axis represents the total ohmic area-specific resistance of the cell; the tail stretched into the positive Z″ domain is caused by the inductance of connecting leads. Identification of the anode and cathode processes requires a study of impedance response to the change in either oxidant or fuel. Figure 9.9 shows an example of impedance responses to the changes from air to He–20% O2 mixture and from air to pure oxygen. Clearly, only the second semicircle has responded, implying that it is related to the cathode process.

9.2

Galvanic current interruption

The galvanic current interruption (GCI) method is another way to measure the internal resistance of an electrochemical cell. It is particularly useful in circumstances where ohmic voltage losses (IR) are significant. The working principle of the method is to rely on a very fast switch to momentarily opencircuit the cell and measure the cell voltage instantly after the interruption. Ohmic IR losses vanish almost instantaneously whereas polarizations associated with activation and concentration processes take a much longer time to equilibrate. In operation, a constant current is applied for a sufficiently long time through the cell to achieve a steady-state condition. The current is then interrupted with a fast switch and the voltage is recorded with a storage oscilloscope or a transient recorder within microseconds of the current interruption. Figure 9.10 illustrates a block diagram of a typical configuration for the GCI method. Because of the very small time constant associated with the electrolyte processes, the voltage decays almost instantaneously (usually

Imaginary part (Ω cm2)

–0.1

Air –0.05

0 O2

He–20% O2 ∆R cconc

0.05 0.25

0.3

0.35

0.4

0.45

0.5

Real part (Ω cm2)

9.9 Identification of anode and cathode processes in the impedance spectrum.

164

Solid oxide fuel cell technology Constant current source WE Solid electrolyte

Electronic switch

Control computer

RE CE

High-speed voltmeter or oscilloscope

9.10 Block diagram of the GCI method with a three-electrode configuration. I=0 Voltage

IR

h

0

Time (ms)

9.11 A schematic of the voltage response from an SOFC to a galvanostatic transient to zero (open circuit); η is the polarization voltage loss.

in less than 0.1 µs) for the ohmic IR losses and no trace on the oscilloscope or the data is recorded, as illustrated by Fig. 9.11. The slow part of the transient for which trace or data are recorded corresponds to polarization losses (non-ohmic part) associated with the electrolyte/electrode interfaces. Note that the instantaneous IR drop after the current interruption should be comparable to Rb + Rgb extracted from the spectrum shown in Fig. 9.2. Depending on the relative difference between the time delay setting on the oscilloscope and the real time constants of the processes, it is often observed that the instantaneous IR drop is higher than Rb + Rgb, possibly owing to the involvement of the charge transfer, another process with a very small time

Performance characterization techniques

165

constant. Therefore, the EIS method, when compared with the GCI, is better suited for obtaining more accurate resistance values for SOFCs.

9.3

Voltage–current curve characterizations

A typical V–i curve of an SOFC can be measured under either a galvanic or a potentiostatic mode. Figure 9.12 illustrates the basic operating principle of measuring the V–i curve of an SOFC. The load bank with a variable resistor provides a constant current passing through the SOFC while the cell voltage is measured. In many cases, the total resistance of the SOFC is much smaller than that of the power leads and connectors; this situation requires a higher voltage to generate the current, particularly for a higher current. This requirement can be accomplished by adding a power booster in series with the SOFC. The typical voltage range for the power booster is between 2 and 5 V, capable of handling up to hundreds of amperes. Higher voltage may potentially damage the SOFC. The characterization of a V–i curve is straightforward. Prior to setting the current, the fuel flow rate and air flow rate must be predetermined according to equations given in Chapter 6 for a desirable fuel utilization and air utilization. The thus-obtained V–i curve represents an iso-Uf and iso-Uo case, which is analogous to the practical operation of an SOFC generator. A representative V–i curve measured under an isothermal and an iso-utilization condition is given in Fig. 9.13; the cell voltage is observed to become flattened out at low current density and come close to the electromotive force (EMF) reflecting the fuel composition of the exit as the cell current approaches zero. The fundamental reason behind this behavior has been discussed in Chapter 7. In contrast, the V–i curve with a near-zero fuel utilization would intersect at the SOFC

Vc

i Power booster

Load bank

9.12 A schematic of an electrical circuit for measuring V–i curves of an SOFC.

166

Solid oxide fuel cell technology Cathode-supported SOFC T = 900 °C, Uf = 85%, Ua = 16% Inlet fuel: 89% H2-11% H2O

0.800

0.750

Vc (V)

0.700

0.650 0.600

0.550 0.500 50

100

150

200

250 300 i (mA/cm2)

350

400

450

9.13 A representative V–i curve measured under isothermal conditions with iso-fuel and iso-oxygen utilizations.

voltage reflecting the fuel composition of the inlet when the cell current becomes zero. This situation is commonly observed in laboratory-scale studies where no fuel utilization is considered owing to the leakage and smaller sample size. In the intermediate current density region, the V–i curve features a straight line, implying domination of the ohmic IR loss. At a higher current density, the cell voltage falls rapidly, indicating that the limitation of mass transport of either reactants or products in the electrodes comes into play.

9.4

‘Helium–oxygen shift’

The method of ‘helium–oxygen (He–O2) shift’ is aimed at determining the concentration polarization resistance of a cathode. The principle of this method is based on the fact that the O2 diffusivity in an He–O2 mixture is much greater than that in an N2–O2 (air) mixture. Therefore, switching from air to a mixture of He–O2 gas at the cathode should reduce the concentration polarization resistance markedly as a result of an enhanced O2 diffusivity. Selection of 20% O2 in the He–O2 mixture eliminates the effect of PO 2 on activation polarization. Furthermore, the PO 2 effect on the electrical conductivity of the cathode and the ceramic interconnect, which contributes to the ohmic resistance, can also be avoided. Therefore, the change induced by the He–O2 shift reflects only the change in the concentration polarization of the cathode. This method has been proven very effective in determining the concentration polarization of a cathode in cathode-supported SOFCs.

Performance characterization techniques

9.4.1

167

Theory

In equation [7.51], the concentration polarization area-specific resistance Rcconc of a cathode is shown at a given T and X O 2, as a function of i and iLM; the latter varies with the effective oxygen diffusivity DOeff2. For simplicity, DOeff2 is expressed as the product of the molecular DO 2 and a coefficient m, a factor that is strongly dependent on the microstructure of a cathode substrate (m = ε /τ) DOeff2 = m × DO 2 = ε DO 2 τ

[9.5]

The values of DO 2 as a function of temperature and gas component can be readily calculated from equation [7.31] and [7.32] and are listed in Appendix 4 for some commonly used binary gas systems in SOFC applications. When equation [7.51] is applied to air, a mixture of roughly 20% O2 and 80% N2, the concentration resistance, Rcconc (air), can be further written as Rcconc (air) = A ×

1 i LM,air × (1 – (1 – X O 2 ) ( i / iLM,air – 1) )

A = RT ln (1 – X O 2) 4F

[9.6] [9.7]

where iLM,air is given by i LM,air = B × m × DO 2 air

B= –

ln(1 – X O 2) × 4 F × Pt RTr2 r ln  2   r1 

[9.8] [9.9]

for a cylindrical geometry with outer and inner radii of r2 and r1, respectively, and B = – ln(1 – X O 2) × 4 F × Pt RTt c

[9.9a]

for a one-dimensional diffusion through a thickness tc of the cathode wall. Similar equations can also be written for the 20% O2–He mixture Rcconc (O 2 He) = A ×

1 1 × i LM,O 2 He 1 – (1 – X O 2)( i / iLM,O 2 He – 1)

i LM,O 2 He = B × m × DO 2 He

[9.10] [9.11]

The difference ∆Rcconc between Rcconc (air) and Rcconc (O2He) can then be expressed as

168

Solid oxide fuel cell technology 0.500 0.450

T = 1000 °C i = 300 mA/cm2

Rcconc (air) (Ω cm2)

0.400 0.350 0.300 0.250 0.200 0.150 0.100 0.050 0.000 0.000

0.100

0.200

0.300

0.400

0.500

∆R cconc (Ω cm2) (a)

9.14 Cathode R cconc in air as a function of ∆ R cconc caused by ‘He–O2 shift’: (a) 1000 °C, 0.30 A/cm2; (b) 950 °C, 0.27 A/cm2; (c) 900 °C, 0.25 A/cm2.

∆Rcconc

 1 1 =A×  1 × – ( i / i LM,air –1) i i LM,O LM,air X 1 – (1 – ) 2 He  O2 ×

1 1 – (1 – X O 2) ( i / iLM,O 2 He –1)

  

[9.12]

The significance of equation [9.12] is that the microstructural factor, m, can be first determined by the measurable ∆Rcconc. The attainment of m can then be utilized to calculate the absolute Rcconc (air), which is a direct measure of gas-diffusion properties. Owing to the complexity of the analytical expression shown in equation [9.12], a graphical determination of Rcconc (air) is adopted for convenient use. Figures 9.14(a), (b), and (c) show the plots of Rcconc (air) of a cylindrical cell as a function of ∆Rcconc under various conditions. For accuracy, tabulated values shown in these figures are also given in Appendix 3. For other operating conditions, similar tables can also be calculated out based on equation [9.12].

9.4.2

Application to cell testing

The key to the ‘He–O2 shift’ is to measure the value of ∆Rcconc. This can be done either by measuring the change in cell voltage under a constant current

Performance characterization techniques

169

0.500

T = 950 °C i = 270 mA/cm2

0.450 0.400

Rcconc (air) (Ω cm2)

0.350 0.300 0.250 0.200 0.150 0.100 0.050 0.000 0.000

0.100

0.200

0.300

conc ∆R c (Ω

0.400

0.500

cm2)

(b) 0.500 0.450

T = 900°C i = 250 mA/cm2

Rcconc (air) (Ω cm2)

0.400 0.350 0.300 0.250 0.200 0.150 0.100 0.050 0.000 0.000

0.100

0.200

0.300

conc ∆R c (Ω

0.400

0.500

2

cm )

(c)

9.14 Continued.

density i or by directly measuring the resistance change of the cathode with the EIS method described earlier after air is switched to 20% O2–He mixture. As there are no additional contributions from EMF, the activation polarization, and ohmic IR loss, the measured voltage change ∆Vc obtained by the former method, is a true representation of the voltage difference caused by ∆Rcconc ∆ Rcconc

=

∆ Vc

i

[9.13]

170

Solid oxide fuel cell technology

For example, if measured ∆Vc = 30 mV at T = 1000 °C and i = 300 mA/cm2, 2 ∆Rcconc = 0.100 Ω cm . From the tables in Appendix 3, Rcconc (air) is found to be approximately 0.116 Ω cm2. The corresponding effective porosity ε is calculated to be approximately 24%. From the EIS method, ∆ Rcconc can be directly measured from the spectrum change before and after the ‘He–O2 shift’. Figure 9.9 gives an example of the impedance spectrum change of a cathode-supported cell after the ‘He–O2 shift’. It not only gives ∆ Rcconc, but also identifies that the second arc at low frequency is related to the concentration polarization process of the cathode.

9.5

Fuel sensitivity

The fuel sensitivity β is defined as the change in cell voltage, ∆Vc, with respect to the change in fuel utilization, ∆Uf, under a constant temperature and current. Mathematically, β = (∂Vc/∂Uf)T,i . β is a proper indicator of the fuel losses that would directly affect the electrical efficiency and performance of an SOFC. It is, therefore, perceived as a valuable tool for the evaluation of fuel losses in a cell or a stack. This section describes the use of β methodology to gauge the leakage current/flux indirectly.

9.5.1

Theory

A generic relationship of β with Uf and leakage flux QL can be derived from the basic equation of energy conservation within an SOFC Vc ( i ) = E ( i ) – iRohm –

∫

i

0

Rc ( i ) di –

∫

i

Ra ( i ) di

[9.14]

0

where cell voltage, Vc(i), cell EMF, or E(i), cathode area-specific resistance Rc(i), and anode area-specific resistance Ra(i) are all considered to be currentdependent. The only current-independent term is the ohmic area-specific resistance Rohm. It is obvious that the differentiation of Vc(i) with regard to Uf under a constant T and i depends on how E(i) and Ra(i) vary with Uf. In what follows, the fundamental relationships are derived individually. The true E(i)–Uf relationship essentially depends on the E(i)–i correlation. In Chapter 8, it is stated that E(i) becomes the EMF reflecting the fuel composition of the exit when the current approaches zero. At higher current density (>200 mA/cm2),* the average representative E can be considered as current-density-independent and can be approximated by equation [7.67].

*

This assumption mandates that fuel sensitivity measurement be conducted under a higher current density.

Performance characterization techniques

171

For H2–H2O fuel, Ein and Eex are rewritten from equation [2.10] with the mass flow rate Q replacing partial pressures of H2 and H2O

 QHo O  E in = 1275 – 0.3171T – 0.0430 T ln o2   QH 2 

(mV)

[9.15]

 QHo 2 O + U f QHo 2 + QL  E ex = 1275 – 0.3171T – 0.0430 T ln  (mV) o  QH 2 (1 – U f ) – QL  [9.16] which leads to –1   ∂E ( i )  RT ×   r ×  1 + QL  + U    = – f   4 F   6.97 × I   ∂U f  T, i  

–1

–1    QL  –1 + 1+ – Uf   6.97 × I    

[9.17] where r = PHo2 O / PHo2 = QHo 2 O / QHo 2 . While Rc(i) is independent of Uf, the variation of Ra(i) with Uf is real and not simple. In a simpler case, Ra(i) can be considered as a current-independent constant, which is often a good estimate for cathode-supported SOFCs owing to its insignificant contribution. Therefore

 ∂Raact ( i )   ∂R conc ( i )   ∂R act ( i )   ∂Ra ( i )  [9.18] +  a ≈  a   =     ∂U f  T, i  ∂U f  T, i  ∂U f  T, i  ∂U f  T, i and  ∂Raact ( i )   ∂Raact ( i )   ∂iex  RT 1  ∂iex   ∂U  =  ∂i  ×  ∂U  = – 2 F 2 ×  ∂U  iex   T, i   T, i ex f f T, i f T, i

[9.19]

Raact

It is assumed that there is a linear relationship between and iex. The correlation of iex–Uf can be derived from equation [7.30] by substituting PH 2 O = PHo2 O + U f × PHo2 + QL /( QHo 2 + QHo 2 O ) and PH 2 = PHo2 × (1 – U f ) – QL /( QHo 2 + QHo 2 O ) into it, from which equation [9.19] can be further elaborated into the following expression with assumption of ( PH 2 / PH* 2 ) << 1

172

Solid oxide fuel cell technology

iH* 2  ∂Raact  3  ∂U  = 16 ×  ( PH* 2 )1/4 f  T, I –3/4 Uf QL  ×  PHo2 (1 – U f ) – × 1 + r 6.97 × I  

U f  –1/4 QL ×  PHo2 O + U f PHo2 + ×  1 + r 6.97 × I  2 QL 1  ×  PHo2 + ×  1 + r 6.97 × I 

[9.20]

Where I is the cell current. The system pressure Pt is assumed to be the atmosphere in the above deduction. The overall analytical expression of β is therefore given by the combination of equations [9.17] and [9.20]  ∂V  β= c  ∂U f  T, I –1 –1 –1 –1       QL  QL   RT = – + U f  +  1+ – Uf   × r × 1+ 4 F    6.97 × I   6.97 × I       

iH* 2 U f  –3/4 QL + 3 × ×  PHo2 (1 – U f ) – × * 1/4 1 + r 6.97 × I  16 ( PH 2 )  U f  –1/4 QL ×  PHo2 O + U f PHo2 + ×  1 + r 6.97 × I  2 QL 1  ×  PHo2 + ×  1 + r 6.97 × I 

[9.21]

A graphical representation of equation [9.21] is given in Fig. 9.15, where T = 1000 °C, i = 0.30 A/cm2, and Acell = 270 cm2. Four levels of fuel-loss values are chosen to calculate the fuel sensitivity β as a function of Uf. As expected, β becomes more sensitive to Uf with a higher fuel loss. For an even simpler case, ( ∂Raact / ∂U f ) T, i = 0 is assumed. Equation [9.21] is simplified into equation [9.17]. The importance of equation [9.21] is that it provides a methodology for obtaining the fuel loss from a measured β–Uf curve. However, the QL value obtained by this method is closely related to i and T. In other words, a QL– i–T relationship can be obtained if QL is measured under different i and T conditions. In order to extract the QL value accurately, a non-linear leastsquares fitting program can be used to fit the experimental data, i.e. the β–

Performance characterization techniques 18.00

T = 1000 °C QL = 30 sccm H2 i = 0.3 A/cm2 Acell = 270 cm2 O PHO = 0.89 atm, PH O = 0.11 atm 2 2 r = 0.11/0.89 C = 3/16 × (iH* )/(PH )1/4 = 0.15 20

16.00 14.00 12.00

–b (mVl %)

173

2

10.00

2

8.00

10 0

6.00 4.00 2.00 0.00 55

65

75 Uf (%)

85

95

9.15 Fuel sensitivity β as a function of Uf calculated with equation [9.21]. C is a constant.

Uf curve, defined by equation [9.21], with QL being the only adjustable parameter. The QL value that yields the least standard deviation is considered to be the choice of QL for the given T and i.

9.5.2

Applications to practical measurements

In order to validate the method further, equation [9.21] is used to fit the experimental data. Figure 9.16 shows an example of the measured β–Uf curve (diamonds) of a cathode-supported cell with a non-linearly fitted curve (line). It is evident that the measured values agree well with the calculated values. The best-fit QL and constant C are 39.68 ± 1.20 sccm H2 and 0.155 ± 0.045, respectively, for a 50 cm-long cell under conditions of T = 1000 °C and i = 0.30 A/cm2.

9.6

Determination of fuel loss

A knowledge of fuel loss is critical to the healthy operation of an SOFC generator, particularly during start-up and shut-down periods when a limited amount of fuel is present. An overlooked fuel loss during these transient periods could easily lead to a depletion of fuel, causing the anode to be oxidized. As shown in Chapter 8, the fuel loss also impacts the maximum electrical efficiency achievable by an SOFC. The sources of fuel losses are physical (molecular) O2 leakage, fuel bypass, and most importantly the electronic leakage in the electrolyte and ionic leakage in the ceramic interconnect, as discussed in Chapter 4. Owing to the fact that molecular O2

174

Solid oxide fuel cell technology

16

T = 1273 K i = 0.30 A/cm2 Acell = 270 cm2 PHO = 0.89 atm, PHOO = 0.11 atm 2 2 r = 0.11/0.89

14

–b (mV/ %)

12 10 8 6

QL = 39.68 ± 1.20 sccm H2 C = (3/16) × (iH* )/(PH*)1/4 = 0.155 ± 0.045 2

2

4 2 0 65

70

75

80

85

90

95

Uf (%)

9.16 Fuel sensitivity β as a function of Uf measured on a cathodesupported SOFC and calculated with equation [9.21].

leakage flux in the cell and the rate of fuel bypass in the generator are generally low, the major contribution to fuel loss is the chemical leakage from the two dense layers (one dense layer if a metal interconnection is used) in an SOFC. This section describes several experimental methods to measure fuel losses directly; they differ from the fuel-sensitivity approach mentioned previously.

9.6.1

Theory

The key to determining fuel losses over the entire surface of an SOFC is to determine the fuel composition at the exit, provided that the inlet fuel composition is known. In what follows, a mixture of H2, H2O, CO, and CO2 is taken as an example for illustrating the principle of the method. Let o o QHo 2, QHo 2 O , QCO , and QCO be the corresponding mass flow rates of fuel 2 components at the inlet, and QH 2, QH 2 O , QCO , and QCO 2 be the corresponding mass flow rates at the outlet. Under an operating condition, either opencircuit voltage (OCV) or loading, the total fuel loss, QL, is given by o QL = ( QHo 2 + QCO ) – ( QH 2 + QCO )

[9.22]

If X H 2, XCO, and X CO 2 represent the dry-gas (water free) fuel compositions at the exit measured by the gas chromatography (GC) method (NB, the GC

Performance characterization techniques

175

method is not adequate for measuring the water content) and assuming the total dry-gas mass flow rate at the exit is Qt′, the mass flow rates of individual dry-fuel components at the exit are expressed by

QH 2 = X H 2 Qt′

QCO = X CO Qt′

QCO 2 = X CO 2 Qt′

[9.23]

According to the mass balance of carbon, the following relation holds for Qt′ Qt′ =

o o + QCO QCO 2 X CO 2 + X CO

[9.24]

Combining equations [9.22], [9.23], and [9.24] leads to QL

 X H 2 + X CO  o o o QL = ( QHo 2 + QCO )–   × ( QCO 2 + QCO )  X CO 2 + X CO 

[9.25]

Another simple way of determining QL is based on the measurement of the OCV of a cell, from which the ratio r of oxide-to-fuel can be readily determined by equation [2.10], rearranged as follows with H2–H2O as the fuel r=

PH 2 O QH 2 O 1275 – OCV – 0.3171T  = = exp  PH 2 QH 2 0.043T  

[9.26]

where OCV is in mV. From the H2 mass balance of QHo 2 + QHo 2 O = QH 2 + QH 2 O QL =

rQHo 2 – QHo 2 O 1+r

[9.27]

Note that the above method is limited to the OCV condition only. In Chapter 4, it is stated that the fuel loss or leakage current usually decreases with the loading current. Therefore, use of QL obtained from the OCV method to compensate the fuel loss during SOFC operation is a conservative approach. The universal measurement of QL is by chemical analysis such as by GC of the exit-gas composition as described above. In addition, any potential molecular leakage can also be detected in the form of N2 by GC and corrected correspondingly for O2. Two examples are given to illustrate the utility of the above equations in the following context.

9.6.2

Applications

The first example utilizes a GC measurement of the exit-gas composition. Figure 9.17 illustrates the experimental set-up, where a small fraction of exit gas is sampled to an on-line GC. In such a configuration, the analysis can take place under any operating condition. For an inlet fuel composition containing 42.7% H2, 18.5% CO, 22.0% H2O, and 16.8% CO2, Fig. 9.18

176

Solid oxide fuel cell technology

Air Sampling tube Cooler Pump

Vent

GC Cell

H2, CO2, CO, H2O

9.17 Schematic of the experimental set-up for a GC measurement during tubular cell testing. 90 80

QL (sccm H2)

70 60 50 40 30 20 10 0 0.55

T = 1000 °C i = 100 mA/cm2 Cell area = 830 cm2 0.60

0.65

0.70

0.75

0.80

Uf

9.18 Plot of GC-derived QL as a function of Uf at 1000 °C and i = 100 mA/cm2.

shows the measured QL as a function Uf at 1000 °C. As expected, QL decreases with Uf as a result of a decreased PO 2 at the fuel/electrolyte interface. Another example demonstrates how to use the OCV method to determine QL. Figure 9.19 shows the measured OCV of an SOFC with a surface area of 270 cm2 as a function of temperature under the flow condition of 25 sccm H2

Performance characterization techniques 12

1175

11 H2 loss

1125

Cell voltage

10

1100

9

1075

8

1050

7

1025

6

1000

5

975

4

950

3 2

925 900

Hydrogen loss (sccm)

1150

Cell voltage (mV)

177

Cell area: 270 cm2

875 600

1 0

650

700

750 800 850 900 Average cell temperature (°C)

950

1000

9.19 Plot of H2 loss as a function of temperature, measured by the OCV method.

and 475 sccm N2. Using equation [9.27], QL is calculated from the measured OCV values as a function of temperature and is shown on the second y-axis of Fig. 9.19. Clearly, the fuel loss decreases exponentially with decreasing temperature. Therefore, an effective way of suppressing fuel loss resulting from ionic and electronic leakage currents is to lower the operating temperature of an SOFC. Note that an iso-potential distribution along the electrode is required to ensure an accurate measurement for a cell with a large surface area. In order to achieve this, a large bus bar made of Ni metal is typically used as a voltage equalizer during the measurement. As the majority of fuel loss occurs at the fuel entrance under iso-potential conditions, QL measured by this method represents an average value over the entire surface area.

9.7

Electrical conductivity and sheet resistance measurements

Electrical conductivity is undoubtedly one of the most important properties of SOFC materials. High electrical conductivity is always desirable in selecting SOFC materials for the purpose of minimizing the resistance. The electrical conductivity of a material with a well-defined geometry, such as a rectangular or a cylindrical bar, is commonly measured by a four-probe configuration as shown in Fig. 9.20, where the two current leads are separated from the two voltage ones. According to Ohm’s law (equation [7.2]), the voltage drop across a given length of the sample that is subject to a constant current is proportional to the resistance of the sample. The measured resistance value

178

Solid oxide fuel cell technology V

I

DC I

A

L

9.20 Schematic of four-probe DC conductivity measurement. Probes V (+) V (–)

I (+)

Centerline

S

S

I (–)

S

d

a

9.21 Schematic of four-probe sheet resistance measurement.

is then converted to electrical conductivity by the known geometrical dimensions of the sample. The separation of voltage and current probes is necessary since it eliminates the interference from the lead resistance, which is particularly important when the studied object has a very small resistance. In order to study the effect of temperature and atmosphere on electrical conductivity, the sample can be mounted on a fixture which enables the whole testing rig to be inserted into a sealed muffle tube located in a hightemperature furnace. The sheet (or in-plane or surface) resistance (defined in Chapter 7) measurement, Rs, uses a slightly different electrode configuration. Figure 9.21 shows a standard set-up for measuring the sheet resistance of a thin film. The four probes, two for the current and two for the voltage, are aligned and separated by a defined equal space of S. The directly measured resistance R = ∆V/I is not equal to Rs, but is related to Rs with a correction factor C Rs = C × ∆V I

[9.28]

Assuming that the length and width of a thin film are a and d, respectively, Smits74 tabulated C values for different a, d, and S as values shown in Table

Performance characterization techniques

179

Table 9.1 Correction factor C for the measurement of sheet resistance with the four-point probe d/S

1.0 1.25 1.5 1.75 2.0 2.5 3.0 4.0 5.0 7.5 10.0 15.0 20.0 40.0 ∞

Circle diameter d/S

2.2662 2.9289 3.3625 3.9273 4.1716 4.3646 4.4364 4.5076 4.5324

a/d = 1

2.4575 3.1137 3.5098 4.0095 4.2209 4.3882 4.4516 4.5120 4.5324

a/d = 2

a/d = 3

a/d ≥ 4

1.4788 1.7196 1.9454 2.3532 2.7000 3.2246 3.5749 4.0361 4.2357 4.3947 4.4553 4.5129 4.5324

0.9988 1.2467 1.4893 1.7238 1.9475 2.3541 2.7005 3.2248 3.5750 4.0362 4.2357 4.3947 4.4553 4.5129 4.5325

0.9994 1.2248 1.4893 1.7238 1.9475 2.3541 2.7005 3.2248 3.5750 4.0362 4.2357 4.3947 4.4553 4.5129 4.5324

9.1. To give an example, if the measured R of a sheet conductor with a = 180 cm and b = 5.2 cm is 0.01 Ω, according to the table, the correction factor C ≈ 3.575. The real Rs, therefore, equals 3.575 × 0.010 = 0.035 75 Ω/SQ.

9.8

Determination of porosity and density of a porous ceramic body

As discussed in Chapter 7, the porosity in electrodes is critical in determining the concentration polarization loss. A greater porosity generally leads to a lower concentration polarization. However, the degree of porosity in the electrode of structural support must also be optimized to provide sufficient mechanical support for the deposition of subsequent layers. Therefore, a balance between electrical performance and mechanical strength has to be retained for electrode-supported SOFCs. In this section, a fast, accurate, inexpensive, and non-destructive experimental method is described for the determination of the porosity of a sintered porous ceramic body. The basis of the method is Archimedes’ law, in which the volume of a studied sample with an arbitrary shape can be measured by either the displacement of the liquid after the sample is submerged or by weighing the submerged sample.75 In practice, the approach includes a three-step weighing procedure as illustrated in Fig. 9.22. The first step is to measure the true mass M1 of the sample in air, followed by a full degassing of the sample before submersing into a liquid with a known density ρliq. After a complete impregnation of the liquid, the sample is again weighed in the suspension in

180

Solid oxide fuel cell technology M2

M1

Dry sample

Liquid rliq

M3

Wet sample

Suspension

9.22 Three-step weighing procedure to determine the density and porosity of a porous ceramic body.

the liquid as M2; the sample is then removed from the liquid and wet surface carefully blotted with a damp cloth to obtain another weight M3. The sample volume Vs and pore volume Vp are then calculated by M3 – M2 ρliq

Vs =

Vp =

M 3 – M1 ρliq

[9.29]

The porosity ε of the porous sample is then given by

ε=

Vp M – M1 = 3 Vs M3 – M2

[9.30]

In addition, the bulk density ρb and true density ρt of the sample can be further calculated by

ρb =

M1 × ρliq M3 – M2

ρt =

ρb M1 = × ρliq M1 – M 2 1–ε

[9.31]

As an example, M1, M2, and M3 of a porous ceramic body are weighed as 20, 10, and 25g, respectively. According to equation [9.30], the porosity is 33.3%. If the liquid used is pure water, the bulk and true densities are calculated based on equation [9.31] to be 1.33 and 2.00 g/cm3, respectively.

9.9

Diffusivity and diffusion conductance measurements

The combination of molecular diffusivity with porosity and tortuosity leads to the effective diffusivity, a parameter that determines the concentration polarization loss of an electrode. If the electrode is the substrate of an SOFC, a direct knowledge of the diffusivity of the diffusing species becomes even more important since it allows the processing of the candidate electrode materials to be optimized in microstructure and serves as a quality tool for production. In this section, we describe a simple methodology to measure

Performance characterization techniques

181

directly the transport properties of a porous substrate. As an example, a cathode substrate with a cylindrical tubular geometry is used for measuring the oxygen diffusion process. Experimentally, one can easily measure the diffusivity of the tubular geometry shown in Fig. 9.23. The net oxygen flux J O 2 (in mol/s) across sectional area A (in cm2) of a tubular geometry is governed by Fick’s first law and is expressed in cylindrical coordinates

J O 2 = – DOeff2 × A × dC dr

[9.32]

As the oxygen diffuses from the inner surface of the tube to the outer surface, the cross-sectional area of the diffusion varies with r in the form of A = 2π × r × L

[9.33]

where L (in cm) is the length of the specimen. Substitution of equation [9.33] into equation [9.32], followed by integrating from r1 to r2, yields JO2 =

2 π × DOeff2 × L × ( C2 – C1 ) ln( r2 / r1 )

[9.34]

where the term 2π × DOeff2 [9.35] ln( r2 / r1 ) is defined as the diffusion conductance (in cm2/s). It is clear that K is associated with the geometry, and therefore is a better indicator of the diffusion capacity K=

R1 o R2

L

9.23 Tubular geometry for an oxygen diffusion conductance measurement.

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Solid oxide fuel cell technology

given by a specific geometry. It has more practical meaning and is directly related to the limiting current density iLM described in Chapter 7. According to equation [9.34], DOeff2 or K can be determined if J O 2, C2, and C1 (in mol/cm3) are known. Experimentally, it can be achieved by the ‘sweeping gas’ method. A schematic of the experimental set-up is shown in Fig. 9.24, where pure N2 at a flow rate of Q (in sccm) is used as the sweeping gas, which is led to an oxygen analyzer that measures the oxygen content C1 in the sweeping gas N2. The net oxygen flux (in mol/s) is then given by JO2 =

4 FE  P Q Q × C1 = × 2 exp  – 60 60 RT  RT 

[9.36]

where E (>0) and T are the voltage and temperature of the oxygen analyzer, respectively; P2 is the partial pressure of oxygen of the outer surface and is kept constant at 0.209 atm. Combining equations [9.34] to [9.36] leads to

K=

exp(– 4 FE / RT ) Q × 60 × L 1 – exp (– 4 FE / RT )

[9.37]

The measured K values can then be either converted into DOeff2 or plotted against the porosity ε values measured by the method described in the last section, from which a correlation between K and ε can be established. One precaution that has to be taken during the experiment is to balance the internal pressure with the ambient pressure outside the specimen tube so that the bulk oxygen flow can be minimized. One way to do this is to maintain a controlled vacuum at the exit of the oxygen analyzer. The balanced condition can be constantly monitored during the measurement with a differential pressure gauge as indicated in Fig. 9.24. Differential pressure gauge

Vacuum

Ambient air O2

O2 sensor

MFC Specimen tube

9.24 Experimental set-up for an oxygen diffusion conductance measurement. MFC, mass flow controller.

N2

10 Steam methane reforming and carbon formation in solid oxide fuel cells (SOFCs)

Abstract: This chapter discusses the thermodynamic equilibrium and kinetic mechanism of the steam methane reforming reaction. The equilibrium compositions and the cell electromotive force are calculated under various temperatures, steam methane ratios, and pressures. The kinetic mechanism of the steam methane reaction is also briefly mentioned. Carbon formation is discussed from the perspectives of thermodynamics and kinetics. Key words: steam/methane ratio, oxygen/carbon ratio, carbon activity, equilibrium composition, kinetics of carbon formation.

10.1

Introduction

Steam reforming of natural gas is a well-established, and also the leastexpensive, industrial process for producing commercial bulk hydrogen as well as the hydrogen used in the industrial synthesis of ammonia. The major advantage of the process is the ability to produce the highest yield of H2 of any reformer type. At high temperatures (700–1100 °C) and in the presence of a metal-based catalyst, often Ni, steam reacts with methane to yield CO and H2 by the reaction CH4 + H2O = CO + 3H2 ∆G

o

= 225 720 – 253.35T

[10.1] (J/mol)

[10.1a]

Additional H2 can be recovered by a lower-temperature gas-shift reaction with the CO produced via CO + H2O = CO2 + H2 ∆G

o

= –35 948 + 31.98T

[10.2] (J/mol)

[10.2a]

In an SOFC, the anode contains a large amount of Ni. The presence of Ni in the fuel environment makes reforming natural gas directly on the anode possible while an oxygen flux is constantly transported from cathode to anode under a load condition. As is shown in equation [10.1a], the steamreforming reaction is highly endothermic and needs a considerable amount of heat supply to sustain the reaction. In contrast, the anodic electro-oxidation of fuels is highly exothermic. Equation [2.13] describes quantitatively how 183

184

Solid oxide fuel cell technology

much heat can be produced by a specific anode reaction. Utilization of the heat produced from fuel electro-oxidation to support the endothermic steamreforming reaction within the stack would considerably improve the overall energy efficiency of an SOFC generator system, and therefore the concept of on-cell reformation or direct utilization of hydrocarbon fuels has drawn worldwide attention in recent years. However, one of the greatest challenges is that the kinetic rate of steam methane reforming is much faster than that of the fuel electro-oxidation. As a result, a localized ‘cold zone’ often occurs near the region of fuel entrance in the stack, often leading to a large temperature gradient across the stack. This becomes a major issue for the normal operation of an SOFC since the thermal stresses induced by the temperature gradient are often the foremost cause of mechanical failure. Another risk of on cell natural gas reforming is the carbon formation due to unfavorable thermodynamic and mostly kinetic conditions in the stack. Therefore, the common practice in the SOFC industry is to pre-reform or partially prereform the natural gas prior to its entering the stack. In order to prevent carbon formation effectively, a fraction of depleted anode gas, rich in steam and CO2, is recycled back to mix with the fresh natural gas. Certainly, the overall efficiency and performance may be somewhat compromised, but the reliability of the stack is significantly enhanced and a long-term steam supply sub-system can be avoided. Realizing that the mechanism of steam reforming, particularly carbon formation, on the Ni-containing anode is a fairly complicated subject, the focus of this chapter will be the basics of steam methane reforming (SMR) and carbon formation from the perspectives of thermodynamics and kinetics.

10.2

Thermodynamics of steam methane reforming

The simple SMR reactions are shown in equations [10.1] and [10.2]. For a generic hydrocarbon, the reaction with steam/carbon ratio S/C = 1/1 can be expressed by CnHm + nH2O → nCO + (m/2 + n) H2

[10.3]

followed by the subsequent water–gas shift reaction CO + H2O → CO2 + H2

[10.3a]

Since the water–gas shift reaction [10.2] is slightly exothermic, the overall steam-reforming reaction is still heavily endothermic. Furthermore, the SMR is accompanied by a volume expansion. Therefore, high temperature and low system pressure favor the SMR by the principle of Le Chatelier. In the studies of SMR performance, the following terms are commonly used:

Steam methane reforming and carbon formation

CH 4 conversion (%) = H 2 yield (%) =

CO yield (%) =

CO2 yield (%) =

CO selectivity (%) = CO2 selectivity (%) =

in out – X CH X CH 4 4 in X CH 4

185

× 100%

XH2 × 100% in X CH 4

X CO in X CH 4

× 100%

X CO 2 × 100% in X CH 4

X CO × 100% X CO + X CO 2 X CO 2 × 100% X CO + X CO 2

[10.4]

where Xi is the molar fraction of species i. CH4 conversion is also referred to as the reforming percentage in the following context.

