Modeling Solid Oxide Fuel Cells: Methods, Procedures and Techniques (Fuel Cells and Hydrogen Energy)

Modeling Solid Oxide Fuel Cells FUEL CELLS AND HYDROGEN ENERGY Volume 1 Series Editor NAROTTAM P. BANSAL NASA Glenn R...

Author: R. Bove | S. Ubertini

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Modeling Solid Oxide Fuel Cells

FUEL CELLS AND HYDROGEN ENERGY Volume 1 Series Editor NAROTTAM P. BANSAL NASA Glenn Research Center Cleveland, OH 44135 USA [email protected]

Aims and Scope of the Series During the last couple of decades, notable developments have taken place in the science and technology of fuel cells and hydrogen energy. Most of the knowledge developed in this field is contained in individual journal articles, conference proceedings, research reports, etc. Our goal in developing this series is to organize this information and make it easily available to scientists, engineers, technologists, designers, technical managers and graduate students. The book series is focused to ensure that those who are interested in this subject can find the information quickly and easily without having to search through the whole literature. The series includes all aspects of the materials, science, engineering, manufacturing, modeling, and applications. Fuel reforming and processing; sensors for hydrogen, hydrocarbons and other gases will also be covered within the scope of this series. A number of volumes edited/authored by internationally respected researchers from various countries are planned for publication during the next few years. Narottam P. Bansal NASA Glenn Research Center

For a list of books published in this series, see final pages.

R. Bove

• S. Ubertini

Editors

Modeling Solid Oxide Fuel Cells Methods, Procedures and Techniques

R. Bove European Commission DG-Joint Research Centre Petten, The Netherlands

ISBN-13: 978-1-4020-6994-9

S. Ubertini DiT – Dipartimento per le Tecnologie University of Naples “Parthenope” Naples, Italy

e-ISBN-13: 978-1-4020-6995-6

Library of Congress Control Number: 2008926212 © 2008 Springer Science+Business Media, B.V. No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper 987654321 springer.com

To Piero Lunghi, without whom we would never have started studying fuel cells. Ciao Piero

Table of Contents

Preface

ix

Acknowledgements

xi

List of Symbols

xiii Part 1. Modeling Principles

1. SOFC in Brief T. Malkow

3

2. Thermodynamics of Fuel Cells W. Winkler and P. Nehter

13

3. Mathematical Models: A General Overview S. Ubertini and R. Bove

51

Part 2. Single Cell, Stack and System Models 4. CFD-Based Results for Planar and Micro-Tubular Single Cell Designs L. Andreassi, S. Ubertini, R. Bove and N.M. Sammes

97

5. From a Single Cell to a Stack Modeling I.B. Celik and S.R. Pakalapati

123

6. IP-SOFC Model S. Grosso, L. Repetto and P. Costamagna

183

7. Tubular Cells and Stacks V. Verda and C. Ciano

207

vii

viii

Table of Contents

8. Modeling of Combined SOFC and Turbine Power Systems E.A. Liese, M.L. Ferrari, J. VanOsdol, D. Tucker and R.S. Gemmen

239

9. Dynamic Modeling of Fuel Cells R.S. Gemmen

269

10. Integrated Numerical Modeling of SOFCs: Mechanical Properties and Stress Analyses H. Yakabe

323

Preface

Current national and international energy policies of industrialized and developing countries opened the way towards an increased exploitation of renewable and alternative energy sources as well as a more efficient use of fossil fuels. Clean energy conversion systems capable of operating efficiently on fossil fuels, as well as on renewable energies, represent the ideal final ring of a sustainable energy chain. The high energy conversion efficiency of solid oxide fuel cells (SOFC), together with their high fuel flexibility, make them an ideal candidate for a better exploitation of fossil fuels, and for an efficient conversion of renewable energy sources into electricity. Nowadays, numerical modeling plays an important role and represents a critical aspect of the research and technology development process in industry. This is mainly related to the increase in competition in all fields of engineering and to the enormous improvements in computer technology in the last two decades. Therefore, numerical modeling is also likely to play a key role in the development of SOFCs. This requires an interdisciplinary approach as a number of different phenomena are involved in SOFC operation. The recent increased interest in mathematical and numerical models of SOFCs is confirmed by the high number of modeling studies published in the literature. Due to the above mentioned multidisciplinary approach, the publication of such studies is largely fragmented, thus being introduced to SOFC modeling approaches, techniques and results is not always straightforward. In fact, at present, the source of information is limited to technical papers, which usually fail to provide basic understanding of the phenomena taking place in an SOFC, or to describe the different approaches needed for different requirements. This situation is in contrast with the demand from scientists, students, and young researchers for an exhaustive source of information on SOFC modeling. This book is intended to be a practical reference for defining mathematical models for SOFC simulations. For this reason, particular attention is given to all the assumptions and simplifications introduced.

ix

x

Preface

Most of the efforts of the authors who contributed to this book were focused on presenting, in a clear and understandable way, different approaches and assumptions employed for modeling SOFC operations. In particular, the book intends to be as transparent as possible with regards to data and formulas used in the implementation phase. Our hope is that the reader finds in this book a useful source of information for enabling the deployment of solid oxide fuel cells. It is our intention to remind the reader that numerical modeling represents a valid tool for analyzing energy systems more cost effectively and more rapidly than with experimental campaigns. However, this tool must be used very carefully, bearing in mind that numerical modeling means reproducing the behavior of a system by solving a set of equations describing the evolution of variables that define the state of the system. This evolution is governed by extremely complex or unknown processes. This means that models are only a simplified representation of real physics, and experimental validation is a necessary step. The book is divided into two parts. Part One introduces the reader to solid oxide fuel cells, the related main thermodynamic principles, and the main equations to be used for modeling purposes. In Part Two, the reader is provided with practical examples of how the general equations defined in part one can be simplified/adapted to specific cases. The examples cover the most widely employed single cell designs, for steady-state and dynamic conditions, and evolve towards stack and system modeling. Finally, Chapter 10 introduces the reader to the problem of mechanical stresses in SOFC, and shows an approach for modeling mechanical stresses induced by the operating conditions. Roberto Bove Stefano Ubertini

Acknowledgements

The editors would like to acknowledge the support of all the contributing authors for their work, mostly conducted by reducing their own spare time. Without their support the present book could never have been in place. During the book review process, we received a solid ground for discussion and improvements from: • • • • • • • •

Marco Biancolini, University of Rome “Tor Vergata” Comas Haynes, Georgia Institute of Technology Jan van Herle, Ecole Polytechnique Fédérale de Lausanne Paolo Iora, Politecnico di Milano Jaroslaw Milewski, Warsaw University of Technology Vincenzo Mulone, University of Rome “Tor Vergata” Alberto Traverso, University of Genoa Francesco Ubertini, University of Bologna

To them, our strong gratitude for their efforts. Finally, we would like to thank Dr. Jon Davies for reading and reviewing all the chapters, and Dr. Marc Steen and Dr. Georgios Tsotridis, who supported the idea of editing a book on solid oxide fuel cell modeling.

xi

List of Symbols

Symbols are defined in the text, when used. The following is a list of the most used ones. ∇ ∇ ∂ a x ηact ηcon ηohm σele,ion σi ρele,ion φion φele ρi αi G H P S T E0 Ea F g H J j J0

divergence gradient partial derivative symbol for indicating vector a position vector activation overpotential concentration overpotential Ohmic loss electronic or ionic conductivity mechanical stress (traction/compression) along the i direction electronic or ionic charge density ionic potential electric potential density of the ith species transfer coefficient Gibbs free-energy change enthalpy change change in pressure entropy change change in temperature open circuit voltage activation energy Faraday constant gravity acceleration enthalpy current density current generation rate exchange current density xiii

xiv

K mi Mi Ni P Pi R T t u Uf Vcell Xi Yi

List of Symbols

equilibrium constant mass flux of the ith species molecular weight of the ith species mole flux of the ith species static pressure partial pressure of the ith species universal gas constant temperature time gas velocity fuel utilisation cell Voltage mole fraction mass fraction of the ith species

Part 1 Modeling Principles

Chapter 1

SOFC in Brief Thomas Malkow Institute for Energy, European Commission, Joint Research Centre, Petten, The Netherlands

The solid oxide fuel cell (SOFC) has existed for almost 100 years. The research on SOFC started as early as the 1930s, with the most prominent work of Baur and his colleagues particularly driven by the discovery of appreciable ionic conductivity of the so called Nernst mass – doped zirconia. However, this type of fuel cell only obtained strong interest from the 1970s. It seemingly holds potential for electrical efficiencies as high as 55%; up to 70% and 90% in hybrid configuration with gas turbines and combined heat & power (CHP) generation, respectively. The SOFC has the potential for application in transportation, too, for example, in vehicular auxiliary power units (APU). Current SOFC technology demonstrates viable manufacture, feasible power ranges and applications. This has been accompanied by developments of new concepts, cell and stack designs, advanced and cost effective processing methods and improved and novel materials. The SOFC, like other fuel cells, is an electrochemical device for the conversion of chemical energy of a fuel into electricity and heat. The fuel, for example, hydrogen is not combusted but electro-oxidized at the anode (fuel electrode) by oxygen ions conducted across the electrolyte according to the following overall reaction H2 (g) + O2− → H2 O(g) + 2e− .

(1.1)

The liberated electrons pass through an external circuit to arrive at the cathode (air electrode) where they reduce molecular oxygen (present in air) to oxide ions. 1 O2 (g) + 2e− → O2− . 2

(1.2)

Water vapour is produced at the anode diluting the fuel. The hydrogen oxidation reaction (HOR) and the oxygen reduction reaction (ORR) occur at the triple phase boundary (TPB) zone where the electrode (electronic phase), electrolyte (ionic R. Bove and S. Ubertini (eds.), Modeling Solid Oxide Fuel Cells, 3–12. © Springer Science+Business Media B.V. 2008

4

T. Malkow

phase) and gas phase (hydrogen, air) are in contact. Generally, the TPB is viewed as a surface occupying a small finite volume which extends into the electrodes. However, these reactions are more complex than envisaged here and stepwise in nature; one step being rate limiting. The steps include gas phase transport (convection and diffusion) from the gas flow in the channels to the porous electrodes, porous media molecular (Fickian and Knudsen) diffusion, species adsorption, desorption, dissociation, surface and bulk diffusion, surface reaction, bulk exchange and charge transfer steps. Often, several pathways are possible; one being usually preferred depending on the prevalent conditions and the properties of the catalyst and support materials. It should be noted that the ORR is often more sluggish than the more facile HOR. A SOFC usually operates at temperatures between 800◦ C (1,472◦F) and 1,000◦C (1,832◦F). It has major advantages such as high power density (typically 200– 2,000 mW/cm2 ), fuel flexibility and the possibility of internal reforming. The latter means hydrogen is generated in the anode volume basically on its porous surface by steam reforming (SR) of, for example, methane (MSR) CH4 + H2 O ↔ CO + 3H2

(1.3)

and subsequent water gas shift (WGS) CO + H2 O ↔ CO2 + H2 .

(1.4)

Both overall reactions are catalyzed by transition metals such as Ni present as particles in the anode. The exothermic electrode reactions provide the heat required in the endothermic SR. In addition, direct electro-oxidation of such a fuel, CH4 (g) + 4O2− → CO2 (g) + 2H2 O(g) + 8e− ,

(1.5)

CH4 (g) + 3O2− → CO(g) + 2H2 O(g) + 6e− ,

(1.6)

CO(g) + O

2−

→ CO2 (g) + 2e

−

(1.7)

may take place under dry conditions though proceeding in elementary steps as does the SR. The SOFC consists of cathode, electrolyte and anode collectively referred to as the PEN – positive electrode, electrolyte, negative electrode. A single cell operated with hydrogen and oxygen provides at equilibrium a theoretical reversible (Nernst) or open circuit voltage (OCV) of 1.229 V at standard conditions (STP, T = 273.15 K, p = 1 atm). With the standard electrode potential E 0 , universal gas constant R, temperature T , Faraday’s constant F , molar concentration x and pressure p, the OCV is given by RT RT xH2 O 0 0 ln ln p. (1.8) + Erev (p, T ) = E (T , p ) − √ 2F xH2 · xO2 4F

1 SOFC in Brief

5

Fig. 1.1 Schematic of the origin of polarization (voltage) losses in a SOFC.

However, the actual measured OCV will often fall slightly below Erev by UL which represents losses in potential due to residual electronic conduction in the electrolyte and possibly also cross over of gases via micro cracks and fissures in the electrolyte. The thermodynamically obtainable cell voltage Vcell at OCV, moreover, depends on the used fuel and particularly, on operation temperature and pressure. For example, the OCV of an atmospheric SOFC operating on hydrogen and oxygen is about 0.908 V at 1,000◦C. Upon drawing current I , the voltage drops further according to Vcell = Erev − UL − I R − (ηact + ηconc ) (1.9) due to charge transfer or activation polarization ηact , electronic and ionic cell resistance R or ohmic losses (IR drop) and concentration polarization (mass transfer) losses ηconc . Ideally, the area specific resistance (ASR) of the electrolyte, the product of its conductivity and thickness, should be approximately 0.15 cm2 or lower in the temperature range of 800–1,000◦C. Then, the ASR of the overall cell current collector assembly can be reasonably limited to about 0.50 cm2 at 1,000◦C and about 1.5 cm2 at 800◦ C. Neither the mass transfer (gas phase transport of reactants to, and adducts from the reaction sites via the porous electrodes from and into the flow channels, respectively) nor the charge transport in the electrolyte are negligible (see Figure 1.1). This is of particular importance at lower operation temperatures where the electrolyte conductivity is reduced. It is also important in long term operation where materials exposure increases associated with the risk of materials degradation (occurrence of inter-diffusion zones, formation of secondary phases, decrease in cell conductance, etc.). Nonetheless, it is significant for stacks operating at high fuel utilizations, which runs the risk of fuel starvation downstream.

6

T. Malkow

Fig. 1.2 Principles, functions and schematic of a flat planar SOFC where the PEN is stacked between two bipolar plates (interconnect with gas flow fields).

By stacking several cells in series or parallel, the voltage and power sought in an application can be attained. It requires however another component, interconnect or current collector. The interconnect material can be ceramic or metallic in nature. The entire build up of the individual cells and interconnect is called the stack. Currently, two major stack designs – tubular and planar – are widely used both with benefits and drawbacks. For example, the tubular SOFC (TSOFC) as developed by Siemens Power Generation (SPG) can be seal-less manufactured. It has however a long current path around the circumference of the single cell tube which basically limits the power density obtainable to typically 200 mW/cm2 at 1,000◦C. A novel high power density (HPD) design consisting of flattened tubes is currently being developed by SPG to overcome this limitation. The HPD design employs internal ribs made of the cathode material to mechanically stabilize the tubes and to shorten the current path. Further improvements may come from the monolithic tubular structure known as DELTA-N HPD. The major benefit of TSOFC is its enhanced tolerance for high thermal stresses which allows better resistance to thermal cycling. In addition, the cathode can be made thinner, thereby reducing polarization. Other major tubular stack geometries include the open-ended tubes design by TOTO, the segmented-inseries thin flat tube integrated planar design by Rolls-Royce (RR), the segmented or staggered tubes design by Mitsubishi Heavy Industries (MHI) and the anode supported micro tubules concept by Acumentrics and Adelan. Planar SOFC, in particular, monolithic designs (MHI) are capable of high (volumetric) power densities most favoured by direct and short current paths across the stack components. The PEN is principally square, rectangular and circular (Ceramic Fuel Cells Limited (CFCL), Mitsubishi Materials Corp., Sulzer Hexis) in shape with active surface areas of 100–200 cm2 (15.5–31 in2 ). A drawback of this design is that it often necessitates the use of high temperature sealants for application at the in-

1 SOFC in Brief

7

terconnect edges to prevent premature air fuel mixing. The sealants could be rigid or compressive. Materials mainly employed are insulating glass and glass ceramics. However, these materials usually contain highly volatile and mobile alkali and alkaline earth metals. Since hermeticity, creep and stability impose a challenge at stack fabrication and operation conditions, some designs aim to avoid sealing all together. In addition to sealants, a planar SOFC often requires the use of contact materials such as ceramic layers at the cathode and metallic meshes at the anode to ensure good adherence between the all ceramic PEN and the interconnect. The few millimetres thick interconnect is sometimes called the bipolar plate as it functions in addition to current collection and heat conduction as gas separator. Furthermore, protective coatings are sometimes applied onto the bipolar plate to prevent excessive corrosion, particularly evaporation of chromium species which are poisonous to cathode activity. The gas supply to the stack is accomplished using either external manifolds, usually made of stainless steel, or internal manifolds (Versa Power Systems, HTceramix). In the latter case, the manifolds are integrated into the cell design and interconnect (Delphi). Typically, TSOFC use co- and counter-flow configurations whereas planar stacks sometimes favour cross flow simplifying manifolds attachment. The flow of air usually provides cooling to a stack in either design as does internal reforming (Sulzer Hexis). The flow regime strongly affects the distribution of gas composition, mechanical stress, stack temperature and ultimately current density. However, the mechanical self support of cells is basically provided by the thickest PEN layer; either one of the electrodes or the electrolyte (Sulzer Hexis, MHI, RR, CFCL). Thick porous ceramic (RR, MHI) and metallic substrates and interconnects (Ceres Power) onto which a thin PEN is applied have also been suggested to provide the mechanical support required. The electrolyte and anode supported cells are nowadays preferred in tubular and planar stacks. The materials requirements for the cathode are high electronic and ionic conductivity (mixed ionic electronic conductor, MIEC), sufficient open porosity (20– 40 vol-%) for gas phase transport, to be catalytically active towards OOR, chemically compatible with other stack components (electrolyte, interconnect, sealant) as well as to have dimensional and thermodynamic stability in oxidizing atmospheres. The preferred choice of material is strontium-doped lanthanum magnetite, La1−x Srx MnO3−δ (LSM) with typically x = 0.10–0.25. It has a thermal expansion of around 11 × 10−6 K−1 between room temperature and operation temperature. Other perovskites of type A3+ B3+ O2− 3 such as rare earth and alkali iron-doped cobaltite (i.e. La0.5 Sr0.5 Mn Co0.8 Fe0.2 O3−δ , BSCF) and lanthanum ferrite (preferably A site doped by Sr, LSF) are mainly used at lower temperatures of 500◦C (932◦F) to 600◦C (1,112◦F). In LSM, oxide ion conduction proceeds via oxygen vacancies incorporated by adding lower valence ions such as Sr (in Kröger Vink notation) 2LaxLa + OxO + 2SrO → 2SrLa + VO·· + La2 O3

(1.10)

8

T. Malkow

to the La sub-lattice providing for the deviation δ from stoichiometry. The electronic conduction in LSM is due to hopping of electron holes between the Mn3+ /Mn4+ valence states. Similarly, the anode must be a mixed conductor too and be sufficiently porous, chemically compatible with the electrolyte and interconnect. It must be stable in highly reducing atmospheres exhibiting oxygen partial pressures as low as 10−19 bar (10−20 atm). Furthermore, the anode must catalyse the HOR in hydrogen fuelled SOFC and should exhibit catalytic activity towards steam reforming and water gas shift in hydrocarbon fuelled SOFC. This is true for anodes containing a transition metal (Ni, Fe, Co, Cr, Cu, Mo, V). In this case and for direct hydrocarbon oxidation, the anode material should moreover resist coking due to gas phase carbon deposition. Basically, carbon formation and subsequent deposition onto the anode material, specifically metallic particles in the anode, is assisted by the Boudouard reaction 2CO(g) ↔ CO2 (g) + C(s) (1.11) and for example, methane decomposition (cracking) CH4 (g) ↔ 2H2 (g) + C(s).

(1.12)

In case of hydrogen-fed SOFC, cermets made of yttria-doped zirconia (YSZ) with 30–50 mol-% nickel are mainly used. Such 30% Ni cermets have a thermal expansion of about 12.5 × 10−6 K−1 while exhibiting electronic conductivities of 500–1,800 S/cm in the temperature and partial pressure range anticipated in SOFC operation. For the latter case, nickel could partly be replaced by copper or Cu-Ni alloys in addition to small amounts of ceria and molybdena, added to circumvent coking of Ni particles. Moreover, Cu is less susceptible to deactivation by sulphur, an impurity often present in hydrocarbon fuels. However, fuel loss, for example at high utilization (large H2 O/H2 ratio), has to be avoided to prevent oxidation of the metal particles in the anode. Usually, such oxidation is accompanied by an undesired volume expansion which can be fatal for ceramic materials. Alternative anode materials include cermets made of ceria usually doped with 10–20% samarium (SDC) and gadolinium (CGO) and single phase MIEC such as titania modified YSZ (YTZ) and chromium doped LSM (LSCM). As opposed to the electrodes, the electrolyte should ideally be a pure ionic conductor, be sufficiently dense, gas tight and as thin as possible not to compromise on the required conductivity. It must also be chemically compatible with anode and cathode materials in addition to be thermodynamically stable in oxidising and reducing environments. The electrolyte shall have high mechanical strength and toughness to withstand dynamic mechanical and thermal loads. For high temperature operations, the ultimate materials choice is so far YSZ containing 8 mol-% (13 wt-%) Y2 O3 (8YSZ) to stabilize zirconia in its face-centred cubic fluorite structure down to room temperature. Such material is polycrystalline in nature and exhibits an ionic conductivity of about 0.1 S/cm at 1,000◦C but falling to 0.03 S/cm at 800◦ C. Its thermal expansion is about 10.5 × 10−6 K−1 , rather close to that of the anode. Importantly, YSZ reacts with LSM at the electrode/electrolyte interface to form insulating La2 Zr2 O7 . Measures to combat pyrozirconate formation include the use of La-deficient LSM with excess of Mn and the application of

1 SOFC in Brief

9

a barrier layer at this interface. In addition, glass impurities such as SiO2 may segregate at grain boundaries thereby increasing electrolyte resistivity. Scandia doped zirconia (SSZ), SDC, CGO are useful alternative electrolytes. In case of the latter two materials, ceria exhibits a Ce4+ /Ce3+ reduction in reducing atmospheres, resulting in an increase in electronic conductivity but accompanied by a deleterious volume expansion. The application of a thin YSZ layer between the doped ceria and the anode may inhibit such an effect. Similarly, operation temperatures below 600◦C would retard such a reduction reaction. Moreover, doped lanthanum gallate (La0.9 Sr0.1 Ga0.8 Mg0.2 O3−δ , LSGM) is suggested as an electrolyte for intermediate temperature SOFC (ITSOFC) operating at 650◦C–750◦C (1,202◦F–1,382◦F). But, LSGM suffers from grain boundary coarsening and it reacts readily with Ni to form volatile metallic phases. Apparently, the use of single phase MIEC electrodes, for example anodes made of SDC, CGO, YTZ, doped LSC and cathodes made of La1−x Srx Co1−y By O3−δ (A = Sr, Ba, Ca and B = Fe, Cu, Ni) is a new trend gaining more and wider acceptance. Their entire electrode surface can be used advantageously for the cell reactions as compared to a much smaller active surface in conventional electrode/electrolyte material pairs. The use of MIEC certainly increases reaction rate though the determining step is likely to change. Interconnects must be good electronic and thermal conductors, chemically compatible with electrode materials, high temperature corrosion resistant in oxidising and reducing atmospheres (including carbon containing gases), and have thermal expansion matching those of other cell components. For planar stacks, the bipolar plate must be dense, have high mechanical strength and toughness and be able to be easily machined (flow channel incorporation). Metallic interconnects such as engineered Cr-based alloys and ferritic stainless steels with thermal expansions of 11.5– 12.5 ×10−6 K−1 form scales of iron oxides and chromia when subjected to SOFC conditions. It hinders electronic conduction particularly for scales which can be several tens of and even more than 100 microns thick. In addition, Cr (oxy-hydroxide) species evaporating from the plate metallic surface, or from these scales, poison the cathode when deposited on it; thereby reduce its activity significantly. Measures to combat these effects include applying protective layers of perovskites onto the bipolar plate. However, additional layers complicate stack manufacture and are likely to introduce additional contacts and issues related to changing materials behaviour during operation. In TSOFC, Sr- and Mg-doped lanthanum chromites, La1−x (Sr, Mg)x CrO3 (LSMC) with 10−5 K−1 thermal expansion are used as the interconnects. Such materials are also used in some planar stack designs. Drawbacks of LSMC are poor sinterability in air below 1,350◦C (2,462◦F) and possible formation of volatile Cr species. PEN fabrication particularly for planar stacks, is usually carried out in layers using popular deposition methods such as screen printing, wet spraying, tape casting and calendering at room temperature followed by (intermediate) firing at high temperatures between 1,300◦C (2,372◦F) and 1,700◦C (3,092◦F), respectively to sinter the PEN and ceramic interconnect. Elevated temperature fabrication techniques include plasma spraying, electrophoresis deposition (EPD) and chemical vapour deposition (CVD) for both principle designs. Moreover, co-fabrication is sometimes

10

T. Malkow

employed to minimize process steps, thereby reducing costs and automating component manufacture. It should be noted that reaction layers may form between the cell assembly components already during PEN fabrication and stack manufacture due to inter-diffusion, which can be deleterious to stack performance. It is now common practice to employ bi-layer composite cathodes composed of a thin functional layer, made of the actual cathode material, and the electrolyte sandwiched between the latter and a thick layer made exclusively of cathode material. In a similar way, anode functional layers can be applied. Such arrangement obviously enhances the respective TPB zones. Furthermore, stack developers increasingly use barrier layers between the PEN components and interconnect and graded cell structures to enlarge stack performance and durability. The former approach protects cell materials from deterioration in environments harmful to otherwise well-performing materials. The use of graded materials primarily aims to increase component functionality. For example, electrode porosity may gradually be dominated by smaller sized, more compact pores near the electrolyte to facilitate molecular diffusion and enhance electrochemical activity. A more coarse and open porous electrode structure may prevail near the gas channel to enhance mass transport (gas phase diffusion and convection). Similarly, the chemical compatibility of the PEN components and interconnect can be tailored; though it certainly necessitates the use of suitable and cost effective co-fabrication and co-firing techniques. However, it would result in material anisotropy. Interestingly, research has started on single chamber SOFC (SC-SOFC) concepts. However, the SC-SOFC exhibits inherently low power density and is therefore primarily of academic interest. It has the potential to relax cell component requirements and probably to ease manufacture. The principle of SC-SOFC is that it is fed by an air fuel mixture which flows onto the PEN contained in a single compartment, avoiding the use of gas separator plates and high temperature sealants. The fluid may flow simultaneously or sequentially along the electrodes. Both electrodes are either built onto the same side of the electrolyte some distance apart or on opposite sides. Low temperature operation would apparently suppress direct combustion of the air fuel mixture provided the electrode materials chosen are highly selective towards their respective catalytic reactions. SC-SOFC stacks may hold prospects in specific applications where the reaction products are the prime focus. More recently, protonic ceramic fuel cells (PCFC) have emerged as an alternative SOFC technology. Such fuel cells have the advantage that fuel is not diluted by water which is produced at the cathode rather than the anode as for conventional SOFC. This is of great interest when operating on hydrocarbon fuels to allow their direct conversion in addition to steam reforming. The electrolyte in the PCFC is a high temperature proton conductor such as barium and strontium zirconates as well as their cerates. Doping of alkaline cerate with rare earth oxides, for example, gadolinia forms oxide ion vacancies 2CexCe + OxO + Gd2 O3 → 2GdCe + VO·· + 2CeO2 and alternatively, electron holes

(1.13)

1 SOFC in Brief

11

1 2CexCe + O2 (g) + Gd2 O3 → 2GdCe + 2h. + 2CeO2 . 2

(1.14)

Subsequent exposure to dry hydrogen 1 H2 (g) + h. → H + , 2

(1.15)

1 H2 (g) + h. + OO → OHO· 2

(1.16)

and water vapour 1 H2 O(g) + 2h. → 2H+ + O2 (g), 2 x ·· H2 O(g) + OO + VO → 2OH·O

(1.17) (1.18)

results in the incorporation of protons, specifically hydroxyl ions, into the oxide anion sub-lattice. High temperature conduction of the protons proceeds via their hoping (displacement, hydrogen bond formation) and turning (molecular orientation, rotational diffusion, hydrogen bond breaking) between two adjacent oxide ions via what is known as the Grotthus mechanism. Notably, doped cerate exhibits conductivities of the order of 10−2 S/cm at temperatures between 500◦ C and 600◦C. But it suffers from carbonate formation in CO2 containing atmospheres in contrast to the more stable zirconate. The latter is thus more suitable as an electrolyte in the PCFC. Remarkably, yttria-doped BaCeO3 exhibits a similar level of proton conductivity down to 400◦ C (752◦F) closing the temperature gap to other low temperature proton conductors such as alkali hydrogen sulphate and phosphate recently suggested for solid electrolyte fuel cells. Nevertheless, dense material fabrication, mechanical strength and toughness remain issues in PCFC for future attention. An interesting solid state electrolyte variety is a mixed proton oxide ion conductor of single phase or as a composite. In obvious contrast to PCFC, such mixed conductors may advantageously enable effective use of mixtures of hydrogen and CO (syngas). The very idea of a SOFC being essentially a multi-component chemical and electrochemical system made of solid state partly porous materials to which a variety of gases can be fed, has significant implications for modelling. It inevitably brings together electrochemistry, fluid mechanics, materials science, electrical and system engineering. One cannot be solely considered without giving due regard to the others even for moderate model predictions. For example, composition of fluids do quite naturally vary along the flow path and locally in time during SOFC operation particularly for large stack components, greater sized stacks and in transport applications and power backup solutions. Hence, changes in the thermo-chemical properties of the fluids, stack temperature and pressure occur and exhibit certain dynamics too. This often leads to rather broad distributions of gas composition, temperature, pressure, mechanical stress and current density in the cells and the entire stack (across and in-plane). In addition, morphology, microstructure and composition of interfaces and materials including their properties and integrity vary with time and locally mainly due to inter-diffusion, precipitation and aggregation of impurities

12

T. Malkow

as well as the formation and growth of secondary phases driven by differences in temperature and partial pressure or chemical potential. Most likely, all these deteriorate the stack and thus the system performance which in turn may affect the former contributions. Note, stack deterioration or degradation can be modelled but shall be evaluated experimentally. This is usually done by measuring the decline in cell or stack voltage, in percentage per 1,000 hours of operation, compared to the voltage of the device originally obtained under first time operation. All aforementioned factors must be accounted for in comprehensive models which seek to evaluate fabrication techniques and materials selection, to identify optimum SOFC operation regimes and to predict SOFC performance at cell, stack and system level including efficiency, fuel utilization and operational life. More simple models may, however, concentrate on a smaller number of considerations to highlight certain effects, for example, to understand fluid dynamics and materials phenomena, to rate performance improvements and to elucidate performance degradation.