10.2.1 Equilibrium composition of a CH4–H2O mixture For a simple mixture of CH4 and H2O, the thermodynamic equilibrium is established among species of CH4, H2, H2O, CO, CO2, and C under a given temperature, system pressure, and S/C ratio. The calculation of equilibrium compositions is based on the principle of minimization of the Gibbs freeenergy change of all possible reactions involved, and it can be easily carried out by commercially available software such as NASA code. Figure 10.1 shows examples of equilibrium compositions as a function of temperature calculated at two fixed S/Cs of 1.0 and 2.0 and a system pressure of 1 atm. For S/C = 1.0, CH4 and H2O react to produce H2, CO, CO2, and solid C as temperature increases, indicating that the SMR reaction is activated by the temperature. From 420 to 820 °C, solid C emerges from the product, accompanied by a peak CO2 content around 500 °C. For S/C = 2.0, however, no solid C is seen over the entire temperature range plotted, viz. 0–1000 °C. CH4 and H2O react to produce only H2, CO, and CO2, with H2 being the primary product as temperature increases. Moreover, as shown in Fig. 10.2, the effect of the S/C ratio on equilibrium compositions of the reformate at 750 °C, a typical temperature for the pre-reformer in an SOFC system, indicates that all CH4 should be reformed fully at S/C ≥ 1.1 (with an S/C resolution of 0.1) into H2, CO, and CO2 without forming solid C. The highest yield is of H2, followed by CO. Both the yields of H2 and CO decrease slightly with

186

Solid oxide fuel cell technology 100 C Equilibrium composition (mol %)

90 80

H2(g)

70 60 CH4(g)

50

H2O(g)

40 30

CO(g)

20 CO2(g)

10 0

Equilibrium composition (mol %)

70

0

200

400 600 Temperature (°C) (a)

H2O(g)

800

1000

H2(g)

60 50 40 CH4(g) 30 CO(g)

20 10

CO2(g) 0 0

200

400

600

800

1000

Temperature (°C) (b)

10.1 Equilibrium compositions as a function of temperature at a fixed S/C ratio and Pt = 1 atm: (a) S/C = 1; (b) S/C = 2.

S/C at the expense of an increased CO2, implying that S/C should be kept lower to increase the production of H2 and CO. The effect of system pressure on SMR is demonstrated in Fig. 10.3, where the CH4 conversion rate (or reforming percentage) is plotted against system pressure under a fixed S/C = 2.0 and three different temperatures. The methane conversion rate is clearly shown to decrease with increasing Pt and decreasing

Steam methane reforming and carbon formation

187

100

Equilibrium composition (mol %)

H2 H2O CH4 CO CO2 C

C

90

T = 750 °C Pt = 1 atm

80 70 60

H2 50 40 30 H2O

CO 20 CH4

10

CO2

0 0.8

1.0

1.2

1.4

1.6 S/C

1.8

2.0

2.2

2.4

10.2 Equilibrium compositions of CH4–H2O mixture at different H2O/ CH4 ratios and 750 °C. 100 S/C = 2.0

CH4 conversion rate (%)

90

80 700 °C 70

60

650 °C

50

600 °C

40 0.0

2.0

4.0

6.0 Pt (atm)

8.0

10.0

12.0

10.3 CH4 conversion rate as a function of system pressure at a fixed S/C and different temperatures.

T. This result is explicable in that the steam-reforming reaction (equation [10.1]) is a volume expansion reaction, which infers that increasing system pressure favors the methanation reaction. The analysis seems to challenge the effectiveness of the pre-reformer and therefore application of the on-cell SMR to a pressurized SOFC system.

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Solid oxide fuel cell technology

10.2.2 Steam methane reforming under load of a solid oxide fuel cell Under load conditions, the SMR and electro-oxidation of fuels occur simultaneously in an SOFC. In such a case, the O/C ratio (oxygen/carbon ratio) as opposed to the S/C ratio, is a better parameter for describing the SMR reaction since the contribution of oxygen flux from the SOFC is also considered. In Chapter 6, an example is given to calculate chemical compositions within the stack of an SOFC fueled by natural gas in the presence of a pre-reformer and recirculation loop of the depleted anode gas, see Fig. 6.1. The O/C ratio is kept constant everywhere in the fuel stream during operation. For a mixture of CH4, CO, CO2, H2, and H2O, the O/C ratio can be mathematically expressed as O = X CO + 2 X CO 2 + X H 2 O X CO + X CO 2 + X CH 4 C

[10.5]

The variations of the O/C ratio depend on how much of the anode gas is recycled (measured by the factor F) and how much of the fuel is utilized (Uf). The O/C ratio should be independent of the cell current owing to the iso-Uf operation. For a fixed Uf operation, the higher the F the higher is the O/C. Figure 10.4 shows the conversion rates of CH4 mixed with the recycled anode gas as a function of temperature. The reforming process is clearly temperature-sensitive, and almost all the CH4 can be reformed above 750 °C. 100 90

CH4 conversion rate (%)

80 70 60

Pt = 1 atm 50

O/C = 1.3 O/C = 1.7

40

O/C = 2.1 O/C = 2.5

30 20 10 450

550

650

750 T (°C)

850

950

1050

10.4 CH4 conversion rate as a function of temperature at Pt = 1 atm and different O/C ratios.

Steam methane reforming and carbon formation

189

For the O/C range shown in Fig. 10.4, F is typically in the range of 0.40– 0.77. At lower temperatures, it appears that a higher O/C gives a lower CH4 conversion rate, but this trend is quickly reversed at high temperatures (>600 °C), i.e. a higher O/C yields a higher reforming rate. The effect of the O/C ratio on the CH4 conversion rate is further illustrated at 750 °C in Fig. 10.5. At all the O/C values plotted, the CH4 conversion rate remains very high. As the O/C increases, more H2O and CO2 will be recycled to mix with the incoming fresh CH4. Such an operation usually leads to a higher F and an increased CH4 conversion rate. The benefit of a higher O/C is clearly the assurance of no carbon formation, but the cell E, which reflects the driving force of the oxygen transport process, is also lowered as a result of increased PO 2 in the fuel stream. Therefore, a delicate balance exists between avoiding carbon formation and retaining good stack performance in a practical SOFC system. Similar to Fig. 10.3, the effect of system pressure Pt on the equilibrium composition of the SMR is illustrated in Fig. 10.6 with O/C = 2.1 and T = 600, 650, and 700 °C as an example. Again, increasing pressure and lowering temperature does not favor the SMR. By elevating the Pt from atmosphere to 10 atm, for example, the CH4 conversion rate is lowered by a factor of 7 at 600 °C.

10.3

Partial pressure of oxygen (PO2 ) and Nernst potential (E) of the steam-reformed methane

As mentioned previously, a mixture of CH4 and H2O at elevated temperatures will react to reach a chemical equilibrium among species of CH4, H2, CO, 100.0

CH4 conversion rate (%)

99.5 99.0 98.5 98.0

T = 750 °C Recycled fuel Cf = 0.85 Pt = 1 atm

97.5 97.0 96.5 1.8

1.4

1.6

1.8

2.0

2.2

2.4

O/C

10.5 CH4 conversion rate as a function of O/C ratio at a fixed temperature and pressure.

2.6

190

Solid oxide fuel cell technology 100 O/C = 2.1 Recycled fuel Cf = 0.85

CH4 conversion rate (%)

90 80 70 60

700 °C

50 40

650 °C

30 20

600 °C

10 0 0

2

4

6 Pt (atm)

8

10

12

10.6 Pressure effect on CH4 conversion rate at a fixed temperature and O/C ratio. 4.0 3.0 2.0 1.5

1 × 10–13

H2O/CH4

1 × 10–17

1.0

1 × 10

0.1

1 × 10–21

0.01

–19

2

PO (atm)

1 × 10–15

1 × 10–23 1 × 10–25 1 × 10–27 500

Carbon formation region

600

700

800 T (°C)

900

1000

1100

10.7 Variations with temperature and S/C ratio of the equilibrium PO2 of a CH4–H2O mixture.

H2O, CO2, and solid C depending on the S/C ratio and the temperature. Exemplary equilibrium compositions of a CH4–H2O mixture under a fixed S/C ratio have been shown in Fig. 10.1 as a function of temperature. The equilibrium PO 2 , which is associated with the compositions, can also be calculated out as an output of the multi-component chemical equilibrium, by which means E can be computed by the Nernst equation E = RT/4F × ln(0.209/ PO 2 ). The calculated equilibrium PO 2 in the CH4–H2O mixture at different S/C ratios is plotted as a function of temperature from 600 to 1000 °C in Fig. 10.7. It can be seen from the plot that PO 2 increases with temperature for

Steam methane reforming and carbon formation

191

every S/C ratio, but at different rates. The rate of the increase is clearly higher at higher S/C values than that at lower values. Note that Fig. 10.7 does not illustrate any phase relationship of the C–H–O system. In fact, the thermodynamic prediction of Fig. 10.1 indicates that carbon formation occurs at an S/C ≤ 1.0 in the vicinity of the temperature range plotted. Carbon formation coincides with the change in the slope of the curves shown in Fig. 10.7 viz. the condition subject to carbon formation produces a different slope from that without carbon formation. The region of carbon formation is shaded in Fig. 10.7. With the equilibrium PO 2 available, E can be easily calculated from the Nernst equation and is shown in Fig. 10.8. The temperature dependence of E is unique. It appears that E increases with T for an S/C ≤ 1.0 and decreases with T at an S/C > 1.0, inferring different reaction mechanisms at S/C values greater and less than 1.0. The fact that carbon forms at an S/C ≤ 1.0 seems to suggest that the positive temperature coefficient of E is associated with carbon formation and can potentially be used as an indicator of carbon formation in a CH4–H2O system. In reality, an SOFC preferentially operates at S/C > 1.5 in order to avoid any potential carbon formation. In this case, the E–T plot should bear a negative slope.

10.4

Kinetics of steam methane reforming

Many mechanistic studies have suggested that the SMR reaction on Ni–ZrO2 cermets follows a similar mechanism to conventional catalysts. The latter mechanism typically includes the following basic reaction steps. 1300

0.01 Carbon formation region

1200 0.1

E (mV)

1100 1.0 1000 H2O/CH4 900 1.5 2.0 3.0 4.0

800 700 500

600

700

800 T (°C)

900

1000

1100

10.8 E value of CH4–H2O mixture as a function of temperature and S/C ratio.

192

1

Solid oxide fuel cell technology

H2O reacts with surface Ni atoms, yielding adsorbed oxygen species and gaseous hydrogen H2O + * = O-* + H2

2 3

where * denotes a surface site. The H2 formed is directly released into the gas phase and/or the gaseous H2 is in equilibrium with adsorbed H and H2 on the Ni surface. CH4 is adsorbed on the surface of Ni atoms, which further reacts with the adsorbed oxygen or dissociates to form chemisorbed radicals CHx (x = 0–3) CH4 + * = CH4-* CH4-* + * = CH3-* + H-* CH3-* + * = CH2-* + H-*

4

The adsorbed oxygen species and the carbon-containing radicals react to form chemisorbed CH2O, CHO, CO, and CO2. CO and CO2 are also the products of dissociation from CHO and CH2O species whereas H2 is the product of adsorbed H species CH2-* + O-* = CH2O-* + * CH2O-* + * = CHO-* + H-* CO-* + O-* = CO2-* + * CHO-* + O-* = CO2-* + H-* CO-* = CO + * CO2-* = CO2 + * 2H-* = H2-* + * H2-* = H2 + *

Step 4 has been determined as the rate-limiting step of the above mechanism, suggesting that the presence of the oxygen may determine to a large extent the reaction kinetics. This in turn stresses the importance of an oxygenconducting support such as CeO2 that could enhance the SMR. Assuming Langmuir adsorption isotherms, the rate of the SMR reaction has been proposed to have the general form76 rCH 4 =

k ( T ) PCH 4 (1 + K H ( T ) PH1/22 + K s ( T ) PH 2 O / PH 2 ) n

[10.6]

where: k(T) is the intrinsic rate constant; Pi is the partial pressure of gas i; KH(T) is the equilibrium constant of the hydrogen adsorption; KS(T) is the

Steam methane reforming and carbon formation

193

equilibrium constant for the steam adsorption; and n is the number of surface sites required for the methane adsorption. KH(T) and KS(T) are assumed to vary linearly with temperature T. The temperature dependence of the intrinsic rate constant, k(T), follows a classical Arrhenius relationship E k ( T ) = k o exp – a   RT 

[10.7]

Fitting of equation [10.6] with experimental data suggests n = 2. Careful examination of equation [10.6] implies that the reforming rate vanishes at PH 2 = 0. The physical explanation is that surface sites are flooded with rapidly adsorbed steam because the absence of hydrogen pushes the SMR reaction to completion. Experiments utilizing purely steam/methane mixtures confirmed a very low reforming rate in the micro-reactor. From a practical viewpoint, this implication suggests the importance of pre-reforming for an SOFC stack. A calculation using equation [10.6] with the pre-reforming percentage as a variable indicates that the reforming rate peaks at 30% prereforming composition for the conditions of T = 850 °C and S/C = 2.0. The first-order dependence on partial pressures of CH4 and H2O is another form of the rate equation widely used for SMR on Ni–ZrO2 cermets m rCH 4 = kPCH Pn 4 H2O

[10.8]

where m > 0 and n < 0 are constants closely related to the order of the reaction. Equation [10.8] is consistent with the most generally accepted ratelimiting step for the reaction, namely the methane chemisorption of step 4 as discussed earlier.

10.5

Carbon formation: thermodynamics versus kinetics

Carbon formation is a well-recognized phenomenon in the field of fuel processing. Sufficient steam or oxygen is the prerequisite to prevent carbon formation in the process. In SOFC applications, a direct electrochemical utilization of natural gas or higher hydrocarbons by the SOFC could considerably improve the system efficiency and simplify the system design. However, it could also present a greater risk of depositing carbon at the triple phase boundaries (TPBs) of Ni-containing anodes, blocking reactive sites for electro-oxidation, if the testing condition is thermodynamically and/or kinetically favorable to carbon formation. The possible carbon formation reactions in the SMR system include 2CO(g) = CO2(g) + C(s)

[10.9]

CH4(g) = 2H2(g) + C(s)

[10.10]

CO(g) + H2(g) = H2O(g) + C(s)

[10.11]

194

Solid oxide fuel cell technology

The thermodynamic possibility of carbon formation can be evaluated from the carbon activity aC defined by a C,CO = K1

2 PCO PCO 2

a C,CH 4 = K 2

[10.12]

PCH 4 PH22

a C,CO–H 2 = K 3

[10.13]

PCO PH 2 PH 2 O

[10.14]

With aC > 1, the system is not in equilibrium and carbon formation occurs. At aC = 1, equilibrium is reached. Note that aC is an indicator of the presence of carbon, not of the amount of the carbon formed. Carbon formation is impossible for aC < 1, which means that any carbon phase is unstable or the anode is carbon free. Another straightforward way of defining the boundary of carbon formation is to plot the temperature at which the carbon forms, TC, as a function of S/C or O/C ratios. TC can be determined by calculating the equilibrium compositions of a steam–methane mixture as illustrated in Fig. 10.1. The thus-calculated domain of carbon formation77 is shown in Fig. 10.9. Clearly, carbon will form within a range of temperatures for a fixed S/C from the perspective of thermodynamics.

1000

T (°C)

800

600 C formation

C free

400

200

0 0.0

0.5

1.0

1.5

2.0

2.5

S/C

10.9 Thermodynamic equilibrium boundaries for carbon formation.

Steam methane reforming and carbon formation

195

The ternary phase diagram of the C–H–O system offers more information about the phase relationships involved. Figure 10.10 shows a calculated C– H–O phase diagram emphasizing the carbon formation region.78,79 The carbon boundaries can be easily identified for a given S/C ratio and T; this is also illustrated in Fig. 10.9. The greatest challenge to a thermodynamic prediction of carbon formation is, however, the fact that the reaction involving the carbon formation is hardly at equilibrium. Carbon is often found to form in a practical system where it is not predicted to form by the thermodynamics. For example, according to Fig. 10.10 the system would be carbon free with an S/C > 1.5 at T = 750 °C. However, in commercial steam-reforming reactors, the S/C is usually kept in the range of 1.8–3.0 for methane in order to avoid coking. Reforming higher hydrocarbons requires an even higher S/C, in the range of 2.5–4.5 for naphtha for instance. The root cause for the observed discrepancy is that thermodynamics assumes both forward and backward reactions are reversible and fast in equilibrium. In reality, this is not always the case. Considering a mixture of CH4, H2O, C, CO, and H2, the forward and backward reactions of the CH4 pyrolysis (equations [10.10]) are kf CH 4 → ← C + 2H 2

[10.15]

kb

C 0 100 CH4 C2H6 20

C2H4

80

CH3OH C2H5OH 40

Carbon deposition region

C3H7OH

60

60 CnH2n 80 CH4

1000 °C 800 °C 600 °C 400 °C 20

200 °C 100 °C

100 H 0

H2O

C CO 0° 100 C 0° 40 80 °C CO2 600 °C 0 0 1

20

H2O 40

60

80

0 100 O

11% H2O, 29–40% CO2, 49–60% H2

10.10 Phase diagram of the C–H–O system. Reproduced by permission of The Electrochemical Society.

196

Solid oxide fuel cell technology

Experiments indicate explicitly that the backward reaction of equation [10.15] for carbon removal is very slow, even with steam at 800 °C, in the absence of a catalyst. Based on the above discussion, it is concluded that carbon formation is dictated mainly by kinetics, not thermodynamics. Such a fact makes prediction of the carbon formation very difficult since the kinetics depends on the mechanism, catalysts, size of the reactor, etc. However, two mechanisms for describing the carbon formation are generally accepted. Mechanism 1 is the solid catalyzed reaction, which leads to the formation of filamentous carbon. Mechanism 2 is the gas-phase pyrolysis, which produces the formation of graphitic-like compounds. Methods for dealing with the gas-phase pyrolysis include avoiding free volume and terminating radicals. A fundamental difference between the gas-phase pyrolysis (graphitic carbon) and the solid catalyzed reaction (filamentous carbon) is that the latter forms carbon inside the metal catalyst, often leading to an internal stress and destruction of the catalyst support, whereas the former usually forms carbon on the surface of the catalyst without creating the internal stress.

11 Poisoning of solid oxide fuel cell (SOFC) electrodes

Abstract: The degradation of SOFC performance is often found to be associated with various mechanisms of electrode poisoning. Good examples include the presence of trace sulfur in the fuel stream that can lead to deterioration of the anode performance, and gas-phase chromium oxide in the air stream that can degrade cathode performance. A commonly accepted mechanism for these poisoning effects is the loss in number of reactive sites for either oxygen reduction or fuel oxidation by the coverage with nonreactive or insulating species at the triple phase boundaries. The electrode overpotential and ohmic IR loss are therefore gradually increased with time, causing the degradation of SOFC performance. In this chapter, the poisoning mechanisms of sulfur, silica, and phosphorus on the anode and chromium on the cathode are discussed. Key words: sulfur poisoning, phosphorus poisoning, silica poisoning, chromium poisoning, performance degradation.

11.1

Sulfur poisoning of the anode

Sulfur poisoning of metal catalysts is probably the most severe poisoning phenomenon encountered in the conversion of synthetic gas (CO + H2) to fuels and chemicals, owing to a significant amount of sulfur-containing compounds such as H2S and COS present in synthetic gas produced by the coal gasification process. An understanding of the sulfur poisoning mechanism of metal catalysts becomes increasingly more important as coal emerges as the most economic, abundant, and secure resource for future energy demand. Chemically speaking, sulfur has a much stronger affinity to metal surfaces than fuel molecules; the availability of fuel molecules to catalyst surfaces is significantly reduced by the coverage of sulfur molecules even at extremely low sulfur concentrations. In commercial practice, the life of supported metal catalysts may be reduced to only a few months or weeks in the presence of only ppm quantities of sulfur contaminants in the feed. Because of the essentially irreversible adsorption of sulfur compounds on metals, regeneration is usually impossible or impractical.80 The sulfur poisoning phenomenon of the Ni-containing anode in an SOFC resembles that of the Ni catalyst in the commercial application of catalytic processes. Deactivation of the Ni-containing anode of an SOFC by trace contaminants of sulfur compounds in the fuel stream is well-recognized by the SOFC community. Numerous studies have been carried out in industries 197

198

Solid oxide fuel cell technology

to investigate the threshold of sulfur level below which a traditional Nicontaining anode can sustain its electrical performance. It has been found from these studies that the upper limit of sulfur content in the form of H2S is 0.1ppm, above which degradation of cell voltage is generally observed. Since the total sulfur content (converted to H2S) in any commercial natural gas is well above this value, a desulfurizer subsystem is required to reduce it below 0.1ppm prior to entering into the stack, which also makes the emission of sulfur compounds into the atmosphere extremely low.

11.1.1 Thermodynamics of the Ni–S–O system The basics of sulfur poisoning on Ni metals is the strong chemical affinity between S species and Ni metal. There are two types of Ni sulfides formed on the surface of Ni metal catalysts: bulk (three-dimensional (3D)) Ni sulfides and two-dimensional (2D) surface sulfides. The latter are often referred to as chemisorbed sulfur species. Sulfide formation can be generically expressed by the following generic reactions 3D Ni sulfide:

nS + mNi(s) = NimSn(s) 2 2(g)

2D surface chemisorption:

1 S = Sad(s) 2 2(g)

[11.1] [11.2]

where m and n are the stoichiometric coefficients of the chemical reactions. The equilibrium partial pressures of S2(g) over a range of 3D Ni sulfides are plotted in Fig. 11.1 as a function of reciprocal temperature. The compound Ni3S2(s) is clearly the most easily formed and therefore the most stable sulfide at a given temperature since it requires the least PS 2(g) among other sulfides to form. Some Ni sulfides exist in both solid and liquid states over the temperature range of interest. The thermodynamic properties pertaining to the solid-to-liquid transition are also considered in the plot. Figure 11.2 further supports the observation where the phase stability domain is illustrated at 800°C. As the temperature decreases, the stability domain of Ni3S2(s) expands towards a lower PS 2(g) as is shown in Fig. 11.3 for 600 °C, suggesting that lower temperature favors the formation of the 3D sulfide Ni3S2(s). In SOFC practice, all forms of sulfur compounds in the natural gas will be steam-reformed into H2S owing to its thermodynamic stability. Therefore, H2S is often used as a measure of the sulfur threshold in single-cell verification tests. Reaction [11.1] can be rewritten as NimSn(s) + nH2(g) = nH2S(g) + mNi(s) with the chemical equilibrium constant

[11.3]

Poisoning of solid oxide fuel cell (SOFC) electrodes

199

0 NiS2(s) –2

Ni3S4(s,l)

NiS(s,l) –6

log PS

2(g)

(atm)

–4

–8

Ni3S2(s,l)

–10

–12 0.700

0.800

0.900

1.000

1.100

1.200

1000/T (K–1)

11.1 Equilibrium partial pressures of S2(g) as a function of reciprocal temperature over a range of Ni sulfides.

4 2

log PS

2(g)

(atm)

0

Ni3S4 NiS

NiS2

–2 –4 Ni3S2

NiO

–6 –8 –10 –12 –16

Ni

–14

–12

–10 log PO

2(g)

–8

–6

(atm)

11.2 The phase stability domain of the Ni–S–O system at 800 °C.

–4

200

Solid oxide fuel cell technology 2 0 NiS2

–2

(atm)

Ni3S4

log PS

2(g)

–4

NiS

–6 NiO –8

Ni3S2

–10 –12 Ni –14 –22

–20

–18

–16

–14

log PO

2(g)

–12

–10

–8

(atm)

11.3 The phase stability domain of the Ni–S–O system at 600 °C.

n o  PH S  K 3D =  2  = exp – ∆G    P RT  H2 

[11.4]

Similarly, reaction [11.2] can be rewritten as Sads(s) + H2(g) = H2S(g)

[11.5]

with the chemical equilibrium constant o  PH S  K 2D = 1 ×  2  = exp – ∆G   RT  θ  pH 2 

[11.6]

where θ is the coverage of sulfur on the surface of Ni metal. ∆Go values of reaction [11.5] are given for different catalyst supports and S coverage81 1. Ni sponge, θ > 0.50 ∆G

o

= –143500 ± 18000 + 35.9T

(J/mol)

2. 5 wt% Ni/γ-Al2O3, θ = 0.64 ∆G

o

= –143100 ± 16300 + 39.6T

(J/mol)

3. 5 wt% Ni/α-Al2O3, θ = 0.75 ∆G

o

= –155200 ± 17700 + 35.9T

(J/mol)

Poisoning of solid oxide fuel cell (SOFC) electrodes

201

With the available ∆Go, the thermodynamic stability of various S species is plotted in Fig. 11.4 in the form of PH 2 S / PH 2 versus reciprocal temperature. It is clearly indicated that higher temperature and lower S chemical potential favors the formation of 2D chemisorbed S species on the surface of Ni metal whereas lower temperature and higher S chemical potential promotes the formation of 3D bulk sulfides. This observation agrees well with those suggested by Figs 11.1 to 11.3. The figure also suggests that chemisorbed sulfur is bonded to Ni more strongly than sulfur in a bulk Ni sulfide. This behavior is also supported by the fact that the Ni–S bond lengths found on all 2D surface sulfides (0.218nm) are smaller than those occurring in stable Ni–S bulk compounds (0.238 nm for NiS(s) and 0.228 nm for Ni3S2(s)). At high temperature and low sulfur concentration, the chemisorbed 2D surface sulfides are close to equilibrium and therefore are reversible. This is an important fact that makes it possible to regenerate Ni-containing anodes that have been poisoned by H2S by flushing sulfur-free fuel into the stack.

11.1.2 Effects of sulfur on adsorptions of H2 and CO on Ni catalysts Since one of the necessary steps in the anodic reaction of an SOFC is the adsorption and spillover processes of H2 and CO molecules on the surface of Ni, investigation of the effects of adsorbed sulfur on the adsorption of H2 and CO can reveal a great deal about the poisoning mechanism. It is generally 100 000 10000 1000 Ni3S2

2

2

PH S /PH (ppm)

100 10 1 0.1

Ni sponge x > 0.5

Ni /γ-Al2O3 x = 0.64

0.01 0.001 0.0001 Ni/α-Al2O3 x = 0.75

0.000 01 0.000 001 0.000 0001 0.75

0.95

1.15

1.35

1.55

1.75

1000/T (K–1)

11.4 The sulfur activity of chemisorbed S and Ni3S2 as a function of reciprocal temperature.

202

Solid oxide fuel cell technology

believed that the sulfur adsorption on an Ni surface may alter the adsorption characteristics for H2 and CO, either by blocking the active surface sites and thus making them inaccessible to the adsorbing H2 and CO molecules (geometric effects) or by structural changes caused by strong Ni–S interactions (electronic effects). It is also possible that sulfur will have direct chemical reactions with the adsorbed H2 and CO molecules instead of having indirect interactions via Ni atoms. The sulfur poisoning of Ni for H2 adsorption appears to follow a simple blocking mechanism or geometric effect. In other words, the ensembles of Ni required for dissociation of H2 molecules are no longer accessible at the surface as a result of a complete coverage of sulfur atoms. Such a hypothesis is strongly supported by the fact that the fraction by which H2 adsorption is reduced in polycrystalline and supported Ni catalysts is generally proportional to the mean fractional coverage of sulfur. The studies on single-crystal Ni also suggest that electronic effects may be important at low coverage since one atom of sulfur can prevent H2 adsorption on four or more Ni atoms. At intermediate coverage, desorption rates of H2 are shifted to lower energies, suggesting that sulfur weakens the bonds between Ni and hydrogen. The effects of sulfur poisoning on the adsorption of CO on Ni are very complex because the nature of the adsorbed species and the adsorption stoichiometry vary considerably with changes in pressure, temperature, sulfur coverage, and catalyst support. Many experimental studies seem to suggest that the CO adsorption on a supported Ni surface can be enhanced by forming tetracarbonyl Ni(CO)4 or subcarbonyl Ni(CO)x (x = 2–3), depending on the CO/Ni(s) ratio, in the presence of adsorbed sulfur. The formed Ni carbonyl is able to migrate to the gas phase and/or the support, thereby exposing new Ni sites for adsorption of more CO. The fact that the nature of CO adspecies on Ni is considerably modified by the presence of adsorbed sulfur has important implications for reactions involving CO as a reactant such as in an SOFC. Although high coverage of sulfur generally causes complete loss of activity, small amounts of sulfur may bring about desirable changes in selectivity and adsorption of CO on an Ni surface.

11.2

Silica poisoning of the anode

Silica is a common constituent of alumina-rich insulation materials widely used in SOFC modules. A higher silica content implies a lower cost and therefore is often preferable. The cost of insulation materials has a significant effect on the final product cost of a large SOFC generator where a significant amount of insulation material is used. However, caution must be exercised in selecting low-cost and low-purity insulation materials that will not compromise the performance of an SOFC generator. Single-cell tests with the anode exposed to high-silica insulation materials have explicitly demonstrated

Poisoning of solid oxide fuel cell (SOFC) electrodes

203

performance degradation over time. It was determined that the upper limit of SiO2(s) content in alumina-based insulation materials is around 3–4 wt%, above which a degradation in performance is noticeable.

11.2.1 Thermodynamics of the Si–O and the Si–O–C systems In reducing atmospheres and at high temperatures, SiO2(s) is reduced into gaseous SiO(g) by losing oxygen to the atmosphere. By taking an H2–H2O fuel mixture as an example, the reduction of SiO2(s) can be expressed as SiO2(s)+ H2(g) = SiO(g) + H2O(g) at initial at equilibrium

a

0

b

a–x

x

b+x

[11.7]

where a and b represent the starting molar fractions (or partial pressures if under atmospheric conditions) of H2 and H2O, respectively, and x is the molar fraction of SiO(g). The chemical equilibrium constant K is then written by o x(b + x ) K = exp – ∆G  =  RT  a–x

[11.8]

The standard Gibbs free energy change ∆Go of reaction [11.7] is given by ∆G

o

= 561622– 209.40T

(J/mol)

[11.8a]

In the case of a negligible x in comparison to a and b, equation [11.8] can be simplified into o x = PSiO = K × a = a × exp – ∆G   RT  b b

[11.9]

Figure 11.5 shows the calculated logarithmic PSiO as a function of reciprocal temperature under two H2–H2O mixtures. A higher fuel-to-oxide ratio leads to a higher PSiO. It is also obvious that PSiO increases sensitively with increasing temperature. Therefore, one of the effective ways of reducing SiO(g) is to lower the temperature to which free solid SiO2(s) is exposed. A similar calculation can also be carried out on a reformed hydrocarbon fuel, which is considered to be a mixture of H2, CO, H2O, and CO2 in the following calculations. The overall chemical reaction can be expressed by 2SiO2(s) + H2(g) + CO(g) = 2SiO(g) + H2O(g) + CO2(g) at initial at equilibrium

a

c

a – x/2 c – x/2

0

b

d

x

b + x/2

d + x/2

[11.10]

204

Solid oxide fuel cell technology 1 × 10–11 1 × 10–13

PH O /PH = 89/11 2

1 × 10–17

PSiO

(g)

(atm)

1 × 10

2

–15

1 × 10–19

PH O /PH = 1/1 2

1 × 10

2

–21

1 × 10–23 0.700

0.800

0.900

1.000

1.100

1.200

1000/T (K–1)

11.5 Partial pressure of SiO(g) produced from SiO2(s) after exposure to H2–H2O mixture as a function of temperature.

The chemical equilibrium constant K associated with the equilibrium molar fraction or partial pressure of SiO(g), x, is therefore given by o ( x ) 2 ( b + x /2)( d + x /2) K = exp – ∆G  =  RT  ( a – x /2)( c – x /2)

[11.11]

The standard Gibbs free energy change ∆Go for reaction [11.10] is given by ∆G

o

= 1 085341 – 383.90T

(J/mol)

[11.11a]

If x is negligible compared to a, b, c, and d, the PSiO = x can be simplified into PSiO = x =

Kac = exp – ∆G o   2 RT  bd

ac bd

[11.12]

The calculated logarithmic PSiO as a function of reciprocal temperature is shown in Fig. 11.6. Two representative gas compositions of an H2–CO– H2O–CO2 mixture in an SOFC stack fueled by reformed natural gas are chosen as an example of the calculation. The trend and magnitude of changes in SiO(g) formation with temperature are very similar to those shown in Fig. 11.5, implying that the equilibrium PO 2 in the fuel stream determines the level of formed SiO(g). Another example of unstable Si-containing compounds in reducing atmospheres is given by SiC(s), which is a standard industrial choice of structural ceramics for high-temperature applications because of its excellent mechanical strength, oxidation resistance, and dimensional stability. The

Poisoning of solid oxide fuel cell (SOFC) electrodes

205

1 × 10–12 Gas 2

PSiO

(g)

(atm)

1 × 10–14

1 × 10–16 1 × 10–18

1 × 10–20

1 × 10–22

H2

CO

H2O

CO2

Gas 1

0.154

0.060

0.466

0.263

Gas 2

0.239

0.099

0.326

0.197

1 × 10–24 0.700

0.800

0.900

Gas 1

1.000

1.100

1.200

1000 /T (K–1)

11.6 Partial pressure of SiO(g) produced from SiO2(s) exposed to H2– CO–H2O–CO2 mixture as a function of temperature.

atmosphere in which SiC(s) is used is primarily oxidizing. There has been interest in using SiC(s) as a structural support for an SOFC stack in a reducing environment. However, the chemical stability of SiC(s) under reducing conditions is questionable, as it may potentially introduce an unwanted SiO(g) into the fuel cell stack as discussed above. The likely chemical reaction of SiC(s) with hydrocarbons can be expressed as follows SiC(s) + H2O(g) + CO2(g) = SiO(g) + H2(g) + 2CO(g) at initial at equilibrium

c

d

0

a

b

c–x

d–x

x

a+x

b + 2x

[11.13]

with the standard Gibbs free energy change ∆G

o

= 388209 – 322.10T

(J/mol)

[11.13a]

and the chemical equilibrium constant K=

2 o PH 2 PCO x( b + 2 x ) 2 ( a + x ) × PSiO = = exp – ∆G  PH 2 O PCO 2  RT  ( d – x )( c – x )

[11.14] Figure 11.7 shows the calculated results for two fuel compositions. For each fuel composition, there appears to be an onset temperature above which the

206

Solid oxide fuel cell technology 1 × 100

(g)

PSiO (atm)

1 × 10–1

Gas 1

1 × 10–2 1 × 10–3

Gas 2

1 × 10–4 1 × 10–5 1 × 10

–6

H2

CO

H2O

CO2

Gas 1

0.154

0.060

0.466

0.263

Gas 2

0.239

0.099

0.326

0.197

1 × 10–7 0.700

0.800

0.900

1.000

1.100

1.200

1000 /T (K–1)

11.7 Partial pressure of SiO(g) produced from exposing SiC(s) to H2–CO–H2O–CO2 mixture as a function of temperature.

formation of SiO(g) becomes less sensitive to temperature. The higher the fuel concentration the higher is the onset temperature. Nevertheless, the level of formed SiO(g) is much greater than that from reducing SiO2(s), as shown in Figs 11.5 and 11.6; it greatly exceeds the tolerance limit for fuelcell operation. Since the conversion reaction equation [11.13] is endothermic, higher temperature favors the formation of SiO(g).

11.2.2 Mechanism of silica poisoning of the anode The mechanism of silica poisoning is understood to be that free solid SiO2(s) exposed to the fuel stream becomes reduced into gaseous phase SiO(g) at elevated temperatures. The formed gaseous SiO(g) is then entrained into the fuel stream and electrochemically oxidized to SiO2(s) at the triple phase boundaries (TPBs) of the anode by O2– transported through the electrolyte. The deposit of SiO2(s) covers the TPBs, which further blocks the anodic reaction, leading to degradation and loss of performance. Consider the following anodic oxidation reaction of SiO(g) in a reducing atmosphere ..

SiO (g) (an) + O O× (el) = SiO 2(s) (an) + VO (el) + 2e ′

[11.15]

If air is used at the cathode 1 O (cat) + V .. (el) + 2e ′ = O × (el) O O 2 2(g)

[11.16]

The overall reaction of electrochemical oxidation of SiO(g) is then expressed by

Poisoning of solid oxide fuel cell (SOFC) electrodes

SiO(g)(an) + 1 O2(g)(air) = SiO2(s)(an) 2

207

[11.17]

with the Gibbs free energy change ∆G

o

= – 806945 + 262.10T

(J/mol)

[11.17a]

According to equation [2.7], the E of reaction [11.17] can be written as E = Eo – 3.372 × 10–5 T + RT ln(PSiO) 2F

[11.18]

where the equilibrium SiO(g) is determined by equation [11.9]. Eo = –∆Go/ 2F; ∆Go is the Gibbs free energy change of equation [11.17a]. The calculated E value as a function of the absolute temperature at PH 2 / PH 2 O = 1 is shown in Fig. 11.8. This figure implies that the formation of SiO2(s) at TPBs of the anode takes place if the cell voltage is below E. In fact, careful examination indicates that the E of equation [11.18] is equal to that of equation [2.10] under the same temperature and PH 2 / PH 2 O. It implies that the electro-oxidation of SiO(g) would inevitably happen once the current is drawn. In other words, the deposited SiO2(s) can also be removed from TPBs of the anode once the cell is open-circuited if the reduction of SiO2(s) into SiO(g) is kinetically fast enough.

11.3

Phosphorus poisoning of the anode

The chemical interaction between Ni and phosphorus was first witnessed in a Westinghouse SOFC generator by tiny holes formed on the Ni-based inner 1.020 1.000 0.980 SiO(g) formation

E (V)

0.960 0.940 0.920 0.900

PH /PH 2

2O

=1 SiO2(s) formation

0.880 0.860 800

900

1000

1100 T (K)

1200

1300

1400

11.8 Cell EMF of the electrochemical oxidation of SiO(g) as a function of temperature.

208

Solid oxide fuel cell technology

liner after exposure to phosphate-bonding cement for a long period of time. The observation seems to suggest that the presence of phosphorus in the fuel stream could potentially invoke chemical reactions with Ni, forming compounds with low melting points and low electrical conductivity within the anode to give performance degradation. In practice, there are two likely sources of phosphorus for the anode. One of them comes from the syngas fuel produced from the coal gasification process. Phosphorus is one of the species that passes through warm gas clean-up. The concentration of phosphorus in the syngas typically varies within a few ppm. Another physical source of phosphorus originates from phosphate-based cements used for bonding various ceramic parts. One type of phosphate binder is AlPO4. In what follows, thermodynamic assessments of Ni–P and Ni–AlPO4 systems are given for the operating conditions of SOFC stacks.

11.3.1 Thermodynamics of the Ni–P system A quick examination of the Ni–P phase diagram, Fig. 11.9, suggests that Ni and P can form many compounds, some of which have lower melting points. The generic chemical reactions between Ni and P can be expressed by n P + mNi = Ni P [11.19] m n (s,1) (s) 2 2(g) The thermodynamic stability of various Ni phosphides is shown in Fig. 11.10, where the equilibrium partial pressure of phosphorus, PP2(g), is plotted 1700

Liquid

1500 FCC_A1 1300

T (K)

Ni12P5(s) 1100 Ni3P(s)

900

Ni2P(s) Ni6P5(s)

700

Ni5P2(s)

500 300

0

0.1

0.2 Mole P/(Ni + P)

11.9 Phase diagram of the Ni–P system.

0.3

0.4

Poisoning of solid oxide fuel cell (SOFC) electrodes

209

0

NiP3(s)

NiP2(s) –10

log PP

2(g)

(atm)

–5

Ni6P5(s) –15

Ni2P(s)

Ni5P2(s)

–20

Ni3P(s) –25 0.700

0.800

0.900

1.000

1.100

1.200

–1

1000/T (K )

11.10 Plot of equilibrium partial pressure of phosphorus, PP2(g) , over a series of Ni phosphides as a function of reciprocal temperature.

as a function of reciprocal temperature. Evidently, Ni3P and Ni5P2 are the most stable phases in the range of SOFC operating temperatures since they require the least PP2(g) to form. This prediction is further confirmed by Fig. 11.11 where the phase stability domain of phosphides is shown at 800°C. Ni5P2 is formed over a wider range of PO 2 and PP2(g); it is most likely the dominant product of an Ni–P chemical reaction. Therefore, it is assumed to be the product of all Ni–P reactions in the following analysis. The chemical potential of phosphorus is represented in this analysis by PP2(g) . It can also be fixed by, for example, a mixture of PH3/H2. In this case, the additional equilibrium P2(g) + 3H2(g) = 2PH3(g) has to be considered in reaction [11.19] [11.20] nPH3(g) + mNi(s) = NimPn(s,1) + 3 n H2(g) 2 A similar thermodynamic analysis can be performed with the phosphorus potential expressed by the ratio of PPH 3 / PH 2.

11.3.2 Chemical reactions between Ni and phosphates AlPO4 One of the mechanisms that is thermodynamically plausible for P transport from phosphate-based cements is the reduction of AlPO4(s), a major component

210

Solid oxide fuel cell technology 0 NiP3 –5 Ni5P2

log PP

2(g)

(atm)

–10 –15

Ni3P

–20 NiO –25 –30

Ni

–35 –40 –40

–35

–30

–25

–20 log PO

–15

–10

–5

0

(atm) 2(g)

11.11 The phase stability domain of the Ni–O–P system at 800 °C.

in the cement, by H2 and CO at high temperatures, which releases P-containing compounds such as PH3(g) that react with Ni metal. The overall chemical reaction in an H2–H2O mixture can be expressed by 2AlPO4(s) + 5H2(g) + 5Ni(s) = Ni5P2(s) + 5H2O(g) + Al2O3(s)

[11.21]

Note that the only Ni–P product in equation [11.20] is taken as Ni5P2(s), the most common compound as suggested in Fig. 11.11. The free Gibbs energy change of reaction [11.21] is given by ∆G

o

= 237178 + 125.58T log10T – 1068.55T

(J/mol)

[11.21a]

and the chemical equilibrium constant is 5 o  PH O  K =  2  = exp – ∆G    P RT  H2 

[11.22]

Equation [11.22] suggests that an elevated temperature and a higher H2/H2O ratio would favor the formation of Ni5P2(s). It also defines the stability domain for the Ni–Ni5P2 system. In order to better illustrate the stability domain, Fig. 11.12 plots equation [11.22] graphically as the upper curve, where the stability regions for Ni and Ni5P2(s) are marked. As shown in the figure, exposure of AlPO4(s) to 97% H2–3% H2O would form Ni5P2(s) at all temperatures

Poisoning of solid oxide fuel cell (SOFC) electrodes

211

0.00

–0.50

Ni

2AIPO4(s) + 5H2(g) + 5Ni(s)

Ni5P2

= Al2O3(s) + 5H2O(g) + Ni5P2(s) –1.00

Gas 2

log Ki

PH O/PH = 3/97 2

–1.50

2

–2.00 Ni –2.50

2AIPO4(s) + 4H2(g) + CO(g) + 5Ni(s) = Al2O3(s) + 4H2O(g) + CO2(g) + Ni5P2(s)

–3.00 0.700

0.800

0.900

1.000

Ni5P2 1.100

1.200

1000/T (K–1)

11.12 Stability domain for Ni and Ni5P2 formed by the reduction of AlPO4 in H2–H2O and H2–H2O–CO–CO2 mixtures. Ki is the chemical equilibrium constant given by either equation [11.22] or [11.24].

plotted. Independent scanning electron microscopy (SEM) studies have confirmed the thermodynamic predictions, as shown in Fig. 11.13 where a deposit of P-containing compounds is clearly seen on the surface of Ni fibers after exposure to 1000 °C and 97%H2–3%H2O fuel. Similarly, the reaction between AlPO4(s) and hydrocarbons can be expressed by 2AlPO4(s) + 4H2(g) + CO(g) + 5Ni(s) = Ni5P2(s) + 4H2O(g) + CO2(g) + Al2O3(s)

[11.23]

With the free Gibbs energy change ∆G

o

= 192575 + 116.94T log10T – 960.19T

(J/mol)

[11.23a]

which leads to the chemical equilibrium constant 4 o  PH 2 O   PCO 2  K=  = exp – ∆G      RT   PH 2   PCO 

[11.24]

The phase stability domain of the Ni–Ni5P2 system in an H2–H2O–CO–CO2 environment is shown in Fig. 11.12 in the lower curve. For the gas 2 composition shown in Fig. 11.6, the formation of Ni5P2 is unlikely. In summary, the consequences of P transport during SOFC operation are twofold. One consequence is the potential reaction with Ni metal forming

212

4000

Overall

Ni (atomic %)

Panel (a), overall

3.83

7.02

96.17

92.98

Panel (b), spot 1

6.27

11.25

93.73

88.75

Panel (b), spot 2

0.91

1.70

99.09

98.30

Ni

Ni

6000

Spot 1

Ni

Ni

Spot 2

1000 500

P

2000 1000

Ni

0

Ni

Counts

1500

Counts

Counts

3000

2000

P Ni

4 6 Energy (keV)

8

10

Ni

P 0

0 2

4000

2

4 6 Energy (keV)

8

10

2

4 6 Energy (keV)

8

11.13 SEM photograph and energy dispersed spectrum (EDS) chemistry of an Ni-fiber surface after exposure to 97% H2–3% H2O at 1000 °C for 10 hours.

10

Solid oxide fuel cell technology

2000

Ni (wt %)

(b)

(a) Ni

P P (wt %) (atomic %)

Poisoning of solid oxide fuel cell (SOFC) electrodes

213

compounds with lower melting points and low electrical conductivity on the surface of the anode, which could in turn increase the anode resistance. Another consequence is the degradation of mechanical bonding strength within the cement after phosphorus is transported. Therefore, the use of phosphorus-bonding cement in an SOFC operating at high temperatures and in a reducing environment should be approached with caution.

11.4

Chromium poisoning of the cathode

One of the greatest challenges to high-Cr metal interconnects widely used in planar SOFCs is the Cr mobility at operating temperatures around 800 °C. A series of gas-phase, high-valent Cr species are thermodynamically stable under SOFC operation conditions, particularly on the cathode side where high PO 2 and PH 2 O are available. It is generally accepted that volatile Cr species are responsible for the observed performance degradation in planar SOFCs using metal interconnects. In this section, the thermodynamics of the Cr–O–H system and mechanisms of Cr poisoning of the cathode are discussed in detail.

11.4.1 Thermodynamics of the Cr–O–H system The high-valent gaseous Cr species formed at high temperatures, high PO 2, and high PH 2 O present themselves in the forms of oxides, hydroxides, and oxyhydroxides. The generic chemical reactions between Cr2O3(s) and gaseous Cr species can be expressed by Cr2O3(s) + (m – 1.5)O2(g) = 2CrOm(g)

[11.25]

2Cr2O3(s) + (m – 3)O2(g) + 2mH2O(g) = 4Cr(OH)m(g)

[11.26]

2Cr2O3(s) + (m + 2n – 3)O2(g) + 2mH2O(g) = 4CrOn(OH)m(g) [11.27] The thermodynamic stabilities of these gaseous species over a solid Cr2O3(s) under conditions of 60% relative humidity (at 25 °C) air and 600–1000 °C are shown in Figs. 11.14 to 11.16, respectively, for oxides, hydroxides, and oxyhydroxides. It can be seen from the figures that CrO3(g), Cr(OH)4(g), CrO2(OH)2(g), and CrO2(OH)(g) are the most abundant species among each group owing to their highest equilibrium partial pressures. Comparing the magnitudes of partial pressures of these species, however, narrows down the major gaseous Cr species formed on the cathode to CrO3(g), followed by CrO2(OH)2(g) and CrO2(OH)(g). In addition, CrO2(OH)2(g) exhibits the lowest activation energy, inferring its dominance at lower temperatures. The effect of H2O(g) on the formation of the Cr species at the cathode is

Solid oxide fuel cell technology –5 CrO3(g)

–10

log Pi(g) (atm)

–15

CrO2(g)

–20 –25 CrO(g)

–30 –35 –40

Cr(g)

PO = 0.209 atm 2

–45 0.700

0.800

0.900

1.000

1.100

1.200

1000/T (K–1)

11.14 Equilibrium partial pressures of gaseous Cr oxides in air as a function of reciprocal temperature.

–13

m(g)

(atm)

–15

log PCr(OH)

214

–17

Cr(OH)4(g)

–19

–21

–23

Cr(OH)3(g)

PO = 0.209 atm 2

PH

2O

= 0.019 atm

Cr(OH)2(g)

(60% RH at 25 °C) –25 0.700

0.800

0.900

1.000

1.100

1.200

1000/T (K–1)

11.15 Equilibrium partial pressures of gaseous Cr hydroxides in air with 60% relative humidity (RH) at 25 °C as a function of reciprocal temperature.

Poisoning of solid oxide fuel cell (SOFC) electrodes –5

PO2 = 0.209 atm PH2O = 0.019 atm (60% RH at 25 °C)

CrO2(OH)(g)

log PCrO

n(OH)n(g)

(atm)

–10

215

CrO2(OH)2(g)

CrO(OH)2(g)

–15

–20

CrO(OH)3(g)

–25

–30

–35 0.700

CrO(OH)(g)

CrO(OH)4(g)

0.800

0.900

1.000

1.100

1.200

1000/T (K–1)

11.16 Equilibrium partial pressures of gaseous Cr oxyhydroxides in air with 60% relative humidity at 25 °C as a function of reciprocal temperature.

0 –5

CrO3(g)

log Pi(g) (atm)

–10 –15 CrO2(OH)2(g)

–20 –25

T = 800°C PO = 0.209 atm

–30

2

–35 1 × 10–10

1 × 10–8

1 × 10–6 1 × 10–4 PH O (atm)

1 × 10–2

1 × 100

2

11.17 Plot of partial pressures of Cr species formed at the cathode side versus partial pressure of H2O at 800 °C.

216

Solid oxide fuel cell technology

further illustrated in Fig. 11.17 at 800°C for CrO2(OH)2(g) and CrO3(g), respectively. CrO2(OH)2(g) surpasses CrO3(g) and becomes the predominant Cr species at higher PH 2 O . On the other hand, PO 2 also plays an equal role in forming gaseous Cr species. Figure 11.18 shows the partial pressures of CrO2(OH)2(g) formed at the anode where very low PO 2 but high PH 2 O are present. Clearly, PCrO 2 (OH) 2 (g) increases with PH 2 O , but at a much lower level in comparison to that formed in the cathode environment. Therefore, it is safe to say that the formation of high-valent Cr species in the fuel environment is negligible. Note that a pure and solid Cr2O3(s) is assumed in the above evaluation, which may be more true for metal interconnects than for ceramic LaCrO3 interconnects. The equilibrium partial pressures Pi(g) of gaseous Cr species over a solid LaCrO3(s) is reduced by the square root of the chemical activity of Cr2O3 in LaCrO3(s) [11.28]

Pi (g) = AK i a Cr2 O 3

where A is a constant determined by the O2 and H2O partial pressures as well as the corresponding stoichiometric coefficients. Ki is the chemical equilibrium constant of the relevant reaction shown in equations [11.25] to [11.27] . The a Cr2 O 3 is often found to be very small owing to the excellent chemical stability of LaCrO3(s) material. Hilpert et al.82 estimated that the volatility of the gaseous Cr species over LaCrO3(s) is reduced by more than three orders of magnitude compared with their volatility over Cr2O3(s). The reduction becomes even larger when dopants such as Ca and Sr are added into LaCrO3(s). –20 –25

T = 800 °C H2–H2O mixture PO varies with H2O/H2 ratio

log PCrO

2(OH)2(g)

(atm)

2

–30 –35 –40 –45 –50 –55 0.01

PH

0.1 (atm)

1

2O

11.18 Plot of partial pressures of Cr species formed in a mixture of H2 and H2O versus partial pressure of H2O at 800 °C.

Poisoning of solid oxide fuel cell (SOFC) electrodes

217

Consequently, it is rare to observe Cr vaporization and therefore poisoning of the cathode with ceramic LaCrO3-based interconnects. The same principle can also be applied to the metal interconnects whose oxide scales are no longer pure Cr2O3(s), but a solid solution or compound such as MnCr2O4(s). The reduction in chemical activity of Cr2O3 from unity to a fraction would certainly decrease the formation of gaseous Cr species and therefore improve the stability of metal interconnects. Such an understanding has become the basis of the latest design of metal interconnect compositions. For example, Crofer22 APU and ZMG232, the two leading ferritic steels for SOFC applications, contain an elevated level of Mn as well as reactive elements such as La and Zr. The result is the formation of a relatively more dense and more stable MnCr2O 4 spinel scale at the service temperature, which considerably impedes the Cr and O diffusion processes across the formed scale and leads to much reduced partial pressures of gaseous Cr species. The adherence of the scale to the substrate has also been improved to yield a lower contact resistance.

11.4.2 Mechanisms of Cr poisoning of the cathode With the thermodynamic assessments of the gaseous Cr species available, understanding the mechanism of cathode Cr poisoning is straightforward. During a typical SOFC operation, the gaseous Cr species formed are first entrained in the air stream and then transported to the cathode/electrolyte interface where they are reduced to solid Cr2O3 via the following cathodic reactions (taking CrO3(g) as an example) .. CrO 3(g) (cat) + 3 VO (el) + 3e ′ = 1 Cr2 O 3(s) (cat) + 3 O O× (el) 2 2 2 [11.29]

The deposition of Cr2O3(s) occurs preferentially at the reactive TPB sites where oxygen molecules, electrons, and oxygen vacancies are available as required by equation [11.29]. Unfortunately, the normal oxygen reduction of a cathode also occurs at these locations and has to compete with the oxygen reductions of gaseous high-valent Cr species for the right to react. The coverage by a catalytically inactive Cr2O3(s) layer at reactive sites undoubtedly slows down the rate of oxygen reduction and therefore increases the resistance. As more and more Cr2O3(s) precipitates out at the TPBs, and the number of reactive sites accessible for oxygen molecules become fewer and fewer, degradation of the performance of an SOFC is observed. .. 1 With the anodic reaction H2(g)(an) + O O× (el) = H2O(g)(an) + VO (el) + 2e′, 2 the overall electrochemical reaction is

218

Solid oxide fuel cell technology

CrO 3(g) (cat) + 3 H 2(g) (an) = 1 Cr2 O 3(s) (cat) + 3 H 2 O (g) (an) 2 2 2 [11.30] with the Gibbs free energy change ∆G

o

= –608152 + 141.50T

(J/mol)

[11.31]

The Nernst potential, E, of reaction [11.30] at equilibrium can be given by o  PH O  E = – ∆G – RT ln  2  + RT ln ( PCrO 3 ) 3F 2 F  PH 2  3F

[11.32]

where PCrO 3 can be calculated from the Gibbs free energy change of reaction [11.25]. A careful examination of equation [11.32] reveals that it is virtually identical to equation [11.18] and is independent of the actual poisoning reaction of either equation [11.17] or [11.30]. The cell E is completely dictated by equation [2.10] of reaction [2.8] . Since E represents the cell voltage under open-circuit voltage (OCV) conditions, it implies that the reduction of high-valent Cr species occurs as soon as the current is drawn. The mass transport of Cr from the metal interconnect to the cathode has also been estimated by Hilpert et al.82 for a typical planar SOFC geometry under operating conditions. The most abundant Cr species, CrO2(OH)2(g), was considered. A conservative consideration revealed a deposition rate of Cr2O3(s) on the cathode surface of 8 × 10–8 kg/(m2 s) at 950 °C. This value translates to a Cr2O3 monolayer of 1018 atoms/m2 after 2.4s. Such a fast deposition rate would result in a rapid deterioration in cathode performance. Chemical reactions between the formed Cr2O3(s) with perovskite cathode materials such as Sr-doped LaMnO3 and Sr- and Co-doped LaFeO3 under the operating conditions of an SOFC is another likely reason for the decay of SOFC performance. The freshly deposited and fine-grained Cr2O3(s) particles possess high surface area and have a great propensity to react with the cathode. In the case of Sr-doped LaMnO3, the formed phase is primarily a Cr–Mn–O spinel solid solution as illustrated by La1–xSrxMnO3(s) + 1 Cr2O3(s) 2 = La1–xSrxMn1–yCryO3(s) + (Cr1–yMny)O1.5–δ(s) + δ/2O2(g) [11.33] For Sr- and Co-doped LaFeO3, SrCrO4(s) appears to be the main product of the reaction La1–xSrxFe1–yCoyO3(s) + 0.5x′ Cr2O3(s) + 5x′/4O2(g) = La1–xSrx–x′Fe1–yCoyO3(s) + x′SrCrO4(s)

[11.34]

Poisoning of solid oxide fuel cell (SOFC) electrodes

219

When there is free SrO segregated from the perovskite phase,* the above reaction becomes even more prevalent. The dissolution of Cr2O3 as well as precipitation of spinel and chromate phases would also result in deterioration of the electrical properties of the perovskite cathode to further degrade the cathode performance.

*

Segregation of free SrO on the surface of a cathode is often found in Sr-doped LaFe(Co)O3 perovskites.

12 Materials for solid oxide fuel cells (SOFCs)

Abstract: Although the SOFC presents a great potential to meet the future demand for efficient and clean energy, development of SOFC technology into a commercial product has proved challenging. Owing to hightemperature operation, the requirements for functional materials are in general more stringent than those of low-temperature fuel cells. Two major barriers identified for commercialization of SOFC technology are reliability and product cost, both of which are closely associated with the materials used in SOFCs. Therefore, it is a rational assertion that the success of SOFC technology largely depends upon the maturity of SOFC materials. In this chapter, we cover SOFC technology from the perspective of materials that have been actively developed and are being engineered. Three major types of SOFCs are classified by the electrolyte material used. For each type of SOFC, the discussion focuses primarily on the electrical, chemical, and thermal properties of their materials. At the end, the fabrication technique for each functional layer is also briefly reviewed. Key words: ZrO2-based, CeO2-based, LaGaO3-based, ceramic and metallic interconnects, fabrication techniques.

12.1

ZrO2-based solid oxide fuel cells

The ZrO2-based SOFC represents the most studied and engineered system in the history of SOFC development. It is technically more mature than other types of SOFCs discussed in this chapter and therefore is likely to be in the first commercialized system. The materials discussed in this section include ZrO2-based electrolytes, LaMnO3-based cathodes, and the Ni–ZrO2 cermet anode. The interconnect material will be reviewed separately in Section 12.4.

12.1.1 ZrO2-based electrolytes As mentioned frequently in this book, the oxide-ion electrolyte must be a conductor of oxide ions, preferentially a fast oxide-ion conductor and an electronic insulator at the operating temperature of an SOFC. Although many oxides exhibit such a conducting behavior at elevated temperatures as pointed out in Chapter 5, the most prominent fast oxide-ion conductors are found in

The majority of this chapter has been published in Chapter 8 of Materials for Fuel Cells, edited by M. Gasik, Woodhead Publishing Limited, 2008.

220

Materials for solid oxide fuel cells (SOFCs)

221

oxygen-deficient compounds with either the fluorite structure (such as ZrO2-, CeO2-, and Bi2O3-based materials) or the perovskite structure (such as LaGaO3-based materials). In this section, ZrO2-based electrolytes are reviewed. Pure ZrO2 has no practical use for an SOFC owing to its low intrinsic oxide-ion conductivity and the large volume contraction (~9%) induced by the monoclinic-to-tetragonal phase transition that takes place at 1170 °C. The favorable cubic structure for oxide-ion conduction is only stable above 2370 °C. However, the cubic structure can be stabilized to room temperature by substituting Zr with alkaline-earth elements such as Ca, Mg, and Sr or rare-earth elements such as Sc and Y. More importantly, doping with 2+ and 3+ cations into the Zr4+ lattice introduces oxygen vacancies, a necessity for oxide-ion conduction, by the principle of charge compensation (Kröger– Vink notation) as shown in Table 12.1. Also included for comparison are the defect reactions for CeO2- and LaGaO3-based electrolyte systems. In order to ensure the ‘quasi charge-neutrality’ condition of equation [4.5a] in the .. .. lattice, the relationships of [ M x′ ] = 2 [VO ] and [ M x′′ ] = [ VO ] must hold for 3+ and 2+ dopants, respectively, where square brackets represent the concentration. This mass relationship appears to suggest that the higher the doping level, the higher the concentration of oxygen vacancies being created and therefore the higher the oxide-ion conductivity. In reality, the isothermal oxide-ion conductivity of a given dopant peaks at a certain doping level. This phenomenon is illustrated in Fig. 12.1, where rare-earth cations are the dopants for ZrO2.83 The peaks occur at 10–11 mol% for Sc and 8–9 mol% for other rare-earth dopants. The reason behind this observation can be interpreted by the interaction .. .. between M Zr ′ or M Zr ′′ and VO , forming associates of 2 M Zr ′ – VO or M Zr ′′ – .. VO. The formation of such clusters is stabilized at lower temperatures, typically increasing the activation energy for oxide-ion migration, ∆Hm (also known as motional enthalpy), by half of the binding energy for the associate, 1 ∆Ht (also known as trapping enthalpy), i.e. Ea = ∆Hm + 2 ∆Ht. Chapter 5 gives a more detailed explanation of this matter. Such a change in activation energy with temperature is often revealed by a kink at a characteristic temperature or a gradual curvature over a temperature range in the Arrhenius plot of oxide-ion conductivity. Above this temperature, the clusters disassociate back .. into free M Zr ′ or M Zr ′′ and VO , and the total activation energy returns to ∆Hm. Since the formation of larger clusters is related to cation diffusion, the aging behavior of oxide-ion conductivity observed at lower temperatures is often linked to the interaction between dopants and oxygen vacancies. For a given doping level, on the other hand, the oxide-ion conductivity was also found to vary considerably with the type of dopant,83 see Fig. 12.2. The highest conductivity observed in Sc-doped ZrO2 is explained by the closest match in ionic radius to the host Zr4+ ion, which minimizes the lattice

222

Divalent dopant

ZrO2-based

CeO2-based

LaGaO3-based

.. ZrO 2 MO  → M ′′Zr + O ×O + VO

.. CeO 2 MO → M ′′Ce + O ×O + VO

.. LaGaO 3 SrO   → 2SrLa ′ + O ×O + VO .. LaGaO 3 MgO   → 2Mg ′Ga + O ×O + VO

Trivalent dopant

.. ZrO M2 O3 →  2 2MZr ′ + 3O O× + VO

.. CeO2 M2O3  → 2MCe ′ + 3O O× + VO

—

Solid oxide fuel cell technology

Table 12.1 Defect reactions illustrating creation of oxygen vacancies of ZrO2–, CeO2–, and LaGaO3 based solid electrolyte systems

Materials for solid oxide fuel cells (SOFCs) 0.30

Ln Sc Yb Er Y Dy Gd

0.25 0.20 s (S/cm)

223

0.15 0.10 0.05 0.05

0.10

0.15

X

12.1 Oxide-ion conductivity of doped ZrO2 at 1000 °C as a function of dopant level (X). Copyright Elsevier.

0.20

0.15

0.10 0.05

Ion migration enthalpy (kJ/mol)

s1000°C (S/cm)

0.25

110 100

3+

s1000°C

Yb

80

Y3+ Dy3+

60

90 Gd3+

80 Zr4+

40

70 20

Er3+

60 0.85 Sc3+

0.90

0.95 1.00 Dopant cation ionic radius (Å)

Association enthalpy (kJ/mol)

Ion migration enthalpy Association enthalpy

0.30

1.05 Eu3+

12.2 Oxide-ion conductivity of doped ZrO2 at 1000 °C, motional enthalpy, and trapping enthalpy as a function of dopant ionic radius. Copyright Elsevier.

elastic strain for an easy oxide-ion pass across the potential saddle point. The closest match in ionic radii between dopant and host ions as well as the smallest charge difference can be used as a general guide in selecting the right dopant for a host in order to achieve the highest oxide-ion conductivity. The temperature dependence of oxide-ion conductivity σ V .. of isotropic O ZrO2-based materials has been described by random-walk theory84 in Chapter 5; it is further elaborated into the Arrhenius form by combining equations [5.5] and [5.7]

224

Solid oxide fuel cell technology

(

σ V .. T = σ Vo .. exp – Ea O kT O

σ VO.. = O

)

[12.1]

w Nq 2 ∆S c(1 – c ) l 2 υ O f   exp  m   k   6 k

[12.2]

For a fluorite unit cell, l = a/2 (a is the lattice parameter), w = 6, f ≈ 1, N = 4a–3, q = 2e, equation [12.2] reduces to 2 ∆S σ VO.. = 4 e × c × (1 – c ) × υ O exp  m   k  ka O

[12.2a]

Equation [12.1] suggests that the activation energy can be obtained from the slope of a plot of ln σ V .. T versus 1/T. Examples of such plots are given in O Fig. 12.3 for 8 mol% Y2O3-doped ZrO2 (8YSZ) and 10 mol% Sc2O3–1 mol% CeO2-doped ZrO2 (10Sc1CeSZ) in the temperature range of 800–1000 °C and air.67 The analytical expressions of the two plots are given by 8YSZ log 10 σ VO.. T (SK/cm) = – 4549 + 5.885 T (K) 10Sc1CeSZ

Ea = 0.91 eV

[12.3]

log 10 σ V .. T (SK/cm) = – 3683 + 5.510 O T (K)

Ea = 0.73 eV

[12.4]

3.00

10Sc1CeSZ 2.00 8YSZ

O

log σV. . T (SK/cm)

2.50

1.50

1.00

0.50 0.70

0.80

0.90

1.00

1000/T (K–1)

12.3 Arrhenius plots of oxide-ion conductivity of 8YSZ and 10Sc1CeSZ electrolytes.

1.10

Materials for solid oxide fuel cells (SOFCs)

225

The lower Ea observed in 10Sc1CeSZ compared with that in 8YSZ is a clear indication of a more favorable lattice environment for oxygen-vacancy migration created by the closer ionic radius match of Sc3+ to Zr4+. The electrical conductivity is not always purely ionic. Depending upon temperature and partial pressure of oxygen, PO 2, the conduction could result from contributions of oxide-ions and electrons, the latter either in the form of electron holes or excess electrons. The electronic properties of ZrO2based electrolytes are vitally important to the loss of fuel and therefore the electrical efficiency of an SOFC. The details on how to evaluate the effect of electronic conductivity on electromotive force (EMF), leakage current density, and electrical efficiency can be found in Chapters 4 and 8.

12.1.2 Cathode materials LaMnO3-based perovskite oxides represent the state-of-the-art cathode materials for ZrO2-based SOFCs. The selection of this material over others such as LaFeO3- and LaCoO3-based perovskites is primarily based on a balanced consideration of electrical conductivity, chemical reactivity, electrocatalytic activity, and thermal expansion coefficient (TEC). LaCoO3-based perovskites are known to be superior in electrical conductivity and electrocatalytic activity compared with LaMnO3-based perovskites. However, the former are more reactive with ZrO2 and have a much higher TEC than doped ZrO2. An adequate balance in material properties is particularly important for cathode-supported SOFCs where a large thermal mismatch between the substrate and the supported layers could lead to catastrophic failure upon thermal cycling. Pure LaMnO3 is neither a good electrical conductor nor a good catalyst for oxygen reduction. Substituting alkaline-earth elements such as Ca or Sr into LaMnO3 has been a common practice to enhance electrical conductivity and electro-catalytic activity while still maintaining a good TEC match to ZrO2 electrolytes. Figure 12.4 shows an example of Arrhenius plots of electrical conductivity measured in O2 of LaMnO3 doped with various levels of Sr.85 The plots agree well with the form of equation [12.1], from whose slope activation energies are determined to decrease with Sr-doping level. The magnitude of 0.03–0.25 eV suggests a predominant small-polaron hopping mechanism for electrical conduction. Moreover, Fig. 12.5 shows the variation of isothermal conductivity with PO 2, further confirming that the doped LaMnO3 is a p-type conductor.85 The generation of p-type small polarons (electron holes) is explicable from the perspective of defect chemistry, by which the presence of Ca ′La or SrLa ′ requires creation of more electron holes in the form of Mn .Mn to balance the local charge. This process is best described by the following defect reactions

Solid oxide fuel cell technology 3.5 La1–x Srx MnO3+d:PO = 1 bar 2

3 2.5

log s (S/cm)

2 1.5 1

x = 0.7 x = 0.5

0.5

x = 0.4 x = 0.3

0

x = 0.2 x = 0.1

–0.5

x = 0.0 –1

0

0.001

0.002

0.003

0.004

1/T (K–1)

12.4 Arrhenius plot of electrical conductivity of La1–xSrxMnO3+δ at PO2 = 1 atm. Copyright Elsevier. 350 300

Conductivity of La1–x Srx MnO3+ d at 1273 K

250 s (S/cm)

226

200 150 100 50 0 –20

x = 0.4 x = 0.3 x = 0.2 x = 0.1 x = 0.0 –15

–10 log PO (bar)

–5

0

2

12.5 PO2 dependence of electrical conductivity of La1–xSrxMnO3+δ at 1000 °C. Copyright Elsevier.

..

LaMnO 3 2SrO  → 2SrLa ′ + 2O O× + VO

..

LaMnO 3 or 2CaO  → 2Ca ′La + 2O O× + VO

[12.5]

1 O + V .. + 2Mn × = O × + 2Mn . Mn Mn O O 2 2

[12.6]

Materials for solid oxide fuel cells (SOFCs)

227

..

with the ‘quasi charge-neutrality’ relationship of 2[VO ] + p = [SrLa ′ ] ([Ca ′La ]) . It is obvious that the concentration of electron holes p or [Mn .Mn ] increases with the dopant concentration [Ca ′La ] or [SrLa ′ ]. The TECs of doped LaMnO3 vary with the type of dopant, the doping level, and more importantly PO 2. Ca-doped LaMnO3 perovskites generally have lower TECs (in the range of 10 × 10–6/K to 11 × 10–6/K) than those of Sr-doped LaMnO3 (11 × 10–6/K to 12 × 10–6/K) for a similar doping level. As a result, Ca-doped LaMnO3 is better suited for a ZrO2-based SOFC with regard to thermal expansion match as the TEC of 8YSZ is 10.4 × 10–6/K. For a given dopant, on the other hand, Fig. 12.6 shows a minimum of TEC occurring at roughly x = 0.15 and 0.20, respectively, for Sr and Ca in La1– 86 xAExMnO3 (AE = Ca and Sr). However, the most important characteristic of thermal expansion behavior for doped LaMnO3 is the change with PO 2 under isothermal conditions, also known as ‘chemical expansion’. This behavior has been described in detail in Chapter 2. The key to the chemical expansion is the dependence of the oxygen stoichiometry δ of doped LaMnO3 on PO 2 . Figure 12.7 shows the measured variations of 3 + δ of La0.8Sr0.2MnO3 with PO 2 at various temperatures.87 As PO 2 decreases, 3 + δ changes from oxygen excess (δ > 0) to oxygen stoichiometry (δ = 0) and finally to oxygen deficiency (δ < 0) before the perovskite decomposes. While it is straightforward to understand oxygen loss under low PO 2, the exact accommodating mechanism for oxygen excess at high PO 2 by the formation of cation vacancies is still debated. One reasonable hypothesis consistent with chemical expansion is the formation of cation vacancies VLa ′′′ via the following defect ′′′ and VMn 88 reaction. 13

TEC (×10–6/°C)

12

Sr

11

Ca

10

9 0

0.1

0.2 0.3 x in La1–x AEx MnO3

0.4

12.6 TECs as a function of doping level in La1–xAExMnO3 (AE = Ca and Sr) measured in air from 50 to 1000 °C. Reproduced by permission of The Electrochemical Society.

228

Solid oxide fuel cell technology 3.2

3.1

La0.8Sr0.2MnO3 + δ

3+d

3 873 K 973 K 1073 K 1173 K 1273 K 1273 K (Kuo) Decomp.

2.9

2.8

2.7 –30

–25

–20

–15

–10

–5

0

log PO (× 105 Pa) 2

12.7 Change of oxygen stoichiometry of La0.8Sr0.2MnO3+δ with PO2 and temperature. Copyright Elsevier.

3 O + 6Mn ′ → 3O × + V ′′′ + V ′′′ + 6Mn × Mn La Mn Mn O 2 2

[12.7]

where Mn ′Mn represents Mn2+, which is equilibrated by the surface charge × into Mn ′Mn and Mn .Mn disproportionation of Mn Mn × 2Mn Mn = Mn ′Mn + Mn .Mn

[12.8] 3+

4+

The creation of cation vacancies with oxidation of Mn to Mn is supported by the results of electron and neutron diffraction89 and density measurement.90 The creation of VLa ′′′ leads to a lattice expansion. With the formation ′′′ and VMn of oxygen vacancies at low PO 2, reduction of Mn3+ to the larger Mn2+ ion also results in lattice expansion. Another important aspect of doped LaMnO3 is any chemical reactivity with ZrO2-based electrolytes and LaCrO3-based interconnections. A general and widely accepted mechanism is the formation of the insulating phases La2Zr2O7 and SrZrO3 (for La1–xSrxMnO3 (LSM) cathode) at the contacting interface.91, 92 The presence of the pyrochlore phase La2Zr2O7 and/or the perovskite phase SrZrO3 at the interface is deleterious as it not only deactivates the catalytic activity for O2 reduction, but also increases the ohmic resistance, to decrease cell performance. A simple physical interpretation of the formation of La2Zr2O7 involves diffusion of Mn into the ZrO2 lattice, leaving La activity elevated in LaMnO3 and therefore promoting the chemical reaction between La and Zr. The study showed that Ca-doped LaMnO3 is chemically less active with ZrO2 than Sr-doped LaMnO3; Ca-doped LaMnO3 begins to react readily with ZrO2 above ca. 1300 °C whereas Sr-doped LaMnO3 begins to react above ca. 1200 °C.