References and Further Reading Appleby A.J. (1996) Fuel cell technology: Status and future prospects. Energy 21(7/8), 521–653. Badwal S.P.S., Foger K. (1996) Solid oxide electrolyte fuel cell review. Ceramics Int. 22(3), 257– 265. Haile S.M. (2003) Fuel cell materials and components. Acta Mater. 51(19), 5981–6000. Quadakkers W.J., Piron-Abellan J., Shemet V., Singheiser L. (2003) Metallic interconnectors for solid oxide fuel cells – A review. Mater. High Temp. 20(2), 115–127. Singhal S.C. (2000) Advances in solid oxide fuel cell technology. Solid State Ionics 135(1/4), 305–313. Singhal S.C., Kendall K. (2004) High-Temperature Solid Oxide Fuel Cells: Fundamentals, Design and Applications, Elsevier, Oxford. Steele B.C.H., Heinzel A. (2001) Materials for fuel-cell technologies. Nature 414, 345–352. Vielstich W., Lamm A., Gasteiger H. (2003) Part. 8 Solid oxide fuel cells and systems. In: Handbook of Fuel Cells: Fundamentals, Technology, Applications, Volume 4: Fuel Cell Technology and Applications, John Wiley & Sons, Chichester, pp. 987–1122. Yamamoto O. (2000) Solid oxide fuel cells: Fundamental aspects and prospects. Electrochim. Acta 45(15/16), 2423–2435. Zhu W.Z., Deevi S.C. (2003) A review on the status of anode materials for solid oxide fuel cells. Mater. Sci. Eng. A 362(1/2), 228–239.

Chapter 2

Thermodynamics of Fuel Cells W. Winkler and P. Nehter Fuel Cell Lab, Hamburg University of Applied Sciences, Berliner Tor 21, D-20099 Hamburg, Germany

Nomenclature A CP e ei F r G H H˙ r H h∗ i I K∗ LHV m ˙ NA n˙ ∗ nel n˙ el nF n˙ n p p0 Pel pel pi

cell area temperature dependent heat capacity elementary charge specific exergy of the component i Faraday constant Gibbs enthalpy or free enthalpy of the reaction enthalpy enthalpy flow reaction enthalpy specific enthalpy related to the standard state current density electrical current non-equilibrium constant lower heating value mass flow Avogadro constant constant molar flow at the fuel cells anode quantity of released electrons related on the utilised fuel molar flow of electrons molar quantity of the supplied fuel molar flow molar quantity total pressure standard pressure electrical power specific electrical power partial pressure of the component i

R. Bove and S. Ubertini (eds.), Modeling Solid Oxide Fuel Cells, 13–50. © Springer Science+Business Media B.V. 2008

14

Ploss Prev Q q R Rm s∗ S r S s T Uf V EN δV ν −Wt −wt x η ϕ λ µ ν ζ

W. Winkler and P. Nehter

power loss reversible electrical power heat specific heat electric or ohmic resistance universal gas constant specific entropy related to the standard state total entropy reaction entropy entropy production absolute temperature fuel utilisation voltage or potential Nernst voltage voltage loss specific volume technical work specific technical work molar concentration efficiency celsius temperature excess air value fuel related specific mass fuel related quantity exergetic efficiency

Indices and abbreviations 0 stoichiometric value A air aB after burner outlet AH air heater AFC air at thermodynamic state of the fuel cell An anode ASR area specific resistance Ca cathode CC Carnot cycle CHP combined heat and power generation ECO economizer EXCO external cooling F fuel FC fuel cell FFC fuel at thermodynamic state of the fuel cell FGC flue gas cooler FH fuel heater G flue gas

2 Thermodynamics of Fuel Cells

GT H2 H2 O HEG HEX HP HPA HPF I i, j INEX irr LP O O2 PH ref rev RG RH SH ST syst

15

gas turbine hydrogen water reversible heat engine (flue gas) heat exchangers high pressure heat pump (air) heat pump (fuel) inlet components intermediate expansion irreversible low pressure outlet oxygen product heater reformer reversible reaction product gas reheater superheater steam turbine system

2.1 Introduction A solid oxide fuel cell is an electrochemical device which converts the Gibbs free enthalpy of the combustion reaction of a fuel and an oxidant gas (air) as far as possible directly into electricity. Hydrogen and oxygen are used to illustrate the simplest case. This allows the calculation of the reversible work for the reversible reaction. Heat must be transferred reversibly as well to the surrounding environment in this instance. During the operation of a SOFC, two effects are identified to reduce the electrical power available from an ideal cell; the first is the ohmic resistance which generates heat, and the second is the irreversible mixing of gases which causes a voltage drop. Generally, this means that an SOFC is not able to convert the complete fuel. In a real SOFC system, heat is exchanged within the SOFC in several ways including fuel processing, air preheating, flue gas cooling, etc. The excess air is commonly required to prevent overheating, while the conversion of hydrocarbons into hydrogen and carbon monoxide often absorbs heat. The released heat of a SOFC can be used within a heat engine such as a piston engine or a gas turbine. These combined SOFC/heat engine cycles are analysed as well.

16


Fig. 2.1 The reversible fuel cell, its energy balance and its system boundary.

2.2 The Reversible Voltage The first and the second law of thermodynamics allow the description of a reversible fuel cell, whereas in particular the second law of thermodynamics governs the reversibility of the transport processes. The fuel and the air are separated within the fuel cell as non-mixed gases consisting of the different components. The assumption of a reversible operating fuel cell presupposes that the chemical potentials of the fluids at the anode and the cathode are converted into electrical potentials at each specific gas composition. This implies thatno diffusion occurs in the gaseous phases. The reactants deliver the total enthalpy ni Hi to the fuel cell and the total enthalpy nj Hj leaves the cell (Figure 2.1). The first law of the thermodynamics gives qFC + wt FC = r H.

(2.1)

The molar reaction enthalpy r H of the oxidation consists of work and heat. The second law of thermodynamics applied on reversible processes yields (2.2) dS = 0 ⇒ qFC = qFCrev = TFC · r S, where the reversible heat exchange with the environment equalises the generated reaction entropy, and we get qFCrev + wt FCrev = r H.

(2.3)

The reaction entropy r S is a result of the different opportunities of the species to save thermal energy between the absolute zero level of temperature and the temperature level of the reactor. Concerning the energy balance of a fuel cell (Figure 2.1), the heat QFCrev has to be transferred reversibly from the fuel cell to the environment. QFCrev is defined as a positive value if the reversible change in entropy is


17

Fig. 2.2 Transport processes within a SOFC.

positive as well and thus the heat is transported from the environment to the fuel cell. The negative reversible work −WtFCrev is transferred from the fuel cell to the environment. Equations (2.3) and (2.4) give the molar reversible work wt FCrev wt FCrev = r H − TFC · r S.

(2.4)

Using the ambient temperature as a reference for the calculation of the Gibbs free enthalpy r G, the reversible work wt FCrev of the reaction is equal to the Gibbs free enthalpy of the reaction wt FCrev = r G = r H − TFC · r S.

(2.5)

The reversible efficiency ηFCrev of the fuel cell is defined as the ratio of the Gibbs free enthalpy r G and the reaction enthalpy r H at the thermodynamic state of the fuel cell. r H − TFC · r S r G . (2.6) ηFCrev = r = H r H Generally, a reversible SOFC system operation is exclusively possible if the system environment is connected reversibly with the ambient state. This is the approach in Section 2.5. A SOFC can be described as an electrical device as well, whereas the electrical effects are explained by thermodynamics. Figure 2.2 shows the transport processes which occur within a SOFC. The oxidation of hydrogen complies to the following equation 1 H2 + O2 → H2 O. 2

(2.7)

18


The hydrogen oxidation within a fuel cell occurs partly at the anode and the cathode. Different models were supposed for the detailed reaction mechanisms of the hydrogen at Ni-YSZ (yttria stabilised zirconia) cermet anodes. The major differences of the models were found with regard to the location where the chemical and electrochemical reactions occur at the TPB (three-phase boundary of the gaseous phase, the electrode and the electrolyte). However, it is assumed that the hydrogen is adsorbed at the anode, ionised and the electrons are used within an external electrical circuit to convert the electrical potential between the anode and the cathode into work. Oxygen is adsorbed at the cathode and ionised by the electrons of the load. The electrolyte leads the oxide ion from the cathode to the anode. The hydrogen ions (protons) and the oxide ion form a molecule of water. The anodic reaction is H2 → 2H+ + 2e−

(2.8)

1 O2 + 2e− → O−2 . 2

(2.9)

and the cathodic reaction is

The oxide ion O2− is conducted through the electrolyte from the cathode to the anode. At the anode, water is formed according to the reaction 2H+ + O−2 → H2 O.

(2.10)

The reaction product H2 O is mixed with the anode gas and its concentration increases with higher fuel utilisations Uf as shown in Figure 2.2. The fuel utilisation Uf is defined as the ratio of the utilised fuel and the maximum available fuel, Uf =

m ˙ FOX m ˙ FAnO =1− , m ˙ FI m ˙ FI

(2.11)

˙ FI is the fuel mass flow at the inlet of the anode where m ˙ FOX is the oxidised fuel, m and m ˙ FAnO is the fuel mass flow at its outlet. A similar definition can be formulated with the molar flow. Equation (2.8) shows that the molar flow of the electrons is twice that of the molar flow of hydrogen. n˙ el = 2n˙ H2 .

(2.12)

The electric current I is a linear function of the molar flow n˙ el of the electrons or the molar flow of the spent fuel – in this case the molar flow n˙ H2 of the spent hydrogen. I = n˙ el · (−e) · NA = −n˙ el · F = −2n˙ H2 · F.

(2.13)

In (2.13) we introduced the elementary charge e e = (1.60217733 ± 0.00000049) · 10−19C. and the Faraday constant F

(2.14)


19

F = e · NA = (96485.309 ± 0.029)C/mol

(2.15)

as the product of the elementary charge and the Avogadro constant NA . Equations (2.13) and (2.15) show that the electric current I is a measure of the oxidised fuel. Thus, the current measurement is a common method to evaluate the fuel spent. This implies that the electrolyte is free of electric leaks and the electrodes are free of any parallel reactions. The reversible power can be written as a product of the reversible voltage VFCrev and the current I or as a product of the molar flow of the fuel n˙ H2 and the Gibbs free enthalpy r G of the oxidation reaction. PFCrev = VFCrev · I = n˙ H2 · wt FCrev = n˙ H2 · r G.

(2.16)

The reversible voltage VFCrev at constant anodic and cathodic gas compositions can be calculated by Equations (2.13) and (2.16) VFCrev =

−n˙ H2 · r G . n˙ el · F

(2.17)

Equation (2.12) shows that the ratio between the molar flow of the electrons and the spent hydrogen is 2. This can be simplified by nel which is the number of the electrons that are released during the ionisation process per utilised fuel molecule, related to the molar flows calculated by Equation (2.11) nel =

n˙ el . Uf · n˙ FI

(2.18)

Finally, the reversible voltage VFCrev of the oxidation of any fuel gas is VFCrev =

−r G . nel · F

(2.19)

The influence of the fuel utilisation on the reversible voltage VFCrev , which is similar to the Nernst voltage EN , can be calculated by the change of the partial pressures of the components within the system [2, 3]. We can write Equation (2.4) more precisely as r G(T , p) = r H (T , p) − T · r S(T , p). (2.20) Using the assumption of the ideal gas, the enthalpy gets independent from the pressure (2.21) r G(T , p) = r H (T ) − T · r S(T , p) and we can write, with dS = (dH − ν dp)/T , for the entropy Sj of any component j T CPj (t) pj 0 Sj (T , p) = Sj + dt − Rm · ln . (2.22) t p0 T0 where CPj is the temperature dependent heat capacity of the component j . The pressure dependence of CPj can be neglected due to the previous assumption of the

20


ideal gas. Using Equation (2.22), we can calculate the reaction entropy δ r S(T , p) for the complete conversion of reactants. pj νj r S(T , p) = r S(T ) = Rm · ln j . (2.23) p0 νj is the fuel related quantity of the component j in the equation of the oxidation reaction. The standard pressure is (1 bar): p0 = 1 bar.

(2.24)

Using Equations (2.21) to (2.23) we get pj νj r r G(T , p) = G(T ) + T · Rm · ln j = r G(T ) + T · Rm · ln(K ∗ ), p0 (2.25) where K ∗ can be considered as non-equilibrium constant which represents the product of partial pressures of reacting species in the gaseous bulk at the anode and the cathode, respectively. The temperature dependent Gibbs free enthalpy r G(T ) at standard pressures can be calculated by T νi · b Hi0 + cp,i (T ) dT r G(T ) = T0

i

−T ·

i

νi · Si0 +

T T0

cP ,i (T ) dT T

.

(2.26)

The conversion of the reaction at the specific temperature, pressure and initial gas compositions is governed by its thermodynamic equilibrium. According to Equation (2.16), the maximum available work of the fuel cell can be determined by the Nernst potential, which represents the electrical potential of the reaction. EN =

−1 −r G(T , p) = el · [r G(T ) + T · Rm · ln(K ∗ )]. el n ·F n ·F

(2.27)

The oxidation of hydrogen H2 , of carbon monoxide CO and of methane CH4 can be analysed as examples by using Equation (2.27): 1 H2 + O2 → H2 O, 2 1 CO + O2 → CO2 , 2 CH4 + 2O2 → 2H2 O + CO2 .

(2.28) (2.29) (2.30)


21

Table 2.1 The reversible oxidation of hydrogen, carbon monoxide and methane. Fuel r H 0 in kJ/mol r S 0 in J/(mol·K) r G0 in kJ/mol r G∗ at 1000◦ C, 1 bar in kJ/mol nel 0 in V EN ∗ at 1000◦ C, 1 bar in V EN ∗ at 25◦ C, 0.1 bar in V EN ∗ at 1000◦ C, 0.1 bar in V EN ∗ at 25◦ C, 10 bar in V EN ∗ at 1000◦ C, 10 bar in V EN

H2

CO

CH4

–241.82 –44.37 –228.59 –185.33 2 1.185 0.960 1.170 0.897 1.199 1.024

–282.99 –86.41 –257.23 –172.98 2 1.333 0.896 1.318 0.833 1.348 0.960

–802.31 –5.13 –800.68 –795.68 8 1.037 1.031 1.037 1.031 1.037 1.031

The reaction enthalpy, the reaction entropy, the Gibbs free enthalpy and the Nernst voltage are calculated for the oxidation of hydrogen, carbon monoxide and methane with the thermodynamic data at the standard conditions 0 (25◦C, 1 bar) as collected in e.g. [1, 5]. The variation of the thermodynamic state of the environment of the reversible cell provides a first idea of the behaviour of real cells under changing operation conditions. It can be assumed that the reaction enthalpy and the reaction entropy depend only slightly on the temperature. Thus, the Gibbs free enthalpy of the reaction can be approximated at higher temperatures with the values of the reaction enthalpy and the reaction entropy at standard conditions (first approximation of Ulrich). The values of the reversible cell voltage are calculated for the standard state 0, at 1000◦C/1 bar and for 25◦ C and 1000◦C at 0.1 and 10 bar. These linearised values of the Gibbs free enthalpy and the reversible cell voltage at different thermo∗ . The water is always assumed to be dynamic states are written as r G∗ and EN in the gaseous phase because of the high operating temperatures of the SOFC. The results can be found in Table 2.1 and Figure 2.3. Equations (2.28) and (2.29) indicate that the molar number of the products caused by the oxidation of hydrogen H2 and carbon monoxide CO are smaller than the total molar number of the reactants. Concerning the oxidation of CH4 , the molar number of the products and reactants are equal. Thus, there is theoretically no change in the entropy for the last case. This is the reason for the low dependency of the Gibbs free enthalpy of the methane oxidation from the temperature. Furthermore, the idealised pressure dependence of the entropy yields no change in the cell voltage caused by the system pressure. The reversible cell voltage resulting from the oxidation of hydrogen and carbon monoxide decreases with a higher system temperature and increases with a higher system pressure. The influence of the fuel utilisation on the voltage reduction can be easily calculated by considering the change of the partial pressures of the components within the system [2]. The oxidation of hydrogen 1 H2 + O 2 → H 2 O 2

(2.31)

22


Fig. 2.3 The reversible cell voltage of different fuels at different states (p, T ) of the environment (linearised model and assumption of ideal gas).

is a good example to illustrate this influence. The partial pressure pi of the component i (2.32) pi = xi · p, can be expressed by xi for the molar concentration of the component i and p for the total pressure of the system. Concerning Equation (2.11), we can express the fuel utilisation as xFI · n˙ AnI − xFO · n˙ AnO Uf = (2.33) xFI · n˙ AnI by the molar flow of the fuel F as the product of the molar concentration x and the total molar flow at the inlet I and the outlet O of the anode side An. Uf is used as a variable and thus the outlet O can be interpreted as a space variable along the axis of the parallel flowing fuel and air defined by a certain Uf to be obtained. The local Nernst voltage EN(Uf ) depends on the local gas concentration and can be thus expressed as a function of Uf as well. For the further calculation all terms have to be expressed by Uf . The molar flow at the anode is constant in this case. n˙ AnI = n˙ AnO = n˙ ∗ .

(2.34)

Thus, Equation (2.33) can be simplified to Uf H 2 = 1 −

xH2 ,O . xH2 ,I

(2.35)


23

The equation of the reaction (2.7) shows that the molar flow n˙ H2 ,U of the utilised fuel is equal to the molar flow n˙ H2 O,O of the produced water at the outlet O n˙ H2 ,U = n˙ H2 O,O

(2.36)

if the used hydrogen is dry (xH2 ,I = 1). These yields Uf H 2 =

n˙ H2 ,U n˙ H O,O = 2 ∗ = xH2 O,O . ∗ n˙ n˙

(2.37)

Following Equation (2.7) we can write for the cathode side n˙ O2 ,U =

1 · n˙ H2 ,U . 2

(2.38)

Technically realised SOFC systems operate with air instead of oxygen and with an excess air λ > 1. The incoming air flow is the inlet flow n˙ Cal of the cathode n˙ Cal =

n˙ ∗ 1 ·λ· . 2 0.21

(2.39)

The outlet flow of the cathode can be expressed by n˙ CaO =

n˙ ∗ 1 1 ·λ· − · n˙ H2 ,U . 2 0.21 2

(2.40)

The related molar oxygen flow at the inlet is n˙ O2 ,I =

1 · λ · n˙ ∗ 2

(2.41)

and the molar oxygen flow at the outlet is respectively n˙ O2 ,O =

1 · (λ · n˙ ∗ − n˙ H2 ,U ). 2

Equations (2.41) and (2.42) can be expressed as a function of Uf n˙ H2 ,U 1 ∗ 1 ∗ λ λ − − Uf H 2 = · n˙ · n˙ CaO = · n˙ · 2 0.21 n˙ ∗ 2 0.21 and n˙ O2 ,O

1 1 ∗ n˙ H2 ,U = · n˙ ∗ · (λ − Uf H2 ). = · n˙ · λ − 2 n˙ ∗ 2

(2.42)

(2.43)

(2.44)

We find now yi easily as a function of the fuel utilisation Uf xH2 ,O = 1 − Uf H2 ,

(2.45)

xH2 O,O = Uf H2

(2.46)

24


and xO2 ,O =

λ − Uf H2 n˙ O ,O = 2 . λ/0.21 − Uf H2 n˙ CaO

(2.47)

With Equation (2.27) the ideal Nernst voltage EN can be expressed as a function of the fuel utilisation Uf with

1/2 Uf H2 λ/0.21 − Uf H2 . G(T ) + T · Rm ln (1 − Uf H2 )[(λ − Uf H2 ) · p]1/2 (2.48) Equation (2.49) shows that the current I is proportional to the fuel utilisation Uf −1 EN = el n ·F

r

Uf =

I . nel · n˙ FI · F

(2.49)

This implies that the electrolyte is free of electric leaks and the electrodes are free of any parallel reactions. The fuel (2 for H2 ) determines the number of transferred electrons nel . The Faraday constant F is a constant value and the fuel inlet flow n˙ FI is the only variable influencing the relation between fuel utilisation Uf and current I . Fuel utilisation Uf and current I deliver the same expression if the fuel flow is kept a constant. Equation (2.48) shows that EN → +∞ for Uf → 0 and EN → −∞ for Uf → 1, respectively. However, the model of the ideal gas gives a good approximation for 0 < Uf < 1 in the real operation regime of a SOFC. This model based on the thermodynamic equilibrium already shows the principal influences of the system pressure p, SOFC temperature ϑSOFC , excess air λ and fuel utilisation Uf on the Nernst voltage EN . The Nernst voltage EN is shown as a function of the fuel utilisation Uf in a SOFC in Figure 2.4 with H2 as a fuel and with the system pressure p as a parameter. The excess air and the SOFC temperature are the fixed parameters. The range of practical interest between Uf = 0.1 and Uf = 0.9 can be well approximated with the model of the ideal gas. The dotted lines show the adaptation of the model for a high fuel utilisation. The amount of the water fraction and the decrement in the hydrogen and oxygen fraction within the SOFC reduces EN between Uf = 0.1 and Uf = 0.9 by about more than 200 mV. An increment of the system pressure from 1 to 10 bar increases EN by about 70 mV, which is caused by the change in the partial pressure of the oxygen. At higher SOFC temperatures, the Nernst Voltage EN decreases as shown by Equations (2.27) and (2.48). The design of the total system strongly depends on the excess air λ. Figure 2.5 shows the Nernst voltage EN as a function of the excess air ratio λ and the system pressure p as a parameter. Higher excess air ratios λ result in an increment of the Nernst voltage EN , but the amount in EN decreases at higher excess air. At excess air ratios λ higher than 2, the increment of EN shows a minor amount. In a range of 1 < λ < 2 the voltage increase of about ≈ 30 mV. The maximum power Pelmax (Equation (2.48)) of a single cell is determined by the Nernst voltage EN,O and the corresponding current IO depending on the fuel utilisation Uf , O at the outlet O.


25

Fig. 2.4 The calculated Nernst voltage EN as a function of the fuel utilisation Uf .

Fig. 2.5 The calculated Nernst voltage EN as a function of the excess air λ.

Pel max = EN,O · IO .

(2.50)

The left hand side of Figure 2.6 illustrates the maximum available power of a single cell. It shows that the maximum available power of a single cell is limited by the lowest Nernst voltage which is governed by the fuel utilisation. The serial connection of a number of cells allows an integration of the curve of EN as shown on the right hand side of Figure 2.6. The integration of each cell voltage results in a higher total voltage and lower total current at a constant fuel utilisation. The

26


Fig. 2.6 The increment in efficiency by cascading single cells.

cascaded cells allow an increment in the maximum available power and maximum available efficiency compared with a single cell. A two dimensional numerical simulation of the cascaded concept, which takes the local distribution of gas species, current, voltage and temperature into account, has shown that this concept is able to operate at higher average cell voltages in principle [6]. This is because the average cell voltage of the cascaded concept is not limited by the lowest Nernst voltage which is given by the gaseous bulk of the reactants. This could be in particular interesting if highest fuel utilisation could be realised. The fuel utilisation is commonly chosen with 85% at Ni-Cermet anodes. A further amount in the fuel utilisation would result in a stronger formation of nickel oxide, which decreases the catalytic activity for the hydrogen oxidation. However, a concept which allows highest fuel utilisation at a constant tendency of the nickel oxide formation could be realised by using an anode gas condenser as proposed in [7]. Generally, the limitation of the cell voltage by the lowest Nernst voltage depends on cell voltage at design point as well. Economic analyses of the optimum cell voltage at the design point have shown that cell voltages in a range of 0.6– 0.7 V are acquired [11]. For this range of cell voltage the cascaded concept is not necessary from a physical point of view as long as the fuel utilisation is lower than 95%. But another benefit of the cascaded concept is given by the reduction in the total current, which could reduce the ohmic losses within the electrodes.

2.3 Power Density at Constant Resistance The power density of fuel cells is an important issue with regard to the design of cells. The results of the reversible calculation can be used as well for these con-


27

siderations. The total power Pel of a single cell at a locally constant cell voltage is given by the integral of the differential power. Pel = VCell · dI. (2.51) The area specific power density pel is used to estimate the required cell area at a specific total power. It can be determined by the quotient of the total power and the total cell area A. The quotient of the total current Itotal and the total cell area is defined as average current density i¯˙total.

VCell · dI Pel = = VCell · i¯total. (2.52) pel = A A It is necessary to consider the relation between the electrochemical conversion of the reacting species and the cell voltage. The conversion of the reacting species in fuel cells is coupled directly with the exchanged electrical current. The principles will be shown again for hydrogen as fuel. The molar consumption of reactants is determined by the following Faradays law Itotal , 2·F Itotal , n˙ H2 O,O − n˙ H2 ,OI = 2·F Itotal , n˙ O2 ,O − n˙ O2 ,O = 4·F n˙ H2 I − n˙ H2 ,O =

(2.53)

where 2 mol electrons per mol hydrogen and 4 mol electrons per mol oxygen are exchanged. This implies again that the electrolyte is free of electric leaks and the electrodes are free of any parallel reactions. The index “I” is used at entry and “O” at the outlet of the cell (Figure 2.7). The fuel utilisation Uf is defined as quotient of the converted hydrogen and the maximum convertible hydrogen according to Equation (2.35). The maximum convertible hydrogen can be expressed as maximum available electrical current Imax , as well Imax = F · 2 · n˙ H2 ,I , (2.54) Itotal = Uf · Imax .

(2.55)

The total electrical current Itotal of the whole cell area A is proportional to the converted part of the maximum available current Imax . It is assumed that the electrical current flow is exclusively directed perpendicular to the cell’s area. With this assumption the ohmic losses in the direction parallel to the electrodes are neglected. In the case of a non-linear locally distribution of the electrical current, it is necessary to calculate the local current density ilocal,

28


Fig. 2.7 Schematic of a single solid oxide fuel cell.

ilocal =

Imax · Uf dI = , dA dA

(2.56)

whereas the differential electrical current is equal to the differential utilisation of the maximum current. The existence of a current is caused by a potential gradient. In case of a fuel cell, the Nernst voltage EN represents the driving force as potential gradient. The Nernst voltage changes with the partial pressure of reacting species, which is caused by the change in entropy. The influence of the fuel utilisation on the Nernst voltage increases with higher operation temperatures. As a result of the transferred species, loss mechanisms occur. In terms of the first law of thermodynamics these losses are well known as polarisation losses. Polarisation losses are sensitively influenced by numerous mechanisms, which are strongly non-linear with respect to a change of the operational parameters like the current density, electrical potentials, temperature, pressure, gas compositions and material properties. These parameters are assumed to be constant in case of a differential cell area. Thus, the loss mechanisms are summarised in a constant area specific resistance ASR [ cm2]. A change of the local overpotential (EN(Uf ) − VCell ) at constant ASR complies with a proportional change in the local current density. EN(Uf ) − VCell dI = . dA ASR

(2.57)

Generally, high fuel utilisations are aspired to achieve high efficiencies, whereas different Nernst voltages occur along the cell area. Thus, the dependency of the fuel utilisation from the electrical current (Equation (2.56)) has to be implemented in Equation (2.57). EN(Uf ) − VCell Imax · dUf = . (2.58) dA ASR As mentioned previously, the ASR is assumed to be constant along the cell area to consider exclusively the influence of the Nernst voltage on the cell performance. The


29

local resolution of the current can be calculated by the integral of Equation (2.58). This approach neglects the change in the Nernst potential, which is caused by the diffusion in the gaseous bulk along the flow direction of reactants.

A=Atotal A=0

1 dA = ASR

Uf =Uf total Uf =0

Imax dUf . EN (Uf ) − VCell

(2.59)

Different conditions under test procedures and practical operation require different calculation procedures for the evaluation of the test results. In this subsection, three cases at different distributions of the Nernst voltage are considered with regard to the solution of the integral cell area and integral fuel utilisation, respectively. Case A: Nernst voltage is constant. Case B: Nernst voltage changes inversely proportional with the fuel utilisation. Case C: Nernst voltage changes according to Equation (2.27). The temperature of the gaseous phases and the solid materials are assumed to be equal and constant along the cell area. Case A: At comparably high maximum currents (Imax I ), the gaseous outlet composition is similar to the inlet composition. Thereby, the Nernst voltage along the isothermal fuel cell is approximately constant. This condition occurs mostly where small cell areas are investigated. n˙ H2 ,O ≈ n˙ H2 I n˙ H2 O,O ≈ n˙ H2 O,I

→

EN ≈ const.

(2.60)

The solution of Equation (2.59) at a constant Nernst voltage shows a linear dependency of the cell voltage from the overpotential. The Ohmic law is similar to the term, where the voltage drop is proportional to the current density represented by the quotient of the utilised part of the maximum current and the cell area. VCell = EN −

ASR · Uf · Imax . A

(2.61)

The cell voltage is governed by Equation (2.61), whereas Equation (2.52) gives the power density at a constant Nernst voltage. Uf · Imax ASR · Uf · Imax pel = EN − · . (2.62) A A Case B: If the fuel is utilised within a range of hydrogen to water pressure ratios (pH2 /pH2 O ) between 0.7 and 0.3, the Nernst voltage changes approximately inversely proportional with the fuel utilisation. Thereby, a linear approach is used to determine the

30


Nernst voltage from the fuel utilisation, EN(Uf ) = EN,I +

EN · Uf , Uf

(2.63)

whereas EN,I is the Nernst voltage at the entry of the anode and EN /Uf is the slope of change in Nernst voltage.

A=Atotal A=0

1 dA = ASR

Uf =Uf total Uf =0

EN,I +

Imax EN Uf · Uf

− VCell

dUf .

(2.64)

The solution of Equation (2.64) is given by

VCell

⎡ ⎞ ⎛ ⎤−1 EN EN Uf · A ⎠ − 1⎦ , = EN,I − · Uf ⎣exp ⎝ uf ASR · Imax

whereas the power density is governed by ⎛ ⎡ ⎛

(2.65)

⎤−1 ⎞ Uf · Imax EN ⎜ ⎠ − 1⎦ ⎟ . (2.66) pel = ⎝EN,I − · Uf ⎣exp ⎝ ⎠· Uf ASR · Imax A EN Uf

·A

⎞

Case C: Concerning practical applications, high fuel utilisations result in low of hydrogen to water pressure ratios at the outlet of the anode, whereas high hydrogen pressures occur at the entry of the anode. Thereby, the non-linear dependency of the Nernst voltage from the fuel utilisation has to be taken into account with respect to the integral solution of Equation (2.59).