Materials for solid oxide fuel cells (SOFCs)

229

The polarization behavior is the most important property of a cathode. It directly determines the performance of an SOFC. Typically, the polarization process of an SOFC originates from two sources: oxygen molecular diffusion and oxide-ion/electron transfer. The former is known as concentration polarization whereas the latter is often referred to as activation polarization. For an anode-supported SOFC, the cathode polarization is mainly dominated by the activation polarization owing to the fact that the cathode layer is very thin. However, both concentration and activation polarizations become significant to the performance of a cathode-supported SOFC. Analytical solutions to the concentration polarization η cconc as a function of current density have been given in equations [7.50] and [7.48a] for a cylindrical tubular cathode and equations [7.50] and [7.52] for a planar cathode. For the activation polarization process, a simple analytical solution is not always straightforward since it involves many elementary steps for a complete O2 reduction. These elementary steps on the cathode have been discussed in Chapter 7, including adsorption of O2 on the surface of doped LaMnO3, – , followed by surface diffusion dissociation of O2 into adsorbed Oads and/or O ads – of Oads and/or O ads to the triple phase boundary (TPB), where electrolyte (e.g. YSZ), cathode (e.g. LSM), and gas meet to complete the ionization of – to O O× (O2–) by electron transfer. Depending upon the rateOads and/or O ads limiting elementary step, the actual analytical equation for describing the activation process can be very different. A common assumption for the LaMnO3based cathode, particularly for cathode-supported SOFCs, is that the last – to O O× – is rate limiting. As a result, step – the ionization of Oads and/or O ads – to a simplified Butler–Volmer equation, assuming a single-electron ( O ads × O O ) rate-limiting step and symmetrical (α = 0.5) electron transfer, is often adopted to express the activation polarization η cact as a function of current density in equations [7.20] or [7.20a]. One of the decisive parameters for activation polarization is the linear TPB density. The greater the linear TPB density, the lower the activation polarization loss. In order to maximize the TPB density, the functional layer of a cathode is, in practice, made of a composite consisting of electrolyte (e.g. YSZ) and cathode (e.g. LSM) with a relative volumetric ratio of 50:50. The microstructural parameters are correlated with the effective charge-transfer resistance Rct(eff) in the low-current-density limit as shown in equation [7.21] from Tanner et al.66

12.1.3 Anode materials In order to facilitate the fuel oxidation reaction taking place at the anode, and collect the current, an electrically conducting catalyst is needed. Metallic Ni is a cost-competitive and performance-effective catalyst for oxidation of Hand C-containing fuels and a good electrical conductor for carrying electrical

230

Solid oxide fuel cell technology

current. However, direct use of Ni catalyst as an anode in the SOFC is not feasible because of the large mismatch in thermal expansion coefficient between Ni and the underlying ZrO2 electrolyte layer, as well as the great propensity of Ni particles to sinter, leading to loss of catalyzing surface area. If an SOFC was made of a pure Ni anode, the cell performance would degrade rapidly and delamination of the anode by the electrolyte would occur upon thermal cycling. In order to circumvent these problems and still use the Ni as a catalyst, a mixture of electrolyte powder (e.g. YSZ), Ni powder, and physical pores was proposed by Spacil16 about four decades ago. This idea proves to be one of the most important inventions in the history of SOFCs, and it continues to be the dominant anode template for today’s developments. The advantages of this composite anode are multiple. First, it alleviates the large thermal expansion mismatch existing between Ni and ZrO2, which increases the sustainability under thermal cycling. Second, it provides a large number of TPB sites for the fuel electro-oxidation reaction, facilitating the anode kinetics. Third, the porous structure ensures that the gaseous reactants diffuse into the TPB sites and the gaseous products diffuse out to the bulk fuel stream with minimal resistance. Fourth, the affinity provided by the ZrO2-based materials in both anode and electrolyte enhances their adherence and avoids interfacial cationic interdiffusion. Fifth, the presence of ZrO2 suppresses Ni particle sintering. This property ensures long-term stability of the anode. An ideal microstructure of the anode should, therefore, consist of an interlinking Ni pathway for current conduction on which a thin layer of ZrO2 is deposited with a percolated pore network. Figure 12.8 shows an example of the structure of an YSZ–Ni composite anode made with an electrochemical vapor deposition (EVD) process invented by Westinghouse. It is the requirement for a multi-phase, porous structure that makes the performance of the anode sensitive to the volumetric ratio of the phases as well as the porosity, thickness, relative and absolute particle size distributions of ZrO2 and Ni particles, and sintering temperature. It is often seen that the anode performance varies greatly from one manufacturer/laboratory to another. However, the general consensus in the SOFC community is that in order to make an optimal anode, the volume ratio of ZrO2/Ni should be in the range of 50/50 to 40/60, ZrO2 particles with d50 ~ 0.5 µm coexist with coarser Ni particles with a bimodal distribution at d50 ~ 0.5 µm and d50 ~ 10 µm with a volume ratio 6:1 for each mode, respectively. The resulting porosity is in the vicinity of 30 vol%. Determined by whether a dense electrolyte film can be achieved for an anode-supported SOFC, the sintering temperature of the anode is typically in the range of 1300–1400 °C. The activation polarization of an anode is generally described by the Butler–Volmer equations [7.20] or [7.20a], provided that the charge-transfer process is the rate-limiting step. For instance, the ScSZ + Ni cermet anode of a Siemens cathode-supported SOFC with H2 as a fuel exhibits a linear

Materials for solid oxide fuel cells (SOFCs)

231

20.00 µm

Ni

  →

→

8YSZ

12.8 An ideal microstructure exemplified by an EVD-made anode.

η–i behavior up to 0.6 A/cm2 in a temperature range of 900–1000 °C; from the slope of the η–i curves, the exchange current density is estimated to be in the neighborhood of 1.0–2.0 A/cm2. However, other elementary steps involved in anode kinetics could also compete to limit the rate of H2 oxidation, making the determination of the mechanism complicated. The elementary steps that are well accepted for anode kinetics are generally analogous to those of the cathode; these were discussed in Chapter 7. Molecules of H2 preferentially adsorb and disassociate on the surface of the catalytic Ni particles into Hads, followed by diffusion along the surface of Ni towards TPB sites where cations are available for oxidizing H species and O2– ions for forming gaseous H2O and releasing electrons to the external circuit via the following reaction ..

2H ads + O O× = H 2 O (g) + VO + 2e ′

[12.9]

If H2 adsorption and surface diffusion steps become the rate-limiting steps, the electrode polarization could vary with partial pressure of H2 and H2O93, 94; in this case, the polarization is generally described by a Langmuir isotherm.95, 96 As with the cathode concentration polarization, η aconc of an anode can also generally be formulated by a combination of Fick’s diffusion equation and the Nernst equation, as has been expressed in equations [7.58] and [7.59]. In comparison to the cathode, however, the magnitude of the anode η aconc is not an alarming issue even for a greater thickness, owing to a much

232

Solid oxide fuel cell technology

higher binary diffusivity of H2–H2O. The elegant combination of electrode substrate with faster diffusing species bestows on the anode-supported SOFC a great advantage over a cathode-supported SOFC with regard to concentration polarization loss. In recent years, a widely discussed and debated topic on the anode is the possibility of directly oxidizing the readily available hydrocarbons by an SOFC stack without undergoing a prior reforming process. The realization of such a process is technically important as it would not only simplify the SOFC system and make it cost competitive, a major issue for the commercialization of SOFC technology, but also increase the overall electrical efficiency. The primary challenge for such an SOFC system is, however, the formation of carbon on the SOFC stack by the mechanism of either solidcatalyzed reaction or gas-phase pyrolysis of hydrocarbons, by which the TPB sites are gradually blocked out. One of the effective approaches to avoid carbon formation is to substitute a transition-metal oxide for Ni as the catalyst for fuel oxidation since Ni is easily poisoned by ppm concentrations of sulfur and is a good catalyst for carbon formation. The challenge is to identify an oxide that is catalytically active and retains the high current density (or oxide-ion flux) of the cell. Since the Ni also provides current collection, the oxide should also be a good electronic conductor or the Ni network should be replaced by another metallic network. This approach was first demonstrated by Murray et al.97 who operated a cell with dry methane as fuel using a CeO2-containing anode below 650°C where the rate of carbon formation on the Ni network is constrained. No carbon formation was reported on the cell during and after the test. In recognizing that Ni is known as a good catalyst for carbon formation as well as for fuel oxidation, direct feed of hydrocarbons (n-butane and toluene in this case) into an SOFC without carbon deposition was further reported by replacing Ni with Cu as the current collecting network;98, 99 Cu is catalytically inactive for cracking the hydrocarbons as well as for fuel oxidation. As Cu is only used as an electronic conductor, not as a catalyst, CeO2 was added as a known ceramic catalyst for promoting oxidation of hydrocarbons to provide the needed catalytic activity for oxidizing fuels. In order to alleviate the dimensional instability of CeO2 in reducing atmospheres and to avoid high sintering temperatures, a porous ZrO2-based skeleton was first made, into which CeO2 and Cu were subsequently impregnated using aqueous solutions of the corresponding salts.100, 101 Since then, direct use of heavier hydrocarbons such as iso-octane,102, 103 decane, and diesel104 has been demonstrated in laboratory-scale SOFC tests. Exceptional sulfur tolerance of the Cu–CeO2 anode has also been reported.105 Alternative oxides that are mixed oxide-ion and electronic conductors in the reducing atmosphere at the anode have also shown sulfur tolerance and show promise as anode materials for an SOFC.37 Unfortunately, no direct utilization of hydrocarbons has been shown so far

Materials for solid oxide fuel cells (SOFCs)

233

on a larger-scale SOFC stack with a reasonable operating lifetime. One of the obstacles is how to deliver the dry hydrocarbons to the stack without dropping carbon on peripheral module components. As indicated in Chapter 10, the carbon formation process is largely dictated by the kinetics, not thermodynamics. Careful control of residence time of hydrocarbons on module components is critical to the successful operation of SOFC generators directly fueled by dry hydrocarbons.

12.2

CeO2-based solid oxide fuel cells

CeO2-based SOFCs are largely limited to low-temperature (e.g. <600 °C) applications where the electronic conduction in CeO2 electrolytes and the propensity for dimensional change due to loss of oxygen (oxygen nonstoichiometry) in reducing atmospheres can be significantly suppressed.

12.2.1 CeO2-based electrolytes One unique characteristic of pure CeO2–δ is the appreciable variation of oxygen stoichiometry, δ, with temperature and PO 2. This is the direct result of the Ce3+/Ce4+ redox couple existent in CeO2–δ. The phases formed under a given temperature and PO 2 vary sensitively with actual O/Ce ratio in the form of homologues of phases CenO2n–2, among which some intermediate phases can also be formed.106 Owing to the fact that volume change is always associated with a phase change of CeO2–δ and that the electrical conductivity of CeO2–δ is generally low, there is no practical use of CeO2–δ as an electrolyte for SOFCs. However, like ZrO2, substitution of alkalineearth or rare-earth cations into the Ce4+ lattice can lead to the stabilization to room temperature of the fluorite cubic phase, a crystal structure favorable for oxide-ion conduction. More importantly, the acceptor doping creates oxygen vacancies, the essential element for oxide-ion conduction to charge compensate the foreign and host cations, as illustrated in Table 12.1. The oxide-ion conductivity of doped CeO2 electrolytes is generally higher than that of ZrO2-based electrolytes, particularly at lower temperatures. A conductivity peak at a specific dopant level for a given type of dopant, or a maximum conductivity for a specific cation at a fixed dopant concentration, resembles closely the behavior of ZrO2-based electrolytes. Figures 12.9 and 12.10 show, respectively, the two maxima mentioned above for CeO2-based electrolytes.107 An interpretation similar to that for ZrO2-based electrolytes can be applied to CeO2-based electrolytes. The conductivity maximum with concentration variation is mainly due to the interaction between dopant M Ce ′ .. (or M Ce ′′ ) and oxygen vacancies VO , leading to the formation of associates .. .. (or clusters) 2 M Ce ′ – VO (or M Ce ′′ – VO ). As additional energy is needed to free a vacancy from the association, the Arrhenius plot of conductivity often

234

Solid oxide fuel cell technology 100

10–2 s (S/cm)

900 °C 800 °C 700 °C 10–4

600 °C

(ZrO2)1–x(CaO)x at 800 °C

10–6

0

0.1

0.2

0.3 X

500 ° C

0.4

0.5

0.6

12.9 Oxide-ion conductivity of (CeO2)1–x(Sm2O3)x as a function of doping level x at various temperatures. Copyright Elsevier.

log sT (S/(cm K))

2.0 1.8

Sm 3+ Gd 3+ Dy 3+ Ho3+ Y 3+ Nd 3+ Yb 3+ Ca 2+ La 3+

1.6

Sr 2+

1.4 0

–0.2

Mg2+ 3+ Ce4+ rc (M ) 0.09

rc (M 2+)

0.10 0.11 0.12 0.13 Radius of dopant cation (nm)

Ba2+ 0.14

12.10 Oxide-ion conductivity of doped ceria at 800 °C against the radii of dopant cations. Copyright Elsevier.

shows a kink (or a curvature) towards a low-temperature regime to reflect the increase in activation energy. On the other hand, the maximum conductivity observed for a specific dopant is found with the closest match in ionic radius of the dopant cation and the host Ce4+, which leads to a minimization of the lattice elastic strain. For CeO2-based electrolytes, the most favorable dopants of the rare-earth group are Gd and Sm. The most appreciable disparity between CeO2-based and ZrO2-based

Materials for solid oxide fuel cells (SOFCs)

235

electrolytes is the magnitude of the electronic conductivity. The presence of × ), considerably a variable Ce valence, namely Ce3+ ( Ce ′Ce ) and Ce4+ ( Ce Ce elevates the excess electronic conduction in reducing atmospheres by the following defect reaction .. × O O× + 2Ce Ce = 1 O 2 + VO + 2Ce ′Ce 2

[12.10]

The excess electrons in the form of small polarons migrate via the Ce4+–Ce3+ redox couple across the lattice, giving electronic conduction. A good compilation of the oxide-ion conductivity for CeO2-based electrolytes by Steele44 for Rhodia-made low-SiO2 (<30 ppm) Ce0.9Gd0.1O1.95 (GDC10) is shown in Chapter 4. It is evident that the clusters disassociate above 400 °C. The electronic conductivity σe′ and characteristic Pe′ of CeO2based electrolytes along with other electrolytes are given in Table 4.1. The significance of electronic conduction at higher temperatures and lower PO 2 confines CeO2-based electrolytes for SOFCs to low-temperature applications, e.g. below 600°C, where the electronic conductivity can be considerably suppressed to an acceptable level. If this effect were not considered, the electrical efficiency of the SOFC would be greatly compromised. In Chapter 4, a case study has been presented to show how electronic conduction affects the PO 2 distribution and leakage current density of CeO2 electrolytes. The appearance of a peak in cell efficiency at low load current is a characteristic of mixed-conducting electrolytes used in SOFCs.

12.2.2 Cathode materials LaCoO3- and LaFeO3-based perovskites are the leading cathode materials for low-temperature CeO2-based SOFCs. In addition to a better thermal expansion match to the CeO2-based electrolyte, these two materials (as well as their mixtures in general) have appreciable mixed oxide-ion and electronic conduction by which the oxygen reduction process extends from the TPB reactive sites into the bulk of the cathode layer. As a result, the polarization losses are typically smaller than those of LaMnO3-based cathode materials that rely largely on the density of the TPB. Another unique characteristic of LaCoO3- and LaFeO3-based materials is that they do not react with CeO2based electrolytes as readily as ZrO2-based ones.108 This advantage can be best utilized by cathode-supported SOFCs, which require a co-sintering process for electrolyte and cathode without causing a substantial performance loss. The common dopants for A-sites of LaCoO3 and LaFeO3 are alkalineearth elements such as Ca and Sr whereas a mutual occupation of Co and Fe on the B-sites has proven to be the best choice. Increasing the doping level generally increases the electrical conductivity in both systems. Figure 12.11

236

Solid oxide fuel cell technology t (°C) 1000

300

100

La1–xSrxCoO3–d

4.0

PO = 1 atm 2

x = 0.5 0.2

0.3

log s (S/cm)

3.0 0.7 0

0.1

2.0

1.0

0.0

1.0

2.0 1000/T (K–1)

3.0

12.11 Temperature dependence of electrical conductivity of La1–xSrxCoO3–δ measured in Po2 = 1 atm. Reproduced by permission of The Electrochemical Society.

shows the effect of the Sr level on the conductivity of LaCoO3 measured at various temperatures and PO 2 = 1 atm.109 The electronic conduction changes from semiconducting to metallic at different dopant levels and temperatures. This change is due to an increase in the Co–O–Co interactions of the σbonding 3d electrons as well as to thermal excitations to a higher spin state on the cobalt ions.110 In contrast, LaFeO3-based perovskites exhibit only semiconductor behavior with a much lower electronic conductivity than that of LaCoO3-based perovskites. On the other hand, the TECs of LaFeO3-based materials have a better match to CeO2-based electrolytes than those of LaCoO3based materials. In order to utilize best the advantages exhibited by these two perovskite materials, a combination of LaCoO3- and LaFeO3-based materials has been systematically studied.111, 112 Figure 12.12 shows the variations of the electrical conductivity with temperature and Sr level of the La1–xSrxCo0.2Fe0.8O3–δ system. At an Sr level greater than x = 0.1, a peak conductivity was observed to occur at a certain temperature that shifts towards a lower temperature as the Sr level increases. This behavior was further explained by Stevenson et al.113 to be related to an increase in oxygen stoichiometry with temperature. Loss of oxygen creates oxygen vacancies,

Materials for solid oxide fuel cells (SOFCs) 550 °C

350

x = 0.4

300

Conductivity (S/cm)

237

250

x = 0.3

200

600 °C

x = 0.2

800 °C

150 100

x = 0.1

1000 °C

50

x=0 0

0

200

400

600 800 Temperature (°C)

1000

1200

12.12 Electrical conductivity of La1–xSrxCo0.2Fe0.8O3–δ as a function of temperature and Sr content in air. Copyright Elsevier.

one of which eliminates two electron holes, therefore leading to a decrease in conductivity. With incorporation of this factor in the pre-exponential term of the Arrhenius equation, the calculated σ–T curves fit well with the measured curve. The composition for the La–Sr–Co–Fe–O system was optimized to La0.6Sr0.4Co0.2Fe0.8O3–δ, which reflects a good balance between conductivity and TEC for CeO2-based electrolytes. The PO 2 dependence of electronic conductivity of LaCoO 3-based perovskites suggests a p-type conduction mechanism. Figure 12.13 shows that σ decreases with PO 2 in the range of 1–10–4 atm, which can be interpreted by the following defect reaction

1 O + V .. + 2Co × → O × + 2Co . O Co O Co 2 2

[12.11]

As PO 2 decreases, the reaction shifts to the left, resulting in the creation of one oxygen vacancy for two electron holes consumed. The above p-type conduction mechanism has also been supported by Seebeck measurements where a positive Seebeck coefficient is typically observed. For LaFeO3based perovskites, a transition from p-type to n-type conduction is observed as PO 2 decreases,114 see Fig. 12.14. Such a transition can be explained with the aid of the following defect reaction for the p-type conduction 1 O + V .. + 2Fe × → O × + 2Fe . Fe Fe O O 2 2

and for n-type conduction

[12.12]

Solid oxide fuel cell technology 1.5

x = 0.1 0

0.2

10–3 s (S/cm)

1.0

0.3

0.5 0.5 0.7

La1–x SrxCoO3–d at 900 °C 0

–4

–2 log PO (atm)

0

2

12.13 Dependence of electrical conductivity of La1–xSrxCoO3–δ on PO2 measured at 900 °C. Reproduced by permission of The Electrochemical Society.

2.0

x = 0.6 log s (S/cm)

238

1.0

1200 °C 1100 °C

0

1000 °C –15

–10

–5 log PO (Pa)

0

5

2

12.14 Dependence of electrical conductivity of La0.4Sr0.6FeO3 on PO2 and temperature. Copyright Elsevier.

Materials for solid oxide fuel cells (SOFCs) .. O O× + 2Fe ×Fe → 1 O 2 + VO + 2Fe ′Fe 2

239

[12.13]

with a charge disproportionation existing in the Fe species 2Fe ×Fe = Fe ′Fe + Fe .Fe

[12.14]

Unlike LaMnO3-based cathode materials, LaCoO3- and LaFeO3-based perovskites tend to lose their lattice oxygen more easily. Figure 12.15 shows the variations of oxygen non-stoichiometry 3 – δ in Sr-doped LaCoO3–δ with temperature and Sr doping level.115 No oxygen excess was observed for all the conditions. In comparison to doped LaCoO3, Fig. 12.16 shows that doped LaFeO3 has a smaller oxygen deficiency under a similar condition.116 Yang and Lin117 proposed the following semi-empirical equation to describe the variation of non-stoichiometry of oxygen δ with PO 2 for the La–Sr–Co–Fe– O system with reasonable agreement to the experimental data

3.0 0.1 0.2

3–d

0.3

0.5

2.9

x = 0.7

PO = 1 atm 2

PO = 0.21 atm 2

2.8

0

400

800

t (°C)

12.15 Dependence of oxygen non-stoichiometry of La1–xSrxCoO3–δ on x and temperature. Copyright Elsevier.

240

Solid oxide fuel cell technology 3.00

3–d

2.90

600 °C 700 °C 800 °C

2.80

900 °C

2.70 1000 °C –20

–15

1200 °C 1100 °C –10 –5 log PO (atm)

0

2

12.16 Dependence of oxygen non-stoichiometry of La0.75Sr0.25FeO3 on PO2 and temperature. Copyright Elsevier.

δ=

3 KPOn2 1 + KPOn2

[12.15]

where K and n are the temperature-dependent constants. The creation of oxygen vacancies by oxygen non-stoichiometry in the cathode atmosphere promotes mixed electronic and oxide-ion conduction, a favorable property for facilitating the electrode kinetics. For example, the ionic and electronic conductivities of La0.6Sr0.4Co0.2Fe0.8O3–δ at 900 °C in air were reported to be 0.23 and 252 S/cm, respectively,118 while for La0.8Sr0.2MnO3 the corresponding conductivities are in the order of 10–3 and 102 S/cm, respectively. However, the number of oxygen vacancies must be controlled to avoid the creation of vacancy-ordering and thermal expansion mismatch with the electrolyte. The presence of a mixed electronic and oxide-ion conduction in LaCoO3and LaFeO3-based systems diversifies the mechanism of the rate-limiting step of the oxygen reduction process. As mentioned previously, the activation polarization of a porous LaMnO3-based cathode is generally dominated by the charge-transfer process that occurs at the TPB. As the solid-state oxygen diffusion in the bulk of a doped LaCoO3 is significantly enhanced by the electronic conductivity, the rate-limiting step for the overall activation polarization changes to surface oxygen exchange (or adsorption) for a dense film118, 119 and becomes a mix of surface exchange and solid-state diffusion for a porous film.120, 121 The reactive zone for oxygen reduction as a result of mixed conduction is extended from the interface to the bulk of the cathode by a few microns.120

Materials for solid oxide fuel cells (SOFCs)

241

12.2.3 Anode materials The constituents of the anode of CeO2-based SOFCs very much resemble those of ZrO2-based SOFCs, i.e. a mixture of CeO2 electrolyte, metallic Ni, and physical pores at a preferential volume ratio. Like LaCoO3-based cathode materials, the introduction of mixed electronic and oxide-ion conductivity in CeO2-based materials as a result of the reducing atmosphere considerably enlarges the TPB reaction zone and makes it an excellent anode material. Therefore, lower electrode polarization is generally observed for a CeO2based anode, suitable for low-temperature SOFC applications. Figure 12.17 compares the performance of YSZ + Ni, ScSZ + Ni, and Sm–CeO2 (SDC) + Ni anodes measured in Siemens SFC. Clearly, SDC + Ni and ScSZ + Ni outperform YSZ + Ni, particularly at lower temperatures. As mentioned before and to be discussed later, even for ZrO2- and LaGaO3-based SOFCs, CeO2based anodes are often selected to boost the electrical performance. However, in these two cases the thermal expansion mismatch and interfacial cationic diffusion between electrolyte and anode have to be addressed. Another noticeable advantage for CeO2-based anodes is the exceptional tolerance to higher sulfur levels in the fuel stream. This is probably the result of a stronger chemical affinity between CeO2 and S than between Ni and S.

0.250

Anode polarization ASR (Ω cm2)

8YSZ + Ni 0.200

0.150

SDC20 + Ni 0.100

10ScSZ + Ni

0.050

0.000 750

800

850

900

950

1000

t (°C)

12.17 Comparison of anode polarization resistance of YSZ + Ni, ScSZ + Ni, and SDC + Ni as a function of temperature.

1050

242

Solid oxide fuel cell technology

Under typical SOFC operating conditions, Ce2O2S is thermodynamically stable and constantly keeps S away from Ni, leaving Ni uncovered by S. In an internal study of Siemens SFC, adding CeO2 into the anode YSZ + Ni could increase S tolerance from 0.1 to 10 ppm without causing voltage degradation.

12.3

LaGaO3-based solid oxide fuel cells

A recent important development in the field of SOFC research is the discovery of LaGaO3-based solid electrolytes. Based on the fact that it has higher oxide-ion conductivity at lower temperatures and shares the same crystal structure with cathode materials, it is widely anticipated that the LaGaO3based material can be an excellent choice of electrolyte for SOFCs operating below 800 °C.

12.3.1 LaGaO3-based electrolytes Oxides with perovskite and related structures have been observed in the past to exhibit moderate oxide-ion conductivity at elevated temperatures. For example, a measurement of oxide-ion conductivity of Ba2In2O5 revealed a first-order transition at 930 °C from the brownmillerite to an oxygen-deficient perovskite structure that demonstrated fast oxide-ion conduction in the perovskite cation framework where oxygen vacancies are present and disordered.18, 61 The perovskite system showing superior oxide-ion conductivity was first reported by Ishihara et al.19 and Feng and Goodenough20 in the system of Sr- and Mg-doped LaGaO3 (LSGM). The oxide-ion conductivity of the optimal LSGM composition reaches 0.15 S/cm at 800 °C, a value that could only be achieved for 8YSZ at 1000 °C, and is stable over time and a broad range of PO 2 values (1–10–23 atm). A comparison of the oxide-ion conductivity of LSGM with other electrolytes has been given in Fig. 5.4. Clearly, LSGM exhibits its superiority over other electrolytes over the plotted temperature range of 650–1000 °C. The phase relationship of Sr- and Mg-doped LaGaO3 systems is not simple. Undoped LaGaO3 has an orthorhombic GdFeO3-type crystal structure with lattice parameters of a = 5.496 Å, b = 5.524 Å, and c = 7.787 Å in space group Pbnm. Upon doping with Sr and Mg on La and Ga sites, respectively, the crystal structure gradually changes to a primitive cubic structure. Huang et al.21 systematically studied the phase relationships in the La–Sr–Ga–Mg– O system. The single cubic-phase region identified, as shown in Fig. 12.18, is fairly narrow with space group Pm3m (221) and lattice parameter a = 3.9103 ± 0.0007 Å. The major impurity phases are LaSrGa3O7 (JCPDS 450637, S.G.: P421m (113)) and LaSrGaO4 (JCPDS 24-1208, S.G.: I4/mmm (139)).

Materials for solid oxide fuel cells (SOFCs) SrO

LaO1.5

243

P-cubic 214 + P-cubic 237 + P-cubic 214 + 237 + P-cubic

(214)

(237)

MgO

GaO1.5

Single cubic-phase region (214)

(237)

La

Ga

3

O

214 + cubic region 237 + cubic region 214 + 237 + cubic region

12.18 Phase diagram of the La–Sr–Ga–Mg–O system. P-cubic, primitive cubic.

The Arrhenius plot of oxide-ion conductivity in LSGM revealed a progressive curvature with temperature. This behavior was explained by Huang et al.21 as being the result of a nucleating process for the formation of ordered .. vacancy clusters between VO and dopant ions SrLa ′ and Mg ′Ga . The proposed model simplified the observed curvature into two straight lines separated by .. .. a characteristic temperature T*. Below T*, the VO – 2SrLa ′ and VO – 2Mg ′Ga clusters form to add a trapping energy ∆Ht to the motional enthalpy ∆Hm as .. .. discussed in Chapter 5. Above T*, all VO – 2SrLa ′ and VO –2 Mg ′Ga clusters dissolve into the oxygen sites and the activation energy reduces to ∆Hm. Based on the proposed model, T* was experimentally determined to be approximately 600 °C for the LSGM electrolyte. The isothermal oxide-ion conductivity of LSGM was also studied by Huang et al.21 by systematically varying Sr and Mg dopant concentrations. Figure 12.19 shows the contour plot of oxide-ion conductivity at 800°C. Based on this plot, the optimal composition of the single cubic LSGM phase

244

Solid oxide fuel cell technology 0.30

0.10

0.08

0.12

0.12

0.12 0.25

0.10

0.12

0.14 0.14

0.12

0.20

0.14 0.14

0.12

y

0.10

0.12 0.14

0.15 0.14 0.10

0.08

0.12 0.12

0.10

0.06

0.08 0.10 0.12 0.05 0.05

0.10

0.10

0.12

0.15

0.10 0.20

0.08 0.04 0.25

0.30

x

12.19 Isothermal oxide-ion conductivity contour of LSGM electrolyte at 800 °C.

for the highest conductivity was found to be La0.80Sr0.20Ga0.83Mg0.17O3–δ, which has an Arrhenius relationship as follows ln σ VO.. T = 14.024 –

0.815 (eV) kT

(S/cm) t > 600 ° C

[12.16]

ln σ VO.. T = 17.569 –

1.076 (eV) kT

(S/cm) t < 600 ° C

[12.17]

Although the LSGM system exhibits excellent oxide-ion conductivity, one of the concerns is the propensity for Ga to be reduced to Ga2+, introducing n-type electronic conduction, via the following defect reaction when it is exposed to reducing atmospheres .. × O O× + 2Ga Ga = 1 O 2 + VO + 2Ga ′Ga 2

[12.18]

Elevating temperature and decreasing PO 2 would promote the reaction. It is also possible that loss of Ga2O3 in LaGaO3 could occur during high-temperature fabrication of LSGM materials via the following reaction (Ga2O3)LSGM = Ga2O(g) + O2(g) or during operation of an SOFC via the following reaction

[12.19]

Materials for solid oxide fuel cells (SOFCs)

2H 2(g) + (Ga 2 O 3 ) LSGM = Ga 2 O (g) + 2H 2 O (g)

245

[12.20]

The consequence of Ga loss would be significant as the single-phase composition would shift into a two-phase region, causing a decrease in oxideion conductivity. As a prospective electrolyte, LSGM has to be stable in a reducing atmosphere. Hilpert and colleagues122, 123 systematically studied the vaporization behaviors of two compositions of LSGM, namely La0.9Sr0.1Ga0.8Mg0.2O3–δ and La0.85Sr0.15Ga0.85Mg0.15O3–δ, using Knudsen effusion mass spectrometry in the temperature range of 1618–1886 K and in a mixture of H2–H2O. They found that the thermodynamic activity of Ga2O3 in La0.9Sr0.1Ga0.8Mg0.2O3–δ is a(Ga2O3) = 2.3 × 10–5 for 1073K. Using available thermodynamic data for reaction [12.20] and known a(Ga2O3), the partial pressure of the formed Ga2O(g) as a function of temperature under a typical SOFC operating condition can be calculated; this is shown in Fig. 12.20. Clearly, the vaporization process of Ga is strongly temperature dependent. For a typical SOFC operating temperature below 1000 °C, the loss of Ga is negligible. However, the loss of Ga cannot be ignored if the processing temperature of LSGM is very high, as in the case of plasma spraying LSGM materials. The electronic conductivity of LSGM material was also investigated by Kim and Yoo45 and Jang and Choi124 using the Hebb–Wagner blocking electrode method. The levels of both p- and n-type electronic conductivities are generally higher than in ZrO2-based electrolytes, but lower than in CeO2-based ones. Increasing the Sr and Mg dopant level decreases both p- and n-type electronic conductivities. The characteristic partial pressures of oxygen for hole- and excess-electron conductions were given,45 respectively, in Table 4.1. The 1 × 10–6

PH O /PH = 3/97

1 × 10–8

2

2

PGa

2O(g)

(atm)

1 × 10–10 1/1

1 × 10–12

85/15

1 × 10–14 1 × 10–16 1 × 10–18 1 × 10–20 1 × 10–22 1 × 10–24 500

600

700

800 t (°C)

900

1000

1100

12.20 Plot of partial pressure of Ga2O(g) as a function of temperature.

246

Solid oxide fuel cell technology

3000

20

1800

T (°C) 1300

1000

700

p-type domain

0 Electrolytic domain

2

log PO (atm)

sp = sn

–20

tion = 0.5

tion = 0.99

n-type domain –40

0.2

0.4

0.6

0.8

1.0

1000/T (K–1)

12.21 Constructed electrolytic domain of La0.9Sr0.1Ga0.8Mg0.2O3–δ. Copyright Elsevier.

electrolytic domain constructed from the measured p- and n-type electronic conductivities is shown in Fig. 12.21, which indicates negligible electronic conduction under a typical SOFC operating condition.

12.3.2 Cathode materials The best cathode materials for LaGaO3-based SOFCs resemble those used for CeO2-based SOFCs, i.e. LaCoO3-based perovskites; other rare-earth Cocontaining perovskites have also been proven to be equally capable. The selection of this class of materials is primarily based on the fact that LaCoO3based perovskites have a good chemical compatibility125 with LaGaO3-based perovskite electrolyte in addition to their excellent electrocatalytic activity for oxygen reduction as described in Section 12.2.2. Such a chemical compatibility has also been indirectly confirmed by Ishihara et al.,126,127 who purposely substituted a small amount of transition metals such as Fe, Co, and Ni into the B-site of LaGaO3 perovskites. It was reported that the oxide-ion conductivity of LSGM could be further enhanced by such a doping although the introduced electronic conduction is a major concern when using these materials as electrolytes. The thermal expansion mismatch between doped LaCoO3 and LSGM is somewhat mitigated by using a thin-film LaCoO3 cathode configuration on a thick LSGM electrolyte support. Huang et al.62,128 also studied other perovskite cathode materials for LaGaO3-

Materials for solid oxide fuel cells (SOFCs)

247

based SOFCs. It was found that SrCo0.8Fe0.2O3–δ (SCF) exhibited the lowest cathode overpotential, followed by La 0.8 Sr 0.2Co 0.8 Ni 0.2 O 3– δ (LSCN), La0.6Sr0.4CoO3–δ (LSCo), La0.7Sr0.3Fe0.8Ni0.2O3–δ (LSFN), and La0.85Sr0.15 MnO3–δ (LSM). Clearly, the mixed conductivity of these materials is critical in determining the catalytic activity of oxygen reduction and therefore the polarization loss. Independent oxygen permeation studies.129,130 have also confirmed that SCF has the highest oxygen flux density, suggesting the highest oxide-ion conductivity present in this material.

12.3.3 Anode materials A logical choice of anode for LaGaO3-based SOFCs is a mixture of LSGM and Ni, analogy to ZrO2 + Ni and CeO2 + Ni for ZrO2-based and CeO2-based SOFCs, respectively. However, a single-cell test revealed an initially low anode overpotential that quickly degraded with power cycle.131 Ni coarsening and reactivity of LSGM with Ni mechanisms were speculated to be the causes of the degradation. A post-mortem microstructural study131 seemed to support the latter mechanism, considering that LSGM would react with NiO to form LaNiO3 and La2NiO4 during the anode fabrication process. In order to solve this instability problem, the LSGM + Ni composite was replaced by CeO2 + Ni, which has been proven to be an excellent anode material owing to the exceptional mixed conduction of CeO2 in reducing atmospheres. Although such a change considerably improved the stability of LaGaO3-based SOFCs, the fundamental reaction between LSGM and Ni at the anode/electrolyte interface remains active. In order to avoid the direct contact between LSGM and Ni, it was proposed that a buffer layer, comprising only doped CeO2, should be placed in between LSGM and the anode. As a result, the electrical performance was increased by a factor of two.132 The use of an Ni-free buffer layer prevents LSGM from reacting with Ni, but the reaction between LSGM and CeO2 still exists. This was demonstrated by the work of Hrovat et al.,133 in which La2O3 in LSGM was found to diffuse easily into CeO2, forming Ce1–xLaxO2–δ solid solution at 1300 °C, which precipitated out the highresistive phase LaSrGa3O7 in the LSGM matrix. Since the CeO2-based anode of LSGM-based SOFCs is typically fabricated at 1300–1350 °C, the above reaction could well occur and form a resistive layer at the anode/electrolyte interface prior to the test. In order to utilize best the concept of a buffer layer while preventing the reaction between LSGM and CeO2, a modified composition of the buffer layer was proposed to counterattack the La diffusion, i.e. use of La0.4Ce0.6O2–δ to replace commonly used Sm0.2Ce0.8O2–δ as the buffer layer.38 The equality in the chemical activity of La in the buffer layer and in LSGM stops the La diffusion, therefore avoiding the formation of LaSrGa3O7. As a result, both the power density and the long-term stability of the cell were significantly improved.38

248

12.4

Solid oxide fuel cell technology

Interconnects and cell-to-cell connectors

The interconnect in SOFCs serves as an electrical connector between two adjacent cells and a separator of cathode and anode gases. Therefore, the interconnect must be: (a) a good electronic conductor and oxide-ion insulator; (b) chemically stable in both oxidizing and reducing atmospheres; (c) thermally matched to the neighboring cathode and anode; (d) physically gas tight. These combined requirements eliminate all but a few metals such as Cr2O3– forming high-temperature alloys and ceramics such as alkaline-earth-doped LaCrO3. At an earlier stage of SOFC development, the high operating temperature (~1000 °C) prohibited the use of metals as interconnects. The only suitable material for high-temperature SOFCs was alkaline-earth-doped LaCrO3 or other Cr-containing perovskites. After the emergence of highperformance anode-supported SOFCs, the operating temperature of SOFCs was significantly lowered to the range where cost-effective, commercially available high-temperature metallic alloys are suitable for use. Metallic interconnects are generally preferable over ceramic LaCrO3-based interconnects in many aspects. First, metallic interconnects are truly electronic conductors and oxide-ion insulators whereas ceramic interconnects are mixed electronic and oxide-ion conductors, particularly at lower PO 2 and high temperatures where oxygen vacancies are available. Second, metallic interconnects are more cost-effective and easy to fabricate than ceramic ones. Third, metallic interconnects are mechanically strong, durable, easily machineable, and, more importantly, dimensionally stable in both oxidizing and reducing atmospheres. Fourth, metallic interconnects conduct heat much more efficiently. This is particularly important for thermal management in the modern planar SOFC design where a metallic interconnect is typically used as the mechanical support of a thin assembly of cathode/electrolyte/anode and a distributor of cathode and anode gases. The heat generated from fuel oxidation must be easily removed by air or the endothermic internal steam-reforming reaction. In contrast, ceramic interconnects are relatively weaker in terms of mechanical strength, are dimensionally unstable owing to loss of oxygen at low PO 2, and conduct heat slowly. Nonetheless, ceramic interconnects are still in use for high-temperature cathode-supported SOFCs by Siemens SFC for the reason that they do not form high-resistance oxide scale, as is the case with metallic interconnects, and they give a very low contact resistance to the current collector, typically Ni felt, with the aid of a plated Ni film as a contact layer. The low contact resistance turns out to be one of the major advantages of ceramic interconnects over metallic interconnects – whose contact resistance is usually high owing to insufficient contact area and formation of a lowconductivity oxide scale. In this section, we will review properties of both LaCrO3-based ceramics and promising metallic interconnects. The influence of cell design on the selection of current collector in SOFC stacks will also be addressed.