A=Atotal A=0

×

1 dA = ASR

Uf =Uf total

(2.67)

Uf =0

Imax 1 r GH2 (T) + T · Rm · ln p − 2·F

pH2 O(Uf ) √ H2 (Uf ) · pO2 (Uf )

·

√

p0

dUf . − VCell

The numerical solution of Equation (2.67) determines the power density in combination with Equation (2.52) as well. A fixed cell area of 1 cm2 , a temperature of 800◦C, a total pressure of 1 bar, an oxygen partial pressure of 0.21 bar and an area specific resistance of 1 cm2 are chosen to compare the cell performances at uniform conditions. The total current of Itotal = 0.3 A and the molar hydrogen fraction at the outlet of the anode xH2 = 0.299 are kept constant for this consideration as well. The molar hydrogen fraction at the outlet is chosen as constant to show the influence of higher Nernst voltages on the power density (Case C) in comparison to the reference Case A at a constant Nernst


31

Table 2.2 Analytical calculation results of the power density. Case A: EN = const B: EN ∼ (xH2 /xH2 O) C: EN = f (xH2 /xH2 O )

n˙ H2 O,I [10−6 mol/s]

xH2 ,I [mol/mol]

Imax [A]

Uf [%]

VCell [V]

pel [W/cm2 ]

500 2.72 2.22

0.3 0.7 0.99

96.4 0.524 0.428

0.31 57.1 70

0.698 0.726 0.746

0.209 0.217 0.224

Fig. 2.8 Local resolution of the Nernst voltage and the molar hydrogen fraction.

voltage. Thus, the hydrogen flow rate and the hydrogen fraction at the entry of the anode are adjusted to obtain the linear and non-linear dependency of Nernst voltage from the fuel utilisation. The calculation results of each case are summarised in Table 2.2. A constant Nernst Voltage (Case A) is obtained at a high hydrogen flow rate, which results in a high maximum current and low fuel utilisation. The local solution of the Nernst voltage and the molar fraction of the hydrogen (Figure 2.8) show a constant local distribution. In this case, the locally constant overpotential of 0.29V results in a cell voltage of 0.698V. Hence, the current and power density obtain locally constant values with ilocal = 0.3 Acm−2 and pel,local = 0.209 Wcm−2 in each section of the cell’s area (Figure 2.9). In Case B, the hydrogen flow rate is chosen with 2.72·10−6 mol/s to approximate a linear change in the molar hydrogen fraction from 0.7 at the entry to 0.299 at the outlet of the anode along the cell area. In this range of the molar fraction, the Nernst voltage changes approximately inversely proportional with the fuel utilisation. Even if the Nernst voltage, the overpotential, the current density and the power density change inversely proportional with the fuel utilisation, the local distribution

32


Fig. 2.9 Local resolution of the power and the current.

of the Nernst voltage and the current density show a slightly non-linear dependency. This is mainly caused by the fact that the area section, which is required to transfer a specific current at a specific overpotential through the cell, increases disproportionately with lower overpotentials. It is further shown that the differential current of Case B is displaced to the entry of the cell, where higher overpotentials occur. To keep the total current constant, the differential current at the outlet of the cell has to be lower than the total or average current density. Even if the Nernst voltage at the outlet of the cell is kept constant, the cell voltage of Case B achieves a value of 0.726 V, which is 28 mV higher than the cell voltage of Case A. In Case C, the hydrogen flow rate is chosen with 2.22 · 10−6 mol/s to obtain a non-linear change of the Nernst voltage at a change of the molar hydrogen fraction from 0.99 at the entry to 0.299 at the outlet of the anode. The Nernst voltage, the overpotential, the current density and the power density change disproportionately with the fuel utilisation in particular at the entry of the anode. This is caused by the change in entropy, which achieves particularly high absolute values at pH2 O /pH2 1 or pH2 O /pH2 1. One of these conditions occurs at the entry of the anode of the Case C. Hence, the differential current is displaced disproportionately to the entry of the cell. This results in an amount of the cell voltage up to 0.746 V, which is 48 mV higher than the cell voltage of Case A. This result is not unexpected from a qualitative point of view. As long as the Nernst voltage is considered as driving force, the cell performance will increase with a higher average Nernst voltage or higher reversible power Prev at a constant ASR and constant total current density. The reversible power given by the Gibbs free enthalpy of reaction can be calculated independently by the local current distribution as follows:


33

Table 2.3 Analytical calculation results of the losses. Case

Prev [W]

E¯ N(Uf ) [V]

VCell [V]

Pel [W]

Ploss [W]

ηB [%]

A: EN = const B: EN ∼ (xH2 /xH2 O ) C: EN = f (xH2 /xH2 O )

0.299 0.307 0.315

0.998 1.026 1.050

0.698 0.726 0.746

0.209 0.217 0.224

0.0899 0.0902 0.0912

69.9 70.6 71.0

Prev = Imax

Uf =Uf total Uf =0

EN(Uf ) dUf = Uf · Imax · E¯ N(Uf ) .

(2.68)

The quotient of the total power and reversible power at operating temperature gives the relation between the converted power at irreversible conditions and the maximum convertible reversible power of the fuel cell. ηB =

Pel Prev − Ploss Ploss = =1− . Prev Prev Prev

(2.69)

Comparing the Cases A, B and C, the total power Pel increases non-linear and disproportionate with a higher reversible power. Thereby, the efficiency ηB changes disproportionately as well. On the one hand this is caused by the change of the average Nernst voltage and on the other hand by different local distributions of the current. Some models use the average Nernst voltage, according to Equation (2.61), to determine the cell voltage and cell power from a given current density and ASR. The results of this common approach are in deviation to the results of the integral determination according to Equation (2.59). Pel,Case B,Case C = Uf · Imax · (E¯ N − ASR · i¯total).

(2.70)

The relative error of the total power calculated by the commonly used average Nernst voltage related to the integral results is errCase A = 0%, errCase B = 0.1% and errCase C = 0.5%. The average Nernst voltage can be used to determine the cell power as long as the conditions of Case A are complied. At conditions according to Case B and Case C the integral determination of the cell’s performance (Equation (2.59)) is recommended. The total required cell area for an isothermal cell with a constant ASR is calculated as follows: Uf 2 A = Imax · ASR (2.71) Uf 1

34


× 1 − 2·F

1 dUf . pH O(U ) √ r GH2 (T ) + T · Rm · ln pH (U )2·√pfO (U ) · p0 − VCell 2 f 2 f Uf Uf = (ASR · i¯total) η¯ˆ

The integral part of Equation (2.59) represents the inverse average overpotential which is proportional to the inverse total current density at constant ASR. Uf Uf = = ¯ηˆ ASR · i¯total

Uf 2

(2.72) Uf 1

× 1 − 2·F

r G

H2 (T )

+ T · Rm · ln

1 pH2 O (Uf ) pH2 (Uf )·

√

pO2 (Uf )

·

√

p0

dUf . − VCell

Furthermore, an average Nernst potential can be defined taking the integral overpotential into account, which is inversely proportional to the total required cell area. η¯ˆ + VCell = E¯ N .

(2.73)

If the electrical performance of the cell Pel , the average cell voltage VCell and the fuel utilisation uf are requested parameters, Pel = Itotal · VCell = Uf · Imax · VCell

(2.74)

the required cell area can be calculated by Equations (2.59) and (2.62). A=

Pel · ASR · Uf · VCell

× 1 − 2·F

Uf 2

(2.75) Uf 1

r GH2 (T) + T · Rm · ln

1 pHO (Uf ) pH2 (Uf ) ·

√

pO2 (Uf )

·

√

p0

dUf . − VCell

The required cell area and the efficiency are calculated in this subsection for an exemplary power of 1 W (Figure 2.10). The efficiency of the cell is defined as the quotient of the cell’s power and the enthalpy of reaction. Figure 2.10 shows that the required cell area increases with higher cell voltages and fuel utilisations. Thus, the fuel consumption decreases due to the higher electrical fuel cell efficiency. This is an opposite effect concerning the investment and fuel cost. The long term degradation of the cell’s power decreases with higher cell voltages and lower fuel utilisations. Hence, the choice of operational parameters at the design point has to be assessed carefully for each application and for the particular state of the art.


35

Fig. 2.10 Required cell area.

Fig. 2.11 The power generating burner model of a SOFC module.

2.4 Thermodynamic of SOFC Systems In real SOFC systems with the associated components like fans, heat exchangers, etc., it is necessary to consider the whole system, see Figure 2.11. The system is defined as a module consisting of SOFCs, which are connected electrically in parallel into stacks supplying a common burner with the depleted fuel. The energy balance of the stacks provides the necessary requirements for the excess air [2].

36


Two different descriptions are possible. The simplest approach is the balance border around the complete module including all stacks and the joint burner from the inlet I of the fuel F and the air A to the outlet aB of the flue gas G after the burner. The more detailed approach is a balance border which surrounds all stacks from the inlet I to the outlets O of the anode side AnO and of the cathode side CaO. The calculation of this “power generating burner” is similar to the calculation of a combustor of a gas turbine or of a furnace of a boiler. The calculation of the mass flows of the module does not differ from any calculation of a conventional oxidation. The energy balance of this simpler approach (from I to aB) gives ˙ FC + Pel + H˙ GaB . H˙ FI + H˙ AI = Q

(2.76)

The total enthalpy flow H˙ FI of the fuel includes the enthalpy of reaction. The enthalpy flow of the incoming air is H˙ AI . These enthalpy flows have to cover the energy output of the SOFC module which consists of the produced power Pel , the generated ˙ FC and the enthalpy flow of the flue gas H˙ GaB . From Equation (2.76) we get heat Q the respective related enthalpies h∗ at the respective mass flows m ˙ m ˙ FI · (LHV + h∗FI ) + m ˙ AI · h∗AI = Q˙ FC + Pel + m ˙ GaB · h∗GaB .

(2.77)

The related enthalpies are used to match all enthalpies with the LHV related to the chemical standard state (1 bar, 25◦C). The related enthalpy is defined by h∗ = h(p, ϑ) − h0 (1 bar, 25◦ C).

(2.78)

These equations are sufficient to calculate the excess air λ for a given heat release −Q˙ FC of the module, or vice versa to calculate the necessary SOFC cooling by the heat release −Q˙ FC for a defined excess air λ as shown below. Only the consideration of the stacks allows a more detailed modelling of the energy balance H˙ FI + H˙ AI = Q˙ FC + Pel + H˙ AnO + H˙ CaO ,

(2.79)

whereas the enthalpy flow of the incoming fuel is H˙ FI = m ˙ FI (LHV + h∗FI )

(2.80)

and the enthalpy flow of the incoming air is H˙ AI = m ˙ AI · h∗AI = m ˙ FI · λ · µA0 · h∗AI .

(2.81)

The air demand µA0 of the stoichiometric specific is defined by the ratio of the stoichiometric air mass flow and the corresponding fuel mass flow. The ratio of all terms on the mass flow m ˙ FI of the incoming fuel allows a generalised consideration. The generated heat is ˙ FI · qFC . (2.82) Q˙ FC = m


37

The produced power is Pel = m ˙ FI · pel .

(2.83)

The fuel utilisation Uf is again defined as Uf = 1 −

m ˙ FAnO m ˙ FI

(2.11)

and the enthalpy flow at the anode outlet can be found as H˙ AnO = m ˙ FI · (1 − Uf ) · (LHV + h∗FAnO ) + m ˙ RG · h∗RGAnO .

(2.84)

The flow of the reaction product gas RG is given by m ˙ RG = m ˙ FI · Uf + m ˙ O2 = m ˙ FI · Uf · (1 + µO2 0 ),

(2.85)

m ˙ O 2 = Uf · m ˙ FI · µO2 0 .

(2.86)

with At the outlet of the anode the gas (RG) consists of the non-utilised fuel and the reaction products CO2 and H2 O. This mass flow is equal to the mass flow of the utilised fuel and of the transferred oxygen by the ion conduction through the electrolyte. The stoichiometric demand of oxygen related to the inlet fuel mass flow is given by the figure µO2 0 . Finally, we get for the enthalpy flow at the anode outlet H˙ AnO = m ˙ FI · (1 − Uf ) · (LHV + h∗FAnO ) + Uf · (1 + µO2 0 ) · h∗RGAnO . (2.87) The enthalpy flow at the outlet of the cathode is given by the difference of the enthalpy flow of the non-depleted air and the enthalpy flow of the oxygen which is transferred to the anode. H˙ CaO = m ˙ AI · h∗ACaO − m ˙ O2 · h∗O2 CaO . Equations (2.81) and (2.86) yield with Equation (2.88) H˙ CaO = m ˙ FI · λ · µA0 · h∗ACaO − Uf · µO2 0 · h∗O2 Ca .

(2.88)

(2.89)

The specific generated and to be released heat qFC can be derived from Equations (2.79) to (2.89) as qFC = Uf · LHV + h∗FI − (1 − Uf ) · h∗FAnO − pel − Uf · [µO2 0 · (h∗RGAnO − h∗O2 CaO ) + h∗RGAnO ] + µL0 · [λ · h∗AI − λ · h∗ACaO ].

(2.90)

The required excess air λ for a fixed heat release qFC can be calculated be rearranging Equation (2.90) as

38


˝ heat engine hybrid system. Fig. 2.12 The reversible fuel cell -U

λ=

Uf · [LHV − µO2 0 · (h∗RGAnO − h∗O2 CaO ) − h∗RGAnO ] + h∗FI µL0 · (h∗ACO − h∗AI )

+

−qFC − pel − (1 − Uf ) · h∗FAnO . µL0 · (h∗ACaO − h∗AI )

(2.91)

The process environment, as shown in Figure 2.1, must be connected reversibly to the ambient state for defining the reversible system. As mentioned previously, we assume that Uf → 0 and the flows consist of unmixed components to assure a reversible process. Figure 2.12 shows the reversible fuel cell–heat engine system which fulfils these requirements. The air and fuel at the ambient state T0 , p0 are brought to the thermodynamic state of the cell T , p by the reversible heat pumps HPA (for air) and HPF (for fuel). The required heat consists of the energy from the environment and of the exergy being supplied by the reversible working heat pumps. The fuel cell FC delivers the flue gas, the work and the heat as given in Equation (2.5). The product flue gas is brought from the state T , p of FC to the ambient state T0 , p0 by the reversible heat engine HEG. The reversible work, which is delivered by HEG, is the exergy of the flue gas with the state T , p. Finally, a Carnot cycle CC is used for the exchanged heat between the FC and the environment. Thus, the FC’s heat can be exchanged reversibly with the CC. The fuel cell delivers the reversible work wt FCrev Equation (2.5) wt FCrev = r G = r H − TFC · r S

(2.92)

and the reversible heat qFCrev qFCrev = TFC · r S.

(2.93)


39

The fuel cell is the isothermal heat source of the Carnot cycle CC and delivers the reversible heat qFCrev . The reversible work wt CCrev of CC is defined by T0 T0 r wt CCrev = qFCrev · 1 − = TFC · S 1 − (2.94) TFC TFC and the reversible heat qFCrev qCCrev = QFCrev ·

T0 = T0 · r S. TFC

(2.95)

The heat pump HPF produces the reversible heat qHPFrev to heat the fuel qHFrev = h∗FFC = wt HPFrev + qHPFrev .

(2.96)

The HPF must be supplied reversibly with the work wt HPFrev equalising the exergy eFFC of the fuel with the thermodynamic state of the fuel cell and the heat qHPFrev from the environment ∗ wt HPFrev = eFFC = h∗FFC − T0 · sFFC = (hFFC − hF 0 − T0 · (sFFC − sF 0 ), (2.97) ∗ qHPFrev = T0 · sFFC .

(2.98)

The definition of the exergy of the fuel with the thermodynamic state of the fuel cell is now shown more detailed. Similar methods are used for the reversible heating of the air and the reversible cooling of the flue gas. The reversible air heating requires the work wt HPArev ∗ wt HPArev = µA · eAFC = µA · (h∗AFC − T0 · sAFC )

(2.99)

and the reversible heat engine HEG for the cooling of the flue gas G produces the reversible work wt HEGrev ∗ wt HEGrev = −(µA + 1) · eGFC = −(µA + 1) · (h∗GFC − T0 · sGFC ).

(2.100)

The total work of the reversible fuel cell–heat engine system can be found as wt systrev = wt FCrev + wt CCrev + wt HPFrev + wt HPArev + wt HEGrev.

(2.101)

Using (2.5), (2.70), (2.73), (2.75), (2.76) and (2.77) we get wt systrev = r H 0 − T0 · r S 0 = r G0 .

(2.102)

The reversible work wt systrev of a coupled fuel cell–heat engine system is independent of the state of the cell and is always equal to the Gibbs free enthalpy of the reaction r G0 at the ambient state [4]. It is assumed that the standard condition is equal to the ambient state to keep the argumentation simple. This result indicates the necessary equipment to utilise the exergy of the fuel.

40


Fig. 2.13 Simplified fuel cell–heat engine hybrid system as a reference cycle.

Fig. 2.14 The system efficiency of the ideal and the real fuel cell–heat engine hybrid system with an exergetic efficiency ζHE = 0.7 and the oxidation of hydrogen.

It is useful to define a simplified process for the analysis, because the three reversible heat engines HPA, HPF and HEG do really nothing than to heat fuel and air by cooling the flue gas, but their total reversible work is negligible. Hence, the simplified process uses a heat exchanger system for the heat recovery instead of HPA, HPF and HEG, as shown in Figure 2.13. The simplified reference cycle is generally not reversible [8]. This is caused by the changes in the specific heat capacities of the different substances with the reaction temperature T that change the reaction enthalpy r H (T , p). A (small) amount of the waste heat of FC must be used to heat the reactants completely. Figure 2.14 shows the influence of the temperature on the efficiency of this cycle. The left side of Figure 2.14 shows the efficiency of the reference cycle. The system efficiency ηsyst is defined hereby as


41

ηsyst =

wt . LHV

(2.103)

The cell temperature TFC is again the temperature T of the process environment. The work wt CC produced by the Carnot cycle CC increases with higher TFC and the work wt FCrev produced by FC decreases with lower TFC as already expected. The work wt syst of the system is independent of TFC (or nearly independent in the case of the simplified process). The FC operates reversibly in both cases but the Carnot cycle CC does not operate completely reversible in the simplified process caused by the fact that a small part of the waste heat of FC is needed to heat air and fuel. The practical benefit of this combined fuel cell–heat reference cycle is the opportunity for using exergetic efficiencies to describe the operation of real cycles with this very simple model. The needed exergetic efficiency ζ is defined as ζ =

wt real . wt rev

(2.104)

Heat engines have usually an exergetic efficiency ζ between 0.7 and 0.8. All here relevant types of real cells have efficiencies between 55% and 65% but there is no significant difference caused by the cell temperature TFC as expected by the thermodynamic considerations [2]. The exergetic efficiency of fuel cells is described in [8] for hydrogen as a fuel. It can be shown that the SOFC has the best exergetic efficiency here. The exergetic efficiency ζFC of the fuel cell can be related on the total fuel feed if the non-utilised fuel can be burnt within an isothermal combustor. The fuel cell and the isothermal combustor are defined as one unit in this case (see Figure 2.11). The system efficiency ηsyst of a hydrogen fuelled combined fuel cell–heat cycle is plotted over the cell temperature TFC in Figure 2.14 on the right side. The exergetic efficiency of the fuel cell ζFC is varied between 0.7 and 1 and the exergetic efficiency of the heat engine ζHE is constant at 0.7. The system efficiency ηsyst increases with the cell temperature TFC for all exergetic efficiencies ζFC < 1 until a maximum is reached. The maximum efficiency ηsyst moves to a higher temperatures for decreasing exergetic efficiencies ζFC . The influence of the Carnot cycle dominates at lower temperatures TFC and lower exergetic efficiencies ζFC . One main result of these considerations is the identification of design options of a hybrid fuel cell-heat cycles with efficiencies of about 80% [8]. This value is a target of the U.S. Department of Energy since 1999 [9, 10]. It seems to be useful to operate the cell at the lowest possible cell temperature TFC close to the maximum of ηsyst for reducing material costs of the fuel cell, the heat engine and the heat exchangers. Further theoretical analyses of the coupled SOFC gas turbine cycle have shown that system efficiencies of 83% in maximum are achievable at high average cell voltages for the distributed range of power generation [11, 26]. It was further shown that the economic optimum cell voltage at design point is in a range of 0.6–0.7 V with today’s design parameters. Thus, the current economic optimum of the system

42


efficiency of this cycle and application is about 70%. This study [11] was based on long term target cost of SOFCs. The necessary fuel processing of natural gas or other hydrocarbons and coal before its use in the SOFC changes the system design. The following investigations have been done for methane as the main component of natural gas to keep the calculations simple. A common type of fuel processing for hydrocarbons is the endothermic steam reforming process as shown for methane in Equation (2.105) with the heat demand Equation (2.106) CH4 + H2 O → 3H2 + CO,

(2.105)

r H (750◦C)ref = +14065.1 kJ/kgCH4 .

(2.106)

The heat is required as well to evaporate the feed water. The use of the waste heat of the cell for these purposes reduces the losses. A general model of a methane fed combined SOFC cycle is based on the reference cycle of Figure 2.13 and is shown in Figure 2.15. It describes the thermodynamic influences on the system’s behaviour as simply as possible [2, 12, 13]. The principles can be transferred to other fuels easily. The SOFC can be modelled as one unit consisting of two parallel operating SOFCs fed with hydrogen and carbon monoxide. The irreversible effects including mixing are described by ζFC < 1. The detailed reasons for these irreversibilities of the SOFC and other components are not necessary to understand the system’s behaviour if they are considered properly in the system. The relation between work and heat within the single components and the temperatures of the heat sources and the heat sinks is the important issue here. The SOFC can be used as heat source of the fuel processing and evaporation. The required temperature levels are TFC ≥ Tref > Tevap .

(2.107)

This can be assumed for any SOFC system. Generally, a reversible heat transport between these different temperature levels is possible for a SOFC and two implemented Carnot cycles to provide the heat demand of the reformer and the evaporator. The engines are: the heat engine HE1, operating between the SOFC and the reformer, and the heat engine HE2, operating between the SOFC and the evaporator. The exergetic efficiencies ζHE (Equation (2.104) of the heat engines describe the irreversibilities of components in a principal reversible process structure. Finally, a third heat engine HE3 is implemented to operate between the SOFC and the ambient state if the waste heat of the SOFC cannot be used completely in the system. The flue gas is divided into two flow streams. The air is heated in the air heater AH by cooling a part of the flue gas stream (FGC). The reforming water is heated in the economiser from the ambient temperature T0 to the evaporator temperature Tevap and the saturated steam is superheated from Tevap to the reformer temperature Tref . The fuel is heated in the fuel heaters FH1 and FH2 from T0 to Tref and finally the products (of reforming) hydrogen and carbon monoxide (+ steam) are heated from Tref to the TSOFC in the product


43

Fig. 2.15 Process model for integrated reforming in SOFC systems. Table 2.4 Standard parameters for analysis of SOFC–heat engine hybrid cycles. SOFC temperature TSOFC reformer temperature Tref evaporator temperature Tevap ambient temperature T0 excess air λ water surplus nW exergetic efficiency SOFC ζSOFC exergetic efficiency heat engine ζHE efficiency of air heater ηAH efficiency of heat exchangers ηHEX

900◦ C 750◦ C 200◦ C 25◦ C 2 2 0.60 0.70 0.90 0.98

heater PH. The required heat is supplied by the cooling of the right hand pass of the flue gas (FGC) from TSOFC to a waste gas temperature > T0 , by the SOFC directly (for PH), by the waste heat of HE1 (for T < Tref ) and by the waste heat of HE2 (for T < Tevap ). An auxiliary burner has to be used for the reforming process if the waste heat of the SOFC cannot cover the heat requirement of the reformer and the evaporator. This auxiliary burner is not shown in Figure 2.15. The heat related efficiencies η Q˙ used η= (2.108) ˙ supplied Q can be used to describe the real heating processes (heat exchanger, burner). This model includes the internal reforming of the SOFC (TSOFC = Tref ) as well. The external reforming is included as well if the heat engine HE1 is replaced by a burner. The parameters listed in Table 2.4 are used for analysis of both systems. The water surplus was fixed at 2 to avoid the carbon formation [14].

44


Fig. 2.16 The influence of the excess air λ and the efficiency ηAH of the heat transfer in the air heater on the system efficiency ηsyst of the SOFC–heat engine hybrid cycle.

The system efficiencies ηsyst of combined SOFC cycles with an integrated and external reforming have been considered [12, 13]. The system efficiencies ηsyst of the systems with external reforming are about 5 to 7% points lower than these of the system with an integrated reforming. The different utilisation of the waste heat of the SOFC within the system is the reason for the differences between the processes with an external and integrated reforming. External reforming systems need an external burner producing additional entropy. Thus, the heat supply of the heat engine HE3 operating between the SOFC and the environment and the related heat loss increase. There is no temperature difference available for a power generation during the heat transport is an internal reforming system. The internal reforming system has thus a slightly lower system efficiency ηsyst than the integrated reforming system. The operation of the air heater under real conditions has a strong influence on the performance of the combined SOFC cycles with an integrated reforming. The excess air λ and the efficiency ηAH of the air heater (Equation (2.108)) are the main design parameters here. The system efficiency ηsyst is shown as a function of the excess air λ in Figure 2.16 with ηAH as a parameter. A basic thermodynamics consideration shows that r G – and thus wt FCrev (Equation (2.5)) and wt sysrev (Equation (2.102)) – is independent of the excess air λ. This result can be used to prove the model because Figure 2.16 shows that ηsyst is independent of λ for ηAH = 1 as expected. ηsyst decreases with higher λ for all ηAH < 1. The influence of ηAH increases with higher λ. The behaviour of the system for ηAH = 0.85 is shown in Figure 2.16. The system efficiency ηsyst decreases slightly with higher λ, because the work of the heat engine HE3 decreases by compensating the increasing heat losses. The other heat engines are still operating at full load conditions. The system efficiency ηsyst decreases strongly for an excess air λ ≈ 3 because the heat engine HE3 stops


45

Fig. 2.17 The influence of the heat engine design on the system efficiency ηsyst of SOFC–heat engine hybrid cycles.

caused by a lack of available heat. The total waste heat of the SOFC is used now to supply the heat engines HE1 and HE2 supplying the reformer and the evaporator respectively. This causes the decrement of the system efficiency ηsyst with higher λ in the region 3 < λ < 6 by a decreasing work produced by HE1 and HE2. In a range of λ > 6 there is no heat engine in operation anymore. The increase heat losses (increasing with increasing excess air λ) have to be compensated by the auxiliary burner. ηsyst decreases to values lower than 50% as shown in Figure 2.16. These results prove that a good heat recovery in the air heater system is mandatory and a high excess λ has to be avoided. The exergetic efficiency of the heat engines (HE1, HE2, HE3) ζHE1, ζHE2 , ζHE3 have a strong influence on the system efficiency ηsyst . The system efficiency ηsyst depending on each ζHE of the three heat engines is shown in Figure 2.17. The maximum difference of about 9% points of ηsyst is observed if ζHE3 (heat sink environment) is varied between 0 and 1. The difference is only about 2% points if ζHE1 (heat sink reformer) is varied. The variation of ζHE2 (heat sink evaporator) finally leads to differences in ηsyst of about 8% points. The order of magnitude corresponds to the difference of the related temperature differences between the SOFC and the heat sinks. The entropy recycling by the integrated reforming process is more thus important to achieve high efficiencies than the power generation of the heat engine HE1. The heat engine between the SOFC and the evaporator is important as well, particularly at good exergetic efficiency ζHE2 . The slope of ηsyst of the variation of ζHE2 seems to be unexpected compared with the variations of ζHE1 and ζHE3. This effect is caused by the amount of the SOFC’s waste heat to supply the evaporator or the reformer. The required waste heat QSOFC to operate the heat engines (HE1, HE2 = HEprocess) and to supply the processes (reformer or evaporator) with the heat Qprocess can be expressed by:

46


Fig. 2.18 Possibilities of system integration in SOFC–heat engine hybrid cycles.

QSOFC =

Qprocess 1 − ζHEprocess + ζHEprocess ·

Tprocess TSOFC

.

(2.109)

The required waste heat is governed by the exergetic efficiency of the heat engine ζHEprocess and the relation of the temperature of the heat sink Tprocess and the temperature of the heat source TSOFC (T in K). Table 2.4 shows that the relation between Tprocess/TSOFC is about 0.4 in the case of the evaporator and about 0.9 in the case of the reformer. The recycling of the anode outlet flow is another interesting option to supply the reformer with steam because no evaporation is needed.

2.5 Design Principles of SOFC Systems The process models which are shown in Figures 2.13 and 2.15 can be realised by different heat engines as shown in Figure 2.18. The actual heat source can be the flue gas or the stack heat, however only the stack generates heat. Obviously a gas turbine (GT) or a waste heat boiler of a steam cycle is able to utilise the heat from the flue gas. The total system integration may utilise the waste heat of the cell directly in both cases, as expressed by the dotted line in Figure 2.18. The different cycles based on the Carnot cycle are another options for direct stack cooling. The Stirling engine might be one option as the latest developments indicate [15]. A further option might be the combination with an AMTEC process [16]. The thermoelectric conversion might be a possibility to extract heat for electricity generation in smaller units for defence applications [17]. Finally, any endothermic process needs a transfer of heat at a certain temperature, and thus a certain supply of entropy [18]. This amount in entropy is a thermodynamic process requirement which is dif-


47

Fig. 2.19 Cooling strategies for SOFC modules by GT cycles.

ferent from e.g. a heat supply for room heating that can be clearly reduced by a better heat recovery and a better insulation. The SOFC-GT system promises a high efficient power generation opportunity [19–21]. Any successful cooling strategy for SOFC systems has to avoid a high excess air at the system’s outlet as shown above (see Figure 2.16). Figure 2.19 shows the possible strategies for a combination with a gas turbine. The SOFC module is divided into sub-modules, whereas the heat of the SOFC module is extracted by cooling the waste air of the first sub-module to the inlet temperature of the cathode of the following sub-module by the power generation by a GT. This intermediate expansion (INEX) can be carried on, whereas the last GT delivers the waste gas for the heat exchangers (HEX) to heat the air and the fuel. Another opportunity is to cool the SOFC by an external cooler (EXCO) fed with the flue gas that has been cooled by the heating of air and fuel. The SOFC module is the heat source for the GT cycle and the air is heated by the flue gas. The main differences between the INEX and EXCO configuration are marked in Figure 2.19 [22, 23]. The waste heat is extracted at one pressure level in the EXCO design, whereas n pressure levels are achieved in the INEX design, depending on the allowable temperature difference of the cathode. The pressure difference at the HEX walls of the air heater is the maximum pressure difference (ambient state – after compressor) in the INEX design. Only the pressure loss in the module is the relevant design pressure difference on the walls in the case of the EXCO design. The size of the heat exchanger of an on one side pressurised INEX design is about 2.5 times smaller than under ambient conditions, but the on both sides pressurised EXCO design has up to 7 times smaller HEX surfaces. The system efficiency of an INEX design with two turbines is about 70% [24], similar to the EXCO design. But the EXCO design has the potential for a combination with a steam turbine cycle (ST) that could be e.g. a Cheng cycle. This leads to a system efficiency of about 75% [25]. Previous studies

48


[25] of the EXCO design included a reheat cycle with an integrated heat exchanger in the SOFC. This design seemed to be too comprehensive. A comparison of both designs shows that the benefit of the EXCO design is the opportunity to reduce the excess air in one process step at one pressure level with small HEXs. But it can be combined with the benefit of the INEX design to allow a simple cascading of GT cycles as needed for a reheat GT-cycle. Furthermore, the reheat SOFC-GT cycle combined with a steam turbine (ST) cycle which reaches slightly more than 80% as the calculated efficiency [26].

2.6 Summary Thermodynamic considerations are applied to understand the processes of energy conversion in SOFCs. The reversible work of a fuel cell, represented by the Nernst voltage, can be calculated by the Gibbs free enthalpy of the reaction. The consideration of the electrical effects shows that the molar flow of the spent fuel is proportional to the electric current and that the reversible work is proportional to the reversible voltage. A coupling between the thermodynamic data and the electrical data is only possible by using the quantities power or heat flow and not by using work and heat. Irreversible losses result in the difference of the efficiency between the reversible and the real processes. These losses can be described by the irreversible entropy production within the components however the system structure itself might be reversible. The consideration of the ohmic losses shows that the irreversible entropy production at a high temperature is smaller than at a low temperature. The effects of the irreversible mixing of reactants and products lead to an irreversible entropy production as well that reduce the cell voltage. A simple consideration is used to estimate the required cell area at a specific power at different cell voltages and fuel utilisations. The results show that the required cell area increases with higher cell voltages and fuel utilisations. Thus, the fuel consumption decreases due to the higher electrical fuel cell efficiency. This is an opposite effect concerning the investment and fuel cost. The long term degradation of the cell’s power decreases with higher cell voltages and lower fuel utilisations. Hence, the choice of operational parameters at the design point has to be assessed carefully for each application and for the particular state of the art. Because a complete conversion of the fuel cannot be achieved in practice within the fuel cell, the SOFC stack can be treated like a power generating burner so as to integrate it easily into a system model. The cooling of the stack depends on the excess air ratio. The combination of a SOFC with a heat engine allows highest electric efficiencies. This is caused by the comparable low entropy production within high temperature fuel cells. Generally, the combination of a reversible fuel cell and a reversible heat engine, as represented by the Carnot cycle, results in a reversible process at any operating temperature of the fuel cell. This combination can be used as refer-


49

ence cycle to analyse the principles of the design of a combined SOFC–heat engine. The integration of fuel processing is another important factor to achieve a high efficiency as long as the embedded fuel processor can be supplied with a certain amount of entropy from the cell heat.

References 1. Wark, K., Advanced Thermodynamics for Engineers, McGraw-Hill, New York, 1995. 2. Winkler, W., Brennstoffzellenanlagen, Springer Verlag, Berlin, 2002. 3. Winkler, W., Thermodynamics, in Solid Oxide Fuel Cells: Fundamentals and Applications, K. Kendall and S. Singhal (Eds.), Elsevier, 2003, Chapter 3, pp. 53–82. 4. Winkler, W., Der Einfluß der Prozeßkonfiguration auf das Arbeitsvermögen von Verbrennungskraftprozessen. Brennstoff-Wärme-Kraft 46(7/8), 1994, 334–340. 5. Bossel, U., Facts and figures, Final Report on SOFC Data, IEA Programme of R, D&D on Advanced Fuel Cells, Annex II: Modelling & Evaluation of Advanced SOFC, Swiss Federal Office of Energy; Operating Agent Task II, Bern, April 1992. 6. Nehter, P., 2-Dimensional transient model of a cascaded micro tubular Solid Oxide Fuel Cell fed with methane, Journal of Power Sources 157, 2006, 325–334. 7. Nehter, P., A high fuel utilizing Solid Oxide Fuel Cell with regard to the formation of nickel oxide and power density, Journal of Power Sources 164, 2007, 252–259. 8. Winkler, W., Analyse des Systemverhaltens von Kraftwerksprozessen mit Brennstoffzellen, Brennstoff-Wärme-Kraft 45(6), 1993, 302–307. 9. U.S. Department of Energy, Broad Agency Announcement (BAA) No. DE-BA26-99FT40274 for research entitled: “Multi-Layer Ceramic Fuel Cell Research”, 7 June, 1999. 10. Williams, M.C., Status of Solid Oxide Fuel Cell development and commercialization in the U.S., in Solid Oxide Fuel Cells (SOFC VI), Proceedings of the Sixth International Symposium, S.C. Singhal and M. Dokiya (Eds.), Electrochemical Society Proceedings, Volume 99-19. The Electrochemical Society, Pennington, NJ, 1999, pp. 3–9. 11. Nehter, P., Thermodynamische und Ökonomische Analyse von Kraftwerksprozessen mit Hochtemperatur-Brennstoffzelle SOFC, Dissertation, Shaker Verlag 2005. 12. Winkler, W., SOFC-integrated power plants for natural gas, in Proceedings First European Solid Oxide Fuel Cell Forum, 3–7 October 1994, Ulf Bossel, Lucerne, 1994, pp. 821–848. 13. Winkler, W., Lay out principles of the integration of fuel preparation in fuel cell systems, in Proceedings 2nd IFCC (International Fuel Cell Conference), NEDO, Kobe, Japan, 1996, pp. 397–400. 14. Gubner, A., Grundlagen der Modellbildung für die Methanbildung in Hochtemperaturbrennstoffzellen, Thesis, University of Applied Sciences Hamburg, 1992. 15. 10th International Stirling Engine Conference 2001 (10th ISEC), 24–26 September 2001, Osnabrück, VDI Gesellschaft für Energietechnik, 2001. 16. Tournier, J.-M., El-Genk, M.S., Huang, L., Experimental investigations, modeling, and analysis of high-temperature deices for space applications, Final Report AFRL-VS-PS-TR-19981108, (U.S.) Air Force Research Laboratory, Kirtland Air Force Base, January 1999. 17. Nowak, R.J., A DARPA perspective on small fuel cells for the military, in Proceedings Solid State Energy Conversion Alliance (SECA) Workshop, Arlington, VA, 29 March 2001. 18. Kikuchi, R., Sasaki, K., Eguchi K., Design of energy and chemical co-production systems using Solid Oxide Fuel Cell technology, in Proceedings 7th International Symposium on Solid Oxide Fuel Cells, Tsukuba, Japan, H. Yokogawa, S.C. Singhal (Eds.), Proceedings Volume 2001-16, The Electrochemical Society, Pennington NJ, 2001, pp. 214–223. 19. Williams M.C., Zeh C.M., Executive summary, in Proceedings Workshop on Very High Efficiecy Fuel Cell/Gas Turbine Power Cycles, October 1995, US Department of Energy, Office of Fossil Energy Morgantown Energy Technology Center, 1995.

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20. George, R.A., Status of tubular SOFC field demonstration, Journal of Power Sources 86, 2000, pp. 134–139. 21. Koyama, M., Komiyama, H., Tanaka, K., Yamada, K., Evaluation of a Solid Oxide Fuel Cell and gas turbine combined cycle with different cell component materials, in Proceedings 7th International Symposium on Solid Oxide Fuel Cells, Tsukuba, Japan, H. Yokogawa, S.C. Singhal (Eds.), Proceedings Volume 2001-16, The Electrochemical Society, Pennington, NJ, 2001, pp. 234–243. 22. Winkler, W., Thermodynamic influences on the cost efficient design of combined SOFC cycles, in Proceedings 3rd European Solid Oxide Fuel Cell Forum, Nantes, P. Stevens (Ed.), Oral Presentations, 1998, pp. 525–534. 23. Winkler, W., Cost effective design of SOFC-GT, in Proceedings 6th International Symposium on Solid Oxide Fuel Cells, Honolulu, USA, S.C. Singhal, M. Dokiya (Eds.), The Electrochemical Society, Pennington, NJ, 1999, pp. 1150–1159. 24. Parker, W.G., Bevc, F.P., SureCELLTM integrated Solid Oxide Fuel Cell / Oxidation Turbine power plants for distributed power applications, in Proceedings 2nd International Fuel Cell Conference, Kobe, Japan, February 5–8, 1996, pp. 275–278. 25. Winkler, W., Möglichkeiten der Auslegung von Kombikraftwerken mit Hochtemperaturbrennstoffzellen, Brennstoff-Wärme-Kraft 44(12), 1992, 533–538. 26. Winkler, W., Lorenz, H., Layout of SOFC-GT cycles with electric efficiencies over 80%, in Proceedings 4th European Solid Oxide Fuel Cell Forum, Lucerne, July 2000, pp. 413–420.

Chapter 3

Mathematical Models: A General Overview Stefano Ubertini1 and Roberto Bove2 – Dipartimento per le Tecnologie, University of Naples “Parthenope”, Isola C4 – Centro Direzionale, 80143 Naples, Italy 2 European Commission, DG Joint Research Centre, Institute for Energy, Westerduinweg 3, 1755 LE Petten, The Netherlands 1 DiT

3.1 Modeling Approaches Before starting the definition of any mathematical model, the questions to answer are: “What is the main purpose of the model? What is it going to be used for?” The answers to the above questions will be the main drivers in choosing the most appropriate approach for the model definition and implementation. A fuel cell operation, in fact, involves a relatively large and complex number of phenomena occurring at the same time, at different scale levels, and in different components of the fuel cell. It is obvious that, when representing such a complex system with mathematical equations, simplifications and assumptions represent the starting point for defining the appropriate equations. Bearing in mind that phenomena occurring in nature are too complex to be completely described by mathematical equations, the required details to be described by the model must be goal-driven, i.e. the complexity of the model, and the related results, must be strictly connected to the main goal of the analysis itself. When, for example, the main purpose of the model is to provide the fuel cell performance, in order to analyze the whole system in which it is embedded, the spatial variation in the physical and chemical variables (such as gas concentration, temperature, pressure and current density, for example) are not relevant; however the performances, in terms of efficiency, electrical and thermal power and input requirements are important [1–4]. When, on the other hand, the model is used as a tool for designing or improving a specific component of the fuel cell, it is important that the model is capable of providing very detailed information on the performance-related variables in that specific component. Examples of such analyses are copious in the literature (e.g. [4–8]), and most of them are developed at single cell level, with particular emphasis on one particular component or cell characteristic. Chan et al. [4], for example, applied an SOFC single cell model for analyzing the effect of the electrodes and R. Bove and S. Ubertini (eds.), Modeling Solid Oxide Fuel Cells, 51–93. © Springer Science+Business Media B.V. 2008

52

S. Ubertini and R. Bove

electrolyte thickness on the cell performance, thus all the phenomena related to a change in the structure thickness were modeled in detail. Lin et al. [5] defined an analytical model for assessing the interconnect rib size effect on the concentration losses. The approach used by Lin et al. [5] is an excellent example of how the goal of the analysis influences the model structure, which is mostly focused on the mass transport limitations of the porous media. Another example is given by Bove and Sammes [6], where the effect of the layout of different current collectors is assessed. Usually, a limitation in the complexity of the model is governed by hardware limitations, and the related memory requirements. Therefore, when a model needs to be very detailed for one particular phenomenon, other phenomena are usually dealt with in lower detail, if not totally neglected. The so-called “micromodels” are models of a particular component, or of a part of a cell component, conducted at molecular or atomistic level. Due to the high level of detail related to the material properties and characteristics, the information provided by such models is usually limited to the specific phenomenon analyzed, and provides only limited indications on the resulting fuel cell performance and operating conditions. However, the results of such models play a fundamental role in understanding, analyzing and designing improved solutions for SOFC. Moreover, the results of such analyses may be used as an input for macro-models, i.e. models conducted at fuel cell level. For example, Bieberle and Gauckler [7] developed an electrochemical model for the Ni, H2 -H2 O-YSZ system (i.e. the anodic triple phase boundary). As a result they identify possible reaction mechanisms and calculate some kinetic parameters, thus providing valuable inputs and information for simulating the entire fuel cell. Moreover a better understanding of atomistic phenomena acting at the anodeelectrolyte interface is provided. Several micromodels can be found in the literature on SOFC components or materials (e.g. [8–15]), and the results should be taken into account when defining a mathematical model of a single cell, or stack. However, the analysis and description of such models is beyond the scope of the present book.

3.2 Physical, Chemical and Electrochemical Equations Regulating SOFC phenomena 3.2.1 Conservation and Constitutive Laws A comprehensive analysis of solid oxide fuel cells phenomena requires an effective multidisciplinary approach. Chemical reactions, electrical conduction, ionic conduction, gas phase mass transport, and heat transfer take place simultaneously and are tightly coupled. In the present section, conservation and constitutive laws are illustrated, while in Section 3.3 they are applied to modeling each single domain.

3 Mathematical Models: A General Overview

53

The mathematical model is based on the conservation of mass, momentum, electrical charge, and energy, coupled with appropriate constitutive laws. Mass conservation of a gas species within an infinitesimal volume can be written as: ∂ρYi + u · ∇(ρYi ) = −∇mi + ωi , (3.1) ∂t where i denotes the generic ith species, ωi is the rate of production or consumption, mi is the mass diffusion flux, Yi is the mass fraction, and u is the gas velocity. For gas species evolving in a porous media, the media porosity needs to be taken into account, thus Equation (3.1) can be re-written: ε

∂ρYi + εu · ∇(ρYi ) = −∇mi + ωi , ∂t

(3.2)

where ε is the porosity of the medium. Momentum conservation for a moving gas can be expressed as: ρ

∂u + ρu · ∇u = −∇P + µ∇ 2 u + ρf, ∂t

(3.3)

where f represents the generic body forces (e.g. gravity), and µ is the dynamic viscosity. Equation (3.3) is a simplified formulation of the Navier–Stokes equations, specifically, near the incompressible limit. This form is usually employed in fuel cells gas flow simulation due to the low Mach number within the gas channels. Due to the infinitesimal dimension of the pore sizes, Equation (3.3) is often inapplicable to porous media. Therefore, the momentum conservation for the fluid flow through the porous electrodes is often substituted by the phenomenologically derived constitutive equations, such as Darcy’s law given by k u = − ∇P , µ

(3.4)

where the constant k, the permeability, has dimensions of area and is usually measured in units of Darcy (d). Equation (3.4) mathematically describes the relationship between velocity and pressure fields for a fluid of viscosity µ injected though an infinitesimal volume of a porous medium by applying a pressure gradient ∇P across the volume. It is appropriate to mention that, although Equation (3.4) was experimentally determined by Darcy, it can also be derived from the Navier–Stokes equations via homogenization techniques [16]. Its counterparts in heat conduction, electrical conduction and diffusion are Fourier’s law, Ohm’s law and Fick’s law, respectively. For charge transport, both electrons and ions need to be considered. In particular, electrons move within the electrodes and the current collectors, while ions transport determine the effective charge transport from one electrode (cathode) to the other (anode), through the electrolyte, as explained in Chapter 1. Using J to indicate the current or ionic density vector, the charge conservation can be expressed as:

54


∂ρele,ion + ∇J = j, (3.5) ∂t where ρele,ion is the electronic or ionic charge density, and j represents the current production rate. Energy conservation can be expressed in different forms. When e defines the energy per unit mass, the following equation for the conservation of energy is derived: ρ

∂e + ρu · ∇e = −∇Q + Sq , ∂t

(3.6)

where Sq is the volumetric heat source term and Q is the heat flux vector only from conduction. If Equation (3.6) is applied to a solid (e.g. the electrolyte), this reduces to: ρc

∂T = −∇Q + Sq , ∂t

(3.7)

where c represents the heat capacity of the solid. When Equation (3.6) is applied to a gas mixture, e can be expressed as: e = cν T +

u2 + g · r, 2

(3.8)

where cν is the specific heat at constant volume, g is the acceleration due to gravity, and r is the unity gravity vector. Considering that the working fluid in fuel cell gas channels is always a mixture of gases, the potential energy, g · r and the kinetic energy, u2 /2, can be neglected. Equations (3.1–3.8), coupled with the appropriate constitutive laws, define the set of governing equations. The first constitutive law provides the mass diffusive flux mi of Equation (3.1) (i.e. gas in a non-porous media). If Fick’s law is used, this flux can be expressed as: mi = −[ρDi ∇(Yi )].

(3.9a)

Or, referring to the molar diffusive flux, its equivalent form: Ni = −

P Di ∇Xi = −cgasDi ∇Xi . RT

(3.9b)

Di is the mass diffusion coefficient, and cgas is the total molar concentration of the gas mixture. Although Equations (3.9a) and (3.9b) can be used for a free-path gas (e.g. gas channel), when a gas is moving within a porous media (i.e. electrode), Equation (3.9) may not be the most appropriate. Different constitutive laws can be employed for describing the diffusive flux within a porous medium. The choice of the most appropriate law depends on the operating conditions and the porous media properties, as further explained in Section 3.3.2. For Equation (3.5), the constitutive law is given by Ohm’s law:


J = −σ ∇φ,

55

(3.10)

where J, σ , and φ are, respectively, current density, conductivity and potential, and can be referred to either the ionic or the electronic current. For Equations (3.6) and (3.7), Fourier’s law can be used to relate the heat flux to the temperature gradient in a continuum medium: Q = −λ∇T ,

(3.11)

where λ is the thermal conductivity coefficient, which, for porous media, accounts for both solid and liquid phases under the hypothesis that they are in thermal equilibrium. In addition, the set of equations should be supplemented by the laws of dependence of viscosity and thermal conductivity on the fluid state. However, in many situations, these are assumed to be constant. Due to the high SOFC operating temperature, anodic and cathodic gases are always far from critical conditions, thus the ideal gas constitutive law can be applied: P = ρRT .

(3.12)

3.2.2 Kinetics of Chemical Reactions The working fluid in an SOFC is usually a reacting mixture of gases composed of the following species: • H2 , H2 O, CO, CO2 , and possible higher hydrocarbons in the anode stream; • O2 , N2 , in the cathode stream. Chemical reactions take place in the gas channels (typically the anode), and, as explained in Chapter 1, electrochemical reactions take place in the triple-phaseboundary (TPB), i.e. a reaction zone very close to the electrode-electrolyte interface. If the cell operates on a hydrocarbon, rather than on hydrogen, the reforming reaction takes place in the anode gas channel: Cn Hm Op + (n − p)H2 O = nCO + (n − p + m/2)H2 .

(3.13)

Furthermore, carbon monoxide reacts with water to generate additional hydrogen and carbon dioxide: CO + H2 O ↔ CO2 + H2 . (3.14) This is called the water-gas shift reaction. The electrochemical reactions acting on the TPB can be quite complex, depending on the gas species present in the anode stream. If only hydrogen is used as fuel, the chemical reactions can be easily expressed as: O2 + 2e → 2O−2 ,

(3.15)

56


H2 + O2− → H2 O + 2e.

(3.16)

Reactions (3.15) and (3.16) take place, respectively, at the cathodic and anodic TPBs. The overall reaction is the production of water, via the oxidation of hydrogen. 1 H2 + O2 → H2 O. 2

(3.17)

If CO and hydrocarbons are present in the fuel, in addition to reactions (3.13) and (3.14), they can be directly oxidized at the TPB. In the case of CO, for example, the following reaction also occurs at the anodic TPB: CO + O2− → CO2 + 2e.

(3.18)

Although direct oxidation of CO and hydrocarbons on Ni-based anodes have been demonstrated [17–19], these reactions are much slower than hydrogen oxidation (reaction (3.16)), thus it is usually assumed that they do not react directly on the TPB, but they contribute to the overall reaction by producing H2 through reactions (3.13) and (3.14). Therefore, in the present chapter, no direct CO and hydrocarbon oxidation is assumed. However, it should be stressed that a recent study of Andreassi et al. [20] demonstrated that, under certain operating conditions, ignoring direct oxidation of CO can lead to a significant underestimation of the voltage. To formulate the problem in a generic form, it can be stated that in an SOFC, the fluid domain is a multi-component system involving chemical reactions, i.e: Ns

rik ci ⇔

i=1

Ns

pik ci ,

k = 1, Nr ,

(3.19)

i=1

where Nr is the number of reactions, Ns is the number of species, rik and pik are the stoichiometric coefficient of the reactants and the products of the k-th reaction and ci is the i-th chemical species. When a reaction is fast enough to be considered at chemical equilibrium (e.g. the shift reaction), the production rate can be computed through the equilibrium constant. Referring to expression (3.19), the equilibrium constant of a reaction k, can be expressed as: p i Pi ik kpk = (3.20) r , i Pi ik where Pi indicates the partial pressure of the i-th chemical species. If a chemical reaction reaches the thermodynamic equilibrium, the production rate of the i-th species is the difference between the initial concentration and the equilibrium concentration provided by (3.19). When applied to the shift reaction (3.14), the equilibrium constant can be computed as [21]: kp,shift =

PCO2 PH2 = exp(4276/T − 3.961). PCO PH2 O

(3.21)


57

Fig. 3.1 Planar SOFC.

When the chemical equilibrium approximation is not possible, ωi must be calculated considering the kinetics of the reaction: −

d[ci ] = k[ci ]n . dt

(3.22)

In expression (3.22), n is the reaction order and k can be calculated through the Arrhenius equation: k = A · exp(−Ea /RT ), (3.23) where A is a pre-exponential factor, and Ea is the apparent activation energy. A and Ea are tabulated in literature for specific reactions, assisted by specific catalysts, or are determined experimentally. If methane is considered for reaction (3.13), a first order reaction rate is usually assumed, i.e. the coefficient n of expression (3.22) is equal to 1. According to Achenbach [22], the activation energy of methane steam reforming is 82 kJ mol−1 and the pre-exponential factor is 4274 mol m−2 bar−1 s−1 . More recent experimental studies report an activation energy of 112 ± 15 kJ mol−1 [23]. According to [24], the discrepancy in the values found in the literature for A and Ea are due to the fact that methane reforming over a Ni-based anode does not follow Arrhenius behavior. For a more detailed formulation of the methane reforming rate, the reader is referred to [24].

3.3 Application of the Equations to Each Domain SOFC can be manufactured in different geometrical configurations, i.e. planar, tubular or monolithic. Regardless of the geometrical configuration, a solid oxide fuel cell is always composed of two porous electrodes (anode and cathode), a dense electrolyte, an anodic and a cathodic gas channel and two current collectors. For the sake of simplicity the planar configuration is taken as reference, as shown in Figure 3.1. In the following, equations defined in Sections 3.2.1 and 3.2.2 are applied to each domain, thus obtaining the system of equations that analytically describe a generic SOFC.

58


3.3.1 Channels Flow Using an Eulerian approach, the description of fluid motion requires the determination of the thermodynamic state, in terms of sensible fluid properties, pressure, P , density, ρ, and temperature, T , and of the velocity field u(x, t) [25–29]. Therefore, in a three-dimensional space for a given thermodynamic system having two intensive degrees of freedom, six independent variables are the unknowns of the so-called “thermo-fluid dynamic problem”, thus requiring six independent equations. The six equations are given by the equation of state and the three fundamental principles of conservation: • Mass conservation; • Newton second law or momentum equation (three equations in a threedimensional space x, y and z); • Energy conservation (first law of thermodynamics). The ideal gas equation of state (3.12) is usually applied. Mass conservation is given by (3.1) and, coupled with Fick’s law (3.9a), can be re-written as: ∂ρYi + u · ∇(ρYi ) = ∇(ρDi ∇(Yi )) + ωi . (3.24) ∂t The reaction rates of the forward and the backward reactions (3.13) and (3.14) allow the calculation of the source term ωi in the species transport Equation (3.24) as well as the source term Sq in the energy Equation (3.6): Sq =

Nr ωi

Hi , Mi

(3.25)

i=1

where Nr is the number of reactions and Hi is the enthalpy change associated with the formation of 1 mole of the i-th species from its constituent elements, the enthalpy of formation. The latter is the heat released or absorbed in a chemical reaction at constant pressure and depends slightly on pressure (usually neglected) and more significantly on temperature. The enthalpies of formation of the main species involved in SOFC reactions at 0 K and 1 atm are summarised in Table 3.1. The enthalpy of formation of the elements in their natural state (i.e. oxygen and hydrogen gases) is zero. The temperature dependence is usually given in polynomial form [30]. The properties of any gas mixture depend on the relative proportions of different molecular species in the mixture. These mixture properties are effectively derived by summing the known properties of the individual molecular species, weighted by their proportions in the mixture. Some properties of the molecular species are available from standard tables (e.g. JANAF tables) or empirical fits to these standard tables. The physical properties of the multicomponent mixture, such as viscosity, specific heats at constant volume and at constant pressure, and laminar thermal conductivity, are usually calculated under the assumption of an ideal mixture. Data and


59

Table 3.1 Enthalpy of formation of the main species involved in SOFC reactions at 0 K and 1 atm [30]. Species

Enthalpy of formulation (kJ/mol)

CO CO2 H2 O CH4

–113.8 –393.1 –238.9 –66.6

estimation methods for the calculation of thermophysical properties for both pure components and mixtures in function of temperature are reported in [31]. An arbitrary constitutive fluid property may be calculated from the property value for the fluid i-th molecular species through the following equation: x= xi Yi . (3.26) i

s Therefore, the overall fluid density is the sum ρ = N i=1 Yi ρ over all species and summing-up Equation (3.24), the continuity equation for the gas mixture can be derived: ∂ρ + ∇(ρu) = ω, (3.27) ∂t where ω is the total mass variation. Considering that chemical reactions do not produce a mass variation, ω is null within the gas channels domain. Therefore, if n is the number of species, the mathematical model is defined by means of n − 1 Equations (3.24) plus Equation (3.27). Regarding the conservation of momentum, the Navier–Stokes equations, governing unsteady compressible viscous flows, are the basis for all models. However, the thermal and chemically reacting flows through the SOFC gas channels are often characterized by relatively slow flow motion, much slower than the speed of sound, and by partially varying mixture density. Density variations may be caused in these cases either internally by heat release of chemical reactions or externally by wall heating and by mass variation, but not by expansion or compression. This situation is characterized by a low Mach number (the flow speed divided by the thermodynamic speed of sound), and is consequently modeled by a simplified version of the Navier–Stokes momentum Equation (3.3). Closure of this system of equations is a non trivial task and closed solutions are available only for exceptional cases. Considering that the gas speed in SOFC gas channels is always very low, it is a common practice to assume a laminar flow in the gas channels [32]. This means that the inertia term ρu · ∇u in Equation (3.3) is negligible compared to the viscous term µ∇ 2 u (i.e. gravity force is negligible), and Equation (3.3) for a mixture of gases becomes: ρ

∂u = −∇P + µ∇ 2 u ∂t

(3.28)

60


and, at steady state: ∇P = µ∇ 2 u.

(3.29)

The laminar flow assumption eliminates the non-linear term in the partial differential equations system (3.3), thus significantly reducing the computational cost. In addition, the present formulation often admits an exact solution. For example, in the case of an incompressible 2D laminar flow between two motionless parallel plates (i.e. planar SOFC configuration of Figure 3.1), Equation (3.29) reduces to: ∂P = 0, ∂y µ

d 2 ux dP , = 2 dx dy

(3.30a)

(3.30b)

d 2P = 0, (3.30c) dx 2 where x is the flow direction (Figure 3.1). Using u = 0 as the boundary condition at y = 0 and y = h, where h is the distance between the plates, Equation (3.30b) gives the following exact solution: u=−

1 P (h − y)y, 2µ l

(3.31)

where l is the length of the channel and P is the total pressure drop. By an analogous argument, the laminar flow in a circular pipe of radius r0 gives the following velocity parabolic profile: u=−

1 P 2 (r0 − r 2 ), 4µ l

(3.32)

where a cylindrical coordinate, r, is used. Temperature distribution within the operating gas is obtained through the energy Equation (3.6), coupled with Fourier’s law (3.11): ∂(ρcp T ) + u · ∇(ρcp T ) = ∇(λf ∇T ) + Sq , ∂t

(3.33)

where λf is the fluid thermal conductivity, and the source term is due to the chemical reactions and can be evaluated through Equation (3.25).

3.3.2 Electrodes Both the anode and cathode are assumed to be made of porous materials providing electronic and ionic conductivity. As a consequence, there is a concomitant trans-


61

Fig. 3.2 Schematic representation of the porous anode structure. Black circles represent electronic conductive sites, while grey circles represent ionic conductive sites.

port of gas species, electrons and ions. For this reason, electrodes are the most complex domains to model. Of particular importance is the triple-phase-boundary, where ions, electrons and gases combine, according to reactions (3.15, 3.16), and, eventually, (3.18). Since the electrode structure presents much lower ionic conductivity compared to the electrolyte, and since the electrode electric conductivity is higher than the ionic one, ions coming from the electrolyte are likely to combine with hydrogen (reaction 3.16) very close to the anode-electrolyte boundary. Similarly, reaction (3.15) takes place very close to the cathode-electrolyte interface. For this reason, several mathematical models consider reactions (3.15) and (3.16) confined at the electrode-electrolyte interface (e.g. [6, 33, 34]). This simplification is introduced in Section 3.4, while in the present section the reaction zone is considered as a finite volume. Figure 3.2 represents a schematic of the anode structure, highlighting the electronic and ionic conductive sites, while Figure 3.3 represents the schematic representation used for modeling purposes. According to the representation of Figure 3.3, the reaction zone is assumed to be a continuum. In order to take into account the non-continuum nature of the TPB, it is possible to add specific corrective factors, with the aim of reducing the extent of reactions (3.15) and (3.16). However, in the present chapter, such correction is not added, and a simplified continuum structure is assumed. Usually, SOFC electrodes are composed of two (or sometimes more) layers, where the first (the porous anode in Figure 3.3) has mainly a structural function, and the second is a functional layer (called the reaction zone in Figure 3.3), with the main aim of promoting the electrochemical reaction.

62


Fig. 3.3 Schematic representations of the anode and the reaction zone (continuum).

By analogy to Figures 3.2 and 3.3, the cathode is represented as being composed of two different layers. It should be noted that for the cathode there is a one-way flux of oxygen from the gas channel to the reaction zone, while for the anode there is a hydrogen flux to the reaction zone and a water flux from the reaction zone to the gas channel, as illustrated in Figure 3.3. To solve the electrical problem at the electrodes, two variables need to be found (i.e., the electrical potential (scalar) and the current density (vector)). By analogy, two variables are also needed to find the ionic flux and the potential distribution. The vector relationship is given by Ohm’s law (3.10), while the scalar relation is provided by expression (3.5), which can be re-written as: ⎧ in the anodic TPB, ⎪ ⎨ −j ∂ρele in the cathodic TPB, (3.34a) + ∇Jele = j ⎪ ∂t ⎩ 0 elsewhere, where ρele is the electronic charge density, and j is the current generation rate, produced in the reaction zone. Since the cell charging time is very short, compared to the other phenomena occurring in the fuel cell (cf. Chapter 9 and [35]), this can be considered at steady state (i.e., the first term on the left hand side of Equation (3.34a) can be neglected). By analogy to (3.34a), but now explicitly showing that charge density is unchanged, the conservation of the ionic charge is expressed as: ⎧ in the anodic TPB, ⎪ ⎨j in the cathodic TPB, ∇Jion = −j (3.34b) ⎪ ⎩ 0 elsewhere. Combining Equation (3.10) with Equations (3.34a) and (3.34b), the following relations are obtained for the electronic and ionic potential:


∇ φele = 2

∇ φion = 2

⎧ j ⎪ ⎪ ⎨ σele

in the anodic TPB,

− j ⎪ σele ⎪ ⎩0

σion ⎪ ⎪ ⎩0

in the cathodic TPB,

(3.35a)

elsewhere,

⎧ j ⎪ ⎪ ⎨ − σion j

63

in the anodic TPB, in the cathodic TPB,

(3.35b)

elsewhere.

Due to the porous structure of the electrodes, the electrical and ionic conductivities are different from those of the dense materials, and are strongly dependent on the porosity. Pores, in fact, can be considered as having infinite resistivity, thus they reduce the overall conductivity. For Ni-based anodes, assuming nickel particles as spherical, Zhao and Virkar [36] proposed the following formula: 1 σeff = 2σNi 1 − χ 2 , (3.36) 2 ln[(1 + 1 − χ ) (1 − 1 − χ 2 )] where χ represents the ratio of the inter-particle neck radius and the radius of the particles. Due to the difficulties of defining the radius of the particles and the radius of the inter-particle neck, the effective conductivity is best determined from experimental data when available. The electron and ionic generation are described by the Butler–Volmer equation:

−α2 nF ηact α1 nF ηact − exp . (3.37) j = j0 exp RT RT In expression (3.37), j0 is the exchange current density, α1 and α2 are the transfer coefficients related to, respectively, the forward and backward reaction, n is the number of electrons transferred per reaction, and ηact is the activation loss. It should be noted that the sum of α1 and α2 is not necessarily equal to unity [37]. However, under the assumption that the reaction is a one-step, single-electron transfer process, the following relations apply: α1 = β,

(3.38a)

α2 = 1 − β.

(3.38b)

Under this assumption, Equation (3.37) can be re-written as:

−(1 − β)nF ηact βnF ηact − exp . j = j0 exp RT RT

(3.39)

Although expression (3.39) is commonly used in literature , some studies [38, 39] show that a multi-step reaction formulation (Equation 3.37) can produce a better representation of experimental data.