Materials for solid oxide fuel cells (SOFCs)

249

12.4.1 LaCrO3-based ceramic interconnects Undoped LaCrO3 has an orthorhombic GdFeO3-type crystal structure with lattice parameters a = 5.477 Å, b = 5.514 Å, c = 7.755 Å in space group Pbnm. Pure LaCrO3 is a poor electronic conductor and therefore has little practical use in SOFCs. Substituting alkaline-earth metals into the La-site considerably increases the electrical conductivity as a result of increased concentration of electron holes by the doping reaction ..

LaCrO 3 2MO  → 2M ′La + 2O O× × VO

[12.21]

1 O + V .. + 2Cr × = O × + 2Cr . O Cr O Cr 2 2

[12.22]

Figure 12.22 shows the Arrhenius plot of the electrical conductivity of the La1–xCaxCrO3–δ system measured in air.134 The electrical conductivity evidently increases with Ca doping as predicted by equations [12.21] and [12.22]. The p-type small-polaron hopping conduction mechanism is supported by Fig. 12.23, in which the electrical conductivity σ is shown to decrease with PO 2 at a given temperature and a given composition. Such behavior has been 12

11

In air

ln sT (SK/(cm))

10

9

8

7

La1–xCaxCrO3–d

x = 0.1 x = 0.2 x = 0.3

6

5 0.5

1.0

1.5

2.0 2.5 1000/T (K–1)

3.0

3.5

12.22 Temperature dependence of electrical conductivity of La1–xCaxCrO3 with x = 0.1, 0.2, and 0.3 measured in air. Reproduced by permission of The Electrochemical Society.

250

Solid oxide fuel cell technology 2.5

log s (S/cm)

1.5

1.0 La0.7Ca0.3CrO3–d 900 °C

0.5

950 °C 1000 °C

0.0

–0.5 –25

1050 °C

–20

–15 –10 log PO (atm)

–5

0

2

12.23 PO2 dependence of electrical conductivity of La0.7Ca0.3CrO3–δ at different temperatures. Reproduced by permission of The Electrochemical Society. 3.05

3–δ ln La1–xCaxCrO3–d

3.00

900 °C 950 °C

2.95

1000 °C 1050 °C

2.90

2.85

2.80 –25

La0.7Ca0.3CrO3–d

–20

–15 –10 log PO (atm)

–5

0

2

12.24 Simulated oxygen non-stoichiometry of La0.7Ca0.3CrO3–δ as a function of PO2. Reproduced by permission of The Electrochemical Society.

empirically expressed in equation [4.76]. Based on the defect model described by equation [12.22], the oxygen non-stoichiometry 3 – δ as a function of PO 2 for a given composition La0.7Ca0.3CrO3–δ was calculated out and is shown in Fig. 12.24. A moderate change in 3 – δ only occurs in the PO 2 range of

Materials for solid oxide fuel cells (SOFCs)

251

∆L / L

(%)

10–20–10–10 atm, beyond which 3– δ tends to be invariant with PO 2. One of the drawbacks resulting from oxygen non-stoichiometry in LaCrO3based materials is the emergence of mixed electronic and oxide-ion conduction. For an interconnect material, this property is highly undesirable as the mixed conductivity would lead to a leaked oxygen flux by which excessive fuel could be utilized without generating useful external electrical current. A detailed analysis of the leakage current density in doped LaCrO3 perovskites has been given in Chapter 4. Lowered fuel efficiency and therefore electrical efficiency is a direct consequence of mixed conduction in the interconnect and electrolyte of an SOFC. The magnitude of oxygen flux density shown in Fig. 4.10 is unacceptable for practical use in an SOFC, even at 900 °C. In practice, a chemical composition has to be formulated that suppresses the mixed conductivity. Another problem arising from PO 2-dependent oxygen stoichiometry is the variation of thermal expansion coefficient with PO 2. This is explicable in that loss of oxygen creates oxygen vacancies accompanied by reduction of Cr4+ . ) to larger Cr3+ (Cr × ) , allowing the lattice to expand. This behavior is (CrCr Cr clearly illustrated in Fig. 12.25, where the dimensional change of La0.79Sr0.20CrO3–δ increases with decreasing PO 2. The differential thermal expansion imposed on both sides of an interconnect leads to a non-uniform strain distribution within the material, resulting in warping and even cracking. The early development of ceramic interconnect-supported monolithic SOFCs was thus significantly limited. This problem is, however, knowingly avoided by depositing a thin film of interconnect on a substantially larger cathode support in the Siemens/Westinghouse design. One of the fabrication-related issues with LaCrO3-based materials is the difficulty of sintering them into dense ceramics. This is due to the refractory nature of LaCrO3-based materials. In order to overcome this problem, excess 0.30

La0.79Sr0.2CrO3–d

0.25

1000 °C

0.20 0.15 0.10 0.05 0.00 –20

–15

–10 log PO (atm)

–5

0

2

12.25 Isothermal lattice expansion of La0.79Sr0.20CrO3–δ with PO2 .

252

Solid oxide fuel cell technology

Ca or deficient Cr in Ca-doped LaCrO3 perovskites was introduced.15, 135 A small excess in Ca level, x = 0.01, for example, considerably lowers the sintering temperature required to achieve dense ceramics.136 The sintering mechanism for a Ca level of x > 0.25 can be interpreted by the formation of a transient liquid phase CaCrO4, which causes an initial rapid sintering shrinkage. Unlike metallic interconnects, LaCrO3-based materials are much more stable with regard to Cr vaporization under SOFC operating conditions.137 However, fabrication of LaCrO3-based interconnects at elevated temperatures needs to consider the loss of Cr. The plasma-sprayed LaCrO3 interconnect in Siemens cathode-supported SOFCs is a good example.

12.4.2 Metallic interconnects Resistance to high-temperature oxidation is an essential requirement for metallic interconnects used in planar SOFCs. This requirement limits the potential candidates to high-temperature superalloys and stainless steels that form either Cr2O3 or Al2O3 scale on the surface of the base metals. The oxidation kinetics in terms of grown scale thickness x is governed by Wagner’s equation of oxidation which assumes dominant bulk diffusion x 2 = kpt =

kg E t t= k go exp  – ox  2 2  kT  ( yO ρ ) ( yO ρ )

[12.23]

where kp and kg are the parabolic rate constants expressed in thickness and mass, respectively; yO is the weight fraction of oxygen in the oxide (yO = Table 12.2 Parabolic rate constants of several commercial Ni-based and Fe-based alloys at 800 °C and scale compositions Alloy

Rate constant kg (× 10–14 g2/(cm4s))

Phase composition in the scale

TEC (× 10–6/K)

Reference

Crofer22 APU E-Brite

7.96

AISI446

13.32

Haynes 230

3.61

Cr5FeY2O3

~0.1

Cr2O3, (Mn, Cr)3O4 Cr2O3, (Mn, Cr, Ni)3O4 Cr2O3

ZMG232

kp = 6.2 × 10–6 µm2/s

Cr2O3, (Mn, Cr, Fe)3O4, SiO2

12.2 (RT–760 °C) 11.8 (RT–500 °C) ~10.9 (20–300 °C) 15.2 (RT–800 °C) 11.8 (RT–1000 °C) 12.5 (30–1000 °C)

138

3.53

Cr2O3, (Mn, Cr)3O4 Cr2O3

RT, room temperature.

138, 139 138 138 140 (141–143

Materials for solid oxide fuel cells (SOFCs)

253

48/152 for Cr2O3); ρ is the density of the oxide scale (ρ = 5.225g/cm3 for Cr2O3); k go and Eox are the pre-exponential term and activation energy of the oxidation process, respectively. In practice, kg is often used to describe the oxidation behavior of an alloy owing to the fact that kg is obtained from firsthand experimental data, and there is no need to know the exact chemical composition of formed scale in order to convert to kp. Table 12.2 compares the parabolic rate constants of several popular commercial Ni- and Fe-based alloys at 800 °C and the compositions of the scales formed. In order to achieve an acceptable resistance to oxidation at a service temperature of 700–850 °C, a minimum of 17 wt% Cr is necessary for any metallic interconnect to be used in planar SOFCs. A higher Cr content is certainly helpful to increase oxidation resistance and lower the TEC, but this elevates the material/fabrication costs and more importantly promotes Cr volatility, which would create excessive spallation and jeopardize the longterm stability of cell performance by Cr poisoning. The latter has been discussed in detail in Chapter 11. Therefore, the upper limit of Cr for the modern choice of metallic interconnects is set to be around 22 wt%. Factors other than high-temperature oxidation resistance also have to be seriously considered during the process of selecting a reliable metallic interconnect. One of these factors is the area-specific electrical resistance, ASR, of the formed oxide scale; ASR is directly related to the scale thickness and conductivity by144 (ASR) 2 =

kp  k po  – E ox + 2 E el   t = t  2 exp  2 kT   σ σ o

[12.24]

where σo and Eel are the pre-exponential term and activation energy of electrical conduction in the scale, respectively. The methodology of measuring σ for a thermally grown oxide scale has been described in detail in reference 144. For a given scale thickness (a fixed oxidation kinetics), a lower conductivity of the scale would obviously contribute to higher resistance and therefore a higher voltage loss under load. This condition eliminates the use of Al2O3forming alloys in comparison to Cr2O3-forming alloys for interconnect applications. The fact that the electrical conductivity, for example at 900 °C, of Al2O3 is 10–6–10–8 S/cm as compared to 10–2–10–1 S/cm for Cr2O3 strongly supports the selection of Cr2O3-forming alloys.145 Another criterion for selecting adequate metallic interconnects is TEC. A typical range of TEC values in an electrode/electrolyte assembly varies from 10.5 × 10–6/K to 12.5 × 10–6/K. To match this TEC, Cr-based alloys (11.0 × 10–6/K to 12.5 × 10–6/K) and ferritic stainless steels (11.5 × 10–6/K to 14.0 × 10–6/K) are the best candidates. A good example of a Cr-based alloy is Ducrolloy (Cr–5 wt% Fe–1wt% Y2O3), which was developed earlier by Plansee GmbH (Tirol, Austria) in collaboration with Siemens AG for high-temperature SOFCs (900–1000 °C). It has an excellent TEC match to the electrode/

254

Solid oxide fuel cell technology

electrolyte assembly. However, the greatest problem associated with high-Cr alloys is the Cr mobility at the service temperature. The volatile Cr species in the high-valence state formed under SOFC conditions were identified by Hilpert et al.82 to be CrO2(OH)2(g) and CrO3(g) in air with and without moisture, respectively. In Chapter 11, Figs 11.14 to 11.16 show the calculated partial pressures of Cr species as a function of temperature in air with 60% relative humidity. These gaseous Cr-containing species can be entrained in the air stream and transported to the cathode/electrolyte interface where they are reduced to solid Cr2O3 (or further react with Mn in the LSM cathode to form MnCr2O4146 and with Sr in the LSCF (La–Sr–Co–Fe perovskite oxides) cathode to form SrCrO4147,148) by the reactions of equations (11.29), (11.33), and (11.34) described in Chapter 11. The deposition of catalytically inactive Cr2O3 at the reactive TPB sites blocks the process of oxygen reduction, leading to performance degradation. Based on the fact that Cr-based alloys are difficult and costly to fabricate and promote Cr migration, ferritic stainless steels appear to have been left as the only suitable metals to fulfill all the requirements for interconnects to be used in planar SOFCs. After this realization, the development of ferritic stainless steels has accelerated in recent years to formulate tailor-made compositions that address the various issues associated with applications in SOFCs. One of the issues with ferritic steels is the adhesion of the scale to the metal substrate. This is often observed by spallation of the scale upon oxidation cycles as growth stresses are created and released. The consequence of the spallation is serious in a sense that the growth of the scale and re-growth after spallation leads to depletion of the scale-forming element such as Cr in the metal. If the concentration of the element decreases to below a critical level, the protective scale can no longer form and rapid oxidation of the base metal (such as Fe) occurs, leading to large oxygen uptakes. In order to mitigate the spallation, surface treatment or alloying by reactive elements (REs) is often used.149–151 In order to understand the scale growth mechanism, the role of the RE in metal oxidation needs to be elucidated. The common mechanism of Cr2O3 scale growth assumes that the outward Cr diffusion along grain boundaries in the Cr2O3 scale is the controlling step. According to the dynamic-segregation theory,152 the presence of an RE on the surface of an alloy inhibits the normal outward short-circuit transport of cations along grain boundaries of the scale because these large RE ions diffuse more slowly than the native ions such as Cr. The inhibition of the outward transport of cations results in a reduction in the parabolic oxidation rate constants. Owing to the large ratio of cation diffusivity to anion diffusivity in Cr2O3, the RE has a much greater effect on the parabolic rate constant in Cr2O3 than in Al2O3. The reduction can be as high as 10- to 100-fold. Another effective way to tackle the spallation problem and reduce the parabolic rate constant is to incorporate Mn into the

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parent steel composition.140, 153 Two examples are Crofer22 APU and ZMG232. The former was invented by Forschungzentrum Julich and manufactured by Thyssen-Krupp VDM, while the latter was developed by Hitachi Metals, Inc. The presence of a limited amount of Mn (~0.50 wt%) in the Cr-containing base metal enhances oxide scale conductivity154 and decreases Cr volatility by preferentially forming a dense and conductive spinel MnCr2O4 top layer and Cr2O3-rich underlayer. The reduction of Cr mobility by a dense scale would undoubtedly improve the stability of the cell performance. Table 12.3 lists the chemical compositions of several commercially available ferritic stainless steels that are widely studied for SOFC interconnects. In addition to an elevated Mn level in Crofer22 APU and ZMG232 steels, RE species such as La and Zr have also been added to improve the scale adhesion. Although these carefully formulated Fe–Cr–Mn ferritic steels form a dense spinel scale that serves as a barrier to outward Cr diffusion, inward O diffusion, and gaseous Cr volatility, the contact resistances between the scale and electrodes remain relatively high owing to insufficient contact areas. In order to minimize the contact resistance, various electronically conductive oxide coatings have been investigated. These oxide coatings include Sr-doped LaMnO3, LaFeO3, LaCrO3, and LaCoO3. The results are sometimes conflicting with regard to the effectiveness of the coating in improving the contact resistance and stability of cell performance.155,156 The different coating application techniques used by different researchers are probably the reason for the variability. However, one fundamental requirement for the contact layer is the electronic conduction. Any involvement of oxide-ion conduction in the layer would transport oxygen in the form of O2– to the surface of the underlying metal, resulting in accelerated oxidation kinetics. For this reason, doped LaMnO3 may be preferable to doped LaCoO3 as a contact layer between cathode and interconnect. Analogous to (Mn, Cr)3O4 spinel, the spinel (Mn, Co)3O4 was also studied as a potential protective and contact layer for ferritic stainless steel interconnects. The results showed that the thermally grown, higher-electronic-conductivity Mn1.5Co1.5O4 spinel effectively provided a barrier to Cr outward diffusion, leading to a reduced Cr mobility and therefore a decreased contact ASR.157 For modern planar SOFC designs, a metallic interconnect provides a sealing platform for separating cathode and anode gases. As a result, the interaction between the sealing materials and metallic interconnects could have a direct impact on the stability of gas tightness over thousands of hours of operation. For example, a study showed that stainless steel AISI446 reacts readily with barium–calcium–aluminosilicate-based glass ceramics under a typical service temperature of 750°C.158 When oxygen is available, the reaction product tends to be BaCrO4 whereas under reducing atmospheres the reaction forms a Cr-rich solid solution by dissolving Cr2O3 into the glass. When designing the metallic interconnect and glass ceramics, the interaction between the two should not be ignored.

256

Alloy

Fe

Crofer22 APU ZMG232 E-Brite ANSI446

Bal. Bal. Bal. Bal.

Bal., balance.

Cr 22.8 21.97 26.0 25.0

Mn

Si

C

Ti

P

S

Al

0.45 0.50 0.01 1.50

– 0.40 0.025 1.00

0.005 0.02 0.001 0.020

0.08 – – –

0.016 – 0.02 0.04

0.002 – 0.02 0.03

– 0.21 – –

Ni – 0.26 0.09 –

Zr – 0.22 – –

La 0.06 0.04 – –

Solid oxide fuel cell technology

Table 12.3 Chemical compositions (%) of commercially available ferritic stainless steels

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12.4.3 Cell-to-cell connectors For a planar SOFC design, the cell-to-cell connector is the interconnect itself, as has been discussed in detail in the previous section. The brief review of cell-to-cell connectors given in this section is aimed primarily at cathode-supported seal-less tubular SOFC design. It is important to realize that the current-collection scheme for cathodesupported seal-less tubular SOFCs is advantageous over anode-supported seal-less tubular SOFCs. This is because the cathode-supported configuration allows for current collection to take place in the fuel environment where metals like Ni and Cu are available for use. Figure 1.2 gives an example of such a current-collection scheme in a Siemens/Westinghouse tubular SOFC bundle. In contrast, current collection in oxidizing atmospheres for an anodesupported configuration requires expensive precious metals like Pt and Ag to survive the oxidizing atmosphere; the latter is also rather mobile during SOFC operation.

12.5

Fabrication techniques

The fabrication technique for each functional layer in an SOFC largely depends on the SOFC design. The selection criteria include cost effectiveness, viability for mass production and automation, processing repeatability, and precision. In this section, the fabrication techniques commonly used in the SOFC industry to make thick substrates and thin films are reviewed.

12.5.1 Substrate A substrate in an SOFC is a mechanical support that provides a vehicle on which subsequent functional thin layers can be deposited. Therefore, it must be mechanically strong, chemically stable, and thermally matched to other layers. The substrate can be electrolyte, cathode, anode, or interconnect, leading to the concepts of electrolyte-, cathode-, anode-, and interconnectsupported SOFC designs. Depending on the type of substrate, it can be either dense or porous. The electrode-supported, thin-film electrolyte SOFC is the most popular design in modern SOFC technology. This is mainly due to the fact that electrolyte conductivity is the lowest among the SOFC functional layers, and reducing its thickness is the most effective way of minimizing the ohmic resistance. The cathode substrate was first used by Westinghouse in the form of a cylindrical tube to support films of electrolyte, anode, and interconnect. The tubular substrate is typically fabricated by extruding a ceramic paste through a mold to a desirable diameter (inner and outer) and length. The paste is a mixture of proprietary cathode powder, organic binder, and pore

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Solid oxide fuel cell technology

12.26 A typical microstructure of a Siemens cathode substrate after sintering. Reservoir

Doctor blade Slip

Carrier material: polymer, metal, glass, stone

Relative carrier motion

12.27 Schematic of tape casting process.

former. After appropriate drying, the tube is finally sintered into a porous body with a well-defined microstructure, porosity, and, most importantly, mechanical strength. Figure 12.26 shows a representative microstructure of such a substrate; a roughly 30–35 vol% of porosity is estimated. In order to ensure the quality of the substrate, chemical analysis, mechanical strength, thermal expansion coefficient, thermal cycling shrinkage, oxygen diffusion conductance, and porosity are closely monitored during batch production. The concept of an anode-supported SOFC was first proposed by de Souza et al. in 1997 and it demonstrated very high electrical performance.17 Since then, the anode-supported configuration has been widely accepted as a standard design by planar SOFC developers, particularly after low-cost ferritic stainless

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259

steel interconnects became available for the reduced temperature range (<800 °C). The commonly used technique for fabricating an anode substrate is conventional tape casting, which is illustrated schematically in Fig. 12.27; this process is well-known in many other industries including electroceramics, paper, plastic, and paint manufacturing. The major advantage of the tape casting process is that it is the best way to form large-area, thin, flat ceramic or metallic parts. In a typical tape casting process, the slip or slurry is poured into a puddle or reservoir behind the doctor blade and the carrier which the slip is to be cast upon is set in motion. The gap between the doctor blade and the carrier defines the wet thickness of the tape being cast. Other variables that come into play include reservoir depth, speed of carrier movement, viscosity of the slip, and shape of the doctor blade (to name but a few). The wet film of slip passes into a drying chamber of some sort, and the solvents are evaporated from the surface, leaving a dry tape on the carrier surface. The slurry or slip is typically prepared by dispersing the anode material (a mixture of YSZ and NiO) in solvents (e.g. 2-butanone/ethanol), after which binders (e.g. polyvinyl butyral), plasticizer (e.g. polyethylene glycol), and deflocculant/wetting agent (e.g. glycerol trioleate) are added. The typical thickness of substrate that can be comfortably made by tape casting is 50–500µm. The in-situ reduction of NiO to Ni ensures the desired porosity for anode gas diffusion. An electrolyte in green state can also be made by the tape casting method, followed by sintering at high temperature into a dense ceramic. Owing to the requirement to be a mechanical support the electrolyte thickness typically ranges from 100 to 200 µm. The assembly of cathode/electrolyte/anode is then laminated with a metallic interconnect channeled for gas distribution. In recent years, porous metal has also been explored as a substrate to support SOFC functional layers.159 The advantages include low manufacturing cost and improved durability. Moreover, the metal alloy is both thermally and electrically conductive, and it allows for a simple sealing strategy. However, one of the major challenges is how to fabricate a dense electrolyte layer without compromising the integrity of the porous metal substrate. In order to mitigate this problem, a vacuum plasma spray technique has been used to deposit a relatively dense electrolyte layer on the porous metal substrate so that the high-temperature densification step is not needed.160

12.5.2 Electrolyte thin film Fabrication of quality electrolyte thin films is the most important step of all the stages in the production of the functional layers in an SOFC. The requirement for a thin and dense coating challenges conventional ceramic fabrication techniques. These techniques can generally be classified into three categories: vapor deposition (e.g. electrochemical vapor deposition

260

Solid oxide fuel cell technology ZrCl4 vapor O2–

1

e′

2 Porous substrate

H2O

H2

Phase 1 – pore closure by CVD ZrCl4 + 2H2O → ZrO2 + 4HCl Phase 2 – scale growth by EVD ZrCl4 + 2O2– → ZrO2 + 2Cl2 + 4e′ H2O + 2e′ → H2 + O2–

12.28 Illustration of the principle of the EVD process. CVD, chemical vapor deposition.

(EVD) and physical vapor deposition (PVD)), thermal spray (e.g. atmospheric plasma spray), and slurry coating (e.g. colloidal deposition, slip casting, screen printing, electrophoresis deposition, wet powder spray, and so forth). The EVD process was first invented in 1977 by Westinghouse Electrical Corporation to produce a thin and dense 8YSZ layer on a porous tubular Cadoped ZrO2 and later cathode substrates.161 The main steps of the process are illustrated schematically in Fig. 12.28. As the gaseous mixture of ZrCl4 and YCl3 is fed to the outer surface of the tubular substrate and steam is introduced to the inner surface at elevated temperatures, the chemical reaction first occurs at the locations where they meet, namely pores, by ZrCl4 + 2H2O → ZrO2 + 4HCl

[12.25]

2YCl3 + 3H2O → Y2O3 + 6HCl

As the gas diffusion process in the substrate is the rate-limiting step, the growth of the initial layer that closes the surface pores is linear with time. As soon as the surface pores are closed, the presence of oxide-ion and electronic conduction in the initially formed 8YSZ, at elevated temperature under a gradient of oxygen chemical potential, comes into play: an oxygen flux that counterbalances the electronic flux is generated. This process can be best described by the following electrochemical reactions ..

ZrCl 4 + 2O O× → ZrO 2 + 2Cl 2 + 2VO + 4e ′ ..

2YCl 3 + 3O O× → Y2 O 3 + 3Cl 2 + 3VO + 6e ′

[12.26]

As a result, a film of 8YSZ continues to grow; the growth rate kp is commonly

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governed by the classical parabolic rate law (Wagner equation) described in Section 12.4 on interconnects and in Chapter 4. The relationship between thickness L and time t can, therefore, be written by L2 = 2kpt

[12.27]

At a typical EVD temperature of 1000–1200 °C, the rate constant for deposition of 8YSZ ranges from 1.1 × 10–5 to 3.8 × 10–3 cm2/s. Considering both initial linear and subsequent parabolic growth rates, to achieve a 40 µm 8YSZ film takes approximately 40 min at 1000 °C and 20 min at 1200 °C. Figure 12.29 shows a representative microstructure of an EVD electrolyte. There is no doubt that the EVD process is capable of producing pinhole-free, thin, and dense electrolyte films. The major drawback of the EVD process with regard to mass production is, however, the high cost associated with labor, materials, waste disposal, and maintenance. After successful demonstration of several SOFC generators by Westinghouse using cathode-supported EVD electrolyte and anode, this commercially prohibitive technique was later replaced by the more productive and automation-friendly atmospheric plasma spray coating technique. The plasma spray process is a proven and mature surface coating technology that is widely used in aerospace and power-generation industries for the application of thermal barrier coatings to gas turbine blades, vanes, combustor cans, and turbine shrouds. The coating significantly improves the turbine

12.29 Microstructure of an EVD electrolyte on a porous cathode substrate.

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blade cooling efficiency, prevents hot corrosion, and most importantly, makes transatlantic flight possible. The plasma spray process features injecting coating materials (metals, ceramics, and polymers) into a high-velocity and high-temperature ionized argon gas stream (plasma). During a very short residence time, the material is melted and propelled towards a substrate where it solidifies and builds up a solid coating. By controlling the motion of the plasma source or the substrate in a designated pattern, the substrate in specific geometries can be completely or partially covered to a desired thickness. Based on the surrounding atmosphere, the plasma spray method can be categorized into conventional atmospheric (APS) and controlled atmospheric plasma spray (CAPS). Application of the APS technique to fabricate an electrolyte film on a cathode substrate of a tubular SOFC was first investigated in the late 1990s by Westinghouse. After extensive parametric studies, the processing conditions were eventually qualified for plasma spraying an 8YSZ layer. Figure 12.30 shows a representative microstructure of plasma-sprayed 8YSZ on a porous cathode substrate. The pores shown are in general found closed. Low-pressure plasma spray (LPPS), one type of CAPS, was also shown to deposit an even denser 8YSZ layer.162 Figure 12.31 shows the microstructure of an LPPS 8YSZ layer. It appears that LPPS is more capable than APS of making a dense electrolyte layer.

12.30 Typical microstructure of an APS electrolyte on a porous cathode substrate.

Materials for solid oxide fuel cells (SOFCs)

263

12.31 Microstructure of an LPPS electrolyte with composite cathode interlayer on a porous cathode substrate.

Other conventional ceramic fabrication techniques used worldwide by SOFC developers for making electrolyte thin films include tape casting, screen printing, vacuum slip casting, wet powder spraying/slurry coating, and dry die-pressing. The thickness of the electrolyte varies, based on the technique, from a few to hundreds of microns. A general review of these manufacturing techniques has been given by Tietz et al.,163 to which interested readers are referred for more information.

12.5.3 Electrode thin films Screen printing is the most popular method adopted by anode-supported SOFC developers for making the cathode layer. It is an attractive technique owing to its maturity, ease of automation, and suitability for producing highquality, large-area coatings. It is the standard route for making ceramic capacitors in the electroceramics industry. The formulation of the ink and the printing material, is analogous to that of the tape casting slip. The ceramic material (e.g. LSM) is first intimately dispersed in the solvent (e.g. terpineol) in the presence of a dispersant (e.g. Hypermer LP1). The binder (e.g. ethylcellulose) and plasticizer (e.g. dioctylphthalate) are then added to achieve the desired viscosity. Typical thicknesses of screen-printed cathode films are approximately 20 µm. After preliminary drying, the resultant film undergoes

264

Solid oxide fuel cell technology 20.00 µm Ni 8YSZ

12.32 Representative microstructure of an anode made using the APS process.

burn-out and calcinations; it finally bonds to the surface of the electrolyte. The solid to organic ratio and the particle size distribution of the cathode material mainly determine the resulting porosity and porosity distribution. Westinghouse/Siemens has demonstrated that the anode layer in a cathodesupported SOFC can be fabricated by the APS route. A powder mixture of metallic Ni, pore former, and YSZ is the feedstock. One of the advantages of the APS anode is that no additional sintering step is necessary after APS processing. A representative microstructure of the APS anode is given in Fig. 12.32. The slurry dip-coating method was also used by Toto as a lowcost process to make the anode coating, followed by drying and sintering steps.

12.5.4 Interconnection thin film and plate Generally speaking, the ceramic LaCrO3-based interconnect is difficult to sinter because of the vaporization of Cr–O species during the sintering process, which leads to enhancement of the evaporation–condensation mechanism of sintering. This process tends to suppress densification and causes coarsening of the powder.164 In order to alleviate the Cr migration problem, A-site excess or Cr deficiency in LaCrO3 were formulated and found to promote sintering by forming multi-phase chromates that melt incongruently in the

Materials for solid oxide fuel cells (SOFCs)

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100 µm

12.33 Representative SEM microstructure of a ceramic interconnect made using the APS process on a porous cathode substrate.

range of 1000–1200 °C. This strategy was also adopted by Westinghouse in designing the compositions of the APS interconnect feedstock. Figure 12.33 shows a representative microstructure of an APS interconnect layer on a porous cathode substrate. In an earlier development of SOFCs, a ceramic LaCrO3-based interconnect thick plate was considered to be a bipolar gas distributor. This design was quickly abandoned owing to the high manufacturing cost and emergence of anode-supported, low-temperature SOFCs. A low-cost, metallic interconnect plate with functionalities of electrical/heat conduction and gas distribution was the obvious selection for anode-supported, low-temperature SOFC developers. Figure 12.34 shows some examples of gas manifolding strategies of planar SOFCs with a metallic interconnect in different geometries. Clearly, machineability of the metals favors these designs.

12.6

Conclusions

More than 150 years have passed since the basic concept of an SOFC was first proposed in the late nineteenth century. It generates mixed feelings to see today that this simple–in–principle and efficient–in–energy conversion device still requires considerable development before it becomes a commercial product. A high product cost that is unable to compete with a conventional

266

Solid oxide fuel cell technology Fuel out

Oxidant out

Oxidant in

Fuel in (a)

Metallic Interconnector

Air

Cathode

To afterburner

Electrolyte Anode Fuel (H2, CO)

Metallic interconnector (b)

12.34 Examples of external (a) and internal (b) gas manifolds used in planar SOFC design.

heat engine is the major hurdle for the commercialization of the SOFC technology. As a system, balance-of-plant (BOP) and module cost contribute to a significant portion of the total cost; they have to be reduced through special design cycles. Advanced materials and revolutionary fabrication techniques could lower the cost of all materials while, at the same time, maintaining performance. ZrO2-based materials are the most studied and mature electrolytes. The relevant fabrication techniques are well-developed and in place to mass produce large-volume electrolyte films. Therefore, they are most likely to be the first electrolyte system in commercial SOFC products operating in the range of 700–1000 °C. Other electrolytes such as LSGM and doped CeO2 could potentially find application in SOFC generators operating below 800 °C and

Materials for solid oxide fuel cells (SOFCs)

267

600°C, respectively. Viable fabrication methods are in great demand for making LSGM films on either cathode or anode substrate without compromising the performance. The operating temperature of CeO2-based SOFCs has to be limited to below 600 °C where the electronic conduction can be reduced to a level that is acceptable for attaining sufficiently high electrical efficiency. As the operating temperature is reduced, the role of cathode performance becomes even more important owing to the fact that the activation energy of cathode polarization is generally greater. For a low-temperature SOFC (<800 °C), the cathode performance dominates the overall SOFC performance. The development of a high-performing cathode composition is therefore critical. Tremendous efforts have been devoted to this area by researchers worldwide. One challenge facing the SOFC community is the fact that a cathode with a high catalytic activity is always associated with a high TEC. In order to alleviate this contradiction, a composite form of cathode consisting of electrolyte and cathode is always chosen to meet the requirements for thermal expansion match and low polarization loss. One of the advantages of an SOFC over other competing fuel cells is the fuel flexibility. The type of fuel with which an SOFC is able to operate is largely dependent upon the anode. Direct utilization of hydrocarbons such as natural gas, propane and other liquid fuels without pre-reforming can significantly reduce the system cost and is therefore highly desirable. The conventional Ni-based anode is prone to promoting carbon formation in handling steam-free hydrocarbons although it is an excellent catalyst for the fuel oxidation reaction. Cu has less activity in cracking hydrocarbon fuels than Ni, but it also has less activity in catalyzing the electro-oxidation of fuels. The introduction of CeO2 into a Cu-containing anode aims at balancing both requirements because CeO2 has been proven to possess catalytic activity for the fuel oxidation reaction. A ceramics-based anode has also recently become a hot topic and is under active study. A suitable ceramic anode composition has been shown with excellent electronic conductivity, sufficient catalytic activity for fuel oxidation, and, more importantly, high resistance to carbon formation and sulfur poisoning.37 The ceramics-based anode, if well developed, could play a significant role in reducing the system cost of the SOFC product. Ceramic LaCrO3-based interconnects find application in cathode-supported tubular SOFCs as a thin film. It is, however, expensive to use ceramic interconnects in planar design as both electrical conductor and gas distributor. The advantage of a ceramic interconnect is the low Cr volatility given by the thermodynamic stability of perovskite materials. As the operating temperature of the anode-supported planar SOFCs is reduced to below 800 °C, lowcost ferritic steels, purposely designed to form dense Mn–Cr spinel scale, have been developed to replace expensive ceramic interconnects. Surface

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Solid oxide fuel cell technology

modification/coating with reactive elements is able to reduce the growth of the scale and improve the adhesion of the scale to the metal base. Two major challenges facing planar SOFC developers using metallic interconnects are Cr mobility and high contact resistance, which are two of the reasons for performance degradation and lower performance in the stack. Functional layer-fabrication methods depend greatly on the SOFC design and cost-effectiveness. Repeatability, quality, productivity, and the ability to automate have to be considered seriously when selecting and designing a specific fabrication process. It should be borne in mind that the SOFC product cost includes cell and bundle/stack, module, and BOP. In many cases, the cost of the module and BOP dominates the total cost, giving more choices for cell making. This could also be a deciding factor in which fabrication techniques and materials are used. The scope of this chapter has been to review the progress in SOFC materials that has been achieved so far. It is hoped that this concise review may help, to some extent, advance the development of SOFC technology.

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40 41 42

43 44 45 46

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Appendix 1: Thermodynamic data of selected chemical reactions and substances

Table A1.1 lists the standard Gibbs free energy changes for selected reactions in the form ∆G

o

= A + BT

∆G

o

= A + BT log T + CT

or and lists the ranges of temperature in which the expression is valid. Table A1.1 The standard Gibbs free energy changes of selected chemical reactions Chemical reaction

∆G

o

H2(g) + 0.5O2(g) = H2O(g)

∆G

C(s) + 0.5O2(g) = CO(g) CO(g) + 1/2O2(g) = CO2(g)

T range, (K)

z*

o

298–2500

2

∆G

o

298–2500

2

∆G

o

298–2000

2

3H2(g) + CO(g) + 2O2(g) = CO2(g) + 3H2O(g)

∆G

o

298–2000

8

C(s) + O2(g) = CO2(g)

∆G

o

298–2000

4

CH4(g) + 2O2(g) = 2H2O(g) + CO2(g)

∆G

o

= –817 106T – 51.205 T log T +173.97T

298–2500

8

SiO2(s) + Si(s) = 2SiO(g)

∆G

o = 706 420 + 53.922 T log T – 519.574T

298–1700

4

SiO2(s) = Si(s) + O2

∆G

o

700–1700

4

Ni(s) + 0.5O2(g) = NiO(s)

∆G

o

298–1725

2

2Cu(s) + 0.5O2(g) = Cu2O(s)

∆G

o

298–1356

2

2Cu(l) + 0.5O2(g) = Cu2O(s)

∆G

o

1356–1503

2

H2(g) + S(s) = H2S(g)

∆G

o

298–1750

2

(J/mol)

= –246 202 + 54.758T = –111 606 – 87.57T = –282 150 + 86.735T = –1021 482 + 251.177T = –393 756 – 0.836T

= 901 208 – 173.47T = –234 122 + 85.15T = –166364 + 70.56T

= –195206 – 16.38 T log T + 142.54T = –168 078 + 30.31 T log T – 5.06 T

z*, Number of electrons transferred during the reaction.

278

Appendix 1

279

Table A1.2 presents the constant-pressure molar heat capacities in the form Cp = a + bT + cT2 + dT3 + eT–2 and includes the ranges of temperature in whch the expressions are valid. Table A1.2 The constant-pressure molar heat capacities of some substances Cp = a + bT + cT 2 + dT 3 + eT –2 (J/(mol K))

Substance C(graphite) CH4(g) CO(g) CO2(g) H2(g) H2O(g) N2(g) O2(g) Air

a

b × 103

c × 106

d × 109

e × 10–6

0.108 68 –0.703 029 25.567 59 24.997 35 33.066 178 30.092 26.092 29.659 26.65

38.903 26 108.477 3 6.096 130 55.186 96 –11.363 417 6.832 514 8.218 801 6.137 261 –39.4

–21.53536 –42.52157 4.054656 –33.6914 11.432816 6.793435 –1.97614 –1.18652 25.9

0 5.862 788 –2.671 301 7.948 387 –2.772 874 –2.534 48 0.159 274 0.095 78 –6.07

–0.147 972 0.678 565 0.131 021 –0.136 64 –0.158 558 0.082 139 0.044 434 –0.219 66 0.747 441

Table A1.3 lists standard molar heat of formation and molar entropies for selected substances. Table A1.3 The standard molar heat of formation and molar entropies of some substances at 298 K Substance C(graphite) CH4(g) CO(g) CO2(g) H2(g) H2O(g) O2(g) N2(g)

o ∆ f H 298 (J/mol)

0 –74 870 –110 500 –393 500 0 –241 800 0 0

o S298 (J/(molK))

5.73 186.01 197.5 213.7 130.46 232.9 205.1 191.5

Appendix 2: The standard heat of combustion of common fuels

The standard heat of combustion, ∆cHo, is the energy released as heat when a compound undergoes complete combustion with oxygen at the standard state. The chemical reaction is typically a hydrocarbon reacting with oxygen to form carbon dioxide, water, and heat. It is traditionally measured with a bomb calorimeter. It may also be calculated as the difference between the heat of formation, ∆fHo, of the products and reactants. Depending on the physical state of the products, the heat of combustion can be expressed in two ways. The higher heating value (HHV) (or gross calorific value or gross energy or upper heating value) is determined by bringing all the products of combustion back to the original pre-combustion temperature, and in particular condensing any vapor produced. This is the same as the thermodynamic heat of combustion since the enthalpy change for the reaction assumes a common temperature of the compounds before and after combustion, in which case the water produced by combustion is liquid. The lower heating value (LHV) (or net calorific value) is determined by subtracting the heat of vaporization of the water vapor from the HHV. This treats any H2O formed as a vapor. The energy required to vaporize the water therefore is not realized as heat. Table A2.1 lists both the HHV and the LHV of some fuels at 25°C. For more common fuels, the temperature-dependent HHV and LHV are tabulated in Table A2.2, where ∆Ho of the oxidation reaction is the heat of combustion o ∆c H . Table A2.1 Heat of combustion and constant-pressure molar heat capacities of hydrocarbons and combustion products at 298 K o

∆ c H298, n (kJ/mol)

o

∆ c H 298, n

Hydrocarbon

n

(HHV)

(LHV)

Cp,298,n (J/(mol K))

H2(g) CH4(g) C2H6(g) C3H8(g) C4H10(g)

0 1 2 3 4

–285.82 –890.57 –1559.8 –2307.2 –3009.6

–241.83 –802.82 –1427.8 –2131.2 –2789.6

28.8 35.6 52.5 73.6 97.5

280

(kJ/mol)

Appendix 2

281

Table A2.1 Continued

Hydrocarbon

n

C5H12(g) C6H14(g) C7H16(g) H2O(g) CO2(g) O2(g)

5 6 7

o ∆ c H 298, n (kJ/mol) (HHV)

o ∆ c H 298, n (kJ/mol) (LHV)

Cp,298,n (J/(mol K))

–3685.0 –4383.0 –5080.9 — — —

–3421.0 –4075.0 –4729.0 — — —

120.2 143.1 166.0 33.6 37.13 54.317

Table A2.2 Thermodynamic properties of oxidation reactions of common fuels H2(g) + 0.5O2(g) = H2O(g) o

T (°C)

∆H

25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 625 650 675 700 725 750 775 800 825 850

–241.826 –242.077 –242.323 –242.566 –242.806 –243.043 –243.277 –243.508 –243.738 –243.964 –244.189 –244.411 –244.631 –244.848 –245.062 –245.274 –245.483 –245.689 –245.892 –246.092 –246.289 –246.482 –246.671 –246.857 –247.04 –247.218 –247.392 –247.562 –247.728 –247.89 –248.048 –248.201 –248.349 –248.493

(kJ/mol)

(LHV)

(J/(K mol))

o ∆G (kJ/mol)

K

log(K)

–44.421 –45.229 –45.964 –46.638 –47.260 –47.836 –48.374 –48.877 –49.349 –49.793 –50.213 –50.609 –50.984 –51.339 –51.677 –51.998 –52.302 –52.593 –52.869 –53.132 –53.382 –53.620 –53.847 –54.063 –54.269 –54.464 –54.651 –54.828 –54.996 –55.157 –55.309 –55.453 –55.590 –55.720

–228.582 –227.461 –226.321 –225.163 –223.989 –222.801 –221.598 –220.382 –219.154 –217.915 –216.665 –215.405 –214.135 –212.856 –211.568 –210.272 –208.968 –207.657 –206.339 –205.014 –203.682 –202.345 –201.001 –199.652 –198.298 –196.939 –195.575 –194.206 –192.834 –191.457 –190.076 –188.691 –187.303 –185.912

1.12E+40 5.89E+36 9.10E+33 3.32E+31 2.45E+29 3.20E+27 6.77E+25 2.15E+24 9.59E+22 5.75E+21 4.45E+20 4.29E+19 5.03E+18 6.98E+17 1.13E+17 2.08E+16 4.33E+15 1.00E+15 2.56E+14 7.11E+13 2.14E+13 6.94E+12 2.40E+12 8.81E+11 3.42E+11 1.39E+11 5.96E+10 2.66E+10 1.24E+10 5.96E+09 2.97E+09 1.53E+09 8.13E+08 4.44E+08

40.05 36.77 33.959 31.522 29.388 27.505 25.831 24.332 22.982 21.76 20.648 19.633 18.701 17.844 17.052 16.318 15.636 15.001 14.407 13.852 13.331 12.841 12.38 11.945 11.534 11.144 10.775 10.425 10.092 9.775 9.473 9.185 8.91 8.647

∆S

o

282

Appendix 2

Table A2.2 Continued T (°C) 875 900 925 950 975 1000

∆H

o

(kJ/mol)

–248.632 –248.767 –248.896 –249.021 –249.142 –249.259

H2(g) + 0.5O2(g) = H2O(l)

T (°C)

25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 625 650 675 700 725 750 775 800 825 850 875 900 925

∆H

o

(kJ/mol)

–285.83 –285.039 –284.242 –283.439 –282.624 –281.791 –280.928 –280.024 –279.067 –278.034 –276.892 –275.603 –274.05 –272.246 –270.445 –268.649 –266.859 –265.072 –263.291 –261.514 –259.741 –257.973 –256.209 –254.45 –252.694 –250.943 –249.196 –247.452 –245.713 –243.977 –242.246 –240.518 –238.793 –237.072 –235.355 –233.642 –231.931

o ∆G (kJ/mol)

K

(J/(K mol)) –55.842 –55.958 –56.067 –56.170 –56.268 –56.361

–184.517 –183.12 –181.72 –180.317 –178.911 –177.503

2.49E+08 1.43E+08 8.37E+07 5.02E+07 3.08E+07 1.92E+07

∆S

o

log(K) 8.395 8.154 7.923 7.701 7.488 7.283

(HHV) ∆S

o

∆G

o

(J/(K mol))

(kJ/mol)

–163.303 –160.755 –158.381 –156.153 –154.04 –152.01 –150.029 –148.067 –146.096 –144.074 –141.942 –139.644 –136.994 –134.039 –131.206 –128.488 –125.875 –123.361 –120.939 –118.603 –116.347 –114.165 –112.054 –110.01 –108.028 –106.104 –104.237 –102.422 –100.657 –98.940 –97.268 –95.638 –94.050 –92.500 –90.988 –89.512 –88.069

–237.141 –233.091 –229.102 –225.17 –221.293 –217.468 –213.692 –209.966 –206.289 –202.662 –199.086 –195.566 –192.107 –188.719 –185.404 –182.158 –178.979 –175.863 –172.81 –169.816 –166.879 –163.998 –161.17 –158.395 –155.669 –152.993 –150.364 –147.78 –145.242 –142.747 –140.295 –137.883 –135.512 –133.181 –130.887 –128.631 –126.411

K

3.55E+41 4.79E+37 2.38E+34 3.33E+31 1.08E+29 7.03E+26 8.11E+24 1.52E+23 4.29E+21 1.73E+20 9.40E+18 6.68E+17 5.99E+16 6.61E+15 8.77E+14 1.37E+14 2.47E+13 5.06E+12 1.17E+12 2.98E+11 8.36E+10 2.56E+10 8.45E+09 3.00E+09 1.13E+09 4.55E+08 1.93E+08 8.57E+07 3.99E+07 1.94E+07 9.82E+06 5.15E+06 2.80E+06 1.57E+06 9.02E+05 5.34E+05 3.25E+05

log(K)

41.55 37.68 34.376 31.523 29.035 26.847 24.909 23.182 21.633 20.237 18.973 17.825 16.778 15.82 14.943 14.136 13.392 12.704 12.066 11.474 10.922 10.408 9.927 9.476 9.054 8.658 8.284 7.933 7.601 7.288 6.992 6.712 6.446 6.194 5.955 5.728 5.511

Appendix 2

283

Table A2.2 Continued T (°C) 950 975 1000

∆H

o

(kJ/mol)

–230.225 –228.521 –226.821

o ∆G (kJ/mol)

K

(J/(K mol)) –86.659 –85.281 –83.932

–124.227 –122.078 –119.963

2.02E+05 1.29E+05 8.36E+04

∆S

o

CH4(g) + 2O2(g) = 2H2O(g) + CO2(g)

T (°C)

25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 625 650 675 700 725 750 775 800 825 850 875 900 925 950 975 1000

∆H

o

(kJ/mol)

–802.557 –802.308 –802.058 –801.815 –801.582 –801.363 –801.16 –800.974 –800.806 –800.657 –800.528 –800.418 –800.328 –800.258 –800.206 –800.174 –800.16 –800.163 –800.184 –800.222 –800.275 –800.343 –800.426 –800.522 –800.632 –800.754 –800.887 –801.031 –801.185 –801.349 –801.522 –801.704 –801.894 –802.091 –802.295 –802.504 –802.718 –802.938 –803.164 –803.394

5.306 5.109 4.922

(LHV) o ∆G (kJ/mol)

K

log(K)

(J/(K mol)) –5.233 –4.43 –3.686 –3.011 –2.407 –1.873 –1.406 –1.002 –0.656 –0.364 –0.123 0.073 0.227 0.342 0.423 0.473 0.493 0.488 0.46 0.411 0.343 0.259 0.16 0.048 –0.076 –0.21 –0.353 –0.502 –0.658 –0.82 –0.988 –1.159 –1.334 –1.512 –1.691 –1.871 –2.052 –2.234 –2.416 –2.599

–800.997 –800.876 –800.775 –800.691 –800.624 –800.57 –800.53 –800.50 –800.479 –800.466 –800.46 –800.46 –800.464 –800.471 –800.481 –800.492 –800.504 –800.516 –800.528 –800.539 –800.549 –800.556 –800.561 –800.564 –800.564 –800.56 –800.553 –800.542 –800.528 –800.509 –800.487 –800.46 –800.429 –800.393 –800.353 –800.309 –800.26 –800.206 –800.148 –800.085

2.20E+140 2.93E+129 1.43E+120 1.24E+112 1.11E+105 6.80E+98 2.06E+93 2.40E+88 8.77E+83 8.52E+79 1.92E+76 9.05E+72 8.09E+69 1.27E+67 3.28E+64 1.32E+62 7.90E+59 6.73E+57 7.88E+55 1.23E+54 2.49E+52 6.38E+50 2.03E+49 7.88E+47 3.66E+46 2.00E+45 1.28E+44 9.41E+42 7.88E+41 7.44E+40 7.86E+39 9.22E+38 1.19E+38 1.69E+37 2.60E+36 4.33E+35 7.78E+34 1.50E+34 3.08E+33 6.74E+32

140.343 129.466 120.154 112.092 105.045 98.833 93.314 88.381 83.943 79.93 76.284 72.957 69.908 67.104 64.516 62.121 59.898 57.828 55.896 54.09 52.396 50.805 49.308 47.896 46.563 45.302 44.107 42.973 41.896 40.872 39.896 38.965 38.076 37.227 36.415 35.637 34.891 34.176 33.489 32.829

∆S

o

log(K)

284

Appendix 2

Table A2.2 Continued CH4(g) + 2O2(g) = 2H2O(l) + CO2(g)

T (°C)

25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 625 650 675 700 725 750 775 800 825 850 875 900 925 950 975 1000

∆H

o

(kJ/mol)

–890.565 –888.232 –885.896 –883.561 –881.22 –878.859 –876.462 –874.005 –871.464 –868.797 –865.934 –862.802 –859.166 –855.054 –850.972 –846.925 –842.911 –838.93 –834.981 –831.065 –827.18 –823.325 –819.501 –815.707 –811.942 –808.204 –804.494 –800.811 –797.154 –793.523 –789.918 –786.338 –782.782 –779.25 –775.74 –772.254 –768.788 –765.344 –761.921 –758.518

∆S

o

(HHV) ∆G

o

(J/(K mol))

(kJ/mol)

–242.997 –235.481 –228.519 –222.041 –215.969 –210.22 –204.716 –199.382 –194.15 –188.926 –183.582 –177.998 –171.793 –165.057 –158.635 –152.508 –146.653 –141.05 –135.682 –130.532 –125.587 –120.832 –116.255 –111.846 –107.594 –103.49 –99.525 –95.690 –91.980 –88.387 –84.905 –81.530 –78.254 –75.074 –71.984 –68.979 –66.056 –63.212 –60.441 –57.742

–818.116 –812.136 –806.337 –800.706 –795.232 –789.905 –784.718 –779.668 –774.749 –769.96 –765.303 –760.783 –756.408 –752.198 –748.153 –744.264 –740.525 –736.929 –733.471 –730.143 –726.942 –723.863 –720.899 –718.048 –715.306 –712.667 –710.13 –707.69 –705.345 –703.09 –700.924 –698.844 –696.847 –694.931 –693.092 –691.331 –689.643 –688.027 –686.482 –685.005

K

log(K)

2.20E+143 1.93E+131 9.74E+120 1.24E+112 2.18E+104 3.28E+97 2.96E+91 1.20E+86 1.76E+81 7.66E+76 8.59E+72 2.19E+69 1.15E+66 1.14E+63 1.99E+60 5.72E+57 2.57E+55 1.72E+53 1.64E+51 2.15E+49 3.79E+47 8.67E+45 2.52E+44 9.11E+42 4.02E+41 2.13E+40 1.33E+39 9.75E+37 8.22E+36 7.90E+35 8.58E+34 1.04E+34 1.41E+33 2.10E+32 3.43E+31 6.08E+30 1.17E+30 2.43E+29 5.39E+28 1.28E+28

143.342 131.286 120.989 112.095 104.338 97.516 91.471 86.081 81.245 76.884 72.934 69.341 66.06 63.057 60.299 57.758 55.41 53.234 51.214 49.333 47.578 45.938 44.401 42.96 41.604 40.328 39.125 37.989 36.915 35.898 34.934 34.018 33.149 32.322 31.535 30.784 30.068 29.385 28.731 28.107

Appendix 2

285

Table A2.2 Continued C + O2(g) = CO2(g)

T (°C)

25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 625 650 675 700 725 750 775 800 825 850 875 900 925 950 975 1000

∆H

o

(kJ/mol)

–393.505 –393.522 –393.534 –393.546 –393.559 –393.574 –393.592 –393.614 –393.639 –393.668 –393.701 –393.738 –393.777 –393.819 –393.864 –393.911 –393.96 –394.01 –394.061 –394.114 –394.166 –394.22 –394.273 –394.327 –394.381 –394.435 –394.489 –394.542 –394.594 –394.647 –394.699 –394.751 –394.803 –394.855 –394.907 –394.959 –395.011 –395.062 –395.114 –395.165

∆S

o

∆G

o

(J/(K mol))

(kJ/mol)

2.879 2.826 2.788 2.756 2.723 2.686 2.645 2.598 2.545 2.488 2.427 2.362 2.295 2.226 2.156 2.084 2.013 1.942 1.872 1.804 1.736 1.671 1.607 1.544 1.483 1.424 1.367 1.311 1.258 1.206 1.156 1.106 1.058 1.012 0.966 0.921 0.878 0.835 0.793 0.752

–394.364 –394.435 –394.505 –394.574 –394.643 –394.711 –394.777 –394.843 –394.907 –394.97 –395.031 –395.091 –395.15 –395.206 –395.261 –395.314 –395.365 –395.414 –395.462 –395.508 –395.552 –395.595 –395.636 –395.675 –395.713 –395.749 –395.784 –395.818 –395.85 –395.881 –395.91 –395.938 –395.966 –395.991 –396.016 –396.04 –396.062 –396.084 –396.104 –396.123

K

1.25E+69 5.79E+63 1.57E+59 1.73E+55 6.01E+51 5.35E+48 1.04E+46 3.92E+43 2.58E+41 2.75E+39 4.43E+37 1.02E+36 3.24E+34 1.35E+33 7.19E+31 4.76E+30 3.83E+29 3.66E+28 4.10E+27 5.29E+26 7.74E+25 1.28E+25 2.33E+24 4.71E+23 1.04E+23 2.48E+22 6.40E+21 1.77E+21 5.21E+20 1.63E+20 5.39E+19 1.88E+19 6.86E+18 2.62E+18 1.04E+18 4.32E+17 1.85E+17 8.24E+16 3.79E+16 1.79E+16

log(K)

69.097 63.763 59.194 55.238 51.779 48.728 46.018 43.593 41.412 39.44 37.647 36.01 34.51 33.13 31.857 30.678 29.583 28.564 27.613 26.723 25.889 25.105 24.368 23.673 23.016 22.395 21.806 21.248 20.717 20.212 19.732 19.274 18.836 18.418 18.018 17.635 17.268 16.916 16.578 16.253

286

Appendix 2

Table A2.2 Continued CO(g) + 0.5O2(g) = CO2(g)

T (°C)

25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 625 650 675 700 725 750 775 800 825 850 875 900 925 950 975 1000

∆H

o

(kJ/mol)

–282.964 –283.117 –283.244 –283.349 –283.434 –283.502 –283.554 –283.593 –283.619 –283.633 –283.636 –283.63 –283.615 –283.592 –283.561 –283.523 –283.479 –283.429 –283.374 –283.314 –283.249 –283.181 –283.109 –283.034 –282.955 –282.874 –282.788 –282.699 –282.607 –282.513 –282.416 –282.316 –282.215 –282.111 –282.006 –281.899 –281.79 –281.68 –281.568 –281.455

∆S

o

∆G

o

(J/(K mol))

(kJ/mol)

–86.466 –86.960 –87.339 –87.630 –87.851 –88.017 –88.137 –88.221 –88.274 –88.302 –88.309 –88.298 –88.272 –88.234 –88.186 –88.129 –88.064 –87.994 –87.919 –87.840 –87.758 –87.674 –87.587 –87.500 –87.411 –87.321 –87.230 –87.138 –87.045 –86.951 –86.857 –86.763 –86.670 –86.577 –86.484 –86.391 –86.300 –86.209 –86.118 –86.029

–257.184 –255.016 –252.837 –250.65 –248.456 –246.258 –244.056 –241.851 –239.645 –237.438 –235.23 –233.022 –230.815 –228.609 –226.404 –224.20 –221.997 –219.796 –217.597 –215.400 –213.206 –211.013 –208.822 –206.633 –204.447 –202.263 –200.081 –197.901 –195.724 –193.549 –191.376 –189.206 –187.038 –184.873 –182.709 –180.548 –178.39 –176.233 –174.079 –171.928

K

1.15E+45 1.68E+41 8.66E+37 1.23E+35 3.97E+32 2.52E+30 2.81E+28 5.04E+26 1.35E+25 5.12E+23 2.62E+22 1.73E+21 1.44E+20 1.46E+19 1.77E+18 2.51E+17 4.08E+16 7.55E+15 1.56E+15 3.58E+14 9.00E+13 2.46E+13 7.27E+12 2.30E+12 7.78E+11 2.79E+11 1.06E+11 4.20E+10 1.75E+10 7.62E+09 3.45E+09 1.62E+09 7.90E+08 3.97E+08 2.06E+08 1.10E+08 5.99E+07 3.36E+07 1.93E+07 1.13E+07

log(K)

45.061 41.225 37.938 35.09 32.599 30.401 28.449 26.702 25.131 23.709 22.418 21.239 20.158 19.164 18.247 17.399 16.611 15.878 15.194 14.554 13.954 13.391 12.862 12.362 11.891 11.446 11.024 10.623 10.243 9.882 9.538 9.21 8.897 8.599 8.313 8.04 7.778 7.527 7.286 7.054

Appendix 2

287

Table A2.2 Continued C2H6(g) + 3.5O2(g) = 2CO2(g) + 3H2O(g)

T (°C)

25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 625 650 675 700 725 750 775 800 825 850 875 900 925 950 975 1000

∆H

o

(kJ/mol)

–1427.804 –1427.322 –1426.866 –1426.444 –1426.057 –1425.709 –1425.399 –1425.130 –1424.901 –1424.711 –1424.559 –1424.445 –1424.367 –1424.325 –1424.315 –1424.338 –1424.390 –1424.472 –1424.580 –1424.714 –1424.871 –1425.049 –1425.247 –1425.464 –1425.697 –1425.946 –1426.208 –1426.483 –1426.769 –1427.066 –1427.373 –1427.688 –1428.012 –1428.342 –1428.677 –1429.018 –1429.363 –1429.713 –1430.067 –1430.425

∆S

o

(LHV) ∆G

o

(J/(K mol))

(kJ/mol)

46.411 47.966 49.324 50.497 51.501 52.350 53.060 53.645 54.118 54.490 54.774 54.977 55.110 55.181 55.196 55.162 55.085 54.971 54.823 54.647 54.448 54.228 53.991 53.739 53.475 53.202 52.922 52.636 52.346 52.052 51.756 51.458 51.160 50.863 50.568 50.274 49.983 49.694 49.408 49.124

–1441.642 –1442.822 –1444.038 –1445.287 –1446.562 –1447.860 –1449.178 –1450.512 –1451.860 –1453.217 –1454.583 –1455.955 –1457.332 –1458.710 –1460.090 –1461.470 –1462.848 –1464.224 –1465.596 –1466.965 –1468.328 –1469.687 –1471.040 –1472.386 –1473.726 –1475.060 –1476.386 –1477.706 –1479.018 –1480.323 –1481.621 –1482.911 –1484.194 –1485.469 –1486.737 –1487.997 –1489.251 –1490.497 –1491.735 –1492.967

K

log(K)

3.896E+252 1.738E+233 4.722E+216 2.149E+202 6.239E+189 5.523E+178 8.408E+168 1.401E+160 1.781E+152 1.290E+145 4.193E+138 5.025E+132 1.884E+127 1.925E+122 4.775E+117 2.604E+113 2.867E+109 5.926E+105 2.158E+102 1.311E+099 1.265E+096 1.861E+093 4.016E+090 1.231E+088 5.202E+085 2.953E+083 2.201E+081 2.107E+079 2.545E+077 3.810E+075 6.964E+073 1.533E+072 4.009E+070 1.233E+069 4.407E+067 1.815E+066 8.529E+064 4.539E+063 2.715E+062 1.813E+061

252.591 233.240 216.674 202.332 189.795 178.742 168.925 160.146 152.251 145.111 138.623 132.701 127.275 122.284 117.679 113.416 109.457 105.773 102.334 99.118 96.102 93.270 90.604 88.090 85.716 83.470 81.343 79.324 77.406 75.581 73.843 72.185 70.603 69.091 67.644 66.259 64.931 63.657 62.434 61.258

288

Appendix 2

Table A2.2 Continued C2H6(g) + 3.5O2(g) = 2CO2(g) + 3H2O(l)

T (°C)

25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 625 650 675 700 725 750 775 800 825 850 875 900 925 950 975 1000

∆H

o

(kJ/mol)

–1559.816 –1556.208 –1552.622 –1549.062 –1545.513 –1541.953 –1538.353 –1534.678 –1530.889 –1526.920 –1522.668 –1518.021 –1512.625 –1506.519 –1500.464 –1494.464 –1488.517 –1482.621 –1476.776 –1470.978 –1465.228 –1459.522 –1453.860 –1448.240 –1442.661 –1437.122 –1431.619 –1426.153 –1420.723 –1415.328 –1409.966 –1404.639 –1399.343 –1394.079 –1388.846 –1383.643 –1378.469 –1373.323 –1368.204 –1363.111

o ∆G (kJ/mol)

K

log(K)

(J/(K mol)) –310.235 –298.611 –287.925 –278.048 –268.842 –260.171 –251.905 –243.926 –236.124 –228.352 –220.415 –212.129 –202.920 –192.919 –183.393 –174.309 –165.634 –157.336 –149.389 –141.767 –134.447 –127.408 –120.632 –114.101 –107.801 –101.718 –95.836 –90.146 –84.636 –79.297 –74.121 –69.097 –64.219 –59.480 –54.871 –50.388 –46.024 –41.773 –37.630 –33.590

–1467.320 –1459.711 –1452.381 –1445.308 –1438.473 –1431.862 –1425.461 –1419.264 –1413.264 –1407.458 –1401.847 –1396.440 –1391.249 –1386.301 –1381.599 –1377.128 –1372.880 –1368.843 –1365.010 –1361.371 –1357.919 –1354.646 –1351.547 –1348.613 –1345.840 –1343.221 –1340.752 –1338.428 –1336.243 –1334.194 –1332.277 –1330.487 –1328.821 –1327.275 –1325.846 –1324.530 –1323.325 –1322.228 –1321.236 –1320.346

1.229E+257 9.342E+235 8.433E+217 2.165E+202 5.419E+188 5.850E+176 1.446E+166 4.971E+156 1.597E+148 3.478E+140 3.952E+133 1.891E+127 3.189E+121 1.638E+116 2.253E+111 7.419E+106 5.316E+102 7.632E+098 2.045E+095 9.616E+091 7.515E+088 9.311E+085 1.754E+083 4.843E+080 1.896E+078 1.023E+076 7.408E+073 7.034E+071 8.578E+069 1.318E+068 2.510E+066 5.830E+064 1.629E+063 5.408E+061 2.108E+060 9.545E+058 4.973E+057 2.954E+056 1.985E+055 1.498E+054

257.090 235.970 217.926 202.335 188.734 176.767 166.160 156.696 148.203 140.541 133.597 127.277 121.504 116.214 111.353 106.870 102.726 98.883 95.311 91.983 88.876 85.969 83.244 80.685 78.278 76.010 73.870 71.847 69.933 68.120 66.400 64.766 63.212 61.733 60.324 58.980 57.697 56.470 55.298 54.175

∆S

o

(HHV)

Appendix 3: Gas diffusion properties of a cylindrical tubular cathode substrate

In Table A3.1 the following conditions apply: r1 = 0.9 cm r2 = 1.1 cm T = 1273 K DO 2 N 2 = 2.375 cm 2 /s DO 2 He = 8.206 cm 2 /s Pt = 1 atm i = 0.3 A/cm2 Table A3.1 Properties of a cylindrical tubular cathode substrate – 1 m

Air iL (A/cm2)

O2He iL (A/cm2)

Air R cconc (Ω cm2)

O2He R cconc ∆R cconc (Ω cm2) (Ω cm2)

ε

0.034 0.035 0.036 0.037 0.038 0.039 0.040 0.041 0.042 0.043 0.044 0.045 0.046 0.047 0.048

0.305 0.314 0.323 0.332 0.341 0.350 0.359 0.368 0.377 0.386 0.394 0.403 0.412 0.421 0.430

1.060 1.091 1.122 1.153 1.185 1.216 1.247 1.278 1.309 1.340 1.372 1.403 1.434 1.465 1.496

5.680 1.980 1.196 0.855 0.665 0.544 0.459 0.397 0.350 0.313 0.283 0.258 0.237 0.219 0.203

0.033 0.032 0.031 0.030 0.028 0.027 0.027 0.026 0.025 0.024 0.023 0.023 0.022 0.022 0.021

0.184 0.187 0.190 0.192 0.195 0.197 0.200 0.202 0.205 0.207 0.210 0.212 0.214 0.217 0.219

5.647 1.948 1.165 0.826 0.637 0.516 0.433 0.372 0.325 0.289 0.259 0.235 0.214 0.197 0.182

289

290

Appendix 3

Table A3.1 Continued m

Air iL (A/cm2)

O2He iL (A/cm2)

Air R cconc (Ω cm2)

O2He R cconc ∆R cconc (Ω cm2) (Ω cm2)

ε

0.049 0.050 0.051 0.052 0.053 0.054 0.055 0.056 0.057 0.058 0.059 0.060 0.061 0.062 0.063 0.064 0.065 0.066 0.067 0.068 0.069 0.070 0.071 0.072 0.073 0.074 0.075 0.076 0.077 0.078 0.079 0.080 0.081 0.082 0.083 0.084 0.085 0.086 0.087 0.088 0.089 0.090 0.091 0.092 0.093 0.094 0.095 0.096

0.439 0.448 0.457 0.466 0.475 0.484 0.493 0.502 0.511 0.520 0.529 0.538 0.547 0.556 0.565 0.574 0.583 0.592 0.601 0.610 0.619 0.628 0.637 0.645 0.654 0.663 0.672 0.681 0.690 0.699 0.708 0.717 0.726 0.735 0.744 0.753 0.762 0.771 0.780 0.789 0.798 0.807 0.816 0.825 0.834 0.843 0.852 0.861

1.527 1.559 1.590 1.621 1.652 1.683 1.715 1.746 1.777 1.808 1.839 1.870 1.902 1.933 1.964 1.995 2.026 2.057 2.089 2.120 2.151 2.182 2.213 2.244 2.276 2.307 2.338 2.369 2.400 2.431 2.463 2.494 2.525 2.556 2.587 2.619 2.650 2.681 2.712 2.743 2.774 2.806 2.837 2.868 2.899 2.930 2.961 2.993

0.190 0.178 0.168 0.159 0.150 0.143 0.136 0.130 0.124 0.119 0.114 0.110 0.106 0.102 0.098 0.095 0.092 0.089 0.086 0.084 0.081 0.079 0.077 0.075 0.073 0.071 0.069 0.068 0.066 0.064 0.063 0.062 0.060 0.059 0.058 0.057 0.055 0.054 0.053 0.052 0.051 0.050 0.049 0.049 0.048 0.047 0.046 0.045

0.020 0.020 0.019 0.019 0.018 0.018 0.018 0.017 0.017 0.017 0.016 0.016 0.016 0.015 0.015 0.015 0.014 0.014 0.014 0.014 0.013 0.013 0.013 0.013 0.013 0.012 0.012 0.012 0.012 0.012 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.009 0.009 0.009

0.221 0.224 0.226 0.228 0.230 0.232 0.235 0.237 0.239 0.241 0.243 0.245 0.247 0.249 0.251 0.253 0.255 0.257 0.259 0.261 0.263 0.265 0.266 0.268 0.270 0.272 0.274 0.276 0.277 0.279 0.281 0.283 0.285 0.286 0.288 0.290 0.292 0.293 0.295 0.297 0.298 0.300 0.302 0.303 0.305 0.307 0.308 0.310

0.170 0.158 0.148 0.140 0.132 0.125 0.118 0.112 0.107 0.102 0.098 0.094 0.090 0.087 0.083 0.080 0.077 0.075 0.072 0.070 0.068 0.066 0.064 0.062 0.060 0.059 0.057 0.055 0.054 0.053 0.051 0.050 0.049 0.048 0.047 0.046 0.045 0.044 0.043 0.042 0.041 0.041 0.040 0.039 0.038 0.038 0.037 0.036

Appendix 3

291

Table A3.1 Continued m

Air iL (A/cm2)

O2He iL (A/cm2)

Air R cconc (Ω cm2)

O2He R cconc ∆R cconc (Ω cm2) (Ω cm2)

ε

0.097 0.098 0.099 0.100 0.101 0.102 0.103 0.104 0.105 0.106 0.107 0.108 0.109 0.110 0.111 0.112 0.113 0.114 0.115 0.116 0.117 0.118 0.119 0.120 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 0.130 0.131 0.132 0.133 0.134 0.135 0.136 0.137 0.138 0.139 0.140 0.141 0.142 0.143 0.144

0.870 0.879 0.888 0.897 0.905 0.914 0.923 0.932 0.941 0.950 0.959 0.968 0.977 0.986 0.995 1.004 1.013 1.022 1.031 1.040 1.049 1.058 1.067 1.076 1.085 1.094 1.103 1.112 1.121 1.130 1.139 1.148 1.157 1.165 1.174 1.183 1.192 1.201 1.210 1.219 1.228 1.237 1.246 1.255 1.264 1.273 1.282 1.291

3.024 3.055 3.086 3.117 3.148 3.180 3.211 3.242 3.273 3.304 3.336 3.367 3.398 3.429 3.460 3.491 3.523 3.554 3.585 3.616 3.647 3.678 3.710 3.741 3.772 3.803 3.834 3.865 3.897 3.928 3.959 3.990 4.021 4.052 4.084 4.115 4.146 4.177 4.208 4.240 4.271 4.302 4.333 4.364 4.395 4.427 4.458 4.489

0.045 0.044 0.043 0.043 0.042 0.041 0.041 0.040 0.040 0.039 0.038 0.038 0.037 0.037 0.036 0.036 0.036 0.035 0.035 0.034 0.034 0.033 0.033 0.033 0.032 0.032 0.031 0.031 0.031 0.030 0.030 0.030 0.029 0.029 0.029 0.029 0.028 0.028 0.028 0.027 0.027 0.027 0.027 0.026 0.026 0.026 0.026 0.025

0.009 0.009 0.009 0.009 0.009 0.009 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006

0.311 0.313 0.315 0.316 0.318 0.319 0.321 0.322 0.324 0.326 0.327 0.329 0.330 0.332 0.333 0.335 0.336 0.338 0.339 0.341 0.342 0.344 0.345 0.346 0.348 0.349 0.351 0.352 0.354 0.355 0.356 0.358 0.359 0.361 0.362 0.363 0.365 0.366 0.367 0.369 0.370 0.371 0.373 0.374 0.375 0.377 0.378 0.379

0.036 0.035 0.034 0.034 0.033 0.033 0.032 0.032 0.031 0.031 0.030 0.030 0.029 0.029 0.029 0.028 0.028 0.027 0.027 0.027 0.026 0.026 0.026 0.025 0.025 0.025 0.024 0.024 0.024 0.024 0.023 0.023 0.023 0.023 0.022 0.022 0.022 0.022 0.021 0.021 0.021 0.021 0.020 0.020 0.020 0.020 0.020 0.019

292

Appendix 3

Table A3.1 Continued m

Air iL (A/cm2)

O2He iL (A/cm2)

Air R cconc (Ω cm2)

O2He R cconc ∆R cconc (Ω cm2) (Ω cm2)

ε

0.145 0.146 0.147 0.148 0.149 0.150

1.300 1.309 1.318 1.327 1.336 1.345

4.520 4.551 4.582 4.614 4.645 4.676

0.025 0.025 0.025 0.024 0.024 0.024

0.006 0.006 0.006 0.006 0.006 0.006

0.381 0.382 0.383 0.385 0.386 0.387

0.019 0.019 0.019 0.019 0.019 0.018

In Table A3.2 the following conditions apply: r1 = 0.9 cm r2 = 1.1 cm T = 1213 K DO 2 N 2 = 2.237 cm 2/s DO 2 He = 7.693 cm 2/s Pt = 1 atm i = 0.25 A/cm2 Table A3.2 Properties of a cylindrical tubular cathode substrate – 2

m

Air iL (A/cm2)

O2He iL (A/cm2)

Air R cconc (Ω cm2)

O2He R cconc (Ω cm2)

conc ∆R c (Ω cm2)

ε

0.034 0.035 0.036 0.037 0.038 0.039 0.040 0.041 0.042 0.043 0.044 0.045 0.046 0.047 0.048 0.049 0.050

0.295 0.304 0.313 0.321 0.330 0.339 0.347 0.356 0.365 0.373 0.382 0.391 0.399 0.408 0.417 0.425 0.434

1.027 1.057 1.087 1.117 1.148 1.178 1.208 1.238 1.268 1.299 1.329 1.359 1.389 1.419 1.450 1.480 1.510

0.568 0.475 0.408 0.357 0.318 0.286 0.260 0.238 0.220 0.204 0.190 0.178 0.168 0.158 0.150 0.142 0.135

0.031 0.030 0.029 0.028 0.027 0.026 0.025 0.024 0.023 0.023 0.022 0.021 0.021 0.020 0.020 0.019 0.019

0.537 0.445 0.379 0.330 0.291 0.260 0.235 0.214 0.196 0.181 0.168 0.157 0.147 0.138 0.130 0.123 0.116

0.184 0.187 0.190 0.192 0.195 0.197 0.200 0.202 0.205 0.207 0.210 0.212 0.214 0.217 0.219 0.221 0.224

Appendix 3

293

Table A3.2 Continued

m

Air iL (A/cm2)

O2He iL (A/cm2)

Air R cconc (Ω cm2)

O2He R cconc (Ω cm2)

∆R cconc (Ω cm2)