64


The exchange current density is the electrode reaction rate at the equilibrium potential (identical forward and reverse reaction rates) and depends on the electrode properties and operation. The typical expression for determining the exchange current density is the Arrhenius law (3.23), where the constant A depends on the gas concentration. Costamagna et al. [40] provide the following expressions for the anodic and cathodic exchange current density, respectively: Eact,an PH2 PH2 O j0,an = γan , (3.40a) exp − Pref Pref RT j0,cat = γcat

PO2 Pref

0.25

Eact,cat . exp − RT

(3.40b)

The activation loss can be quantified as: ηact = Vrev − |φele − φion |,

(3.41)

where Vrev is the reversible potential which can be computed with the Nernst equation, subdividing the potential on the anode and cathode sides: RT PH2 O ln , (3.42a) Vrev,an = E0 − nF PH2 Vrev,cat =

RT PO2 1/2 ln . nF Pref

(3.42b)

Considering the porous media as a continuum, the energy equation can be written as: ∂T + ρcu · ∇T = ∇(λ∇T ) + Sq . (3.43) ρc ∂t An average thermal conductivity and heat capacity of the coexisting solid and gas phases is required. The effective thermal conductivity and heat capacity, in fact, depend on those of the gas and of the solid, on the porosity, and on the concentration of the gas species. Considering the gas phase in equilibrium with the solid phase, Gurau et al. [41] proposed the following expression for the effective thermal conductivity: 1 λ = −2λs + . (3.44) (ε (2λs + λg )) + ((1 − ε)/3λs ) The volumetric heat source of Equation (3.43) is the sum of the following heat sources: 1. Ohmic resistance – the ionic and electronic currents, when passing through the electrodes, generate the so-called Joule heating. 2. Chemical reactions – cf. (3.13) and (3.14), which take place mostly in the anode gas channel;


65

3. Electrochemical reactions – cf. (3.15) and (3.16), taking place in the reaction zone and releasing or consuming heat according to the change in entropy of the reaction (reversible heat). 4. Activation over-potential due to the activation energy – the current flow generates an electrochemical loss that is translated into a heat source. While the Joule heating effect and the chemical reaction (1 and 2) take place in the entire electrode domain, the last two heat sources are localized in the TPB. The expressions for the heat source term for the anode and cathode are, respectively:

Sq,an

⎧ σ ∇φ · ∇φele + σion ∇φion · ∇φion + ⎪ ⎪ ⎨ ele ele Nr = ωi ⎪ ∇φ · ∇φ +

Hi σ ⎪ ele ele ele ⎩ M i=1

Sq,cat =

T SH2 O + j ηact,an in the TPB, (3.45a) elsewhere,

i

σele ∇φele · ∇φele + σion ∇φion · ∇φion + σele ∇φele · ∇φele

j 2F

j 4F

T SO2 + j ηact,cat in the TPB, elsewhere,

(3.45b)

where SH2 O is the entropy change associated with reaction (3.16), SO2 with (3.15), and the term Nr i=1 (ωi /Mi ) Hi is the heat source due to the reforming and shift chemical reactions in the anode. This last term was already defined in (3.25), while the values of SH2 O and SO2 cannot be directly calculated by thermodynamic correlations, because they involve the entropy associated to the ionic and electronic phase. Duan et al. [42] attempt to define SH2 O and SO2 , starting from the experimental data of Kanamura et al. [43] for SO2 , while SH2 O is calculated as the difference between the entropy associated to the overall reaction (3.17) and

SO2 . Furthermore, Ito et al. [44, 45] defined a procedure for calculating SH2 O and

SO2 , starting from Feebeck coefficients, which can be experimentally determined. Although the methodology for calculating the above mentioned entropy change results to be well established, at present, there is still a lack of experimental data for the derivation of Seebeck coefficients of SOFC materials. In expression (3.45a), it is assumed that the anodic chemical reactions (3.13) and (3.14) take place in the entire anodic domain, but the TPB. This assumption is due to the small dimensions of the TPB, compared to the anode dimension. Due to the structure of the electrodes, the velocity, pressure and species distribution need to be studied using the theory of mass transport in porous media. This topic has been extensively studied over the years, and reported in the literature. Examples of detailed studies are given by Bird et al. [29], Froment and Bishoff [46], Bear and Buchlin [47], and Bear [48]. The mass transport equation needs to take into account the effect of the porosity of the medium. Moreover the molar flux is mainly due to convection in the gas channel, whereas in the porous media, diffusion is the main cause [49]. The fundamental governing equation of the i-th gaseous species for a mixture of gases through porous electrodes is given by Equation (3.2). The reaction rate is given by (4.46a) and (4.46b) for the anode and (3.46c) for the cathode, respectively:

66


ωH2 ,an =

−j/2F

in the TPB,

Depending on the gas

ωH2 O,an = ωO2 ,cat =

elsewhere, in the TPB,

j/2F Depending on the gas

species1

species1

−j/4F

in the TPB,

0

elsewhere.

elsewhere,

(3.46a)

(3.46b)

(3.46c)

The mass diffusive flux mi of Equation (3.2) generally depends on the operating conditions, such as reactant concentration, temperature and pressure and on the microstructure of material (porosity, tortuosity and pore size). Well established ways of describing the diffusion phenomenon in the SOFC electrodes are through either Fick’s first law [21, 34. 48, 50, 51], or the Maxwell–Stefan equation [52–55]. Some authors use more complex models, like for example the dusty-gas model [56] or other models derived from this [57, 58]. A comparison between the three approaches is reported by Suwanwarangkul et al. [59], who concluded that the choice of the most appropriate model is very case-sensitive, and should be selected, according to the specific case under study. Fick’s law Fick’s law, which describes the steady-state bi-molecular diffusion, is the simplest and most used form. It is described in Section 3.2, but it is also reported below for the reader’s convenience: Ni = −Di cgas∇Xi ,

(3.47)

where Di is the diffusion coefficient of species i in the mixture (i.e. in the case of a binary diffusion, Di = Dij , the binary diffusion coefficient). Maxwell–Stefan model Fick’s law is derived only for a binary mixture and then accounts for the interaction only between two species (the solvent and the solute). When the concentration of one species is much higher than the others (dilute mixture), Fick’s law can still describe the molecular diffusion if the binary diffusion coefficient is replaced with an appropriate diffusion coefficient describing the diffusion of species i in the gas mixture (ordinary and, eventually, Knudsen, see below). However, the concentration of the different species may be such that all the species in the solution interact each other. When the Maxwell–Stefan expression is used, the diffusion of 1

This value needs to be obtained by analogy to the gas channel (Section 3.3.1). It is equal to 0 if the fuel cell is operated on H2 or if reactions (3.13) and (3.14) take place exclusively in the anodic gas channel.


67

each gas species is linked to the other species composing the gas mixture [59]: n j =1,j =i

Xj Ni − Xi Nj = −cgas∇Xi , Dij

(3.48)

with n indicating the number of gas mixture components, and Dij the binary diffusion coefficient. Molecular versus Knudsen diffusion and the Dusty-gas Model In general, the diffusive transport in porous media includes free molecular or Knudsen flow and a continuum flow. The previous two models consider, in principle, only molecular diffusion, which is dominant for large pore sizes and high system pressures. However, when the molecules collide frequently with the pore wall, which is to say that their mean free path is larger than the pore size (pore diameter between 2 and 50 nm), Knudsen diffusion becomes significant. The model that takes into account Knudsen diffusion is the Dusty-gas model, which considers the actual porous medium as an aggregation of giant molecules of an additional gaseous species (particles of “dust”). The general form of this diffusion model is: n Ni + Di,k

j =1,j =i

Xj Ni − Xi Nj = −cgas∇Xi , Dij

(3.49)

where Di,k , indicates the Knudsen diffusion coefficient [56]. Knudsen diffusion can be taken into account also when Fick’s law or the Maxwell–Stefan rate of mass transport are employed [34, 59] by combining the molecular diffusion, Di,m , and the Knudsen diffusion as follows: Di =

1 1 + Di,m Di,k

−1 (3.50)

or using the parallel pore model [57]: Di =

1 1 − ξi,m Xi + Di,m Di,k

ξi,m = 1 −

Mi Mm

−1 ,

(3.51)

0.5 ,

(3.52)

where Mi is the molecular weight of the i-th component, and Mm is the average molecular weight. For a multicomponent flow the ordinary or molecular diffusion coefficient for each species can be calculated on the basis of its binary coefficient with each of the others [60]:

68

S. Ubertini and R. Bove Table 3.2 Constants in expression (3.55) and (3.56). Constant

Value (non-dimensional)

C A1 A2 A3 A4 A5 A6 A7 A8

0.001858 1.06036 0.1561 0.193 0.47635 1.03587 1.52996 1.76474 3.894411

1 − Xi Di,m = . Xj

(3.53)

j =i Dij

All the proposed expressions require the knowledge of the diffusion coefficients. The binary diffusion coefficient determination is non trivial and it is usually made experimentally. The Knudsen diffusion coefficient may be computed according to the kinetic theory of gases: 4¯r 2RT 1/2 Di,k = , (3.54) 3 πMi where r¯ is the mean pore radius and Mi is the molecular weight of the diffusing gas. If not available experimentally, the binary diffusion coefficient, Dij , may also be calculated using elementary kinetic theory (Chapman–Enskog equation proposed by Cussler [61]). A first order approximation for Dij is given by Yakabe et al., as follows [57]: [T 3 (Mi + Mj )/Mi Mj ] Dij = C , (3.55) P σik2 D where C is a non-dimensional constant, σik is the average of the characteristic lengths of species i and k, and D is the collision integral given by D =

A1 TNA2

+

A3 A e 4 TN

+

A7 A5 + . exp(A6 TN ) exp(A8 TN )

(3.56)

Constants C and Ai are reported in Table 3.2, while TN is defined as: TN =

kT , εik

(3.57)

where k is the Boltzmann constant and εik is the average of the characteristic Lennard–Jones energy of species i and k. Suitable values of gases characteristic lengths and Lennard–Jones energies are given in [62].


69

The above equations for the diffusion coefficients do not take into account the volume fraction of porosity and the tortuous nature of the path through porous bodies. When the transport occurs through a porous body, as in fuel cell electrodes, effective diffusion coefficients accounting for the interaction of gaseous species with the porous matrix must be employed. Different theoretical approaches for the determination of the effective diffusion have been proposed in the literature. The Bruggemann correction allows the evaluation of these coefficients, through the following expression [47]: Di,eff = εa Di , (3.58a) where a is an empirical coefficient, usually taken as 1.5, Di is the ordinary diffusion coefficient, and Di,eff is the diffusion coefficient in the porous medium. Some authors (e.g. [4, 21, 54]), use an expression that takes into account the porous tortuosity (τ ): ε Di,eff = Di . (3.58b) τ The diffusion coefficient dependence on temperature is often found to be well predicted by [63]: T 1.5 Di = Di,eff . (3.59) Teff Mass transport within the electrodes is of particular importance in determining the reflection of the porous media structure on the fuel cell performance. In fact, the main results of mass transport limitation is that the reactant concentrations (H2 and CO for the anode, and O2 for the cathode) at the reaction zone are lower than in the gas channel. When applying Equations (3.40) and (3.42), the result is that the lower the concentration of the reactants, the lower the calculated cell performance. The loss of voltage due to the mass transport of the gas within the electrodes is also referred to as concentration overpotential. Simplified approaches for determining concentration overpotential include the calculation of a limiting current, i.e. the maximum current obtainable due to mass transport limitation (cf. Appendix A3.2).

3.3.3 Electrolyte In the cathodic TPB, due to reaction (3.15), oxygen ions are produced, and flow through the electrolyte towards the anodic TPB. For this reason, within the electrolyte there is no production/consumption of ions. Therefore, phenomena acting in the electrolyte are intrinsically easier to deal with than those occurring in the other parts of the fuel cell. Equation (3.5) becomes ∇Jion = 0. (3.60) Combining Equations (3.60) and (3.10), the Laplace equation, is obtained: ∇ 2 φion = 0.

(3.61)

70


When ions flow from one side of the electrolyte to the other, there is ohmic loss and generated heat, due to the Joule effect. The energy conservation equation within the solid is ρc

∂T = ∇(λ∇T ) + Sq , ∂t

(3.62)

and the heat generation is given by Sq = σ ∇φion · ∇φion.

(3.63)

3.3.4 Current Collectors Phenomena acting in the current collectors, also referred to as interconnects and endplates, are similar to those taking place within the electrolyte. In this component, only electrons flow, thus only the equations required to model the electrical problem and the temperature distribution are needed. Considering that the current collectors are usually made of highly electrically conductive materials, the relative ohmic resistance, depending on the purpose of the model, is usually ignored. As a consequence, even the heat generation, that is due only to the Joule effect, can be ignored. When these are not ignored, the following equations apply for modeling the electric current density and potential: ∇Jele = 0,

(3.64)

∇ 2 φele = 0,

(3.65)

∂T = ∇(λ∇T ) + Sq , ∂t Sq = σ ∇φele · ∇φele .

ρc

(3.66) (3.67)

The current collector is in direct contact with one of the electrodes, with the gas, and, with the surroundings (depending on the geometry, and, in the case of the planar configuration, where the current collector is an end-plate, or a bipolar plate). Heat is exchanged by conduction with the electrode, while for convection with the gas and, with the surroundings. Finally, due to the high temperature, radiation can also play an important role [42, 64, 67], although a recent publication by Daun et al. [42] reports negligible effects on the temperature distribution.


71

3.4 Simplification of the Model: Triple-Phase-Boundary Reactions as Boundary Conditions As previously explained, the triple-phase-boundary is the site where ions, electrons and gas coexist, thus enabling reactions (3.15) and (3.16) to take place. As can be observed in the schematic representation given in Figure 3.3, the triple-phaseboundary represents a small portion of the entire electrode domain. Moreover, since the electronic conductivity of the electrodes is much higher than the ionic conductivity, ion migration within the electrode domains is very limited in space, thus reactions (3.15) and (3.16) are likely to take place very close to the electrode-electrolyte interface. For this reason, a simplification of the model can be introduced, i.e. it is possible to consider the reactions confined at the electrode-electrolyte interface. Under this assumption, there is no reaction taking place within the electrode domain, and, consequently, there is no ionic flux within the electrode. Therefore, Equation (3.34a) at steady-state simplifies to: ∇Jele = 0. (3.68) Equation (3.68), combined with Ohm’s law (3.10) gives: ∇ 2 φele = 0.

(3.69)

Since the electrochemical reactions are supposed to take place at the electrodeelectrolyte interface, then the Butler–Volmer equation, regulating the electrochemical kinetics, sets the boundary condition, whilst j (production rate) in Equation (3.37) is replaced with J (current density produced), as explained in detail in Section 3.7.2. For the energy conservation, Equations (3.45a) and (3.45b) are simplified to: Sq,an,cat = σ ∇φele · ∇φele .

(3.70)

For species conservation, the production rate is zero, thus Equation (3.2) reduces to: εu · ∇(ρYi ) = −∇mi .

(3.71)

3.5 A Further Simplification: Lumped PEN Structure Another simplification of the model, widely used when modelling SOFC operation, is to consider the Positive electrode/Electrolyte/Negative electrode (PEN), as a lumped structure. Figure 3.4 shows a schematic representation of a planar SOFC, when the PEN structure is considered in a 1D domain.

72


Fig. 3.4 PEN structure in a 1D domain.

When this simplification is used, each point of the PEN, in a specific time, needs to be characterized by a current density, a voltage, and a temperature. Voltage and current density are related by the following relationship: V = OCV − ηohm − ηact − ηcon .

(3.72)

OCV represents the open circuit voltage, i.e. the voltage difference between the two current collectors, when no current flows. Under the assumption that no gas crossover from one electrode to the other takes place, and assuming that there is no electronic transport within the electrolyte, the Nernst equation can be employed to calculate the OCV: − G − G0 RT PH2 O OCV = , (3.73) = − ln ne F ne F 2F PH2 PO0.5 2

where G0 represents the change of Gibbs free energy associated with reaction (3.17) at standard pressure (cf. Chapter 2). ηohm is the voltage reduction due to ohmic losses and depends on the resistivity of the cell materials, as well as on the cell geometry. The general expression for the ohmic loss is given by Ohm’s law in a finite form: ηohm = Rohm · I,

(3.74)


73

Fig. 3.5 Mass transport within an SOFC.

where I is the current passing through a conductor characterized by a resistance Rohm given by the following expression: Rohm =

1 l , σ S

(3.75)

where l is the length and S the cross-section of the conductor. Since in fuel cell modeling, for several reasons, it is more convenient to use the current density J , rather than the current I , it is a common practice to use an area resistance form rohm : 1 (3.76) rohm = l. σ From Equations (3.75) and (3.76), it is clear that the resistance associated with a specific conductor depends on the path of the electrons or ions, flowing through it. For this reason, when a PEN structure is used for modeling the fuel cell, the calculation of the PEN resistance is required. In Appendix A3.1, a general approach for calculating an equivalent resistance, as well as results for selected geometries are reported. The overpotentials due to the activation barriers at the electrodes, ηact , are calculated through the Butler–Volmer Equation (3.37), but replacing j with J . In order to clarify the meaning of the concentration losses, Figure 3.5 represents, in a 2D domain, the mass transport phenomena within an SOFC. For sake of simplicity, we will refer only to the anode side. As described by Equations (3.47–3.59), the gas concentration of H2 and H2 O in the gas channel is different from that in the reaction zone. Diffusion phenomena occurring in the porous media, in fact, are based on the existence of a concentration gradient, as expressed by Equations (3.47–3.49), i.e. gas concentration variation in the space domain enables the mass transport. Since both OCV and activation overpotential depend on some gas species concentration, and since in a PEN lumped structure the mass transport cannot be directly modeled within the domain thickness, the so-called concentration losses are introduced. They represent the voltage reduction due to the fact that the gas species concentration reacting in the reaction zone is different from that used in the calculation (i.e. the concentration relative to the gas channel, also referred to as the “bulk”).

74


In calculating the concentration losses, however, the effect of mass transport on the activation losses is usually neglected, thus they are defined as: bulk react bulk PH2 PH2 O PO2 RT RT + ln ln ηcon = OCVbulk − OCVreact = react , (3.77) bulk react 2F 4F P PH2 O PH2 O2 where the superscripts “bulk” and “react” refer to, respectively, the gas channel and the reaction zone. In order to evaluate the concentration of the interested gas species in the bulk, specific mass transport models, derived from Equations (3.47–3.59) need to be defined. Appendix A3.2 reports an example calculation. Concentration polarization losses are sometimes expressed as a function of the limiting current, il , which is usually taken as a measure of the maximum rate at which a reactant can be supplied to an electrode: i RT ηcon,el = ln 1 − . (3.78) 2F il As explained in Appendix A3.2, (3.77) and (3.78), under certain conditions, are equivalent. Due to the lumped structure assumption, the energy Equations (3.6) and (3.7) do not provide any information of the temperature distribution along the thickness of each single domain of the PEN. For example, referring to Figure 3.1, temperature distribution along the y-direction is not computed. Lumped values of the temperature representing the anode, cathode, electrolyte, and the gas in the gas channels are provided. The word “lumped” in this case should not create misunderstanding. The use of the PEN structure approach, in fact, does not mean that a temperature distribution cannot be obtained, but it simply implies that its distribution along the cell thickness is not represented. For more details on the model application, the reader is invited to read Chapters 4–10.

3.6 Equations Discretization The governing equations that model a SOFC are highly non-linear and self-coupled, which make it impossible to obtain analytically exact solutions. Therefore, the equations must be solved by discretization thus converting them to a set of numerically solvable algebraic equations. The appropriate solution algorithm to solve a system of partial differential equations strongly depends on the presence of each term and their combinations. The three most commonly used computational techniques to discretize a system of partial differential equations are: • Finite Difference Method (FDM) [68]; • Finite Volumes Method (FVM) [69, 70];


75

• Finite Elements Method (FEM) [71–73]; each of which allows transforming the continuous problem into a discrete problem with a finite number of degrees of freedom. The basic idea of the FDM is to replace the individual terms in partial differential equations with finite difference forms at a discrete set of points (the mesh). In both the FEM and the FVM the spatial domain is divided into a finite number of sub-domains (elements or volumes) and the attention is focused on these subdomains rather than on the nodes (as it is for the FDM). In the FEM and the FVM, in fact, the integral form of the conservation equations is taken into account and the integration is made over the prescribed finite volumes. It is important to underline that not always does the finite volume over which the integration is made coincide with a sub-domain of the mesh. Central to FVMs and FEMs is the concept of local conservation of the numerical fluxes. The integration of the convective term of a conservation Equation over a finite volume leads, in fact, to the flux of the generic variable through the finite volume boundary surface, by applying the well known Gauss theorem. The main difference between a FEM and a FVM is that in the first, a function is assumed for the variation of each variable inside each element while in the latter this function is always equal to 1. Another important feature of FVMs and FEMs is that they may be used on arbitrary geometries, using structured or unstructured meshes. As shown above, independently from the numerical method, the spatial continuum is divided into a finite number of discrete cells, and finite time-steps are used for dynamic problems. In order to perform space discretization, the domain over which the governing equations apply is filled with a predetermined mesh or grid. The mesh is made up of nodes (i.e. grid points) and/or elements at which the physical quantities (i.e. unknowns) are evaluated. Neighbouring points are used to calculate derivatives. Mesh generation is a very complex task for applied problems and many different approaches to it have been developed and are currently under study [74–77]. First of all, computational grids are classified as structured or unstructured meshes, even if each of these classes comprises a broad list of meshing techniques. The simplest structured mesh is the Cartesian mesh, where nodes are distributed regularly at equal distance from one another. Usually FDMs are applied to structured Cartesian meshes. More generally, a grid is classified as structured when only the physical location (coordinates) of each node must be stored since the identity of neighbouring mesh points is known implicitly. This comprises also multiblock structured grid generation schemes (a collection of several structured blocks connected together). The main advantage of structured grids is the simplicity in terms of application development, computation and visualization. The automatic connectivity information implies that structured grids require the lowest amount of memory for a given mesh size and the related simulation is faster. However, structured grids may represent a severe limitation for practical engineering purposes especially when the geometry is complex and there is a need for high resolution in particular regions of the domain.

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In an unstructured mesh each node can have a different number of neighbours and elements have different shapes and sizes. Therefore connectivity information must be explicitly defined and stored. The unstructured grid approach, that has gained popularity with the enormous advancements of computer technology, allows handling complex geometries with a lower number of elements and a much easier realization of local and adaptive grid refinement. [77] A detailed description of mesh generation theory goes beyond the present book and the reader can refer to literature [74–77]. However, it is important to underline that the accuracy and the stability of any numerical computation are significantly influenced by the particular meshing strategy. In the case of dynamic (unsteady) problems, even after the space discretization, we still have to solve a set of ordinary differential equations in time. Therefore, the second step is to discretize the temporal continuum. This is usually done by a finite difference approximation with the same properties of a FDM in space. Depending on the instant in which the information is taken, the time-discretization leads to: • an explicit scheme, when the solution at time step n + 1 depends only on the known solution at time step n (time-marching solution); • an implicit scheme, when the time-discretization step leads to a nonlinear algebraic system of equations that must be solved to calculate the solution at time n + 1. Associated with numerical problems is the concept of stability. A numerical scheme is stable when a solution is reached even with large time-steps (unsteady problems) or iteration steps (algebraic system of equations iteratively solved). Therefore the size of the time-step or of the iteration-step is dictated by stability requirements. It must be kept in mind that stability does not mean accuracy: an implicit scheme of a dynamic problem is unconditionally stable but the solution obtained with large values of the time step may not be realistic.

3.7 Boundary and Interfacial Relations Any governing model equations have to be supplemented by initial and boundary conditions, all together called side conditions. Their definition means imposing certain conditions on the dependent variable and/or functions of it (e.g. its derivative) on the boundary (in time and space) for uniqueness of solution. A proper choice of side conditions is crucial and usually represents a significant portion of the computational effort. Simply speaking, boundary conditions are the mathematical description of the different situations that occur at the boundary of the chosen domain that produce different results within the same physical system (same governing equations). A proper and accurate specification of the boundary conditions is necessary to produce relevant results from the calculation. Once the mathematical expressions of all boundary conditions are defined the so-called “properly-posed problem” is reached. Moreover, it must be noted that in fuel cell modeling there are various


77

intrinsically coupled phenomena (fluid, thermal, chemical and electrochemical) as well as different physical systems to be modeled. This means that the definition of the boundary conditions incorporates both external and interface boundary conditions. The type of partial differential equations and the chosen discretization technique both determine the form of the boundary conditions, which are usually classified either in terms of their mathematical description or in terms of the physical type of the boundary. For steady state problems the following two types of spatial boundary conditions are identified: • Dirichlet boundary condition, when the generic variable on the boundary assumes a known and constant value; • Neuman boundary condition, when the derivatives of the generic variable on the boundary are known and this yields an extra equation. Physical boundary conditions depend on the specific problem to be solved. In the fluid problem within the gas channels the following boundary conditions are observed: • stationary solid walls (sidewalls) where the velocity components vanish at the walls and the heat flux must be defined; • interface between the gas channel and the porous media where mass and energy crosses the boundary surface in either direction; these conditions are therefore usually given imposing the continuity of mass and energy fluxes; • open flows (inlet and outlet) where the fluid enters or leaves the domain; fluid velocity or pressure values together with, or the mass flow rate should be defined as well as the specific enthalpy for the energy equation; the initial molar or mass fraction of the different species must be also imposed at inlet. In the present section, boundary and interfacial conditions are presented for the three modeling approaches given in Sections 3.3, 3.4, and 3.5. Since the modelling approaches described in Sections 3.4 and 3.5 can be considered simplifications of the model presented in Section 3.3, boundary conditions are presented first for Section 3.3, and, consequently, for Sections 3.4 and 3.5.

3.7.1 Boundary Conditions for Each Sub-Domain In the present section, boundary conditions for the equations presented in Section 3.3 are defined. For a better understanding of such conditions, each single domain of Figure 3.1 is analyzed. Channel flow In order to solve Equations (3.24), (3.28) and (3.33), a number of uncoupled conditions equal to the number of unknowns need to be specified at the boundaries

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Fig. 3.6 Schematic representation of the electrode.

(boundary conditions). These are the mass fractions Yi (continuity equation), the fluid temperature T (energy equation) and a characteristic of the flow field (momentum equation), i.e u or P . If these change over time (as in transient problems), then these boundary parameters need to be so updated over the period of time of the solution. Taking the case of Figure 3.1 as a reference (planar configuration), the molar fraction needs to be specified at the inlet, while at the channel boundaries, there is no variation. At the electrode/channel flow interface and at the outlet, continuity is imposed. The velocity of the fluid is zero at the channel walls, while it is specified at the inlet and outlet. Alternatively, it can be convenient to specify the pressure (usually this is specified at the outlet). Finally, boundary conditions for the inlet and outlet thermal flux has to be specified, as well as the heat flux continuity at the channel/electrode and channel/interconnect interfaces. Electrodes According to the schematization of Figure 3.1, the electrodes are in contact with the electrolyte on one side, and with the gas and the current collector on the other side. Although the tubular and monolithic configurations present a different geometry, it is always possible to identify these three boundary surfaces [78]. The electrode domain of an SOFC is reported in Figure 3.6. However, by analogy, similar boundary conditions can be provided for the tubular or monolithic configurations. Surfaces S1, S2, S4 and the other perimeter surface not represented in Figure 3.6 (because it is hidden) are insulated, thus the boundary conditions for the electrical (Equations (3.10) and (3.34)), thermal (Equation (3.43)), and mass transport equations (Equations (3.2) and (3.47), or (3.48) or (3.49)) are, respectively: Jele · n = 0,

(3.79)

Q · n = 0,

(3.80)


79

mi · n = 0,

(3.81)

u · n = 0.

(3.82)

Alternatively, an estimation of the heat, mass and current loss can be provided. The estimation of these quantities is usually affected by a large uncertainty, thus the ideal case is usually considered. Surface S3 represents the boundary between the electrode and the electrolyte. Since the electrolyte is only ion conductive, the electric current leaving this boundary is zero, while continuity of the ionic current density is imposed: Jele · n = 0,

(3.83)

(Jion · n)electrode = − (Jion · n)electrolyte .

(3.84)

Heat is exchanged between the electrode and the electrolyte by conduction. The boundary condition sets the continuity of the heat flux: (Q · n)electrode = − (Q · n)electrolyte .

(3.85)

In absence of any form of leakage, there is no mass transfer between the electrodes and the electrolyte, i.e.: mi · n = 0, (3.86) u · n = 0.

(3.87)

Surface S5 is the boundary between the current collector and the electrode. Within this surface, there is continuity of the electrical current, and of the heat flux. At the same time this surface represents insulation for the ions and for the gas species: (Jele · n)electrode = − (Jele · n)interconnect ,

(3.88)

Jion · n = 0,

(3.89)

(Q · n)electrode = − (Q · n)interconnect ,

(3.90)

mi · n = 0,

(3.91)

u · n = 0.

(3.92)

Surface S6 is in direct contact with the gas. In this case there is a mass continuity with the channel flow, insulation for the electric and ionic current current, while the heat flux is composed of two terms, one considering the heat flux exiting the gas channel, plus one heat source represented by the radiative heat from the current collector: (u · ∇(ρYi ) + ∇mi ) · ngaschannel = −(εu · ∇(ρYi ) + ∇mi ) · nelectrode , (3.93) Jele · n = 0,

(3.94)

Jion · n = 0,

(3.95)

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Fig. 3.7 Current collector/gas channel of a flat planar SOFC.

Q · n = h(TS6 − Tgas) + Qrad−surf .

(3.96)

In Expression (3.96), Qrad−surf is the radiation heat transferred between S6 and any other visible surface of the electrode. Murthy and Fedorov [65] noted that the surface-to-surface approach (as in Equation 3.96) could lead to some temperature prediction mistakes. According to [65], more accurate results can be expected considering the absorption, emission or scattering in the media. On the present topic, there are still ongoing studies and a common position about the effect of considering the absorption, emission or scattering in the media is still a matter of clarification (see for example [79] and the apparently opposite position of [42] and [65]). Current Collectors Since the current collectors have the main role of transporting electricity to/from the electrodes, this implies that there is continuity of the electrical current at the electrode/current collector boundary. Referring to Figure 3.7, this surface is represented by surface S4. In addition to current continuity, heat flux has also continuity through this surface. The equations setting the continuity conditions are (3.88) and (3.90). Surfaces S1, S2, S3 and the forth perimeteral face (not represented in Figure 3.7) are thermally and electrically insulated. For these faces, the following conditions apply: J · n = 0, (3.97) Q · n = 0.

(3.98)


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Surface S5 is electrically insulated, thus expression (3.97) applies. In analogy with (3.96), the condition for the heat flux is: Q · n = h(TS5 − Tgas) − Qrad−surf .

(3.99)

As for surface S4, the continuity conditions for boundary S6 (hidden in Figure 3.7), need to be applied. Electrolyte Referring to Figure 3.1, the electrolyte can be regarded as a parallelepiped, where the surfaces constituting the perimeter are insulated: J · n = 0,

(3.100)

Q · n = 0.

(3.101)

For the surfaces on the top and bottom, continuity with the two electrodes needs to be expressed for the ionic flux and the heat flux. These two conditions are given by Equations (3.84) and (3.85), respectively.