ε

0.051 0.052 0.053 0.054 0.055 0.056 0.057 0.058 0.059 0.060 0.061 0.062 0.063 0.064 0.065 0.066 0.067 0.068 0.069 0.070 0.071 0.072 0.073 0.074 0.075 0.076 0.077 0.078 0.079 0.080 0.081 0.082 0.083 0.084 0.085 0.086 0.087 0.088 0.089 0.090 0.091 0.092 0.093 0.094 0.095 0.096 0.097 0.098 0.099

0.443 0.452 0.460 0.469 0.478 0.486 0.495 0.504 0.512 0.521 0.530 0.538 0.547 0.556 0.564 0.573 0.582 0.590 0.599 0.608 0.617 0.625 0.634 0.643 0.651 0.660 0.669 0.677 0.686 0.695 0.703 0.712 0.721 0.729 0.738 0.747 0.755 0.764 0.773 0.782 0.790 0.799 0.808 0.816 0.825 0.834 0.842 0.851 0.860

1.540 1.571 1.601 1.631 1.661 1.691 1.722 1.752 1.782 1.812 1.842 1.873 1.903 1.933 1.963 1.993 2.024 2.054 2.084 2.114 2.144 2.175 2.205 2.235 2.265 2.295 2.326 2.356 2.386 2.416 2.446 2.477 2.507 2.537 2.567 2.597 2.628 2.658 2.688 2.718 2.748 2.779 2.809 2.839 2.869 2.899 2.930 2.960 2.990

0.129 0.123 0.118 0.113 0.109 0.105 0.101 0.097 0.094 0.091 0.088 0.085 0.083 0.080 0.078 0.076 0.074 0.072 0.070 0.068 0.067 0.065 0.064 0.062 0.061 0.059 0.058 0.057 0.056 0.055 0.054 0.053 0.052 0.051 0.050 0.049 0.048 0.047 0.046 0.046 0.045 0.044 0.043 0.043 0.042 0.041 0.041 0.040 0.040

0.018 0.018 0.018 0.017 0.017 0.016 0.016 0.016 0.015 0.015 0.015 0.015 0.014 0.014 0.014 0.014 0.013 0.013 0.013 0.013 0.012 0.012 0.012 0.012 0.012 0.012 0.011 0.011 0.011 0.011 0.011 0.011 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009

0.111 0.105 0.100 0.096 0.092 0.088 0.085 0.082 0.079 0.076 0.073 0.071 0.068 0.066 0.064 0.062 0.061 0.059 0.057 0.056 0.054 0.053 0.051 0.050 0.049 0.048 0.047 0.046 0.045 0.044 0.043 0.042 0.041 0.040 0.039 0.039 0.038 0.037 0.037 0.036 0.035 0.035 0.034 0.034 0.033 0.032 0.032 0.031 0.031

0.226 0.228 0.230 0.232 0.235 0.237 0.239 0.241 0.243 0.245 0.247 0.249 0.251 0.253 0.255 0.257 0.259 0.261 0.263 0.265 0.266 0.268 0.270 0.272 0.274 0.276 0.277 0.279 0.281 0.283 0.285 0.286 0.288 0.290 0.292 0.293 0.295 0.297 0.298 0.300 0.302 0.303 0.305 0.307 0.308 0.310 0.311 0.313 0.315

294

Appendix 3

Table A3.2 Continued

m

Air iL (A/cm2)

O2He iL (A/cm2)

Air R cconc (Ω cm2)

O2He R cconc (Ω cm2)

∆R cconc (Ω cm2)

ε

0.100 0.101 0.102 0.103 0.104 0.105 0.106 0.107 0.108 0.109 0.110 0.111 0.112 0.113 0.114 0.115 0.116 0.117 0.118 0.119 0.120 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 0.130 0.131 0.132 0.133 0.134 0.135 0.136 0.137 0.138 0.139 0.140 0.141 0.142 0.143 0.144 0.145 0.146 0.147 0.148 0.149 0.150

0.868 0.877 0.886 0.894 0.903 0.912 0.920 0.929 0.938 0.946 0.955 0.964 0.973 0.981 0.990 0.999 1.007 1.016 1.025 1.033 1.042 1.051 1.059 1.068 1.077 1.085 1.094 1.103 1.111 1.120 1.129 1.138 1.146 1.155 1.164 1.172 1.181 1.190 1.198 1.207 1.216 1.224 1.233 1.242 1.250 1.259 1.268 1.276 1.285 1.294 1.303

3.020 3.050 3.081 3.111 3.141 3.171 3.201 3.232 3.262 3.292 3.322 3.352 3.383 3.413 3.443 3.473 3.503 3.534 3.564 3.594 3.624 3.654 3.685 3.715 3.745 3.775 3.805 3.836 3.866 3.896 3.926 3.956 3.987 4.017 4.047 4.077 4.107 4.138 4.168 4.198 4.228 4.258 4.289 4.319 4.349 4.379 4.409 4.440 4.470 4.500 4.530

0.039 0.038 0.038 0.037 0.037 0.036 0.036 0.035 0.035 0.035 0.034 0.034 0.033 0.033 0.032 0.032 0.032 0.031 0.031 0.031 0.030 0.030 0.030 0.029 0.029 0.029 0.028 0.028 0.028 0.027 0.027 0.027 0.027 0.026 0.026 0.026 0.026 0.025 0.025 0.025 0.025 0.025 0.024 0.024 0.024 0.024 0.023 0.023 0.023 0.023 0.023

0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.005

0.030 0.030 0.030 0.029 0.029 0.028 0.028 0.028 0.027 0.027 0.026 0.026 0.026 0.025 0.025 0.025 0.024 0.024 0.024 0.024 0.023 0.023 0.023 0.023 0.022 0.022 0.022 0.022 0.021 0.021 0.021 0.021 0.020 0.020 0.020 0.020 0.020 0.019 0.019 0.019 0.019 0.019 0.018 0.018 0.018 0.018 0.018 0.018 0.017 0.017 0.017

0.316 0.318 0.319 0.321 0.322 0.324 0.326 0.327 0.329 0.330 0.332 0.333 0.335 0.336 0.338 0.339 0.341 0.342 0.344 0.345 0.346 0.348 0.349 0.351 0.352 0.354 0.355 0.356 0.358 0.359 0.361 0.362 0.363 0.365 0.366 0.367 0.369 0.370 0.371 0.373 0.374 0.375 0.377 0.378 0.379 0.381 0.382 0.383 0.385 0.386 0.387

Appendix 3

295

In Table A3.3 the following conditions apply: r1 = 0.9 cm r2 = 1.1 cm T = 1173 K DO 2 N 2 = 2.056 cm 2/s DO 2 He = 7.162 cm2/s Pt = 1 atm i = 0.25 A/cm2 Table A3.3 Properties of a cylindrical tubular cathode substrate – 3 m

Air iL (A/cm2)

O2He iL (A/cm2)

Air R cconc (Ω cm2)

O2He R cconc ∆R cconc (Ω cm2) (Ω cm2)

ε

0.034 0.035 0.036 0.037 0.038 0.039 0.040 0.041 0.042 0.043 0.044 0.045 0.046 0.047 0.048 0.049 0.050 0.051 0.052 0.053 0.054 0.055 0.056 0.057 0.058 0.059 0.060 0.061 0.062 0.063 0.064 0.065 0.066

0.289 0.297 0.306 0.314 0.323 0.331 0.340 0.348 0.357 0.365 0.374 0.382 0.391 0.399 0.408 0.416 0.425 0.433 0.442 0.450 0.459 0.467 0.476 0.484 0.493 0.501 0.510 0.518 0.527 0.535 0.544 0.552 0.561

1.005 1.034 1.064 1.093 1.123 1.152 1.182 1.211 1.241 1.270 1.300 1.330 1.359 1.389 1.418 1.448 1.477 1.507 1.536 1.566 1.595 1.625 1.655 1.684 1.714 1.743 1.773 1.802 1.832 1.861 1.891 1.920 1.950

0.640 0.524 0.443 0.383 0.338 0.302 0.273 0.249 0.229 0.211 0.197 0.184 0.172 0.162 0.153 0.145 0.138 0.131 0.125 0.120 0.115 0.110 0.106 0.102 0.098 0.095 0.092 0.089 0.086 0.083 0.081 0.079 0.076

0.031 0.030 0.028 0.027 0.027 0.026 0.025 0.024 0.023 0.023 0.022 0.021 0.021 0.020 0.020 0.019 0.019 0.018 0.018 0.017 0.017 0.017 0.016 0.016 0.016 0.015 0.015 0.015 0.014 0.014 0.014 0.014 0.013

0.184 0.187 0.190 0.192 0.195 0.197 0.200 0.202 0.205 0.207 0.210 0.212 0.214 0.217 0.219 0.221 0.224 0.226 0.228 0.230 0.232 0.235 0.237 0.239 0.241 0.243 0.245 0.247 0.249 0.251 0.253 0.255 0.257

0.609 0.494 0.414 0.356 0.311 0.276 0.248 0.225 0.205 0.189 0.175 0.162 0.151 0.142 0.134 0.126 0.119 0.113 0.108 0.102 0.098 0.094 0.090 0.086 0.083 0.080 0.077 0.074 0.072 0.069 0.067 0.065 0.063

296

Appendix 3

Table A3.3 Continued m

Air iL (A/cm2)

O2He iL (A/cm2)

Air R cconc (Ω cm2)

O2He R cconc ∆R cconc (Ω cm2) (Ω cm2)

ε

0.067 0.068 0.069 0.070 0.071 0.072 0.073 0.074 0.075 0.076 0.077 0.078 0.079 0.080 0.081 0.082 0.083 0.084 0.085 0.086 0.087 0.088 0.089 0.090 0.091 0.092 0.093 0.094 0.095 0.096 0.097 0.098 0.099 0.100 0.101 0.102 0.103 0.104 0.105 0.106 0.107 0.108 0.109 0.110 0.111 0.112 0.113 0.114 0.115

0.569 0.578 0.586 0.595 0.603 0.612 0.620 0.629 0.637 0.646 0.654 0.663 0.671 0.680 0.688 0.697 0.705 0.714 0.722 0.731 0.739 0.748 0.756 0.765 0.773 0.782 0.790 0.799 0.807 0.816 0.824 0.833 0.841 0.850 0.858 0.867 0.875 0.884 0.892 0.901 0.909 0.918 0.926 0.935 0.943 0.952 0.960 0.969 0.977

1.980 2.009 2.039 2.068 2.098 2.127 2.157 2.186 2.216 2.245 2.275 2.305 2.334 2.364 2.393 2.423 2.452 2.482 2.511 2.541 2.570 2.600 2.630 2.659 2.689 2.718 2.748 2.777 2.807 2.836 2.866 2.895 2.925 2.955 2.984 3.014 3.043 3.073 3.102 3.132 3.161 3.191 3.220 3.250 3.280 3.309 3.339 3.368 3.398

0.074 0.072 0.070 0.069 0.067 0.065 0.064 0.062 0.061 0.060 0.058 0.057 0.056 0.055 0.054 0.053 0.052 0.051 0.050 0.049 0.048 0.047 0.046 0.045 0.045 0.044 0.043 0.043 0.042 0.041 0.041 0.040 0.039 0.039 0.038 0.038 0.037 0.037 0.036 0.036 0.035 0.035 0.034 0.034 0.034 0.033 0.033 0.032 0.032

0.013 0.013 0.013 0.013 0.012 0.012 0.012 0.012 0.012 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.007 0.007 0.007 0.007

0.259 0.261 0.263 0.265 0.266 0.268 0.270 0.272 0.274 0.276 0.277 0.279 0.281 0.283 0.285 0.286 0.288 0.290 0.292 0.293 0.295 0.297 0.298 0.300 0.302 0.303 0.305 0.307 0.308 0.310 0.311 0.313 0.315 0.316 0.318 0.319 0.321 0.322 0.324 0.326 0.327 0.329 0.330 0.332 0.333 0.335 0.336 0.338 0.339

0.061 0.059 0.058 0.056 0.055 0.053 0.052 0.051 0.049 0.048 0.047 0.046 0.045 0.044 0.043 0.042 0.041 0.040 0.040 0.039 0.038 0.037 0.037 0.036 0.035 0.035 0.034 0.034 0.033 0.032 0.032 0.031 0.031 0.030 0.030 0.030 0.029 0.029 0.028 0.028 0.027 0.027 0.027 0.026 0.026 0.026 0.025 0.025 0.025

Appendix 3

297

Table A3.3 Continued m

Air iL (A/cm2)

O2He iL (A/cm2)

Air R cconc (Ω cm2)

O2He R cconc ∆R cconc (Ω cm2) (Ω cm2)

ε

0.116 0.117 0.118 0.119 0.120 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128 0.129 0.130 0.131 0.132 0.133 0.134 0.135 0.136 0.137 0.138 0.139 0.140 0.141 0.142 0.143 0.144 0.145 0.146 0.147 0.148 0.149 0.150

0.986 0.994 1.003 1.011 1.020 1.028 1.037 1.045 1.054 1.062 1.071 1.079 1.088 1.096 1.105 1.113 1.122 1.130 1.139 1.147 1.156 1.164 1.173 1.181 1.190 1.198 1.207 1.215 1.224 1.232 1.241 1.249 1.258 1.266 1.275

3.427 3.457 3.486 3.516 3.545 3.575 3.605 3.634 3.664 3.693 3.723 3.752 3.782 3.811 3.841 3.870 3.900 3.930 3.959 3.989 4.018 4.048 4.077 4.107 4.136 4.166 4.195 4.225 4.255 4.284 4.314 4.343 4.373 4.402 4.432

0.032 0.031 0.031 0.030 0.030 0.030 0.029 0.029 0.029 0.029 0.028 0.028 0.028 0.027 0.027 0.027 0.027 0.026 0.026 0.026 0.026 0.025 0.025 0.025 0.025 0.024 0.024 0.024 0.024 0.024 0.023 0.023 0.023 0.023 0.023

0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.005 0.005

0.341 0.342 0.344 0.345 0.346 0.348 0.349 0.351 0.352 0.354 0.355 0.356 0.358 0.359 0.361 0.362 0.363 0.365 0.366 0.367 0.369 0.370 0.371 0.373 0.374 0.375 0.377 0.378 0.379 0.381 0.382 0.383 0.385 0.386 0.387

0.024 0.024 0.024 0.024 0.023 0.023 0.023 0.022 0.022 0.022 0.022 0.021 0.021 0.021 0.021 0.021 0.020 0.020 0.020 0.020 0.020 0.019 0.019 0.019 0.019 0.019 0.018 0.018 0.018 0.018 0.018 0.018 0.017 0.017 0.017

Appendix 4: Molecular diffusivities of selected binary gas systems

The molecular diffusivities given in Table A4.1 are calculated from equation [7.31] for the binary gas systems of O2–N2 (air), O2–He, and H2–H2O at temperatures of interest and Pt = 1 atm. Table A4.1 Molecular diffusivities of selected binary gas systems

298

t (°C)

D O 2N 2 (cm2/s)

D O 2 He (cm2/s)

D H 2H 2 O (cm2/s)

25 800 850 900 950 1000

0.204 1.799 1.947 2.056 2.237 2.375

0.743 6.183 6.679 7.162 7.693 8.206

0.924 8.294 8.930 9.579 10.248 10.937

Commonly used physical constants

e F h k NA R

Elementary charge Faraday’s constant Planck’s constant Boltzmann constant Avogadro’s number Gas constant

Atmosphere

1 atm

1.60218 × 10–19 C 9.64853 × 104 C/mol 6.62607 × 10–34 Js 1.38065 × 10–23 J/K 6.02214 × 1023 mol–1 8.3145 J/mol K 82.06 atm cm3/mol K 1.987 cal/mol K 1.987 btu lb/mol K 83.145 bar cm3/mol K 10.740 Pa ft3 lb/mol K 1.013 25 bar 101.325 kPa 760 mmHg

299

Nomenclature

Greek symbols Symbol

Meaning

Usual unit Section

α

A real parameter of constant phase element (CPE) impedance Transfer coefficient of an anodic reaction Transfer coefficient of a cathodic reaction Chemical expansion coefficient Thermal expansion coefficient Symmetry factor of a chargetransfer reaction Fuel sensitivity Compressibility Oxygen non-stoichiometry Diffusion distance of the oxygen species Cubic-field splitting of octahedral site Ψt and Ψe Intra-atomic spin–spin exchange energy responsible for Hund’s highest multiplicity rule Electrostatic component of ∆c Porosity of a porous body Strain tensor DC electrical efficiency Thermodynamic efficiency Porosity of an anode Porosity of a cathode

degree

9.1

None

7.2

None

7.2

K–1 K–1 None

2.6 2.6 7.2

volt/% m2/N None

9.5 2.6 All 9.1

eV

3.3, 3.4

eV

3.1, 3.3, 3.4, 3.5

eV None None None None None None

3.3 7.3, 9.8 2.6 8.1 8.1 7.2 7.2

αa αc αt β γ δ ∆c ∆ex

∆M

ε

εo εa εc 300

Nomenclature

εF εij εI εmax εV επ , εσ

η η ηact η aact η aconc ηa(x) η cact η cconc

η ctot ηc(x) ηe′

ηh

•

ηk ∇η k

η V .. O

θ θo θs κ λ λ

Cell efficiency Fermi energy level One-electron energy of perturbation H′ acting on orbital Ψj Current (or Faradaic) efficiency Maximum DC electrical efficiency Voltage efficiency Nearest cation-neighbor εij for electron transfer between π-bonding, σ-bonding, cation orbitals Carnot efficiency Polarization voltage loss Activation polarization loss Anode activation polarization loss Anode concentration polarization loss Local anode overpotential Cathode activation polarization loss Cathode concentration polarization loss Cathode total polarization loss Local cathode polarization loss Electrochemical potential of free electrons Electrochemical potential of electron holes Electrochemical potential of species k Gradient of electrochemical potential of species k Electrochemical potential of oxygen vacancies Radian corresponding to half of the interconnect width Phase angle of AC signal Fraction of occupied surface sites Fraction of unoccupied surface sites Dielectric screening parameter Mean free path length A real constant in constant phase element

301

None eV eV

8.1 3.1 3.1

None None None eV

8.1 8.3 8.1 3.3

None volt volt volt volt

volt volt J/mol

8.1 9.2 7.2 7.2, 7.5 7.3, 7.5, 12.1 7.5 7.2, 7.5 7.3, 7.5, 12.1 7.4 7.5 4.3

J/mol

4.5

J/mol

4.1

J/(mol cm)

4.1

J/mol

4.3

degree × π/180 degree None None

7.3

volt volt volt

None Å

9.1 7.2 7.2 5.2 7.3 9.1

302

Nomenclature

λπ, λσ, λs µe′

µh µk µ k* •

µ ko µ O2 µ v* (cond)

µ v* (sol) µ vo (sol)

µ V .. O

ν oh ξ ρ ρb

•

ρgb ρk ρliq ρt ρV .. O σ σA σe σe′

σ eo′

σh

•

Covalent-mixing parameters for π-bonding and σ-bonding O-2p orbitals and O-2s orbitals Chemical potential of free electrons Chemical potential of electron holes Chemical potential of species k Chemical potential of neutral species k Chemical potential of species k at standard state Chemical potential of oxygen molecules Chemical potential of the vacancy condensate Chemical potential of the vacancy solid solution Chemical potential of the vacancy solid solution at standard state Chemical potential of oxygen vacancies

None

3.3, 3.4

J/mol

4.1

J/mol J/mol J/mol

4.5 4.1 4.1

J/mol

4.1

J/mol

4.3

J/mol

5.4

J/mol

5.4

J/mol

5.4

J/mol

4.3, 4.5

Phonon frequency

Hz

4.5

Thermodynamic factor Density of the oxide scale Bulk density of a porous sample Bulk Resistivity Grain-boundary resistivity Resistivity of k species Density of a liquid True density of a porous sample Resistivity of oxide-ion Total electrical conductivity Lennard-Jones diameter of a spherical molecule Electronic conductivity Electrical conductivity of free electrons Pre-exponential term of Arrhenius form of electrical conductivity of free electrons Electrical conductivity of electron holes

None g/cm3 g/cm3 Ω cm Ω cm Ω cm g/cm3 g/cm3 Ω cm S/cm Å

4.1 12.4 9.8 5.4 5.4 7.1 9.8 9.8 5.4 4.2, 5.1 7.3

S/cm S/cm S/cm

5.1 3.5, 4.1, 4.3 4.2, 4.7

S/cm

4.2, 4.5

Nomenclature

303

σAB

Collision parameter for molecular diffusion

Å

7.3

σ oh

Pre-exponential term of Arrhenius form of electrical conductivity of electron holes Protonic conductivity Electrical conductivity of species k Electrical conductivity of an electronic load Electrical conductivity of oxide ion Pre-exponential term of Arrhenius form of electrical conductivity of oxygen vacancies Anti σ-bonding Electrostatic potential Electrical field Electrical field in the direction of x Electric potential difference Electrostatic potential at x = 0 Electrostatic potential at x = L Symmetrized oxygen 2pπ, 2pσ, and 2s orbitals A factor =– zF /∆ c H To atomic orbitals at site at Ri, Rj octahedral-site π-bonding, σ-bonding, ligand-field cation orbitals Tortuosity of a porous body Time for an electron transfer Time constant Frequency of an oxygen jump Attempt frequency of an oxygen jump Angular frequency Peak angular frequency Oxygen breathing-mode angular frequency Collision integral for molecular diffusion

S/cm

4.2

S/cm S/cm S/cm

5.1 4.1, 7.1 4.3, 4.5

S/cm S/cm

All 4.2, 12.1

volt volt/cm volt/cm volt volt volt None

3.5 4.1 4.1, 7.1 5.3 2.1 7.5 7.5 3.3

volt–1 None None

8.2 3.1 3.3

None second second Hz Hz

7.3 3.2 9.1 5.3 5.3, 12.1

Hz Hz Hz

9.1 9.1 3.2, 3.4, 3.5 7.3

•

σi σk σL

σ V .. O σ Vo .. O

σ* φ ∇φ ∇xφ ∆φ φ1(0) φ2(L) φ π, φ σ , φ s χ Ψi, Ψj Ψt Ψe τ τh τo υ νO ω ωmax ωo ΩD

None

304

Nomenclature

Roman symbols Symbol

Meaning

a

Usual unit Section

Lattice parameter Hopping distance of electron holes a k ( a C , a Cr2 O 3 ) Activity of species k (C, Cr2O3)

Å Å None

Activity of neutral species k Electrolyte/electrode interfacial area Pre-exponential factor of Arrhenius conductivity equation Effective cell surface area Area-specific resistance Energy parameter of the nearestneighbor bij b for electron transfer to cation from anion Energy integral of equation [3.1] for overlapping M-d and O-2p, 2s orbitals b for electron transfer between cations via an anion bridge Expectation value for an electron transfer from orbital Ψi at site at Ri to an overlapping orbital ΨI at site at Ri Rate of fuel bypass Total molar fraction of oxygen vacancies Concentration of species k Concentration gradient of species k Concentration of neutral species k Concentration of mobile oxygen vacancies Surface concentration of oxygen species Concentration of lattice oxygen Concentration of mobile oxygen vacancies Correction factor for sheet resistance

None cm2

12.1 4.5 2.2, 4.1, 10.5, 11.4 4.1 5.1, 7.3

SK/cm

5.4

cm2 2 Ω cm eV

All All 3.1

eV

3.3

eV

3.3, 3.4

eV

3.3, 3.4

eV

3.1

None None

6.4 5.2, 12.1

mol/cm3 mol/cm4 mol/cm3 mol/cm3

4.1 4.1 4.1 5.2

mol/cm3

7.2, 9.1

mol/cm3 mol/cm3

4.2 4.2

None

9.7

ak* A

Acell ASR b ba bca bcac bij

B c ck ∇ck ck* cm cO

cO × O c V .. O

C

Nomenclature

Cf Cb Cgb Cp Cp,n ∆Cp,n

C1

C2

Ca ′La Ce ′Ce × Ce Ce Co •Co × Co Co • CrCr

× CrCr

CH4-* CH3-* CH2-* CH2O-* CHO-* CO-* CO2-* CPE d dn dn dpore D˜

Fuel consumption Capacitance resulting from the bulk Capacitance resulting from the grain boundary Constant-pressure molar heat capacity Constant-pressure molar heat capacity of CnH2n+2 Change of constant-pressure molar heat capacity of oxidation reaction of CnH2n+2 Oxygen concentration at inner surface of a porous cylindrical cathode tube Oxygen concentration at outer surface of a porous cylindrical cathode tube La lattice substituted by Ca Trivalent Ce lattice (free electron) Tetravalent Ce lattice (regular Ce lattice) Tetravalent Co lattice (electron hole) Trivalent Co lattice (regular Co lattice) Tetravalent Cr lattice (electron holes) Trivalent Cr lattice (regular Crlattice) Adsorbed CH4 on Ni surface Adsorbed CH3 on Ni surface Adsorbed CH2 on Ni surface Adsorbed CH2O on Ni surface Adsorbed CHO on Ni surface Adsorbed CO on Ni surface Adsorbed CO2 on Ni surface Constant phase element Grain size d-orbital manifolds Incremental moles of ions Mean pore diameter Chemical diffusivity

305

None µF µF

6.4, 7.5 9.1 9.1

J/(Kmol)

2.3, 6.6

J/(Kmol)

6.6

J/(Kmol)

6.6

mol/cm3

9.9

mol/cm3

9.9

None None None

4.5, 4.6 12.2 12.2

None

12.2

None

12.2

None

4.5, 4.6, 12.4 4.5, 4.6, 12.4 10.4 10.4 10.4 10.4 10.4 10.4 10.4 9.1 7.2 3.3 2.1 7.3 4.1

None None None None None None None None Ω

µm None mole µm cm2/s

306

Nomenclature

D* * Dactual * Drandom

Do Da De′ Dh• Dk DO DAB ( DO 2 N 2 ) eff D AB ( DOeff2 )

DH 2 DH 2 –H 2 O DHeff2 –H 2 O

DK,A ( DK,O 2 ) DK,H 2 DO 2 DO 2 He DV .. O

e e′ e(t) EMF or E E

Ea EA EDC Eex Eg

Tracer (self) diffusivity Tracer diffusivity under actual conditions Tracer diffusivity of the atom making successive independent jumps Pre-exponential term of Arrhenius diffusivity equation Ambipolar diffusivity Particle diffusivity of free electrons Particle diffusivity of electron holes Particle diffusivity of species k Surface diffusivity of oxygen species Mass diffusivity of A (O2) through B (N2) Effective mass diffusivity of A through B (O2 through N2) Molecular hydrogen diffusivity, Mass diffusivity of H2 through H2O Effective mass diffusivity of H2 through H2O Knudsen diffusivity of species A (O2) Knudsen diffusivity of H2 Molecular oxygen diffusivity, Mass diffusivity of O2 through He Particle diffusivity of oxygen vacancies Elementary charge Free (excess) electrons Time-dependent AC voltage signal Electromotive force or Nernst potential Average Nernst potential Change of electromotive force Amplitude of AC signal Activation energy Electron affinity of O– ion DC bias applied to AC signal Nernst potential of the exit fuel versus air Energy gap

cm2/s cm2/s

4.1 4.1

cm2/s

4.1

cm2/s

5.2

cm2/s cm2/s cm2/s cm2/s cm2/s

4.1 4.1 4.2 4.1 7.2, 9.1

cm2/s

7.3

cm2/s

7.3

cm2/s cm2/s cm2/s

7.3 7.3 7.3

cm2/s

7.3

cm2/s cm2/s cm2/s cm2/s

7.3 7.3 9.4 4.5

C None volt volt

All All 9.1 2.1

volt volt volt J/mol eV V volt

7.5 2.5, 7.6 9.1 All 3.1 9.1 7.5

eV

5.2

Nomenclature

Eih Ein •

EI EM ENi Eo Eox E(x) ∆E

f fe , f t fn F Fe •Fe

Fe ′Fe Fe ×Fe FOR dG ∆G o ∆G ∆Gm ∆Gt

•

h H′

Hopping energy of electron holes Ionization energy of cation Nernst potential of the inlet fuel versus air Total energy required to transfer an electron from a cation to an O– ion Madelung energy Nernst potential of the Ni oxidation versus air Electromotive force at standard state Activation energy for metal oxidation process’ Local Nernst potential Energy to transfer electron from 2p orbital on oxygen atom to empty orbital on cation Geometrical or correlation factor Frequency Atomic d orbitals that σ-bond, π-bond, in an octahedral site f-orbital manifolds Recycling factor of anode gas Faraday’s constant Tetravalent Fe lattice (electron hole) Divalent Fe lattice (free electron) Trivalent Fe lattice (regular Felattice) Fuel oxidation reaction An increment of Gibbs free energy Change of Gibbs free energy Change of Gibbs free energy at standard state Change of motional Gibbs free energy of oxide ion Change of Gibbs free energy for trapping oxygen vacancies into clusters Electron holes Perturbation of potential energy at site at Rj by presence of atom at site at Ri

307

eV eV volt

4.5 3.1 7.5

eV

3.1

eV volt

3.1 2.2

volt

2.1, 8.1

J/mol

12.4

volt eV

7.5 3.3, 3.4

None Hz None

4.1 9.1 3.3

None None C/mol None

3.1 10.2 All 12.2

None None

12.2 12.2

None J/mol J/mol J/mol

3.5 2.1 2.2 2.1

J/mol

5.2

J/mol

5.4

None eV

All 3.1

308

Hads HR ∆H ∆Hb ∆Hc ∆He′ ∆Hg

∆ H h• ∆Hm ∆Hp ∆HPL

Nomenclature

Hydrogen atoms adsorbed on the surface Ni metal Haven ratio Change of enthalpy An energy barrier for a vacancy to hop Coulomb component of ∆Ht Motional enthalpy of free electrons Energy gap between occupied and empty oxygen sites of an ordered array Motional enthalpy of electron holes Motional enthalpy of oxide-ion Polaronic activation enthalpy

Enthalpy change of power leads due to temperature change ∆Hr Relaxation energy to equalize energies of occupied and empty oxygen site ∆Hre Lattice relaxation energy at a dopant site ∆Ht Trapping energy of an oxide ion or an oxygen vacancy o ∆H T ,air (∆Hair) Change of standard enthalpy of air at T ∆HT,dry-air Change of enthalpy of dry air at T ∆ H To ,exh Standard enthalpy change of exhaust at T ∆HT,hum-air Change of enthalpy of humid air at T ∆H To , n Standard enthalpy change of species n at T ∆ c H To , n ( ∆ c H To ) Standard heat of combustion of species n at T Standard heat of combustion of ∆ c H To ,NG natural gas at T o ∆ c H 298, Standard heat of combustion of n species n at 298 K o ∆ f H 298, Standard enthalpy of formation of n species n at 298 K H-* Adsorbed H species on Ni surface Adsorbed H2 species on Ni surface H2-*

None

12.1

None J/mol eV

4.1 2.3, 6.6 5.2

eV J/mol eV

5.2, 5.4 4.2 5.2, 5.4

J/mol J/mol eV J/mol

4.2 All 3.1, 3.4, 5.4 6.6

eV

5.2

eV

5.2, 5.4

eV

All

J/mol

6.6

J/mol J/mol

6.6 6.6

J/mol

6.6

J/mol

6.6

J/mol J/mol

6.6, 8.1, 8.2 6.6

J/mol

6.6

J/mol

6.6

None None

10.4 10.4

Nomenclature

H(s)Ni HHV HPR HS HTC IHS iave ie′ iex * iex

ih iH* 2 •

ik iL iLM iLM, air

i LM,O 2 He iO 2– iV .. O i(k) i(x) i1(x) i2(x) i3(x) Ic Ie′ Iion or I V .. O

Ik IL I(t) IS j

Adsorbed H species on Ni surface Higher heating value of a fuel Heat production rate High-spin cation dn configuration Hydrogen-to-carbon ratio Inverse hyperbolic sine Average current density Current density of free electrons Exchange current density Cathode- and temperaturedependent characteristic constant Current density of electron holes Anode- and temperaturedependent characteristic constant Current density of species k Current density of an electrical load Limiting current density Limiting current density of a cathode with air Limiting current density of a cathode with 20% O2–He Current density of oxygen ions Current density of oxygen vacancies Local current density of finite element k Local current density Local in-plane (lateral) current density in anode Local in-plane (lateral) current density in cathode Local current density in electrolyte Critical current at which cell voltage shows bend-over Free electronic current Ionic current or oxygen vacancy current Current of species k Load current Time-dependent current Intermediate-spin cation dn configuration Flux density

309

None J/mol J/s None None None A/cm2 A/cm2 A/cm2 A/cm2

7.2 8.1 2.3 3.3 6.5 7.2 7.5 4.3 7.2 7.2

A/cm2 A/cm2

4.5, 4.6 7.2

A/cm2 A/cm2

4.1, 7.1 4.3–4.6

A/cm2 A/cm2

7.3 9.4

A/cm2

9.4

A/cm2 A/cm2 A/cm2

4.3 4.3–4.6 7.5

A/cm2 A/cm2

7.5 7.5

A/cm2

7.5

A/cm2 A

7.5 7.6

A A

8.1 6.2, 8.1

A A A None

7.1 6.4, 8.1 9.1 3.3

mol/(s cm2) 4.1

310

jk j k* jm (n) jO jO 2 jN2 J Jo k* ko [k] kad,b kad,f kb kf kg k go kp k(T) K Kad KH KS K2D

K3D Kn lx

ly

Nomenclature

Flux density of species k Flux density of neutral species k Flux density of species m (n) Flux density of oxygen atoms Flux density of oxygen molecules Flux density of nitrogen molecules Flux of fuel at utilization Uf Flux of fuel at full utilization Neutral species k Pre-exponential term in equation [10.7] Concentration of species k Rate constant of backward adsorption reaction Rate constant of forward adsorption reaction Rate constant of backward reaction Rate constant of forward reaction Parabolic rate constant in mass Pre-exponential term Parabolic rate constant in thickness Intrinsic rate constant of steam methane reforming Chemical equilibrium constant Diffusion conductance Chemical constant of adsorption reaction Equilibrium constant of the hydrogen adsorption Equilibrium constant for the steam adsorption Chemical equilibrium constant of sulfur surface adsorption reaction on Ni Chemical equilibrium constant of Ni sulfide reaction Knudsen number Distance between the nearestneighbor oxygen sites in the direction of x Distance between the nearestneighbor oxygen sites in the direction of y

mol/(s cm2) mol/(s cm2) mol/(s cm2) mol/(s cm2) mol/(s cm2) mol/(s cm2) mol/s mol/s None cm/s

4.1 4.1 7.3 7.2 7.3 7.3 6.2 6.2 4.1 10.4

mol/cm3 cm/s

4.2 7.2

cm/s

7.2

cm/s cm/s g2/(cm4 s) g2/(cm4 s) cm2/s cm/s

10.5 10.5 12.4 12.4 12.4 10.4

Depending All 9.9 cm2/s Depending 7.2 Depending 10.4 Depending 10.4 None

11.1

None

11.1

None Å

7.3 5.3

Å

5.3

Nomenclature

lz

L LHV LS m

Distance between the nearestneighbor oxygen sites in the direction of z Length or thickness

Lower heating value of a fuel Low-spin cation dn configuration Maximum number of species in natural gas Molecular weight of A MA Molecular weight of B MB M Ce ′′ (or M Ce ′ ) Divalent (or trivalent) cation M substituted on Ce lattice M Zr ′′ (or M Zr ′ ) Divalent (or trivalent) cation M substituted on Zr lattice Weight of a porous sample in air M1 Weight of a porous sample in M2 suspension in a liquid Weight of a porous sample M3 saturated with liquid MIEC Mixed oxide-ion/electron conductor Mn •Mn Tetravalent Mn lattice (electron hole) Mn ′Mn Divalent Mn lattice (free electron) × Mn Mn Trivalent Mn lattice (regular Mn lattice) Mn-4s Manganese 4s orbital n Number of carbons in hydrocarbons Concentration of excess electrons N Number of oxygen sites per unit volume on which oxygen can move Number of cells in an SOFC stack Avogadro’s number NA Total concentration of species k Nk Normalization constant Nπ Normalization constant Nσ Oxide ion O2– Oxygen molecule O2 Interstitial oxide ion O 2– i (O 2– ) s Surface-charged oxygen species 2– Surface oxygen ions (O )s Oad(s) (Oads) Adsorbed oxygen species – O ′ad (s) (O ads ) Negatively charged oxygen species

311

Å

5.3

cm J/mol None None

4.3–4.7, 7.1 6.6, 8.1 3.3 6.2

g/mol g/mol None

7.3 7.3 12.1

None

12.1

g g

9.8 9.8

g

9.8

None None

3.5 12.1

None None

12.1 12.1

None None cm–3 cm–3

3.1 6.2 4.1 5.2

None mol–1 cm–3 None None None None None None None None None

6.2, 8.1 4.5 4.1 3.3 3.3 All All 5.2 3.5 3.5 7.2, 12.1 7.2, 12.1

312

O O× O O× (el) O-*

Nomenclature

Lattice oxygen ions Lattice oxygen ions in electrolyte Adsorbed oxygen species on Ni surface O-2p Oxygen 2p orbital O/C Oxygen to carbon ratio OCV Open-circuit voltage ORR Oxygen reduction reaction p p-orbital Concentration of electron holes Partial pressure of water at pA atmospheric pressure Partial pressure of water at pWS saturation P Power Pressure o Atmospheric pressure P Pb-6p Lead 6p orbital Rate of energy used to cool power Pair-PL leads Rate of energy input in air Pair PCH 4 Partial pressure of CH4 Partial pressure of CO PCO Partial pressure of CO2 PCO 2 PCrO 3 Partial pressure of CrO3 Rate of energy produced or DC PDC electrical power Specific oxygen partial pressure of Pe′ free electrons Rate of energy input in the exhaust Pexh Rate of energy input in fuels Pf Specific oxygen partial pressure of Ph electron holes PH* 2 Anode- and temperaturedependent characteristic constant PH 2 (PH 2 ( x )) Partial pressure of H2 (local) PH 2 (6) = PHo2 Bulk partial pressure of H2 PH 2 O ( PH 2 O ( x )) Partial pressure of H2O (local) PH 2 O (6) Bulk partial pressure of H2O = PHo2 O Ploss Rate of energy lost to the surroundings Partial pressure of oxygen PO 2 •