3.7.2 Boundary Conditions when the Triple-Phase-Boundary is Reduced to the Electrode/Electrolyte Interface When the electrochemical reaction of hydrogen and oxygen is considered at the electrode/electrolyte interface (cf. Section 3.4), all the boundary conditions described in the previous section are still valid, except those of the electrodes. To avoid repetitions, only the boundary conditions related to the electrode domains are reported in the present section. As illustrated in Section 3.4, the reaction zone is here considered as a surface at the electrode/electrolyte boundary. Referring to Figure 3.6, this surface is S3. The boundary condition for Equations (3.68) and (3.69) is given by the Butler– Volmer equation:

−(1 − β)nF ηact βnF ηact J · n = ±J0 exp − exp , (3.102) RT RT where the sign “+” refers to the anode, and “−” to the cathode. Expressions (3.40a) and (3.40b) for the exchange current density are still valid if j (the current production rate) is replaced with J (the current density flowing at the boundary). Expression (3.41) is used to quantify the activation loss, and it is here reported for the reader’s convenience: ηact = Vrev − φelectrode − φelectrolyte, (3.41)

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Fig. 3.8 Schematic showing both ohmic and activation losses, and the modeled discretized potential jump at the anode-electrolyte interface.

Fig. 3.9 Schematic showing both ohmic and activation losses, and the modeled discretized potential jump at the cathode-electrolyte interface.

where Vrev is the reversible potential and φelectrode and φelectrolyte are referring to the electric potential at the boundary of the electrode and the ionic potential of the electrolyte at the boundary, respectively. The ideal voltage Vrev can be computed with the Nernst Equation ((3.42a) and (3.42b)). Figure 3.8 depicts a qualitative description of the resulting potential jump taking place at the anode-electrolyte interface. Since electrons are produced at the anode-electrolyte interface, they proceed from this interface toward the current collector above the anode as shown in Figure 3.8. (The reader is reminded that it is a common convention to consider the “electric current” direction as opposite to that of electron flow.) Due to ohmic losses, a potential decrease takes place as the current flows within the anode. At the cathodeelectrolyte interface, a mass flux occurs, due to reaction (3.16). Additionally, the reaction at the cathode consumes electrons, thus a potential jump is also established between the cathode and the electrolyte. The ideal potential jump is provided by the Nernst equation, however, due to the activation loss, the real potential jump is lower than the ideal one, as reported in (3.41). In analogy with Figure 3.8, Figure 3.9 represents the voltage variation across the electrolyte and the cathode.


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Heat production associated with the electrochemical reactions is also assumed to be confined at the electrode-electrolyte surface, thus the resulting thermal energy produces a discontinuity of the heat flux. The heat generated within this surface, in fact, represents a heat source for the electrode and the electrolyte domains. The sum of the inward heat fluxes is equal to the heat generated as a result of the electrochemical reactions. As explained in Section 3.3.2, the heat is generated by the increase in entropy, associated with the electrochemical reaction (reversible heat), and to the activation irreversibilities. Therefore, the boundary conditions for Equation (3.7) are: (Q · n)electrode + (Q · n)electrolyte = −

J T SH2 O − J ηact an , 2F

J T SO2 − J ηact cat , 4F where Equation (3.103) refers to the anode, and (3.104) to the cathode. The boundary conditions for the mass transport Equation (3.71) are: (Q · n)electrode + (Q · n)electrolyte = −

mH2 · n =

J·n MH2 , 2F

(3.103) (3.104)

(3.105)

J·n MH2 O , (3.106) 2F J·n mO2 · n = − MO2 . (3.107) 4F Expressions (3.105) and (3.106) refer to the anode, while expression (3.107) refers to the cathode. mH2 O · n = −

3.7.3 Boundary Conditions for the PEN Structure-Based Model In this case, no interfacial conditions need to be specified. Only external boundary conditions in terms of operating conditions (gas pressure, gas temperature, chemical composition) and heat transfer from the PEN to the surrounding (adiabatic condition, isothermal condition or a specific heat flux) are required.

Appendix A3.1. Calculation of Ohmic Resistance As explained in Section 3.5, the electrical resistance of a conductor depends on the conductor characteristics (particularly, conductivity and geometry) and on the path of current that passes through it. When a PEN structure is considered for modelling the fuel cell, Ohm’s law is used in a finite form, i.e. expression (3.74). This expression assumes that the resistance of the PEN is known.

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Fig. A3.1 Schematic representation of a conductor with a rectangular shape.

When a planar SOFC is considered, expressions (3.75) and (3.76) are employed for each layer (i.e. anode, cathode and electrolyte), thus obtaining the PEN resistance: 3 3 1 rohm = ρi li = li , (A3.1) σi i=1

i=1

where i = 1, 2, 3 is referred to the anode, cathode and electrolyte. Note that here ρi represents the resistivity of the conductive element. Eventually, the resistance of the current collectors can also be added, if relevant to a specific case. Expression (A3.1) assumes that both the electronic and ionic currents flow perpendicularly through the electrodes and/or the electrolyte, i.e., describing a straight direction from one electrode to the other (i.e., the situation of Figure A3.1, for both the electrodes and the electrolyte). This is the most desirable situation for a fuel cell, because the ohmic resistance is minimized. However, due to the disposition of the current collectors, this situation is not achievable, and a deviation along the plane of the cell is unavoidable. For this reason, the resistance generated if current flows in a straight line is usually named ‘cross-plane’ resistance while the term ‘in-plane’ resistance refers to the contribution due to the directional deviation. Even for the planar configuration, where expression (A3.1) is usually employed, small in-plane resistance is present (Figure A3.2). However, in planar SOFC, the resulting in-plane resistance has a modest contribution to the overall voltage reduction, compared to the cross-plane resistance, thus Equation (A3.1) is usually employed. If the current path is quite complex (i.e., for example, there is a direction change) or if the cross-section of the conductor is not constant, both cross-plane and in-plane contributions need to be evaluated. Bossel [80] provides a second order differential equation that can be applied to arbitrary SOFC geometries: φ ∇ 2φ = 2 , (A3.2) L where L is the characteristic length of the conductor, i.e.:


85

Fig. A3.2 Detail of the current path in a planar SOFC.

L=

ρelectrolyte lelectrolyte . ρanode / lanode + ρcathode / lcathode

(A3.3)

Solving Equation (A3.2), it is possible to correlate the potential drop between two ends of the PEN structure with the ohmic resistance. Bossel [80] reports the resulting ohmic resistance for the specific cases of Figure A3.3. This case is of particular importance because it represents the configuration of the Integrated Planar SOFC (IP-SOFC) developed by Rolls-Royce Fuel Cell Systems, the tubular configuration developed by Siemens Power Corporation, and the segmented configuration. The analytical solution of (A3.2) for the configuration of Figure A3.3 leads to the following expression of the resistance: rohm = CK{cothJ + B[K − 2 tanh(K/2)]}, where C=

3 i=1

ρi li =

3 1 li , σi i=1

(A3.4)

(A3.5)

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Fig. A3.3 SOFC with ideal current collectors at two opposite edges.

Fig. A3.4 Schematic representation of a tubular SOFC for calculating the equivalent resistance [21]. (Reproduction under permission of Elsevier BV).

x , L σan lan σcat lcat B= . σan lan 2 1+ σcat lcat K=

(A3.6)

(A3.7)

An alternative approach for evaluating the ohmic resistance is to represent the PEN structure with an equivalent electrical circuit, and to apply the well known theory on the electrical circuit for calculating the resulting equivalent resistance. Campanari and Iora [21], for example, use the schematic representation of Figure A3.4 for calculating the equivalent resistance of a tubular SOFC.


87

Due to the cell symmetry, the equivalent ohmic resistance of a half tube is calculated first. Consequently, the cell equivalent ohmic resistance is calculated by considering two of the resulting half-tube resistances in a parallel configuration. There are several techniques that can be employed for calculating the equivalent resistance of Figure A3.4. For their employment the reader is referred to specific books of electrical circuits. Finally, another possibility for evaluating the ohmic resistance is through the numerical solution of the Laplace equation within the SOFC domain [81].

Appendix A3.2. Calculation of Concentration Loss As explained in Section 3.3, concentration loss represents the voltage reduction due to mass transport of the gas species through the electrodes. The main result of the mass transfer in a porous electrode, in fact, is that concentration of the gas species in the reaction zone is different from that in the gas channel. The mathematical expression for the concentration loss is given by Equation (3.77) bulk react bulk PH2 PH2 O PO2 RT RT ηcon = OCVbulk − OCVreact = ln ln + , (3.77) react 2F 4F POreact PHbulk O PH 2 2

2

where the superscripts “bulk” and “react” refer to the gas channel and the reaction zone, respectively. In order to calculate the concentration loss, according to Equation (3.77), a relationship between partial pressure of a gas species at the reaction zone and at the bulk is required. This relationship is provided by the equations regulating mass transport in porous media, as defined in Section 3.3.2. The analysis starts from the main assumption that diffusion within the electrodes occurs primarily along the electrode thickness and that the electrochemical reaction takes place at the electrode/electrolyte interface. The analysis is here conducted at steady-state. Under these assumptions, Equation (3.2), combined with Equation (3.47), after some calculation, becomes [82]: ∂Xi + Xi Nj , ∂yi n

Ni = −cgas Di

(A3.8)

j =1

where y is the direction of the electrode thickness, and the term Ni includes also the convective term. Considering hydrogen diffusion, and knowing that NH2 = −NH2 O , (A3.8) can be written as: P DH2 O eff ∂XH2 . (A3.9) NH 2 = − RT ∂y

88


The flux of H2 is regulated by the amount of electricity produced by the fuel cell, i.e. J . (A3.10) NH2 = 2F By substituting Equation (A3.10) into (A3.9), and integrating through by the anode thickness (la ), the following equation is obtained:

react XH 2

bulk XH 2

dXH2 =

la

−

0

J RT 2F P DH2

dy.

(A3.11)

eff

Therefore, the mole fraction at the reaction zone is react bulk = XH − XH 2 2

J RT la . 2F P DH2 eff

(A3.12)

The same equations, applied to water lead to react bulk XH = XH + 2O 2O

J RT la . 2F P DH2 O eff

(A3.13)

Oxygen flow is calculated analogously NO2 = −

P DH2 RT

eff

∂XO2 + XO2 NO2 , ∂y

(A3.14)

where

J . (A3.15) 4F Substituting (A3.15) into (A3.14), and integrating through by the cathode thickness lc , the oxygen concentration at the reaction zone is as follows: J RT lc react bulk XO . (A3.16) = 1 + (X − 1) exp O2 2 4F DO2 eff P NO2 =

Considering that the partial and total pressure are related by Pi = P Xi

(A3.17)

from (3.77), the following expression for the concentration loss is derived: 1 − (J RT lan /2F DH2 , eff PHbulk ) RT 2 ln ηcon = − 2F 1 + (J RT lan /2F DH2 O, eff PHbulk ) 2 1 RT 1 J RT lc ln . (A3.18) − − 1 exp − bulk bulk 4F 4F DO2 eff P XO XO 2

2


89

The right hand side of Equation (A3.18) shows two terms, the first representing the anodic concentration polarization and the second the cathodic one: 1 − (J RT lan /2F DH2 , eff PHbulk ) RT 2 ηcon,an = − ln , (A3.19) 2F 1 + (J RT lan /2F DH2 O,eff PHbulk ) 2O ηcon,cat

RT 1 1 J RT lc ln . =− − − 1 exp bulk bulk 4F 4F DO2 eff P XO XO 2 2

(A3.20)

Very often in the literature, Equation (A3.18) is given as a function of the limiting current [83, 84]. The general form is [83] J νj RT 3 ln j =1 1 − , (A3.21) ηcon = − nF JL,j where nF represents the charge transferred per reaction, j is referred to H2 , H2 O, and O2 , and vj is the stoichiometric coefficient of the j -th gas species in the reaction. This expression can be obtained by derivation of the current density from Equations (A3.12), (A3.13) and (A3.16). For hydrogen, for example, from (A3.12) J can be expressed as J =

2F P DH2 RT la

eff

bulk react (XH − XH ). 2 2

(A3.22)

The maximum current density is obtained in the ideal case where there are no reactreact = 0. This theoretical current is ants at the electrode/electrolyte interface, i.e. XH 2 called the limiting current: JL,H2 =

2F P DH2 RT la

eff

bulk XH . 2

(A3.23)

From (A3.22) and (A3.23) the following equation can be derived: react XH 2 bulk XH 2

=1−

J . JL,H2

(A3.24)

Following the same procedure for H2 O and O2 , and substituting in (A3.17) and (3.77), Equation (A3.21) is obtained.

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Part 2 Single Cell, Stack and System Models

Chapter 4

CFD-Based Results for Planar and Micro-Tubular Single Cell Designs L. Andreassi1 , S. Ubertini2 , R. Bove3 and N.M. Sammes4 1 Dipartimento di Ingegneria Meccanica, University of Rome “Tor Vergata”, Italy 2 DiT – Dipartimento per le Tecnologie, University of Naples “Parthenope”, Italy 3 European Commission, Joint Research Centre, Institute for Energy, The Netherlands 4 University of Connecticut, USA

4.1 Introduction In the present chapter, the mathematical model defined in Chapter 3 is applied for two particular geometries, namely an anode-supported disk-shaped single cell and an anode-supported micro-tubular single cell. The equations implemented are those defined in Sections 3.2–3.4, i.e. in a partial differential form, for each cell component. This approach is also referred to as Computational Fluid Dynamic (CFD). In order to illustrate the capabilities of the model, in terms of assessment of particular phenomena taking place within the fuel cell, one particular problem is analyzed for each geometry. In particular, for the disk-shaped cell, emphasis is put on the effect of the gas channel configuration on the gas distribution, and, ultimately, on the resulting performance. For the tubular geometry, three different options for the current collector layouts are analyzed.

4.2 Disk-Shaped SOFCs The fuel cell analyzed in the present section is a disk-shaped anode-supported SOFC, currently produced by H.C. Starck/InDEC B.V. As illustrated in Figure 4.1 [1], the anode material is a cermet of nickel oxide doped with yttrium stablilized zirconia (NiO/8YSZ). The cathode is composed of two layers: one made of 8YSZ with strontium-doped LaMnO3 (8YSZ/LSM) and one of LSM. The electrolyte consists of a dense 8YSZ material. The main geometrical characteristics of the fuel cell are reported in Table 4.1. The single cell is embedded in an alumina cell housing. Its internal surfaces are in contact with the fuel cell through a metallic mesh, thus they act as gas channels because they guide the gas from the internal part of the disk towards the outlet. In this way, the single cell is arranged in a sealess configuration, i.e. there is no gas R. Bove and S. Ubertini (eds.), Modeling Solid Oxide Fuel Cells, 97–122. © Springer Science+Business Media B.V. 2008

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L. Andreassi et al. Table 4.1 Main geometrical characteristics of the SOFC single cell. Cell diameter Active area Anode thickness Cathode thickness Electrolyte thickness

80 mm 50 cm2 525–610 µm 30–40 µm 4–6 µm

Fig. 4.1 Anode, cathode and electrolyte layers of the single cells [1].

sealing. Through the use of two alumina tubes perpendicular to the fuel cell plane, anodic and cathodic gases come in contact with the anode and cathode, respectively, in a small circular region, where the flux configuration becomes radial. As previously stated, current is collected for each electrode on its surfaces through a mesh interposed between the electrode surface and the pins delimiting the gas channels (cf. Figure 4.3). The mesh is made of platinum on the cathode side and nickel at the anode side. As shown in Figure 4.2, the cell housing is embedded in an electric furnace, which guarantees the desired fuel cell operating temperature. The electric load applied to the cell is varied through the use of an electric load bank. Anodic and cathodic gas mixtures are obtained from laboratory-quality hydrogen, oxygen and nitrogen. Before entering the single cell housing, hydrogen is saturated at room temperature, and then heated-up to 110◦ C to avoid water condensation in the gas line. Since each gas channel is configured as a channel grid, it does not allow the gas to follow a pure radial direction. Thus, the effect of non-uniform gas distribution needs to be quantified. The deviation of the flux from a purely radial configuration can lead to several consequences on the operational conditions of the fuel cell, thus affecting the resultant performance. First of all, if the gas is not well distributed within the cell surface, the reaction rate varies from area to area. This implies the existence of preferred zones for the electrochemical reaction, and, consequently, of local high current density, thus reducing the overall cell voltage. Secondly, different reaction rates throughout the cell causes temperature gradients and, consequently, thermal stresses, which can cause mechanical failure of the cell [2–4]. Finally, the existence of (some) preferred zones for the electrochemical reaction implies that part of

4 CFD-Based Results for Planar and Micro-Tubular Single Cell Designs

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Fig. 4.2 Single cell test rig set-up.

the cell surface is not completely exploited. Such phenomena can be identified and quantified, only if the numerical model is implemented in a 3D environment.

4.2.1 Mathematical Model The presented model [5] is based on the following assumptions: 1. 2. 3. 4. 5.

steady state conditions; negligible ohmic losses within the current collectors; laminar and incompressible flow in the flow channels; negligible radiation heat exchange; electrochemical reactions confined to the electrode-electrolyte interface.

Assumption (2) is justified by the high electrical conductivity of the current collectors, compared to those of the electrodes. Assumption (3) derives from the low gas speed in the SOFC gas channels, where the density variation is not related to compressions/expansions [6]. Assumption (4) is introduced because in anode-supported SOFCs, radiative heat transfer can be neglected compared to convective heat transfer phenomena [7, 8]. Finally, assumption (5) is made for simplicity considering the high electronic conductivity of the electrode, compared to the ionic conductivity, which reduces the ability of oxygen ions to migrate through the electrodes. In the following, the equations defined in Chapter 3, and pertinent to the present model, are reported. Considering the single cell cylindrical geometry and the ribs

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Fig. 4.3 Smallest non-repeating geometrical element considered for the numerical simulation.

orientation, the set of mathematical equations describing the electrochemical and physical phenomena is solved on a quarter of the disk, which represents the smallest non-repeating geometrical element, as reported in Figure 4.3. The fuel flow in the gas channels is modeled by applying the equation of state and the principles of mass and momentum conservation. From Equation (3.27), considering that no reaction takes place within the gas channel (at the anode side only humidified hydrogen is provided), and that the fluid flow is regarded as incompressible (assumption (3)), the mass conservation equation becomes: ∇u = 0.

(4.1)

By neglecting body forces applied to the fluid, momentum conservation (Equation (3.3)) is simplified to: ρu · ∇u = −∇P + µ∇ 2 u.

(4.2)

Convective and diffusive mass transport is considered in the gas channels, while in the porous media the convective term is neglected. Diffusion is modeled through Fick’s law, as described in Chapter 3. Charge transport is modeled by Ohm’s law (Equation (3.10)) and the charge conservation equation (Equation (3.68)), while the current density distribution at the electrode/electrolyte interface is modeled through the Butler–Volmer equation (Equation (3.102)). It should be noted that, contrarily to Section 3.7, Equation (3.102) is here derived from Equation (3.37) rather than Equation (3.39), because the former allows for a better agreement between experimental and simulated results. Equations (3.40)–(3.42) are used to model, the exchange current density, the activation overpotential, and the ideal potential drop at the electrode/electrolyte interface, respectively. Heat transfer is modeled through Equation (3.6), and the appropriate heat terms for each domain.


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4.2.2 External Boundary Conditions The boundary conditions presented in this section are those representing an experimental campaign performed on a single cell. These external boundary conditions for voltage, temperature, species concentration and gas speed are defined as follows: Inlet: Inlet conditions are defined on the basis of the operating conditions, and are summarized in Table 4.2. Lateral: Periodic conditions are considered on the lateral boundary as the analyzed geometry is the smallest non-repeating geometrical pattern of the whole cell (see Figure 4.3). Top/bottom: No slip condition is imposed to the velocity field and the SOFC ceramic support is considered to be at thermal equilibrium with the external furnace thus giving a constant temperature value. Due to the high conductivity of the current collector, the electrical potential at the top and bottom is considered uniform. It should be noted that the boundary condition for Equations (3.61) and (3.69) can be given either providing the electrical voltage established between the two electrodes, or by providing the electrical current flowing through the fuel cell. In the present study, the cell voltage is considered as the independent variable, i.e. φ = 0 and φ = Vcell is considered at the anode and cathode respectively. Outlet: Convective fluxes are considered at the cell outlet. The convection and radiation heat exchange mechanisms between the solid structure and the external furnace are expressed as: (4.3) qconv = hc (TF − T ), qrad = f σrad (TF4 − T 4 ),

(4.4)

where hc is the convective heat transfer coefficient [W m−2 K−1 ], TF is the furnace temperature [K], f is the emissivity, and σrad is the Stefan–Boltzmann constant [W m−2 K−4 ]. Another heat source is given by the combustion of the non-oxidized fuel with oxygen at the cell outlet. In the following, this combustion is referred to as “afterburning”. Afterburning: The afterburning heat source is evaluated considering a uniform combustion around the cell, and therefore a uniform heat production: qafterburn = (H2out · LHVH2 )/Sanode ,

(4.5)

H2out = [H2in − (I /2F )],

(4.6)

where H2out and H2in being the hydrogen flow rate at the outlet and inlet, respectively, LHVH2 the lower heating value of hydrogen, Sanode the active surface of the cell, I the electrical current, and F the constant of Faraday.

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L. Andreassi et al. Table 4.2 Boundary condition at the inlet. Hydrogen volume flow rate (mole fraction) Oxygen volume flow rate (mole fraction) Reference pressure Fuel inlet temperature Air inlet temperature Furnace temperature

24 Nl/h (97%) 12 Nl/h (20%) 1.013e5 Pa 750◦ C 800◦ C 800◦ C

4.2.3 Solution Algorithm A finite element computational package, COMSOL MULTIPHYSICS, is used to solve the non-linear system of equation described in the previous chapter. This is a modular commercial code able to analyze fluid dynamics, mass and energy balance coupled to chemical reaction kinetics, transport phenomena (including ionic and multicomponent diffusion) and heat transfer phenomena. An innovative solution technique is considered in order to perform an efficient management of computer resources and to reduce the computational time. In fact, even if the thermal, electrochemical and fluid dynamic phenomena are fully coupled, the numerical convergence is reached in an iterative way. This technique allows the separate solution of the electrochemical submodel, the fluid dynamic submodel and the thermal submodel: the electrochemical submodel is firstly solved starting from a tentative velocity and a temperature profile, thus evaluating species distribution and electrochemical variables. These data are exchanged with the fluid dynamic submodel and the thermal submodel. This process is repeated until convergence is reached. The solution algorithm is presented in Figure 4.4 [5].

4.2.4 Computational Mesh The mesh is composed of 43785 elements and is sketched in Figure 4.5. The results have been checked for grid sensitivity and the trends are observed to be consistent in all the investigated cases in terms of the polarization curve produced, as the numerical results are included within a difference of 3% when increasing the grid detail level.

4.2.5 Model Validation Validation is performed in two steps: first an experimental polarization curve, obtained with a fixed inlet gas flow rate, is compared with the calculated values, thus allowing the determination of some unknown parameter values (model calibration). Afterwards, three polarization curves, obtained at constant fuel and oxygen utiliza-


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Fig. 4.4 Solution algorithm.

tion, are compared with the model results (model validation). In this second phase, the values of the parameters obtained in the calibration phase are not changed. In the numerical model calibration phase, the unknown parameters are those contained in Fick’s law and in the Butler–Volmer equation, i.e. the diffusion coefficients representing the porous micro-structural characteristics (ε and τ ), and the electrochemical kinetics parameter (A and Ea ). It should be noted that the calibration pro-

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Fig. 4.5 Computational mesh.

Fig. 4.6 Numerical and experimental polarization curve.

cess is performed by varying the parameters within typical values found in the literature. The resulting parameters are reported in Table 4.3. The polarization curve, resulting from the simulation is compared with the experimental data (Figure 4.6). It is worth underlining that the experimental and numer-

4 CFD-Based Results for Planar and Micro-Tubular Single Cell Designs Table 4.3 Model input parameters. Gas thermo-physic properties [9] Oxygen viscosity Nitrogen viscosity Hydrogen viscosity Water viscosity Oxygen specific heat at constant volume Nitrogen specific heat at constant volume Hydrogen specific heat at constant volume Water specific heat at constant volume Oxygen thermal conductivity Nitrogen conductivity Hydrogen thermal conductivity Water thermal conductivity

1.668e-5+3.108e-8*T [Pa s] 1.435e-5+2.642e-8*T [Pa s] 6.162e-6+1.145e-8*T [Pa s] 4.567e-6+2.209e-8*T [Pa s] 2.201e4+4.936*T [J kmol−1 K−1 ] 1.910e4+5.126*T [J kmol−1 K−1 ] 1.829e4+3.719*T [J kmol−1 K−1 ] 2.075e4+12.15*T [J kmol−1 K−1 ] 0.01569+5.690e-4*T [W m−1 K−1 ] 0.01258+5.444e-5*T [W m−1 K−1 ] 0.08525+2.964e-4*T [W m−1 K−1 ] –0.01430+9.782e-5*T [W m−1 K−1 ]

Fuller binary diffusion coefficients [9] Dij =

0.143·10−6 T 1.75 1/2 1/3 1/3 pMij (νj +νj )

Oxygen Fuller diffusion volume Nitrogen Fuller diffusion volume Hydrogen Fuller diffusion volume Water Fuller diffusion volume

16.3 18.5 6.12 13.1

Porous media micro-structure [10] Porosity Tortuosity Porous radium

30% 10 0.5e-6 m

Electrokinetic parameters [10] Ideal potential Anodic preexponential coefficient Cathodic preexponential coefficient Anodic activation energy Cathodic activation energy Anode charge transfer coefficient Cathode charge transfer coefficient

0.8961 V 1.6e9 3.9e9 120 J mol−1 120 J mol−1 θa = 2, θc = 1 θa = 1.4, θc = 0.6

Electrical conductivities [9] Anode electrical conductivity Electrolyte electrical conductivity Cathode electrical conductivity

9.5e7/T*exp(-1150/T) [Ohm−1 m−1 ] 3.34e4*exp(-10300/T) [Ohm−1 m−1 ] 4.2e7/T*exp(-1200/T) [Ohm−1 m−1 ]

Solid parts thermal conductivities [11] Anode thermal conductivity Electrolyte thermal conductivity Cathode thermal conductivity

3 W m−1 K−1 2 W m−1 K−1 3 W m−1 K−1

Heat exchange coefficients Thermal convective coefficient Thermal radiative emissivity

5 W m−2 K−1 0.3

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Fig. 4.7 Numerical-experimental Uf -J profiles.

Fig. 4.8 Flow field arrows on the cathode side, when the oxygen flow rate is 12 Nl h−1 , and the cell voltage is 0.8 V at 360 mA cm−1 .

ical polarization profiles are mostly coincident (the error percentage is everywhere lower than 3%), clearly indicating a proper calibration phase. Model validation phase consists in establishing the range of validity and the accuracy of the code in describing fuel cell behaviour under different operating con-


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(a)

(b) Fig. 4.9 (a) H2 concentration along the gas channels; (b) O2 concentration along the gas channels. The operating conditions are: oxygen flow rate 12 Nl h−1 , hydrogen flow rate 24 Nl h−1 , cell voltage 0.8 V at 360 mA cm−1 .

ditions. For this reason, numerical and experimental data are compared when varying both fuel flow rate and electrical current, so that the resulting fuel and oxidant utilization is kept constant. Figure 4.7 presents the comparison between numerical and experimental results for three current density-voltage curves at constant fuel utilization coefficient (Uf = 0.3, 0.4, and 0.5), Uox = 0.4 being the oxidant utilization coefficient. The agreement between numerical and experimental data is excellent in most cases. Larger differences can be observed at high current densities for Uf = 0.3 and 0.5, while the numerical curve at Uf = 0.4 is perfectly predicted by the numerical code. The percentage errors between numerical and experimental

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(a)

(b) Fig. 4.10 (a) H2 concentration at anode/electrolyte interface; (b) O2 concentration at cathode/electrolyte interface. The operating conditions are: oxygen flow rate 12 Nl h−1 , hydrogen flow rate 24 Nl h−1 , cell voltage 0.8 V at 360 mA cm−1 .

data are within 5% and less than 2% for Uf = 0.4, thus demonstrating a wide range of validity of such a numerical approach.

4.2.6 Analysis of the Results Flow field: Flow field streamlines are reported in Figure 4.8, which shows that the presence of the ribs strongly affects the flow field. In fact, preferential flow directions can be observed along the x and y directions. In the central zone, due to the cell geometry, the flow velocity decreases, thus creating non-uniform gas distribution. However, such non-uniformity does not produce any significant misdistribution of H2 concentration (Figure 4.10).


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Fig. 4.11 Hydrogen concentration along the 0◦ , 22.5◦ , and 45◦ angular direction.

Species concentration: Hydrogen, and oxygen concentration along the gas channels are reported in Figure 4.9. As expected, the highest values are located at the cell inlet, while progressive O2 and H2 concentration decreases are observed moving towards the cell outlet. In Figure 4.10, hydrogen and oxygen concentration at the electrode/electrolyte interface are shown. While a lower concentration of O2 is observed under the ribs, this is not the case for H2 . This gas distribution can be explained considering that the diffusion in the porous electrodes is mainly affected by the following two effects: • cross-plane diffusion; • in-plane diffusion. Under the ribs, the species diffusion in the direction orthogonal to the cell plane, i.e. cross-plane diffusion, is obviously impossible. Hence, the reactant concentrations on the electrode/electrolyte site under the ribs are driven only by in-plane diffusion. Due to the strong diffusion property of H2 , in-plane diffusion allows H2 to penetrate under the ribs, while in the case of O2 , a relevant concentration reduction is noticed. The effect of the ribs is significant at all operating conditions, but it becomes predominant at high fuel utilization (not shown in the figures). For a better understanding, a slice view of the reactants concentration along the 0◦ , 22.5◦ and 45◦ angular directions is given in Figures 4.11 and 4.12. Moreover, comparing the two figures, it can be observed that in-plane diffusion is less pronounced at the cathode than at the anode because of the lower diffusivity of O2 , compared to H2 .

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Fig. 4.12 Oxygen concentration along the 0◦ , 22.5◦ , and 45◦ angular direction.

Finally, in order to give a complete overview of the species concentrations within the investigated domains, a three dimensional representation of the hydrogen, oxygen and water concentration is reported in Figures 4.13, 4.14 and 4.15, respectively. Current density: Figure 4.16 shows the current density distribution at the anode/electrolyte interface. The current density is not uniform, as it is affected by the hydrogen and oxygen distributions and by the electrolyte resistance which is, in turn, dependent on the temperature. Because of the previously discussed reasons, as emphasized in Figure 4.17 where a 2D representation is shown, the produced current is smaller under the ribs than elsewhere. Furthermore, around the ribs, it is possible to observe that the produced current is characterized by a local increase. This effect is related to the local flow deceleration which, in turn, causes a local increase in the species concentrations together with a greater species diffusion perpendicular to the cell plane. Temperature distribution: In Figure 4.18 the temperature distribution in the cell median plane is reported for two operating conditions. It can be seen that the fuel cell does not operate at uniform temperature, but temperature variations up to 25◦C are detectable. The maximum temperature is predicted towards the cell outlet. This information is of primary importance in the SOFC as it allows the estimation of thermally-induced stresses due to steady state thermal gradients. This non-uniform temperature distribution is enhanced by the unreacted hydrogen undergoing the afterburning reaction. Comparing the two operating conditions,


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Fig. 4.13 Hydrogen concentration distribution (slice view).

it can be observed that the temperature rise at the cell outlet is higher for lower current densities, which implies poorer reactant consumption and higher after-burn thermal flux.

4.3 Micro-Tubular SOFCs The most analyzed tubular SOFC in the literature is the Siemens-Westinghouse system. In this case, single cells are stacked together using a nickel felt placed along the length of the tube which connects the cathode of one cell with the anode of the contiguous one. Chapter 7 is entirely dedicated to the simulation of Siemens Power Generation tubular cells, while, in this section, micro-tubular anode-supported cells are considered. Specifically, the cells developed and manufactured at the University of Connecticut are considered. The anode is made of Ni/8% mole Yttria stabilized Zirconia (YSZ), the electrolyte of YSZ, and the cathode of La0.8 Sr0.2 MnO3 -δ (LSM). The main purpose of the analysis described here is to assess the effect of different current collector layouts on the overall performance. More details of the present analysis can be found in [12], while in the present section, the most relevant results are reported.

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Fig. 4.14 Oxygen concentration distribution (slice view).

Fig. 4.15 Water concentration distribution (slice view).


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Fig. 4.16 Current density distribution (cell voltage 0.8 V, total current 18 A).

4.3.1 Geometry Description and Main Assumptions The micro-tubular SOFCs considered are depicted in Figure 4.19. Specifically, Figure 4.19 shows the anode (supporting structure), the anode plus the electrolyte, and the final single cells. More details about the production process, the cell properties and characteristics can be found in [13–15]. The choice of the tubular geometry is related to its main advantages in terms of quick start-up, mechanical resistance, and possibility of realizing sealless stacks. The reduced size drastically reduces thermal shocks and the related mechanical stress, due to quick load changes and start-up/shut-down. The choice of the tubular geometry, however, introduces a number of possibilities for the current collector layouts, and different cell performance is expected in each case. Three possibilities are here analyzed: 1. Current collectors at the two tube ends. 2. Current collectors at the two tube ends and a metal mesh outside the cell. 3. Current collectors at the two tube ends and a metal mesh outside the cell and one inside.

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Fig. 4.17 Current density distribution (2D representation).

Solution 1 is the easiest to realize, but a high in-plane resistance is expected, due to the fact that the current, in order to flow from one tube end to the other, flows along the cell. Better performance can be expected from solutions 2 and 3. Due to the radial symmetry of the three configurations, a 2D model is employed. Figure 4.20 depicts the geometry considered for case 1, i.e. current collectors at the tube ends (not in scale). In cases 2, an additional 16 pieces of conductive material (nickel), with a square section of 1.5 × 1.5 mm, are considered on the cathode (external surface of the tube). In case 3 the conductive materials are present both inside and outside the tube. It should be noted that in case 2 and 3, cylindrical symmetry is not obvious anymore. However, it is reasonable to assume that voltage losses are mainly proportional to ohmic in-plane losses, i.e. to the length of the current path along the tube. In case 2 and 3, this distance is constant in each longitudinal tube cross-section, therefore a cylindrical symmetry is assumed. The mathematical model defined in Chapter 3, and implemented in Section 4.2 for the planar configuration, is also employed in the present analysis, with no major modification, except for the assumption of iso-thermal conditions, thus no further illustration of the model is given in the present section.


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(a)

(b) Fig. 4.18 (a) Temperature distribution (cell voltage 0.8 V, current 18 A); (b) temperature distribution (cell voltage 0.6 V, current 33 A).

4.3.2 Model Calibration The numerical model is implemented for the three current collector configurations previously mentioned. Since the model uses semi-empirical parameters, this is first calibrated and then validated, through a comparison with experimental data. For collecting the current, a silver wire is wrapped around the cathode, while a nickel spring is placed in contact with the anode (case 3 previously defined). The tests are performed using pure hydrogen at a constant flow rate. Voltage is varied by the use of a load bank, and the relative current is measured. The temperature of 800◦ C is

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Fig. 4.19 Anodes, anodes+electrolytes and complete cells (from the top to the bottom).

Fig. 4.20 Schematization of the tube in a 2D domain for the case 2. Table 4.4 Results of the model calibration. Voltage (V) 1.013 0.9 0.8 0.7 0.6 0.55

Experimental Current Density (mA/cm2 )

Simulated Current Density (mA/cm2 )

Error (%)

0.00 69.91 142.12 221.68 303.53 345.59

0.00 68.90 136.21 218.96 315.31 357.84

0.00 1.44 4.14 1.22 3.88 3.50

ensured by an electrical furnace. Figure 4.21 and Table 4.4 show the comparison between the simulated and the experimental data, while Table 4.5 reports the main electrochemical properties used for the simulation.


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Table 4.5 SOFC electrochemical properties. Eact,an Eact,cat γan γcat σanode σcathode σelectrolyte

130000 (kJ/kmole) 130000 (kJ/kmole) 7.3 · 109 7.3 · 109 20300 (S/m) 7000 (S/m) Ai exp(−Bi /T ) Ai = 11000 (S/m), Bi = 10300

Fig. 4.21 Comparison between experimental and simulated performance.

4.3.3 Analysis of the Results Using the data of Table 4.5, the other two configurations are also simulated. Figure 4.22 depicts the comparison between the J-V curves, and Figure 4.23 the relative power densities. As is clearly visible from Figures 4.22 and 4.23, although the configuration of case 1 is desirable in terms of manufacturing and stack assembling simplicity, the relative performance is unacceptable. As expected, the configuration with two meshes, one inside and one outside, produces the best performance, since the in-plane resistance is minimized. On the other hand, this is the configuration that is hardest to manufacture, especially in relation to the internal mesh. Case two seems to be a good compromise between performance and manufacturability. Figure 4.24 depicts the current distribution for case 1. As expected, the current runs mostly along the cylinder, rather than radially, i.e. in-plane resistance is expected to be relevant, due to the long length of the cell, compared to the thickness. During the single cell design phase, much effort has been put in realizing thin layers for minimizing the cell resistance. When current is collected at the two ends, however, these efforts result in a performance reduction, rather than enhancement.

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Fig. 4.22 Current-voltage comparison for the three cases.

Fig. 4.23 Power density comparison.

When the current is collected through an internal and an external mesh (case 3), the current path is fairly straight, as visible from Figure 4.25. In this case, the tubular configuration operates in a condition that is very close to that of the planar configuration. The in-plane resistance is minimized, but, the manufacturability is more complex. Finally, for case 2, the current path is reported in Figure 4.26. In this situation, the current still flows along the tube, as for case 1, but, as visible from the arrows dimension and from the stream lines, the electrons leaving the cathodic current collector, at the tube end on the left hand side, are far fewer, compared to case 1. This is because the external mesh collects a large part of the total current flowing through the cell.


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Fig. 4.24 Current path for case 1 (only the anode is visible).

Fig. 4.25 Current path for case 3.

The reduction of the current flowing along the cell is further shown in Figure 4.27, where the current distribution along the cathode-electrolyte interface is depicted, when the cell voltage is 0.7.

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Fig. 4.26 Current path for case 2 (only the anode is visible).

Fig. 4.27 Current density (A/m2 ) distribution along the cathode/electrolyte interface for case 2 and V = 0.7.

4.4 Concluding Remarks Two SOFC geometries; an anode-supported disk, and an anode-supported microtube were analyzed using computational fluid dynamics (CFD). The effect of gas


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channel configuration and current collector configuration were examined for the two systems respectively. The results for the anode-supported disk were as follows; the J/V curves fitted the model well using three different fuel ulitizations; the presence of ribs affects the flow field particularly at high fuel utilizations; the flow velocity decreases in central zone, thus creating non-uniform gas distribution; the current density distribution is not uniform and is affected by hydrogen and oxygen distributions, electrolyte resistance and thus temperature; temperature differences of up to 25◦ C are detectable, with a maximum temperature at the cell outlet. The overview of the results for the micro-tubular cases showed that if the current collector was placed only at the two ends, then very poor electrochemical results were realized; with wire outside (and potentially inside) the cell, the electrochemical results and current path were far superior. Thus, CFD is a very powerful tool to analyze SOFC systems, and this chapter shows just two possible scenarios.

References 1. H.C. Starck product information. Downloadable from http://www.hcstarck.com. Last access February 2007. 2. Nakajo A., Stiller C., Härkegård G. and Bolland O., 2006. Modeling of thermal stresses and probability of survival of tubular SOFC. Journal of Power Sources 158(1), 287–294. 3. Selimovic A., Kemm M., Torisson T. and Assadi M., 2005. Steady state and transient thermal stress analysis in planar solid oxide fuel cells. Journal of Power Sources 145(2), 463– 469. 4. Yakabe H., Ogiwara T., Hishinuma M. and Yasuda I., 2001. 3-D model calculation for planar SOFC. Journal of Power Sources 102(1/2), 144–154. 5. Andreassi L., Bove R., Rubeo G., Ubertini S. and Lunghi P., 2007. Experimental and numerical analysis of a radial flow solid oxide fuel cell. International Journal of Hydrogen Energy 32, 4559–4574. 6. Bove R. and Ubertini S., 2006. Modeling solid oxide fuel cell operation: Approaches, techniques and results. Journal of Power Sources 159(1), 543–559. 7. Daun K.J., Beale S.B., Liu F. and Smallwood G.J., 2006. Radiation heat transfer in planar SOFC electrolytes. Journal of Power Sources 157, 302–310. 8. Damm D.L. and Fedorov A.G., 2004. Spectral radiative heat transfer analysis of the planar SOFC. In Proceedings of the ASME IMECE, Anaheim, CA, November 13–19, 2004. Paper No. IMECE2004-60142. 9. Perry R.H. and Green D.W., 1997. Perry’s Chemical Engineers’ Handbook, 7th ed. McGrawHill. 10. Costamagna P., Selimovic A., Del Borghi M. and Agnew G., 2004. Electrochemical model of the integrated planar solid oxide fuel cell (IP-SOFC). Chemical Engineering Journal 102, 61–69. 11. Ferguson J.R., Fiard J.M. and Herbin R., 1996. Three-dimensional numerical simulation for various geometries of solid oxide fuel cells. Journal of Power Sources 58(2), 109–122. 12. Bove R. and Sammes N., 2005. The effect of current collectors configuration on the performance of a tubular SOFC. Proceedings of the Solid Oxide Fuel Cells IX, Quebec City, Canada, S.C. Singhal and J. Mizusaki (Eds.), Electrochemical Society, Vol. 1, pp. 780– 789. 13. Du Y. and Sammes N.M., 2004. Fabrication and properties of anode-supported tubular solid oxide fuel cells. Journal of Power Sources 136, 66–71.

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14. Sammes N.M. and Du Y., 2003. The mechanical properties of tubular solid oxide fuel cells. Journal of Materials Science 38, 4811–4816. 15. Pusz J., Smirnova A., Mohammadi A. and Sammes N.M., 2007. Fracture strength of microtubular solid oxide fuel cell anode in redox cycling experiments. Journal of Power Sources 163(2), 900–906.

Chapter 5

From a Single Cell to a Stack Modeling Ismail B. Celik and Suryanarayana R. Pakalapati Mechanical and Aerospace Engineering Department, West Virginia University, Morgantown, WV 26506-6106, USA

Nomenclature A B as C Cp Cf cr e E E0 Ecor Em F f G H h hconv I I i i0 K k lw L l M m ˙ m ˙ m ˙ surf n Nu

area (m2 ) perimeter of the channel (m) surface area of solid matrix for unit volume of mixed media (m−1 ) product of mass flow rate and specific heat (J/s K) specific heat at constant pressure (J/kg K) friction factor condensation rate constant (s−1 ) energy per unit mass (J/kg) open circuit potential (V) potential at standard state conditions (V) corrected potential (V) estimated modeling error Faraday’s constant (Coulomb/mol), view factor (no units) flux of a conserved scalar Gibbs free energy (kJ/Kmol) dimensionless height (m) enthalpy (J/kg), grid size (m) heat transfer coefficient (W/m2 K) current (A) current density vector current density (A/m2 ) exchange current density (A/m2 ) permeability thermal conductivity (W/m K), reaction rate coefficient (mole/m3 s) width of control volume (m) cell length (m) characteristic length (m) molecular weight (gm/mole) mass flow rate (kg/s) mass flux per area (kg/m2 s) net mass flux through surface (kg/s) number of participating electrons, number of samples Nusselt number

R. Bove and S. Ubertini (eds.), Modeling Solid Oxide Fuel Cells, 123–182. © Springer Science+Business Media B.V. 2008

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P P0 Pr Q˙ P q˙ R Re Rg rw S s˙ S2 T s t u u x j xk Xk X¯ Yk

pressure (Pa) reference pressure (Pa) Prandtl number convective heat transfer rate (W) pressure (N/m2 ) rate of heat generation (W) resistance ( m2 ), volumetric transfer current (A/m3 ) Reynolds number universal gas constant (kJ/Kmol·K) condensation rate of water (kg/m3 s) net generation (source or sink) (J/m3 sec) rate of production or consumption of a specie due to chemical reactions (mole/m3 s) variance temperature (K) entropy per mole (kJ/Kmol·K), volume fraction of liquid time (s) velocity in x-direction (m/s) velocity (m/s) length of control volume in x-direction (m) mass fraction of species j in kth phase mole fraction mean mass fraction

Greek letters α transfer coefficient αH2 fraction of total current produced from hydrogen oxidation ε porosity, emissivity β ratio of heat generated by the cell to the power generated δ error φ conserved, intrinsic quantity per unit mass of the continuum material diffusion coefficient (m2 /s) ηact activation loss (V) ηconc concentration loss (V) ηohm ohmic loss (V) ϕ electric potential at a point (V) µ viscosity (N·s/m2 ) υ stoichiometric coefficient ν degrees of freedom ρ density (kg/m3 ) σ Stefan–Boltzman constant (W/m2 K4 ) rate of formation and destruction of specie k (kg/m2 s) ωk ∀ volume (m3 ) Subscripts and superscripts 1 surface 1 2 surface 2 a anode b (superscript) backward b (subscript) bulk fluid c (superscript) convection, cathode, computations c (subscript) cell calc calculation

5 From a Single Cell to a Stack Modeling cond conv d dis e ext echem eff elec f (superscript) f (subscript) g gen h I in iter k, l l m mod n nd net num out p prod r ref s sat SMR surf used WGS w wv xs

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conduction convection diffusion discretization east face of control volume, experiments, electrochemical reactions. extrapolated, external electrochemical effective electrical forward fuel channel gas generated enthalpy interface inlet iteration different phases liquid mass, membrane modeling north face of control volume normal diffusion net amount numerical outlet pore produced reforming reaction reference solid (or specie s), south face of control volume saturation Steam Methane Reforming reaction surface of control volume used or consumed Water Gas Shift reaction wall, west face of control volume water vapor cross-sectional area

5.1 Introduction A single fuel cell is like a small battery which generates relatively small power in the order of 1 W. The amount of power generated is of course proportional to the total current which is the integral of current density (A/cm2 ) over the cross-sectional area of the cell. Again, just like in case of batteries, fuel cells are connected in series and parallel in order to build a stack capable of producing the power needed from that particular unit. Examples of commonly used designs of planar and tubular solid oxide fuel cell stacks are depicted in Figures 5.1b and 5.1d respectively. A modifica-

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tion of tubular design, Flat Tube SOFC, is shown to produce higher power densities while maintaining the seal-less feature of the tubular design (Lu et al., 2005). In general, the larger the surface area is the larger the power generated. However, like all power generating devices, to reduce losses and peripheral costs associated with large units, the smaller units are preferred. This usually translates into higher power density or higher efficiency. For example one would need about 1000 small button cells of 2.0 cm in diameter, stacked together to produce circa 1 kW power. Assuming the thickness of this cell is about 2 mm, the total volume will be approximately 628 cm3 (= 0.022 ft3 = 6.28 × 10−4 m3 or 0.628 liters). Considering a relatively larger planar fuel cell of 10 cm × 10 cm × 0.5 cm in size, and each cell generating about 25 W, one would need 40,000 cells to produce 1 MW with a volume of 2 m3 . The first stack will generate about 1 kW and the second stack about 1 MW of heat as by-product that has to be carefully managed. The aforementioned examples assume that single cells can be stacked together with linear scalability, i.e. if each cell produces 1 W ten cells will produce exactly 10 W. This, of course, is not a valid assumption in most cases. In a real stack each cell experiences different electrical, thermal and structural constraints due the neighboring cells. The interaction between the cells in a stack is through these constraints. Since the bottom and top cells in the stack have only one neighboring cell, they are different from the other cells in the stack, thus making it difficult to achieve uniformity among the cells. In addition, the inherent asymmetry in the fuel cell operation (different flows on fuel and air side) and inevitable differences in gas flow rates among the cells of the stack, render uniform stack operation practically impossible. Thus the voltage and temperatures usually vary from cell to cell in a stack. In an SOFC based power unit, several stacks (also known as bundles in case of tubular cells) are connected together to form a single module (see Figure 5.1c). As was in the case of cells in a stack, the stacks in a module also interact with each other through temperature, electric potential and mechanical stresses. In this regard the primary objective in stack modeling is to investigate scalability, and possible failure scenarios as the cells are stacked together in certain arrangements. This is done by computing the temperature distribution and current density distribution in three dimensions to detect hot spots which may cause structural mismatch, high thermal stresses due to thermal gradients, and fatigue as a result of cyclic application of high thermal stresses. The information gathered from such three dimensional calculations can also be used for optimization of operating conditions to achieve relatively smaller stacks with higher power density and consequently, higher efficiencies. Calculation of fuel and air flow distribution via manifolds to each cell in a stack is an important topic in stack modeling, however, this topic is extensively represented in computational fluid dynamics applications to other relevant topics such as compact heat exchangers, pipe networks, etc., hence it is not considered further in this chapter. First, an overview of various modeling strategies is presented. Then, further details of modeling hierarchy are presented, including governing equations, starting from one-dimensional models and extending to fully three-dimensional models. In addition, a simple one dimensional thermal model which computes the temperat-

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Fig. 5.1 Commonly used SOFC designs (Celik, 2006). (a) Tubular SOFC, (b) 24 cell tubular SOFC stack, (c) a tubular SOFC module with 48 stacks, (d) 28 cell internally manifolded stack design by Versa Power Systems.

ures inside the stack given the cell flow rates and voltages is also presented. In the applications section, examples from literature are presented to illustrate the depth of information that can be obtained from these models and how it can be used in performance improvement and/or structural analysis. Also, an Appendix is provided at the end of the chapter which is devoted to the critical issues of calculation verification and model validation procedures: without these two being an integral part of a study, the results from computations can not be reliable.

5.2 An Overview of Modeling Strategies Numerical modeling of a physical process involves formulating relationships (i.e. equations) between the important process variables and then solving them numerically to predict the behavior of the process for different sets of input conditions that can be controlled. The mathematical relationships are derived from the physical laws that govern the specific process in consideration. Due to the complex nature

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of exact physics, simplifying assumptions are usually made to reduce the number of variables and/or to obtain simpler equations. Also, at times empirical methods are used to model some processes for which underlying physics is either not fully known or is complex. The quality of results obtained from a numerical model depends upon the plausibility of the assumptions made. Numerical predictions usually contain various errors. The errors that arise from assumptions on and/or approximation of physical laws are called modeling errors. In addition to these, there could be numerical errors that arise from solving the equations by discretization or using iterative methods. In order to ensure that the sum of the errors is in tolerable limits, it is best to verify and validate a numerical model by comparing the results with the experiments. Once the validity of a numerical model has been established it can be used with some confidence to simulate other cases. The main advantages of numerical modeling are speed, and relative ease with which detailed parametric studies and other tests can be conducted, once the model is programmed and verified. However, experiments are still needed to validate the numerical models. For best results at low cost, an optimum balance should be maintained between experiments and modeling. Numerical modeling is particularly valuable when experimental investigation is difficult due to instrumentation or other problems. It is for these reasons that numerical modeling is widely used in fuel cell research and development. As a research tool, fuel cell modeling can be used to understand the processes that occur inside the fuel cells and to identify the critical ones which limit the others. Also, the effects of various parameters on different processes inside the fuel cell can be studied and underlying mechanism can be understood. Such knowledge is useful in devising better design concepts. Fuel cell modeling has also become a design tool lately. Commercial Computational Fluid Dynamics (CFD) software packages now come with a designated module for fuel cell modeling. As a design tool cell modeling can be used to predict the performance of a particular design of a fuel cell under various operating conditions. Such a study usually gives the information such as safe operating conditions, key parameters which affect the efficiency, etc. Also, modeling can be used to determine the most appropriate geometric proportions for fuel cells by conducting a parametric study with various geometries. Fuel cell models are available in published literature for a range of applications from detailed modeling of reaction kinetics inside the fuel cells to modeling the environmental and economical impact of incorporating fuel cell technology into power infrastructure. Given the wide scope of applications, the models are developed for specific purposes at different levels of complexity and detail. Though there is no clear-cut delineation, fuel cell models are usually categorized into component/electrode-, cell-, stack- and system-level models. Even though the main focus of this chapter is stacklevel modeling, cell level models are also reviewed briefly since they form the basis and backbone of stack modeling. Cell/stack models can also be classified as zero-, one-, two- and three-dimensional models, and steady and transient models. Detailed one- and multi-dimensional cell models are covered in this chapter. However, the zero-dimensional lumped cell model covered here is only a thermal model which calculates temperature given the operating conditions and the voltage produced by the cell. References for more detailed zero-dimensional models (i.e. lumped mod-


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els) are provided through out the chapter for interested readers (see also Chapters 8 and 9). When applied to a stack, zero-dimensional (0-D) cell models can give onedimensional (1-D) distributions and 1-D cell models give two-dimensional (2-D) distributions. Fuel cells are operated in stacks for all practical applications. Hence their analysis in the context of stack is more important than as a single unit. Some models (e.g. Zitney et al., 2004; Elizalde-Blancas et al., 2007a) assume that all the cells in the stack are identical thus a single cell is simulated and its outcome is multiplied by the number of cells to obtain the stack results. Such assumption is also implicitly present in the zero and one dimensional models for the stacks that are commonly used in system level modeling (for example, Ghezel-Ayagh et al., 2004; Kolavennu et al., 2004; Beil and Seume, 2006). However it was shown by several authors (e.g. Burt et al., 2004a; Pakalapati et al., 2007) that the temperature and performance vary from cell to cell. Variation in performance among cells within a stack can result from asymmetry in fuel cells. A natural asymmetry exists in Solid Oxide Fuel Cells (SOFCs) attributed mostly to a difference in the flow rates of the air and fuel gas channels. This asymmetry can cause non-uniform temperature distributions primarily due to the non-uniform cooling effect of air channels. Numerical simulations by Koh et al. (2002) found temperature variations in the upper and lower regions of a molten carbonate fuel cell stack resulted more from the influence of external heating than from the cell reaction. Such temperature variations, in turn, produce cell-to-cell voltage variations. This cell-to-cell variation was also numerically predicted by Burt et al. (2004a, 2004b, 2003a, 2003b) within a stack of five cells. Costamagna et al. (1994) reported differences in voltage output because of non-uniform distribution of the feeding gas along the planar fuel cell stack. Experimental studies conducted by Gubner et al. (2003) and Maggio et al. (1996) also revealed that cells in a stack do not necessarily operate uniformly. In addition, variations from cell to cell may be triggered by uneven distribution of the fuel and/or air flow among the cells. These variations could be critical for the safe and efficient operation of a stack and should be taken into account in numerical modeling. Thus it is important to model the stack as a set of thermally communicating cells with different fuel and air flow rates. A special issue with solid oxide fuel cell modeling is that there are two significantly different designs – Planar and Tubular. Figure 5.2 shows the difference between these two designs. Tubular design is developed by Westinghouse to overcome the problem of high temperature gas seals that are required for the planar solid oxide fuel cells. Planar fuel cell design is preferred for their relatively higher power density.

5.2.1 Effects of Radiation In SOFC modeling the effect of radiation is usually overlooked. We would like to particularly emphasize this important issue in cell/stack modeling. High temperature fuel cells (SOFCs and MCFCs) usually operate at temperatures in the range

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Fig. 5.2 Designs of Solid Oxide Fuel Cells (after Minh, 1995).

of 700–1200◦C utilizing a variety of fuels (i.e. hydrogen gas, hydrocarbons, and carbon monoxide) (Billingham et al., 2000; Yuan et al., 2003; Krotz, 2003). At these elevated temperatures thermal radiation emitted from the solid elements of the fuel cell may constitute a noticeable portion of the heat transfer within the stack. In the literature there are some studies (Hirata and Hori, 1996; Costamagna and Honegger, 1998; Achenbach, 1994b; Ma, 2000; Aguiar et al., 2002; Yakabe et al., 2000) in which radiation heat transfer was treated in various ways. However, in most other models, the treatment of radiation heat transfer was often neglected. Burt et al. (2003b, 2005) were one of the first groups to include radiation in their models. These studies showed that neglecting radiation will lead to prediction of a more non-uniform temperature distribution in large stacks. Aguiar et al. (2002) developed a 2-D model for the internal indirect reformer, and coupled it with a 1-D model for the SOFC. The SOFC model combined the porous anode and cathode electrodes with the electrolyte as a single solid structure (PEN, Positive electrode, Electrolyte, Negative electrode assembly). Aguiar et al. included radiation between the PEN and reformer using the assumption of two long concentric cylinders. Their results show that radiative heat transfer accounted for up to 79% of the total heat transfer between the solid structure and the reformer. Hirata and Hori (1996) consider radiative heat transfer between the PEN and the separator plate in a similar manner but for a MCFC stack. In Costamagna and Honeggar (1998) and Achenbach (1994b) radiation was considered between the stack and the surrounding shell as part of the boundary condition for the stack, but the radiation between individual PEN and separator plate was neglected. Yakabe et al. (2000) considered a single SOFC cell in a counter-flow configuration using a 3-D model. However, no radiation model was used, because the temperature was considered to be uniform everywhere in the cell. Virkar et al. (2000), like Yakabe et al. (2000), also used a uniform temperature in their study which focused on comparison of electrolyte vs. electrode supported cell and the impact of composite electrodes. Ma (2000) neglected radiation heat transfer


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effects because channels were considered to be thin and the cells were considered to be at nearly the same temperature. Recent work by Murthy and Fedorov (2003) and VanderSteen and Pharoah (2004) discuss radiation modeling in single solid oxide fuel cells. Murthy and Fedorov considered radiation heat transfer in a solid oxide fuel cell with a single rectangular fuel and air channel. They found that a comparison of results with and without inclusion of radiation showed a decrease of the temperature and smaller temperature gradients in the streamwise direction when radiation was included. The discrete ordinates (DO) method was found to be accurate but computational costs and memory requirements made the method not feasible. The authors then developed a simplified Rosseland/two-flux approximation resulting in a ten-fold reduction of CPU time. Agreement was good between both methods except for cases where increased optical thickness of the gas channel caused the approximations to fail. VanderSteen and Pharoah (2004) considered radiation heat transfer with and without participating gases in a single anode gas channel. The Monte Carlo approach was used where photons introduced at a source were tracked through multiple interactions with gas and surfaces until they lost sufficient energy. In their study 2 million photon trajectories were used. The Monte Carlo approach allowed for consideration of the participation of gases like CO2 and H2 O in the anode gas channel. Their study found that while radiation did significantly affect the overall temperature on the channel, the participation of the gases did not have any significant influence. The above brief literature review shows that the importance and the effects of radiation heat transfer have yet to be fully realized in cell/stack modeling.

5.2.2 Reduced Order Models versus Detailed CFD Models Throughout the literature many examples (e.g., Calise et al., 2005; Jiang et al., 2006; Murthy and Federov, 2003; Yuan et al., 2003; Gemmen et al., 2000b; Virkar et al., 2000; Yakabe et al., 2000; Costamagna and Honegger, 1998; Ferguson, et al., 1996; Achenbach, 1994b) have been found where specialized computational models were developed for the simulation of fuel cells. Some of these models were developed to facilitate fuel cell studies as computationally efficient as possible. Such simplicity in modeling is imperative for simulation of multi-cell stacks. In order to do this, some assumptions were made to reduce the order or complexity for which some governing equations or specialized models are solved. These less complex approaches result in what can be described as a reduced order method or model (ROM). The simplest case of a ROM would be lumped models or zero-dimensional models where the fuel cell is modeled as a single set of control volumes: one for each component, e.g. air gas channel, fuel gas channel, PEN, interconnect, etc. (see for example Elizalde-Blancas et al., 2007a). This hides most of the details of what occurs inside the fuel cell but allows for fast simulation times. Lumped models are appropriate for use in system modeling applications where the fuel cell interacts with other devices such as heat exchangers, combustors, turbines, etc. This kind

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of application needs to capture the general operating behavior of the fuel cell and requires the computation to be done rather quickly. One-dimensional models result from simplifying the fuel cell such that variations are accounted for along a single selected direction. The NETL 1-D planar single cell model (Gemmen et al., 2000a) considered the variations in the streamwise direction of the cell. Thus variables such as species concentration, temperature, flow velocity, etc., were allowed to vary as a function of distance along the channels. One-dimensional models might also be stack models where each cell is modeled in a 0-D manner (see for example Elizalde-Blancas et al., 2007a). Following the same logic a 2-D model can be developed by considering variations only within a plane thus neglecting changes in the third direction. An example of this kind of model is the Pseudo 2-D Fuel Cell Stack Model presented in Burt et al. (2004a, 2004b, 2003a, and 2003b) which was an extension of a 1-D planar single cell model into a stack model hence a two-dimensionality was provided. It is still referred to as a pseudo 2-D model because the model does not account for 2-D variations within individual components. Pseudo 2-D models are extension of 1-D models to 2-D; whereas pseudo 3-D models are essentially 3-D but with some 1-D or 2-D approximations. Detailed CFD models of fuel cells (see Chapters 3 and 4), on the other hand, use continuum assumption to predict the 3-D distributions of the physical quantities inside the fuel cells. These models are more complex and computationally expensive compared to reduced order models especially due to the disparity between the smallest and largest length scales in a fuel cell. The thickness of the electrodes and electrolyte is usually tens of microns whereas the overall dimensions of a fuel cell or stack could be tens of centimeters. Though some authors used detailed 3-D models for cell or stack level modeling, they are mostly confined to component level modeling. In what follows, we present the governing equations for some of these models.

5.2.3 One-Dimensional Models One-dimensional models usually treat the fuel cell as a set of layers (e.g. interconnect, air channel, electrodes, electrolyte and fuel channel) and estimate the variations of the physical quantities inside each of these layers along the flow directions. Here a 1-D model developed by Gemmen et al. (2000a) and extended by Burt et al. (2003a, 2005) is presented. Four layers are considered in this model viz. interconnect, air channel, PEN (Positive electrode Electrolyte, Negative electrode) assembly and fuel channel. Each layer was further divided into control volumes. The variations in the streamwise (x) direction are explicitly calculated, those in the vertical (y) direction are accounted for via integral approximations, and those in the transverse (z) direction are ignored. Figure 5.3 depicts the control volume approximation used for mass conservation of a gas channel and is similarly defined for the other conservation equations. Figure 5.4 shows thermal fluxes and source terms relevant for a typical PEN control volume. Each control volume of the fuel and air gas chan-


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Fig. 5.3 Gas channel control volume for mass conservation.