None None None

All 7.2 10.4

None None None None None cm–3 atm or Pa

3.1 10.2 All 3.5 3.1 4.2, 4.5 6.6

atm or Pa

6.6

watt atm atm None J/s or watt

All All 2.5, 8.3 5.4 6.6

J/s or watt atm atm atm atm J/s or watt

6.6 10.4 All All 11.4 6.6

atm

4.2

J/s or watt J/s or watt atm

6.6 6.6 4.2

atm

7.2, 9.5

atm atm atm atm

All 7.4, 9.5 All 7.4, 9.5

J/s or watt

6.6

atm

All

Nomenclature

PO′ 2

PO′′2 PO*2 PO 2(b) = PO 2 (1) PO 2(f) = PO 2(6) PO 2( L) PO 2( x) PO 2(0)

PO 2(2)

PO 2(3)

PP2(g) PPH 3(g) PS 2(g) PSiO Pt q

Qair QC QCO o QCO

QCO 2 o QCO 2

Partial pressure of oxygen at cathode/electrolyte interface Partial pressure of oxygen at anode/electrolyte interface Cathode- and temperaturedependent characteristic constant Bulk partial pressure of oxygen in air Bulk partial pressure of oxygen in fuel Interfacial partial pressure of oxygen at x = L Local partial pressure of oxygen Interfacial partial pressure of oxygen at x = 0 Partial pressure of oxygen at the interface of activation and concentration layers of the cathode Partial pressure of oxygen at the interface of activation layer of the cathode and electrolyte Partial pressure of P2(g) Partial pressure of PH3(g) Partial pressure of S2(g) Partial pressure of SiO(g) System pressure Electrical charge carried by mobile species Heat Mass flow rate of air

313

atm

4.2, 8.1

atm

4.2, 8.1

atm

7.2

atm

4.7, 7.3, 7.4 4.7, 7.4

atm atm

atm

4.3, 4.5, 4.7 4.3, 4.5, 4.3, 4.5, 4.7 7.3

atm

7.3

atm atm atm atm atm C/particle

11.3 11.3 11.1 11.2 All 4.1

atm atm

J sccm or slpm Thermal energy transferred between J system and the cold reservoir Mass flow rate of CO at the exit sccm or slpm Mass flow rate of CO at the inlet sccm or slpm Mass flow rate of CO2 at the exit sccm or slpm Mass flow rate of CO2 at the inlet sccm or slpm

2.3 6.3, 6.6 8.1 6.4, 9.6 6.4, 9.6 6.4, 9.6 6.4, 9.6

314

Nomenclature

Qf

Mass flow rate of fuel

sccm or slpm

Qfo

sccm or slpm J

QH 2

Mass flow rate of fuel at 100% utilization Thermal energy transferred between system and the hot reservoir Mass flow rate of H2 at the exit

QHo 2

Mass flow rate of H2 at the inlet

QH 2 O

Mass flow rate of H2O at the exit

QHo 2 O

Mass flow rate of H2O at the inlet

QL QLchem QLphys QO 2

Total leakage flux Chemical leakage flux Physical leakage flux Mass flow rate of oxygen at utilization Ua Mass flow rate of oxygen at full utilization Mass flow rate of dry-gas at the exit Oxide-to-fuel ratio at the exit Oxide-to-fuel ratio at the inlet Rate constant of steam methane reforming Inner radius of the cathode tube Outer radius of the cathode tube Total resistance or area-specific resistance Anode resistance, area specific Anode activation polarization resistance, area specific Anode concentration polarization resistance, area specific Distance from the center of the saddle-point to the nearest cation site Resistance resulted from the bulk, area specific Cathode resistance, area specific

QH

QOo 2

Qt′ r ro rCH 4 r1 r2 R Ra Raact Raconc

Rb

Rc

2.3, Chapter 6, 8.1 2.3, 6.2 8.1

sccm or slpm sccm or slpm sccm or slpm sccm or slpm sccm H2 sccm H2 sccm H2 sccm or slpm sccm or slpm sccm or slpm None None mol/(scm2)

All All 10.4

cm cm Ω or

7.3 7.3 5.1

Ω cm

2

6.4, 9.6 6.4, 7.6, 9.6 6.4, 7.6, 9.6 6.4, 9.6 6.4, 7.6 6.4 6.3 6.3 6.3 9.6

Ω cm

2

Ω cm

2

4.4, 7.5 7.2, 9.5

Ω cm

2

7.3, 9.5

Å

5.2

Ω cm

2

9.1

Ω cm

2

4.4, 7.5

Nomenclature

Rcact Rcconc

Rcconc (air) = Rcconc (O 2 N 2 ) Rcconc (O 2 He) ∆ Rcconc

Rcell Rct Rct(eff) Rdiff Re Rgb Rk Rs Rohm(x) or Rohm R1 R2 R3 s (s) S ∆S Sad(s) ∆S m ∆S t

∆ STo

315

Cathode activation polarization resistance, area specific Cathode concentration polarization resistance, area specific Cathode concentration polarization resistance in air, area specific

Ω cm

2

7.2

Ω cm

2

7.3

Ω cm

2

9.4

Cathode concentration polarization resistance in 20% O2–He, area specific Change in cathode concentration polarization resistance, area specific Total resistance of a cell Intrinsic charge-transfer resistance, area specific Effective charge-transfer resistance Diffusion-related resistance, area specific Resistance resulted from the electrode, area specific Resistance resulting from the grain boundary, area specific Ohmic resistance of species k Sheet resistance

Ω cm

2

9.4

Ω cm

2

9.4

2

All 7.2, 9.1

Local or total ohmic resistance, area specific Sheet resistance in anode Sheet resistance in cathode Resistance of electrolyte, area specific s-orbital A surface site Air stoich Entropy Change of entropy Chemisorbed sulfur species Change of motional entropy of oxide-ion Change of entropy for trapping oxygen vacancies into clusters Standard entropy change at T

Ω cm

Ω Ω cm Ω cm

2

Ω cm

2

7.2 9.1

Ω cm

2

9.1

Ω cm

2

9.1

Ω Ω/SQ 2

Ω Ω Ω cm

2

7.1 7.1, 7.5, 9.7 5.1, 7.5 7.5 7.5 7.5

None None None J/(K mol) J/(K mol) None J/(K mol)

All 7.2 6.3 All 2.3 11.1 5.2, 5.3

J/(K mol)

5.4

J/(K mol)

8.1

316 o S298 S/C SMR t

ta tc te′ tk t V .. O

T T* Tair Tc TC Texh Tf TH TJT TN Top Tt TEC TPB u

uh u oh• •

uk u V .. O U ∆U

Ua Ueff

Nomenclature

Standard-state entropy at 298 K J/(K mol) Steam-to-carbon ratio None Steam methane reforming None Time second Thickness cm Wall thickness of an anode cm Wall thickness of a cathode cm Transference number of free None electron Transference number of species k None Transference number of oxide None vacancy Absolute temperature K Temperature for short-range K or °C ordering Air inlet temperature of an SOFC K Curié temperature K Temperature of the cold reservoir K Temperature of carbon formation K Exhaust temperature of an SOFC K Fuel inlet temperature of an SOFC K Temperature of the hot reservoir K Jahn–Teller distortion temperature K or °C Antiferromagnetic transition K or °C temperature Operating temperature K or °C Temperature for long-range K or °C ordering Thermal expansion coefficient K–1 Triple phase boundary None Electrical potential volt Modulus in constant phase element Drift mobility of species k cm2/(volt Drift mobility with zero occupancy cm2/(volt of the polarons Drift mobility of small polarons cm2/(volt Drift mobility of oxide vacancy cm2/(volt Intra-atomic energy splitting of eV successive redox energies Difference in intra-atomic energy eV splitting of successive redox energies Oxygen utilization None Effective U, which includes any eV ∆ex or ∆c

2.3 10.2 10.2 All All 7.3 7.3 4.1 4.1 4.3 All 5.4 6.6 3.4 8.1 10.5 6.6 6.6 8.1 3.4 3.4 3.5 5.2, 5.5 All 7.2 7.5 9.1 s) 4.5 s) 4.5 s) 4.1 s) 5.2 3.1, 3.4 3.3, 3.4 6.3 3.3, 3.4

Nomenclature π σ U eff , U eff

Uf U π, U σ vk v V .. O vx ∆V

VA Vc (Vc(k)) Vco Vc′ Vm Vmax VO× .. VO .. VO (el) VOC Vohm(x) Vp Vs VS w

w ′max δw ′max W

Wπ , W σ x xS Xe′

Ueff for π-bonding, σ-bonding electrons Fuel utilization U for π-bonding, σ-bonding electrons Drift velocity of species k Drift velocity of oxygen vacancy Drift velocity in the direction of x Change of volume Voltage drop Partial molar volume of species A Cell voltage (local cell voltage) Leakage-free cell voltage Voltage drop across the ceramic interconnect in the presence of mixed conduction Molar volume Maximum cell voltage Neutral oxygen vacancy Oxygen vacancy Oxygen vacancy in electrolyte Open-circuit voltage Local ohmic voltage drop Pore volume Sample volume Stack voltage Coordination number (nearest neighboring site) The work (other than expansion) An increment of work (other than expansion) Tight-binding electron bandwidth Width of a sample Work W for π-bonding, σ-bonding electrons Distance Specific humidity at saturation Molar fraction of oxygen at which σ V .. = σ e ′

317

eV

3.3, 3.4

None eV

All 3.3, 3.4

cm/s cm/s cm/s cm3 volt cm3/mol volt volt volt

4.1 5.2 5.3 2.3, 2.5 All 2.5 6.6, 7.5 7.6 8.1

mol/cm3 volt None None None volt volt cm3 cm3 volt None

4.5 8.3 3.5 All 7.2 5.1 7.5 9.8 9.8 6.6, 8.1 5.3, 5.4

J J

2.1 2.1

eV cm J eV

3.1 7.1 8.1 3.3, 3.4

cm None None

All 6.6 4.7

None None

10.2 2.5, 7.3

O

Xi X m ( X O 2)

Molar fraction of species i Molar fraction of species m (O2)

318

X O 2 (1)

y yn yO YSZ z zk

|Z| Z′ Z″ ZCPE ZD ZW

Nomenclature

Molar fraction of oxygen in bulk air Molar fraction of the dopant Molar fraction of hydrocarbons in natural gas Weight fraction of oxygen in oxide Yttria-stabilized zirconia Valence Number of electrical charges of species k Modulus of the impedance Z Real-part impedance Imaginary-part impedance Constant phase element (CPE) impedance Diffusion impedance Warburg impedance

None

7.4

None None

4.5 Chapter 6 12.4 All 2.1 4.1

None None None Ω Ω Ω Ω Ω Ω

or Ω cm2 or Ω cm2 or Ω cm2

9.1 9.1 9.1 9.1 9.1 9.1

Index

activation energy, 101 activation polarisation, 101–10, 229 anode, 108–10 activation polarisation overpotential against current density of 900°C, 109 charge-transfer, 109–10 cathode, 101–7 activation polarisation overpotentials against 900°C, 106 activation potential as a function of current density, 106 charge-transfer, 101–7 oxygen-reduction process, 101 oxygen-reduction process scheme, 102 system pressure of LSM/YSZ cathode at 900°C, 106 ageing phenomenon, 78 air, flow, current and oxygen utilisation, 87–8 aliovalent doping, 71 ambipolar diffusivity, 44, 46 anode phosphorus poisoning, 207–213 silica poisoning, 202–7 sulfur poisoning, 197–202 APS see atmospheric plasma spray Archimedes’ law, 179 Arrhenius equation, 48, 193, 237, 243 Arrhenius form, 223 Arrhenius plots, 75, 76, 77 atmospheric plasma spray, 259, 261, 262 Aurivillus phase, 82 Avogadro’s number, 58 ‘bell-and-spigot’ SOFC stack, 2

Bimevox, 82–3 Bi2MoO6, ideal structure, 83–4 δ-Bi2O3, stabilised, 78 brownmillerite, 79, 81, 242 bulk transport theory, 41–6 charge neutrality, 42–3 correlations between different diffusivities, 44–6 flux density and current density, 41–2 Ohm’s law and Nernst–Einstein equation, 43–4 Butler–Volmer equation, 103–4, 229, 230 carbon filamentous, 196 formation, 183–96 phase diagram of C–H–O system, 195 thermodynamic equilibrium boundaries, 194 thermodynamics vs kinetics, 193–6 graphitic, 196 Carnot cycle, 4, 143–5 isentropic gas compression, 145 isentropic gas expansion, 144 reversible isothermal gas compression at cold temperature, 144–5 reversible isothermal gas expansion at hot temperature, 144 temperature-entropy, 144 cathode chromium poisoning of cathode, 213–19 cylindrical tubular substrate gas diffusion properties, 289–97 cylindrical tubular cathode substrate – 1, 289–92

319

320

Index

cylindrical tubular cathode substrate – 2, 292–4 cylindrical tubular cathode substrate – 3, 295–7 cell efficiency, 145–8 efficiency variations at different temperatures, 146 temperature-efficiency relation, 147 cell-to-cell connector, 6, 256–7 CeO2-based SOFCs, 233–41 anode materials, 240–1 anode polarisation resistance of YSZ + Ni, ScSZ + Ni, and SDC + Ni, 241 cathode materials, 235–7, 239–40 defect reactions illustrating creation of oxygen vacancies, 222 dependence of oxygen nonstoichiometry La0.75Sr0.25FeO3 on PO2 and temperature, 240 La1–xSrxCoO3–δ on x and temperature, 239 electrical conductivity La1–xSrxCo0.2Fe0.8O3–δ as function of temperature and Sr content in air, 237 PO2 and temperature dependence of La0.4Sr0.6FeO3, 238 PO2 dependence of La1–xSrxCoO3–δ, 238 temperature dependence of La1– xSrxCoO3–δ, 236 electrolytes, 233–5 oxide-ion conductivity of (CeO2)1–x(Sm2O3) as function of doping level x, 234 of doped ceria at 800 °C against radii of dopant cations, 234 (CeO2)1–x(Sm2O3), 234 cerium, 76–8 charge-neutrality law, 45 charge-transfer resistance, 105–10 cathode microstructural effect, 105–7 pressure effect, 107 chemical diffusivity, 44, 45 chemical equilibrium constant, 198, 200, 203, 204, 205, 210, 211, 216 chemical expansion, 227 coefficient, 19–21

chromium poisoning, of cathode, 213–19 equilibrium partial pressure gaseous Cr hydroxides in air, 214 gaseous Cr oxides in air, 214 gaseous Cr oxyhydroxides in air, 215 mechanisms, 217–19 partial pressures of Cr species formed at cathode side, 215 in H2 and H2O mixture, 216 thermodynamics of Cr–O–H system, 213, 216–17 cold zone, 184 colloidal deposition, 259 combined heat and power, 4 common diffusivity, 46 commonly used physical constants, 299 concentration polarisation, 110–25, 229 anode, 121–5 area specific resistance on current density at different H20/H2 ratios, 124 area specific resistance on current density at different porosities, 124 area specific resistance on current density at different temperature, 123 area specific resistance on current density at different thickness, 123 cathode, 113–20 limiting current density at 1000 °C and 900 °C, 119 O2 diffusion in polar coordinates, 114 polarisation loss, 118 polarisation loss at 1000 °C, 119 polarisation resistance, 118 wall thickness on polarisation loss at 1000 °C, 120 Knudsen diffusion, 111–12 mass flux density, 112–13 molecular diffusion, 111 conductivity-related diffusivity, 44 constant phase element impedance, 158 controlled atmospheric plasma spray, 262 Crofer 22, 217 Crofer 22 APU, 254, 256 Curié temperature, 31

Index current density, 131–6 axial density distribution, 131–3 lateral density distribution, 134–6 normalised distribution in electrolyte layer of SOFC, 137 pathway in SOFC, 134 de Moiver’s theorem, 159 Debye length, 42 decane, 232 diesel, 232 diffusion conductance, 181 oxygen diffusion conductance measurement, 181 experimental set-up, 181 diffusivity, 180–1 direct current electrical efficiency maximum effect of temperature, 152 two different inlet fuel compositions, 152 SOFC, 141–55 cell efficiency, 145–8 thermodynamic efficiency and Carnot heat-engine efficiency, 143–5 system pressure, 154–5 as a function of load current, 155 as a function of power, 155 dominant bulk diffusion, 252 dry die-pressing, 262 Ducrolloy, 253 dynamic-segregation theory, 254 electrical conductivity, 177–8 four-probe direct current conductivity measurement, 178 La1–xSrxCo0.2Fe0.8O3–δ as function of temperature and Sr content in air, 237 La1–xSrxMnO3+δ, 226 PO2 and temperature dependence of La0.4Sr0.6FeO3, 238 PO2 dependence of La0.7Ca0.3CrO3–δ, 249 PO2 dependence of La1–xSrxCoO3–δ, 238 and sheet resistance measurement, 177–9 temperature dependence of La1–x CaxCrO3, 249

321

temperature dependence of La1–x SrxCoO3–δ, 236 electrochemical impedance spectroscopy, 156–63 frequency response analyser, 157 single solid oxide fuel cell, 162–3 anode & cathode processes identification, 163 cathode-supported, impedance spectrum, 162 symmetrical solid electrolyte system, 157–62 constant phase element, 159 equipment set-up, 161 equivalent electrical circuit of a thick-film solid electrolyte cell, 158 equivalent electrical circuit of a thin-film solid electrolyte cell, 161 impedance spectrum thick-film solid electrolyte cell, 158 impedance spectrum thin-film solid electrolyte cell, 160 electrochemical vapour deposition, 3, 230, 259, 261 electromotive force, 7, 68, 150, 225, xii characteristic partial pressures of oxygen, 49 current density distribution, 131–6 axial, 131–3 lateral, 134–6 Eº of various electro-oxidations of fuels, 12 temperature coefficients, 17 effect of concentration, 11–14 effect of electronic conduction in electrolyte, 47–50 and Gibbs free energy change, 10–11 graphical representation of equation, 50 one dimensional distribution, 127–9, 131 current density at 1000°C and fuel consumption 0.85, 131 tubular cell segment with finite elements, 129 various average current densities, 130 various fuel consumptions at 0.100 A/cm2, 132

322

Index

various fuel consumptions at 0.305 A/cm2, 132 YSZ-electrode SOFC modelling, 130 pressure coefficient, 17–19 temperature coefficient, 16–17 variations of E of CO2–CO fuel with temperature and CO2/CO ratio, 14 variations of E of H2O–H2 fuel with temperature and H2O/H2 ratio, 13 electron diffusivity, 46 electron energy transfer integral, 23 electron hole conduction, 49 electron–electron coulomb repulsions, 25 electronic leakage current density application to electrolyte, 55–7 as function of load current density of CeO2-based electrolyte, 57 pressure effect in the electrolyte, 63–6 as function of load current iL, 65 PO2 profiles across SOFC layers under two different system pressures, 64 specific partial pressure of oxygen of plasma-sprayed ScSZ electrolyte, 64 variations of interfacial PO2 with load current iL, 65 solved interfacial partial pressures of oxygen, 56 electrophoresis deposition, 259 EMF see electromotive force energy, in SOFC system balance, 92–7 direct current electrical power, 96 distribution, 93 lower heating valve of industrial natural gas 13AO, 95 rate of exhaust, 96–7 rate of fuels, 93–5 rate of process air, 95–6 energy conservation law, 92 enhancement factor, 46 enthalpy, xii motional, 48, 70, 72, 221 trapping, 221 entropy, xii equilibrium molar fraction, 204 EVD see electrochemical vapour deposition

exhaust temperature, 97 Faraday’s law, 85, 114–15 Fe–Cr–Mn ferritic steels, 256 Fermi energy, 29 Fick’s diffusion equation, 231 Fick’s first law, 45, 103, 112 fluorite cubic phase, 233 fluorites, 33, 74–9 fuel calculating stack composition of reformed natural gas, 90–2 chemical compositions of incoming and reformed natural gas, 92 common, standard heat of combustion, 280–8 standard heat of combustion and constant-pressure molar heat capacities, 280–1 thermodynamic properties of oxidation reactions, 281–8 composition of an industrial natural gas 13AO, 87 consumption, 88–90 flow, current and utilisation, 86–7 flow diagram of fuel in natural fasfuelled SOFC generator, 90 liquid, 267 mass flow rate, 86 oxidation reaction, 33 total fuel flow, 86 fuel loss, 173–7 applications, 175–7 gas chromatography measurement, 176 total leakage flux, 176 theory, 174–5 fuel sensitivity, 170–3 β methodology, 170–3 practical measurements, 173 fuel utilisation on cathodesupported SOFC, 174 theory, 170–3 fuel utilisation, 173 galvanic cell, heat effects, 14–16 galvanic current interruption, 163–5 SOFC voltage response, 164 three-electrode configuration, 164 gas binary systems, molecular diffusivities, 298

Index gas cell, 1 gas chromatography, 91, 174–5 gas diffusion properties cylindrical tubular cathode substrate, 289–97 properties of cylindrical tubular cathode substrate – 1, 289–92 properties of cylindrical tubular cathode substrate – 2, 292–4 properties of cylindrical tubular cathode substrate – 3, 295–7 gas-phase pyrolysis, 196, 232 Gd0.1Ce0.9O2–δ, 34 Gibbs free energy, 70, 185, 203, 204, 205, 207, 218, xii change and electromotive force, 10–11 Gibbs phase rule, 13 grain boundary resistivity, 77 Haven ratio, 44–5 Hebb–Wagner blocking electrode method, 245 helium–oxygen shift, 166–70 cell testing, 168–70 theory, 167–8 cathode concentration polarisation resistance, 168–9 Hitachi Metals, Inc, 254 Hund’s highest multiplicity rule, 25, 27 hydrogenic model, 71 in-plane resistance see sheet resistance interconnects, 247–56 LaCrO3-based ceramic interconnects, 248–52 isothermal lattice expansion of La0.79Sr0.20CrO3–δ, 251 oxygen non-stoichiometry of La0.7Ca0.3CrO3–δ as function of PO2, 250 PO2 dependence of electrical conductivity of La0.7Ca0.3 CrO3-δ, 249 temperature dependence of electrical conductivity of La1–x CaxCrO3, 249 metallic interconnects, 252–4, 256 chemical compositions of commercially available ferritic stainless steels, 255

323

parabolic rate constants of several commercial Ni-based and Febased alloys, 252 intragrain resistivity, 77 ion, 86 ionic leakage current density application to interconnect, 62–3 as function of load current density of 30 mol% Ca-doped LaCrO3, 63 and partial pressure of oxygen, 62 irreversible thermodynamics, 42 principles, 41 iso-octane, 232 Jahn–Teller distortions, 31 Jahn–Teller ion, 31 Jahn–Teller stabilisation, 31 Knudsen diffusion, 111–12 Knudsen effusion mass spectrometry, 244–5 Kröger–Vink notation, 38, 47, 57, 69, 103, 221 LaCrO3, 2 LaCrO3 perovskite, 2 LaGaO3-based SOFCs, 242–7 anode materials, 247 cathode materials, 245–6 constructed electrolytic domain of La.0.9Sr0.1Ga0.8Sr0.2O3–δ, 246 defect reactions illustrating creation of oxygen vacancies, 222 electrolytes, 242–5 isothermal oxide-ion conductivity contour of LSGM electrolyte, 244 partial pressure of Ga2O(g) as function of temperature, 245 phase diagram of La–Sr–Ga–Mg–O system, 243 Langmuir isotherms, 108–9, 192, 231 LaSr2Fe2CrO9–δ, 34 La0.4Sr0.6FeO3, 238 La0.75Sr0.25FeO3, 240 La1–xSrxCo0.2Fe0.8O3–δ, 237 La1–xSrxCoO3–δ, 236, 238, 239 La1–xSrxMnO3, 35 Le Chatelier principle, 184 leakage current density, 150, 225 Lennard-Jones potential, 111

324

Index

limiting current density, 115 low-pressure plasma spray, 262 LPPS see low-pressure plasma spray Madelung energy, 24 mass-action law, 58 maximum cell voltage, 150–1 two different inlet fuel compositions, 151 mixed ionic/electronic conductor, 32 molecular diffusion, 111 gas binary systems, 298 molten carbonate fuel cell, 5 NASA code, 185 natural gas, 267 industrial composition, 87 lower heating valve, 95 reformed calculating stack fuel composition, 90–2 chemical compositions, 92 Néel temperature, 30 Nernst equation, 11, 190, 191, 231 Nernst mass, 1 Nernst potential, 218 Nernst–Einstein equation, 43–4, 47 Nernst–Einstein relationship, 70 Nernst-Planck equation, 103 Ni–Ni5P2 system, 211 ohmic polarisation, 98–101 current pathways, 99 ohmic resistance, 257 Ohm’s equation, 98–9, 135 Ohm’s law, 43–4 Onsager cross-coefficients, 41 open-circuit voltage, 54, 218 oxide-ion conductivity, 223, 233 oxide-ion electrolytes, in SOFCs, 67–84 fluorites, 74–9 cubic fluorite structure, 74 doped ceria, 76–8 pyrocholores, 79 stabilised δ-Bi2O3, 78 stabilised zirconia, 74–6 O2- ion energies and oxygen-vacancy energies, 72 other oxides, 82–4 Bimevox, 82–3 ideal structure of Bi2MoO6, 83–4

oxides with isolated tetrahedral polyanions, 83–4 perovskites, 79, 81–2 Arrhenius plots of total conductivity in air, 79 conductivity, 82 structure of cubic perovskite and brownmillerite, 80 phenomenology, 69–72 polarisation curve for electrochemical power cell, 68 quality criteria, 67–9 random-walk theory, 73 oxide-ion transference number, 54 oxide-to-fuel ratio, 89 oxygen steady-state partial pressure distribution application to electrolyte, 50–4 application to interconnect, 57–61 calculated PO2 profiles across CeO2-based electrolytes, 54 calculated profiles across the thickness of 30 mol% Ca-doped LaCrO3 interconnect, 61 charge-neutrality diagram, 52, 59 constants used for calculation, 59 profiles of ionic transference number of CeO2-based electrolytes, 55 transport of charged particles in LaCrO3-based interconnect, 59 transport of charged species in electrolyte, 52 stoich, 88 stoichiometry, 227 vacancy, 86, 225 oxygen-reduction reaction, 34 parabolic rate constant, 254 parabolic rate law, 260 particle diffusivity, 44, 60 Pauli exclusion principle, 25 perovskite, 20, 27, 33, 34, 40, 57, 81, 218, 219, 242 perturbation theory, 27 phenomenology, 69–72 phosphorus poisoning, of anode, 207–213 chemical reactions between Ni and phosphates AlPO4, 209–11, 213 equilibrium partial pressure of

Index phosphorus over a series of Ni phosphides, 209 phase diagram of Ni–P system, 208 phase stability domain of Ni–O–P system at 800°C, 210 SEM photograph and EDS chemistry of Ni-fiber surface, 212 stability domain for Ni–Ni5P2 system in H2–H2O–CO–CO2 mixture, 211 thermodynamics of Ni–P system, 208–9 physical vapour deposition, 259 poisoning see specific type of poisoning porosity, 179, 257 three-step weighing procedure for porosity & density, 180 praseodymium, 76–8 pressure, 153 effect on electrical efficiency, 154 fuel utilisation and variations of maximum cell voltage, 153 pressure coefficient, 17–19 propane, 267 proton exchange membrane, 4 pyrocholores, 79 ‘quasi-neutrality’ condition, 42, 46, 58, 60, 221, 227 random-walk theory, 72, 73 redox couples, 25 redox energies, 25 Rhodia-made low-SiO2, 235 Rolls-Royce, 9 Ruddlesden–Popper phases, 38 scanning electron microscopy, 211 screen printing, 259, 262, 263 Seebeck coefficient, 237 Seebeck measurement, 237 sheet resistance, 99, 178–9 correction factor C, 179 four-probe sheet resistance measurement, 178 Siemens, 2 Siemens AG, 253 Siemens cathode-supported SOFCs, 252 Siemens SFC, 241, 248 Siemens/Westinghouse, 251, 257 silica poisoning, of anode, 202–7

325

cell EMF of electrochemical oxidation of SiO, 207 mechanism, 206–7 partial pressure of SiO produced from exposing SiC to H2–CO–H2O–CO2 mixture, 206 produced from SiO2 after exposure to H2-H2O mixture, 204 produced from SiO2 exposed to H2–CO–H2O–CO2 mixture, 205 thermodynamics of Si–O and Si–O–C systems, 203–6 single-cell test, 198, 202, 247 slip casting method, 259 slurry coating, 262 slurry dip-coating method, 263–4 small dielectric polarons, 26 small-polaron hopping theory, 58 Sm0.2Ce0.8O1.9–δ, 78 Sm-doped CeO2, 33 SOFC see solid oxide fuel cell solid catalysed reaction, 196, 232 solid oxide fuel cell, 1–9 advantages, 4–5 applications, 5, 32–40 anode, 32–5 cathode, 35–8 interconnect, 38–40 brief history, 1–4 carbon formation, 183–96 thermodynamics vs kinetics, 193–6 components and functionality, 6–7 current, fuel flow, utilisation, and energy balance, 85–97 air flow, current and oxygen utilisation, 87–8 calculating stack fuel composition of reformed natural gas, 90–2 energy balance in SOFC system, 92–7 fuel consumption, 88–90 fuel flow, current and utilisation, 86–7 designs, 7–9 cell configuration in anodesupported planar stack, 8 cell-to-cell connections in cathodesupported tubular stack, 8 ‘segmented-in-series’ design, 9 electrical efficiency and power, 141–55

326

Index

direct current electrical efficiency, 141–8 effect of system pressure, 154–5 efficiency, fuel utilisation and electrical power, 148–9 fuel utilisation and voltage at fixed electrical efficiency and 1000°C, 149 maximum direct current electrical efficiency, 150–3 variations of efficiency of different fuels, 149 electronic properties, 23–40 applications, 32–40 electronic state formation in MnO, 24 4fo/4fn energies of CeO2, 26 formation of electronic states in MnO, 24 general considerations, 23–5 high-spin vs low-spin and intermediate-spin states, 28 ligand-field d-electron energies, 26–9 localised vs itinerant d electrons, 29–31 overpotentials of non-porous cathode films, 37 redox 4fn energies and polaronic conduction, 25–6 Ruddlesden–Popper An+1MnO3n+1 structures, 39 splitting between the t5 and t6 configurations signals, 32 unit-cell volume and Néel temperature, 30 fabrication techniques, 257–65 electrode thin film, 263–4 electrolyte thin film, 259–62 external and internal gas manifolds used in planar SOFC design, 266 interconnection thin film and plate, 264–5 principle of EVD process, 260 representative of anode made using APS process, 264 representative of ceramic interconnect made using APS process, 265 Siemens cathode substrate after sintering, 258

substrate, 257–9 tape casting process schematic, 258 materials, 220–68 CeO2-based, 233–41 interconnects and cell-to-cell connectors, 247–57 LaGaO3-based, 242–7 ZrO2-based, 220–33 nomenclature Greek symbols, 300–3 Roman symbols, 304–18 oxide-ion electrolytes, 67–84 fluorites, 74–9 other oxides, 82–4 perovskites, 79, 81–2 phenomenology, 69–72 quality criteria, 67–9 random-walk theory, 73 performance characterisation techniques, 156–82 diffusivity and diffusion conductance, 180–2 electrical conductivity and sheet resistance measurement, 177–9 electrochemical impedance spectroscopy, 156–63 fuel loss determination, 173–7 fuel sensitivity, 170–3 galvanic current interruption, 163–5 helium–oxygen shift, 166–70 porosity and density of a porous ceramic body, 179–80 voltage–current curve characterisations, 165–6 poisoning of electrodes, 197–219 chromium poisoning of cathode, 213–19 phosphorus poisoning of anode, 207–213 silica poisoning of anode, 202–7 sulfur poisoning of anode, 197–202 porous cathode substrate APS electrolyte, 262 EVD electrolyte, 261 LPPS electrolyte with composite cathode interlayer, 263 steam methane reforming, 183–96 kinetics, 191–3

Index partial pressure of oxygen and Nernst potential, 189–91 thermodynamics, 184–9 technology, xi–xii commonly used physical constants, 299 nomenclature, 300–18 thermodynamics, 10–22 effect of concentration on EMF, 11–14 EMF and Gibbs free energy change, 10–11 heat effects in a galvanic cell, 14–16 pressure coefficient of EMF, 17–19 temperature coefficient of EMF, 16–17 thermal and chemical expansion coefficients, 19–22 transport of charged particles, 41–66 application to electrolyte, 50–7 application to interconnect, 57–63 effect of electronic conduction in electrolyte on EMF, 47–50 general bulk transport theory, 41–6 pressure effect on electronic leakage current density in the electrolyte, 63–6 voltage losses, 98–140 activation and concentration polarisation of cathode, 125–7 activation polarisation, 101–10 concentration polarisation, 110–25 electromotive force and current density, 127–36 interfacial PO2 profile across all functional layer, 125 leakage flux effect, 136–40 Ohmic polarisation, 98–101 working principle, 6 Sr2Co0.8Fe0.2O3–δ, 35, 38 stainless steel ANSI446, 256 steam methane reforming, 183–96 CH4 conversion rate as function of O/C ratio at fixed temperature and pressure, 189 as function of system pressure at fixed S/C and different temperatures, 187 as function of temperature at Pt = 1 atm and different O/C ratios, 188

327

pressure effect at fixed temperature and O/C ratio, 190 equilibrium compositions CH4–H2O mixture at different H2O/CH4 ratios, 187 CH4–H2O mixture at different S/C ratio and temperature, 190 as function of temperature at fixed S/C ratio, 186 Nernst potential valve of CH4-H2O mixture, 191 kinetics, 191–3 partial pressure of oxygen and Nernst potential, 189–91 thermodynamics, 184–9 equilibrium composition of CH4H2O mixture, 185–7 reforming under load of SOFC, 188–9 Stefan-Maxwell equation, 113–14 sulfur poisoning, of anode, 197–202 effects of sulfur on adsorptions of H2 and CO on Ni catalysts, 201–2 equilibrium partial pressures of S2, 199 phase stability domain of Ni–S–O system at 600°C, 200 at 800°C, 199 sulfur activity of chemosorbed S and Ni3S2, 201 thermodynamics of Ni–S–O system, 198–201 sweeping gas, 182 Tafel equation, 104–5 tape casting method, 258, 262 terbium, 76–8 thermal cycling shrinkage, 257 thermal expansion behaviour, 227 thermal expansion coefficient, 19–21, 33, 257 thermodynamic efficiency, 142 and Carnot heat-engine efficiency, 143–5 calculated efficiency of various fuels, 143 Carnot cycle as heat engine, 144 thermodynamic factor, 46 thermodynamics data of selected chemical reactions and substances, 278–9

328

Index

constant-pressure molar heat capacities, 279 standard Gibbs free energy changes, 278 standard molar heat of formation and molar entropies, 279 solid oxide fuel cell, 10–22 entropy changes of fuel oxidation reactions as function of temperature, 15 isothermal chemical expansion and oxygen stoichiometry varying with PO2, 22 isothermal expansion curve of doped LaMnO3 measured at 1000°C, 22 thermal expansion and chemical expansion curves of LSCF, 21 Thyssen-Krupp VDM, 254 total ohmic resistance, 68 tracer diffusivity, 44 triple phase boundary, 32, 34, 35, 36, 38, 193, 206, 229 vacuum plasma spray technique, 259 vacuum slip casting, 262 voltage–current curve, 136–40, 165–6 calculated critical current as a function of leakage flux, 139 effect of leakage flux, 139 electrical circuit, 165 isothermal, and iso-fuel and iso oxygen utilisations, 166 Wagner’s equation, 252, 260 Westinghouse, 2, 230, 257, 259, 261, 262, 264

Westinghouse Electrical Corporation, 259 Westinghouse SOFC generator, 207 Westinghouse/Siemens, 263 wet powder spray, 259, 262 yttria-stabilised zirconia, 33, 35, 38, 77 Zener polaron, 31 zirconia, stabilised, 74–6 ZMG232, 217, 254, 256 ZPLOT, 161 ZrO2-based SOFCs, 220–33 anode materials, 229–33 Arrhenius plots of oxide-ion conductivity of 8YSZ and 10Sc1CeSZ electrolytes, 224 cathode materials, 225–9 change of oxygen stoichiometry of La0.8Sr0.2MnO3+δ, 228 defect reactions illustrating creation of oxygen vacancies, 222 electrical conductivity of La1xSrxMnO3+δ Arrhenius plot, 226 PO2 dependence, 226 electrolytes, 220–1, 223–5 ideal microstructure exemplified by an EVD-made anode, 231 oxide-ion conductivity of doped ZrO2 at 1000°C, motional enthalpy, and trapping enthalpy, 223 at 1000°C as function of dopant level, 223 TECs as function of doping level in La1–xAExMnO3, 227 ZVIEW, 161–2