Fig. 5.4 Electrolyte control volume for energy conservation.

nels is required to satisfy the governing equations for mass, momentum, and energy. Energy and electrochemistry equations are solved for control volumes in the PEN. The following governing equations for mass, momentum, and energy were solved in the air and fuel channel: ∀

∂ρ + (ρuAxs)w − (ρuAxs)e = m ˙ surf , ∂t

∂(uρ) + (uρuAxs )w − (uρuAxs )e = Fx , ∂t d(eρ) ∀ = (eρuAxs )w − (eρuAxs )e = Q˙ conv , dt where m ˙ surf = (m ˙ xlw )s − (m ˙ xlw )n . Specie mass conservation was satisfied using ∀

∀∂(ρYk ) + (ρYk uAxs )w − (ρYk uAxs)e = ω˙ k xlw . ∂t

(5.1) (5.2) (5.3)

(5.4)

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In Equations (5.1–5.4) it is assumed that changes in the x-direction are small therefore diffusion terms are neglected. Note that this assumption may not always be valid. The energy equation (5.3) is used to determine the temperature, and the current density is determined by an electrochemical model (Gemmen et al., 2000a). The current density and temperature are used to calculate appropriate fluxes which are introduced as source (or sink) terms for each of the conservation equations. The molar flux of a given species k is obtained from the current density using: ω˙ k =

−iden , nk F

(5.5)

where nk is the number of electrons per mole of reactant k. The PEN and separator plate are considered to be made of solid material; therefore only the energy equation (that essentially reduces to the heat conduction equation) was solved in these regions which was simplified from Equation (5.3) to: ∀

d(eρ) = Q˙ conv + Q˙ rad + Q˙ mass + Q˙ gen = Q˙ net + Q˙ gen . dt

(5.6)

The radiative and convective heat flux through the surface of the control volume, and the thermal energy transported by mass-flux, are all included in Q˙ net (see Equation (5.1) and Figure 5.4), and the heat source, Q˙ gen , is obtained from ohmic heating and heat associated with change of entropy resulting in the following expression: ˙ gen = (iden )2 R + T s ω˙ H2 . Q

(5.7a)

The total entropy change per mole, s is obtained from s = ¯s 0 + Rg ln

rR , rP

(5.7b)

where Rg is the ideal gas constant, ¯s 0 is the change in entropy per mole of reactant at standard conditions, and rR and rP are the reactant and product activities respectively. Convection and radiation heat fluxes are calculated using the equations: Q˙ conv = hconv A(Ts − Tb ),

(5.8)

Q˙ rad = AF12 (ε1 σ T1 − ε2 σ T2 ).

(5.9)

In Equation (5.8), hconv is the convection heat transfer coefficient, A is the surface area, Ts is the solid surface temperature and Tb is the bulk fluid temperature. In Equation (5.9), F12 is the view factor from surface 1 to 2, σ is the Stefan–Boltzman constant, ε1 , ε2 are emissivities, T1 , T2 are temperatures of surfaces 1 and 2 respectively. Pressure, P , is calculated from the ideal gas law: P = ρRg T .

(5.10)


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The electrochemistry model is based on the assumption that the overall chemical reaction occurring in the fuel cell is: 1 H2 (g) + O2 (g) → H2 O(g). 2

(5.11)

Calculation of the cell potential starts with the Nernst Equation which considers the mole fraction of the H2 , O2 , and H2 O species: Rg T Rg T P [XH2 ][XO2 ]1/2 ln + ln 0 . (5.12) E = E0 + 2F [XH2 O ] 4F P The pressure is assumed to be the same for both the anode and cathode gas channels. The reversible potential at standard state conditions is obtained from the change in the standard Gibbs free energy. E0 = −

G0 . nF

(5.13)

The corrected cell potential, Ecor , is obtained by subtracting the ohmic (ηohm ), concentration (ηconc ), and activation (ηact ) losses (i.e. overpotentials) from the ideal Nernst potential, E: (5.14) Ecor = E − ηohm − ηconc − ηact . The overpotentials are related to the current density. The activation over-potential is defined by an empirical relation represented by a limiting form of the Butler–Volmer equation: ηohm = iden Rnet , (5.15) Rg T iden ln 1 − , (5.16) ηconc = − nF iL Rg T iden ln . (5.17) ηact = nαF i0 A quasi-steady gas flow approximation was used whereby the gas flow was determined from empirical steady state relations; e.g., a steady state friction coefficient equation. This allowed large time steps to be used with the time marching scheme to reach a steady state solution. More details about the mathematical model can be found in previous work (Burt et al., 2003a, 2003b, 2004a, 2004b; Gemmen et al., 2000a).

5.2.4 Two-Dimensional Models As mentioned previously, fuel cells are intrinsically three dimensional in that significant variation of key cell parameters are to be expected in all three spatial dir-

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ections. However some simplifying assumptions can be made in case of co-flow or counter flow planar fuel cells, and also for the case of tubular fuel cell of the kind designed by Siemens–Westinghouse. In case of the planar fuel cells it is assumed that the changes that occur in the gas flow direction and in the direction normal to the gas-flow, i.e. the primary current flow direction are much larger compared to those in the transverse direction normal to the gas flow direction. In such situations, the governing equations can be reduced to two-dimensions thus greatly simplifying the analysis. The tubular design (see Figure 5.1b) also lends itself to two-dimensional (strictly speaking 3-D axi-symmetric) analysis if the changes in the circumferential direction can be neglected. Examples of such models can be found in Burt et al. (2004a, 2004b), O’Hayre et al. (2006) and others for planar cell applications, and Nishino et al. (2006), Sanchez et al. (2006), Li and Suzuki (2004) and others for tubular cell applications. Since most fuel cells are intrinsically three dimensional and the 2-D models are a sub-set of pseudo 3-D models we shall not dwell on these in much detail.

5.2.5 Pseudo 3-D Fuel Cell Models We define this category of models as those that treat the thermal analysis of solid regions (including the porous electrodes) as three dimensional and the fluid regions (air and fuel channels) as essentially one dimensional. One-dimensional fluid models are those that treat the flow in the channels as essentially plug flow which changes only in the primary flow direction along the channels. Pseudo 2-D models are those that attempt to include some effects of flow variation over the cross-section of the channels by assuming some profile shape, such as the parabolic velocity profile for laminar, fully developed channel flow. These are by far the most commonly employed models in analysis of cell/stack design and development. They provide the optimum information but at the same time are relatively simple to implement. Note that in such models convection in the porous electrodes is usually neglected, but it can be included using a simplified version of Darcy’s law (see Section 5.3). Furthermore, the PEN (positive electrode, electrolyte, and negative electrode) is sometimes lumped into one single layer, and sometimes the electrolyte is simply represented as an interface (see e.g FLUENT model presented in Section 5.3). Applications of such models can be found in publications such as Achenbach (1994b), Ferguson et al. (1996), Masuda et al. (2006), and Suzuki et al. (2006). An example of a pseudo 3-D model following the model of Celik et al. (2003) and Pakalapati (2003, 2006) and Pakalapati et al. (2006) will be presented here. This model is applicable to an SOFC. However, those for other types of cells, such as the PEMFC, are very similar except that a careful account of the liquid must be included either by solving extra transport equations for the liquid phase or by some empirical/algebraic relations that essentially keeps track of the liquid content in the cell. The details of the full 3-D transient governing equations for fuel cells can be found in Chapter 3. These equations also include the transient terms. Most models


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in the literature target modeling of the steady state operation of the stacks. However, under start-up and shut down conditions or during variations in load following an increase or decrease in the demand some significant transients can occur (see e.g. Gemmen et al., 2004). Then, the time dependent terms must also be included in the governing equation being solved. Even in a highly variable transient situation, it can be shown that the time scales for fluid dynamics are much shorter than those for thermal processes, i.e. changes in temperature. Thus assumptions are made in many cases that only the transients in thermal behavior are important as response to changes in the total load and/or voltage demanded from a given system. When transients are important, appropriate precaution should be given to the time- marching scheme being used. It should be stable and robust so that large time steps can be used to reach steady state operation in a reasonable execution time. The model described in this section does retain the time derivatives, and the numerical scheme used is designed to account for both accuracy and robustness. The model presented here is a combination of a 1-D model for gas channels and a 3-Dmodel for solid and porous regions. The species transport equation is written as: ∂ j j j j j eff (εp ρp xp ) + ∇ · (εp ρp ueff p xp ) = ∇ · (εp p ∇xp ) + εp ρp Sp + asp fI,p . (5.18) ∂t The above equation is written for the gas phase of the porous regions in terms of j the mass fraction of j th species in pore (gas) phase p, xp . The first term on the right hand side represents the total diffusion resulting from concentration gradients. The effective diffusion is modified such that the diffusion term includes molecular j diffusion terms as well as the Knudsen diffusion term. The source term, Sp includes j chemical reaction rates, Schem (i.e. mass source or sink per unit mass) due to ionization or other chemical reactions. The convection inside the pores is neglected. Due to the electrode reactions involving ions at the electrolyte-electrode interface, the interface transfer term, should include the transfer of ions through the active surfaces. According to Grens and Tobias (1964), the interfacial source term can be calculated as: Mj υj s j ρk fI,k = i , (5.19) nF k where υj is the stoichiometric coefficient of the j th species in the electrochemical reaction, F is the Faraday constant, n is the number of electrons involved in the reaction, Mj is the molecular weight, and iks is the interfacial current which is determined from electric potential field. The energy equation is: ∂T ∂ ∂ (ρCp T ) = k + ρSh . (5.20) ∂t ∂x ∂x Inside the porous regions the temperature is assumed to be the same in gas and solid regions (local equilibrium). The source term includes ohmic heating, which is distributed throughout the current conducting regions and heat produced due to the

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Fig. 5.5 Control volume used for 1-D gas channel model.

electrochemical reactions near the active electrolyte/electrode interfaces. m m Sh = Selec + Sechem .

(5.21)

The ohmic heat source (Ferguson, 1996) is given by: m sechem = σpeff ∇ϕp · ∇ϕp ,

(5.22)

m is the algebraic sum of heat where σpeff is the effective electric conductivity. sechem released from all electrochemical reactions taking place in the continuum. In the present chapter, the heat due to electrochemical reactions is assumed to be produced at the anode/electrolyte interface. Inside each computational cell at this interface, the heat source term is taken proportional to the current density and thus the amount of hydrogen used; the relationship is given by: m sechem =m ˙ reac H2 T Sreac .

(5.23)

The electric current equation applicable in solid regions is given by: ∇ · I = ∇(σpeff ∇ϕ) = s.

(5.24)

The electric potential ϕ is assumed to be continuous throughout the electrodes and electrolyte except at the electrode/electrolyte interfaces. These discontinuities are usually modeled by Nernst’s law. The model to calculate the potential jumps at each electrolyte/electrode interface is described in Celik et al. (2005). The source term in Equation (5.24) is non-zero only near the electrode/electrolyte interfaces to account for the potential jumps. For the 1-D gas channel model, the specie, temperature, and velocity distributions inside the gas channels are assumed to be varying only in the direction of gas flow. The control volume employed, shown in Figure 5.5, encompasses the whole


139

cross-section of the gas channel in x- and y-directions (x-direction is normal to the plane of the paper) and is one grid length deep in z-direction. Let φ be the crosssection averaged value of a conserved scalar expressed in per unit mass basis. The conservation equation for φ can be written as ∂ ∂ ∂ ∂φ ˙ net (ρAφ) = − (Aρuφ) + A − Q˙ nd (5.25) φ + ASφ , ∂t ∂z ∂z ∂z where A is the cross-sectional area of the channel, u is the velocity, is the effective diffusion coefficient, Q˙ nd φ is the normal flux (across the channel walls) of the scalar, and S˙φnet is the net source. The conservation equation for any particular scalar can be derived from Equation (5.25) by substituting appropriate variables, constants, and expressions. In the present case, the normal flux of the scalar, Q˙ nd φ , is calculated from the three-dimensional solution inside the solid and porous regions as = εp q˙φnd da. (5.26) Q˙ nd φ wall surface

The mass conservation equation for a gas channel can be obtained by substituting φ = 1 in the general scalar transport equation, Equation (5.25), resulting in: ∂(ρAu) ∂(ρA) ˙ nd =− −Q m. ∂t ∂z

(5.27)

For the specie conservation equation, the general scalar φ is replaced by the specie mass fraction Xs in Equation (5.25). ∂(ρAXs ) ∂(ρAXs ) =− − Q˙ nd s . ∂t ∂z

(5.28)

Here, the diffusion in the direction of flow is neglected as the transport process is dominated by convection. This may not be a valid assumption for relatively short channels. The normal diffusion flux of the specie, Q˙ nd s , is calculated using Equation (5.26) where q˙snd is given by q˙snd = Ksc (Xs − Xsw ).

(5.29)

In Equation (5.29), Ksc is the mass transfer coefficient between the gas channel and the porous electrode surface (channel wall) and Xsw is the mass fraction of the specie inside the porous electrode near the surface. The total normal flux of mass into the porous electrode, as used in Equation (5.26), is given by ˙ nd Q˙ nd (5.30) Q m = s . all s

The momentum equation can be obtained by substituting velocity for the generic scalar in Equation (5.25),

140


Cf ∂ dP ∂(ρAu) = − (ρAu2 ) − Q˙ nd − (1 − εp )B ρ|u|u, mu+A ∂t ∂z dz 2

(5.31)

where Cf is the friction factor. Note that the factor (1 − εp ), which represents the fraction of wall surface with solid interface, takes into account the area on the surface of the channels, where there is suction or injection. The source terms in Equation (5.31) are the contributions from pressure gradient and friction losses. Also, note that the viscous diffusion in the flow direction is neglected. For the energy equation, the generic scalar in Equation (5.25) is replaced by enthalpy using the ideal gas approximation, dh = Cp dT . ∂Cp T ∂ ∂ ∂ (ρACp T ) = − (ρAuCp T ) + A ∂t ∂z ∂z ∂z − Q˙ nd s Cps T − (1 − εp )Bhconv (T − Tw ) all s

+A

Cf DP + (1 − εp )B ρ|u|u2 . Dt 2

(5.32)

Here, Cps is the constant pressure specific heat of the particular specie, s, and hconv is the convection heat transfer coefficient between channel and wall. The source terms in Equation (5.32) are the contributions of convection heat transfer to walls, pressure work and frictional heating. Again, here thermal diffusion in the axial direction resulting from mass diffusion is neglected. Equations (5.18) through (5.32), along with the electrochemistry model completely describe the transient operation of a fuel cell. When solved simultaneously, they produce 3-D distributions of scalars inside the solid and porous regions and 1-D variations inside the channels.

5.3 Fully Three-Dimensional Models In contrast to the pseudo 3-D models, truly multi-dimensional models use, in general, finite element or finite volume CFD (Computational Fluid Dynamics) techniques to solve full 3-D Navier–Stokes equations with appropriate modifications to account for electrochemistry and current distribution. The details of electrochemistry may vary from code to code, but the current density is calculated almost exclusively from Laplace equation for the electric potential (see Equation (5.24)). Inside the electrolyte, the same equation represents the migration of ions (e.g. O= in SOFC), elsewhere it represents the electron/charge transfer. In what follows, we briefly summarize a commonly used multi-dimensional model for PEM fuel cells because of its completeness and of the fact that it also addresses most essential features of SOFC modeling. The example is the fuel cell module of commercial CFD software FLUENT. As was already mentioned, it is a set of additional subroutines (on top of Navier–Stokes equations and other transport models in CFD) to account for electrochemistry inside


141

a fuel cell and its influence on heat and mass transfer (see FLUENT Manual for more details). In this model two electric potential equations are solved: one (Equation (5.33)) is for the electric potential inside solid and (solid phase of) porous regions that causes the electron transport in these regions and one (Equation (5.34)) for the electric potential in the electrolyte (membrane) that enables the transport of protons (oxide ions in SOFC). ∇ · (σs ∇ϕs ) + Rs = 0,

(5.33)

∇ · (σm ∇ϕm ) + Rm = 0.

(5.34)

Here the subscripts ‘s’ and ‘m’ denote solid (electrodes and current collectors) and membrane (electrolyte) respectively. Note that these two equations can be treated as only one equation with variable σ and source terms. The R’s are the volumetric transfer currents due to electrochemical reaction which are non-zero only in the catalyst layers and can be calculated from the Butler–Volmer equation for anode and cathode sides as: −αc ηa F [H2 ] γan αa ηa F a Ra = iref − exp , (5.35) exp [H2 ]ref RT RT −αc ηc F [O2 ] γcut αa ηc F c Rc = iref + exp , (5.36) − exp [O2 ]ref RT RT Here subscripts ‘a’ and ‘c’ denote anode and cathode respectively, iref is the reference exchange current density, γ is the concentration dependence exponent, [ ] and [ ]ref represent the local species concentration and its reference concentration, respectively. Anode transfer current, Ra , is the source in the electric potential equations at the anode/electrolyte interface with positive sign on membrane (electrolyte) side and negative sign on solid (anode) side. Similarly, near the cathode interface, the source on membrane (electrolyte) side is negative of the cathode transfer current, Rc and that on solid (cathode) side is positive of Rc . The activation over-potentials, in Equations (5.35) and (5.36) are given by ηa = ϕs − ϕm ,

(5.37)

ηc = ϕs − ϕm − EN .

(5.38)

The source terms in species conservation equation due to the electrochemical reactions also need to be added near the active interfaces which are given in terms of the transfer currents as MH2 Ra , (5.39) SH2 = − 2F MO2 SO2 = − Rc , (5.40) 4F MH2 O SH2 O = Ra . (5.41) 2F

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Similarly, the source term in the energy equation is sum of the electrical resistance heating, heat of formation of water, electrical work, and heat release due to phase change (condensation of water vapor). Sh = I 2 Rohm + hreac + (ηa Ra + ηc Rc ) + hphase .

(5.42)

To model the condensation and transport of the liquid water, the following governing equation in terms of volume fraction of liquid water, s, is solved. ∂(ερl s) + ∇ · (ρl Vl s) = rw . ∂t

(5.43)

Here the subscript ‘l’ denotes liquid water. The rate of condensation, rw is calculated from: Pwv − Psat MH2 O , [−sρl ] . (5.44) rw = cr max (1 − s) RT Here, cr is the condensation rate constant, Pwv is the partial pressure of water vapor, Psat is the saturation pressure, MH2 O is the molecular weight of the water vapor. A corresponding source term has to be added to water vapor conservation equation and pressure correction equation (mass source). The liquid water velocity is assumed to be same as the gas velocity inside the gas channel. However, inside the porous region, the convection term is replaced by capillary diffusion term and the equation becomes Ks 3 dPc ∂(ερl s) + ∇ · ρl ∇s = rw . (5.45) ∂t µ1 ds Here, Pc is the capillary pressure which is a function of s, permeability K, and surface tension. Equations (5.33–5.45) along with Navier–Stokes Equations and species equations constitute a fully 3-D description of a PEMFC. When the membrane is replaced by a solid, non porous electrolyte conducting oxide ions instead of protons, the above model essentially becomes a model for SOFCs. In an SOFC, of course, there is no condensation; hphase = 0 and Equations (5.43–5.45) would not be necessary.

5.4 Internal Reforming Cell and stack modeling of SOFCs will not be complete without introducing some basic internal reforming concepts. When methane is used as a fuel in SOFCs, it is first reformed by reacting with steam to produce carbon monoxide and hydrogen which are then electrochemically oxidized. Steam reforming of methane is an endothermic reaction and can be performed either externally or internally to the SOFC stack. In external reforming an independent reactor is used for steam methane reforming reaction, which needs heat input in addition to the reactants. In internal reforming the reforming is carried out inside the SOFC stack where the waste heat


143

produced during the electrochemical oxidation is used to sustain the endothermic reforming reaction. However, careful thermal management is required to avoid under cooling of the stack. Internal reforming could either be done directly inside the SOFC stack on the anodes or in a separate reactor attached to the stack, using the heat from the stack itself. The former is known as direct internal reforming and the latter is indirect internal reforming. Also some times the fuel fed to the SOFCs is partially pre-reformed and the rest of the reforming is achieved inside the stack. To model direct internal reforming in SOFC stacks additional source terms need to be added to the specie and energy equations in addition to including the equations for the extra species. The source terms account for a set of chemical reactions, the chemical mechanism, that is assumed to occur. A simple and commonly used mechanism for steam methane reforming on SOFC anode is: CH4 + H2 O ⇐⇒ CO + 3H2 ,

(5.46)

CO + H2 O ⇐⇒ CO2 + H2 ,

(5.47)

=

CO + O ⇐⇒ CO2 ,

(5.48)

H2 + O= ⇐⇒ H2 O.

(5.49)

Equations (5.46) and (5.47) are referred to as steam methane reforming (SMR) reaction and Water Gas Shift (WGS) reaction respectively. These reactions are homogenous reactions that occur everywhere inside the anode, whereas Equations (5.48) and (5.49) only occur at the active triple phase boundary. Treatment of source terms due to electrochemical oxidation of H2 (Equation 5.49) is already covered in the previous sections and the treatment is similar for electrochemical oxidation of CO (Equation 5.48). Specie source terms due to homogenous reactions in the mechanism are given by: f

b s˙CH4 = −kSMR [CH4 ][H2 O] + kSMR [CO][H2 ]3 ,

(5.50)

f

b [CO][H2 ]3 s˙H2 O = −kSMR [CH4 ][H2 O] + kSMR f

b − kWGS[CO][H2 O] + kWGS [CO2 ][H2 ],

(5.51)

f

b [CO][H2 ]3 s˙CO = kSMR [CH4 ][H2 O] − kSMR f

b − kWGS [CO][H2 O] + kWGS [CO2 ][H2 ],

(5.52)

f

b [CO][H2 ]3 s˙H2 = 3kSMR [CH4 ][H2 O] − 3kSMR f

b + kWGS [CO][H2 O] − kWGS [CO2 ][H2 ], f

b s˙CO2 = kWGS [CO][H2 O] − kWGS [CO2 ][H2 ].

(5.53) (5.54)

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Here k denotes reaction rate constant, superscripts f and b stand for forward and backward reactions respectively, subscripts SMR and WGS stand for steam methane reforming and water gas shift reactions respectively, and the square brackets represent the mole fraction of the specie. The heat sources due to these reactions, to be added to the energy conservation equation are given by: q˙SMR = s˙CH4 HSMR ,

(5.55)

q˙WGS = s˙CO HWGS .

(5.56)

Here HSMR denotes the change in enthalpy per mole of CH4 during steam methane reforming reaction and HWGS denotes the enthalpy change per mole of CO during water gas shift reaction. Here, we note that the issue of partitioning the total current produced between H2 and CO oxidation reactions (Equations (5.48) and (5.49)) is not a fully resolved problem (see e.g. Suwanwarangkul et al., 2006; Aguiar et al., 2002; Gemmen and Trembly, 2006; Nishino et al., 2006).

5.5 Zero-Dimensional Model Finally, a steady 1-D lumped stack model is introduced which uses a 0-D lumped approach for each cell in the stack. The model takes the current and power produced by each cell in the stack as input and predicts the 1-D temperature distribution across the cells of the stack. Such models have the advantage of faster calculation time and are thus better suited for initial design calculations and control system modeling. In this model, each fuel cell is divided into three components, air channel, fuel channel and solid region (electrodes, electrolyte and the interconnect). The control volumes used for air and fuel channel components are shown by the dashed lines in Figure 5.6. The specie concentrations at the exit of air and fuel channels could be calculated using the mass and specie balances for these control volumes which are in the form prod m ˙ in xsin − m ˙ used +m ˙s s xsout = , (5.57) m ˙ out prod ˙ in − m ˙ used + m ˙s . (5.58) m ˙ out = m s all s

all s

Here m ˙ in and m ˙ out denote the mass flow rate of the mixture entering from the inlet and leaving from the outlet respectively. Rate of consumption and rate of prod and m ˙ s . These rates include production of each species ‘s’ is denoted by m ˙ used s the flux of reactants, which take part in electrochemical reactions, across the channel/electrode interfaces and also the consumption and production of species due to methane reforming reaction on the anode side. Both hydrogen and carbon monoxide electrochemistry was considered and it was assumed that αH2 , the fraction of the current that is produced from H2 oxidation, is known. Thus the specie consump-


145

Fig. 5.6 Control volumes and flux and source assumptions used for the 0-D cell model.

tion/production due to electrochemical reactions can be calculated as: m ˙ used H2 =

αH2 I MH2 ; 2F

prod

m ˙ H2 O =

αH2 I MH2 O ; 2F

m ˙ used CO =

(1 − αH2 ) MCO ; 2F

(1 − αH2 )I 1 MCO2 ; m MO2 . ˙ used (5.59) O2 = 2F 2F Here I is the total current, F is the Faraday constant, M is the molecular weight. The energy equations for the air and fuel control volumes shown by dashed lines in Figure 5.6 are given by: prod

m ˙ CO2 =

Caout Taout + COout2 Ta = Cain Tain + Q˙ a ,

(5.60)

˙ f − Q˙ r = 0. Tc − Ceused Tf + Q (5.61) Here, C denotes the heat capacities, i.e. the product of mass flow rate times corresponding specific heat, T denotes temperature, and Q˙ denotes rate of heat transfer or generation. Subscripts a, f , c, r, e denote air channel, fuel channel, cell (solid reprod

Cfin Tfin − Crused Tfin + Cr

prod

Tfin − Cfout Tfout + Ce

146


gions), reforming reaction and electrochemical reactions respectively. Superscripts in, out, used, and prod denote inlet, outlet, used (consumed) and produced respectively. The underlined terms take into effect the contribution of the reforming reaction to the energy equation. A more detailed derivation of Equation (5.61) is given ˙ f are calculated as the fractions in Elizalde et al. (2007b). Heat sources Q˙ a and Q of net heat given off to the gas channels by the solid regions of the cell. Q˙ a = α Q˙ net c ,

(5.62)

Q˙ f = (1 − α)Q˙ net c ,

(5.63)

˙ net where Q c is the summation of the heat generated due to the electrochemical reactions, net heat loss due to convection from the side walls (also from the top or ˙ net bottom walls in case of the end cells), Q conv and net conduction loss to the adjacent net cells, Q˙ cond ; i.e. ˙ gen ˙ net ˙ net (5.64) Q˙ net c = Qc − Qconv − Qcond , where

Q˙ net cond =

i=neighbor cells

kA (Tc − Ti ), l

Q˙ net conv = hconv Aext (Tc − T∞ ), ˙ gen Q c

= βP .

(5.65) (5.66) (5.67)

Here, k is the effective thermal conductivity, A is the effective contact area between the adjacent cells, l is the characteristic conduction length scale, hconv is the convection heat transfer coefficient, Aext external surface area of the cell exposed to the ambient air, T∞ is the ambient temperature and P is the cell power. The characteristic conduction length is calculated as the volume of the bipolar plate divided by the cell normal area. Factor β is an empirical constant which is the ratio of the heat generated to the power produced by the cell, i.e. (1 − η), η being the efficiency. When radiation is considered, Q˙ net rad should be included in Equation (5.64). The heat transfer relationships between the gas channels and the solid regions are given by: hconv,a Aa (Tc − Ta ) = Q˙ a , (5.68) hconv,f Af (Tc − Tf ) = Q˙ f .

(5.69)

In the above equations, hconv,a , Aa , hconv,f , Af , are the overall convective heat transfer coefficients and interface areas respectively between the cell and air or fuel channels. Finally, following relations are used for the average air channel and fuel channel temperatures out T in + 2Tc + Tair Ta = a , (5.70) 4 Tf =

Tfin + 2Tc + Tfout 4

.

(5.71)


147

Substituting Equations (5.68) and (5.70) into Equation (5.60) and simplifying yields ˙a Cain Tain + Q˙ a + Caout Tain + 2 Q h∗a Ta = . (5.72) out out 2Ca + CO2 Equations (5.68–5.72) and (5.61) form a set of simultaneous equations for the unknown temperatures, Tc , Ta , Taout , Tf , Tfout, and the heat distribution factor α for one cell of the stack. Writing similar equations for all cells in the stack will result in a larger system of simultaneous equations. The equations for neighboring cells are coupled through heat conduction terms. Cell power, voltage, heat generation factor, utilizations of hydrogen and methane and inlet temperatures and concentrations of fuel and air for each cell are the input parameters for the model.

5.6 Applications of Stack Modeling 5.6.1 Pseudo Two-Dimensional Model Example Modeling an entire fuel cell stack at sufficient grid resolution and reasonable computational time is a challenge. One approach to this challenge is to run parallel calculations with the help of say Beowulf computer clusters. In this regard a plausible strategy is to parallelize the computation by domain decomposition such that each cell in the stack runs in a separate processor. This is the approach taken by Burt et al. (2003a, 2003b, 2005) from which we present some examples in this section. The first set of results presented in this section is obtained for co-flow configuration using the Pseudo 2-D SOFC Stack Model (Burt et al., 2003a, 2003b; Gemmen et al., 2000a), based on the 1-D model introduced in Section 5.2.3. The computational approach models each fuel cell using a simplified transient single cell model where the variations in the stream-wise direction are accounted directly, thus making it a pseudo 2-D model in the stream wise and stacking directions when more than one cell is simulated. The single fuel cell model treats the fuel cell as four distinct layers namely interconnect, air, fuel and PEN (Positive electrode Electrolyte Negative electrode) assembly. The fuel-cell stack simulation was accomplished by dividing the stack into computational domains using domain decomposition with each cell being treated as a separate process (see Figure 5.7) on a distributed commodity PC-based Beowulf computer cluster. Communication between domains or processes was accomplished using the Message Passing Interface (MPI) library. The necessary temperatures, time step, and termination bit were communicated using MPI library calls. Simulations were performed for stacks with a various number of cells in order to study the effect of the stack size on cell to cell variations. The details of geometry, properties and operating conditions of the single cell under consideration are given in Tables 5.1 and 5.2. Figure 5.8 depicts temperature profiles in the vertical direction

148


Fig. 5.7 Domain decomposition for a fuel cell stack where each cell is treated as an individual process on a separate computer processor. Table 5.1 Physical dimensions of single fuel cell with anode supported electrolyte. SOFC Component

[m]

Cell Length Grid Length, x Fuel Gas Channel Height, y Air Gas Channel Height, y Electrolyte Thickness, y Anode Electrode Thickness, y Cathode Electrode Thickness, y Separator Thickness, y

1.0E-01 5.0E-02 1.0E-03 3.0E-03 1.0E-05 1.0E-03 2.5E-05 7.5E-04

near the middle of the stream wise direction (at x/L = 0.55, i.e. node 10) for a 5, 20 and 40 cell stacks. Adiabatic boundary conditions were imposed at the top and the bottom of the stack. The cell to cell variation in the temperature seen in Figure 5.8 is mainly due to the intrinsic asymmetry of the fuel cell operation principle with more cooling from the bottom air channel and less cooling from the top fuel channel. Due to this asymmetry, the bottom cell is the coldest and the top cell is the hottest in the stack. The temperature variation is nearly linear for small five cell stacks; however, for the larger 20 and 40 cell stacks, interior cells appear to have nearly uniform temperature with cells towards the top and bottom being influenced by the top and bottom cells. Variations in temperature, like those observed near the top and bottom of the stack in Figure 5.8c, can cause significant thermal stresses. Figure 5.9 depicts the cell to cell voltage variation for 5 cell and 20 cell stacks. Similar trends are observed in the cell voltages as were seen in the temperature profiles. A relatively small cell-to-cell voltage variation of 0.7% and 1.1% was ob-


149

Table 5.2 Material properties and model parameters (increasing stack size case). Cell Heat Capacity [J kg−1 ·K−1 ] Cell Density [kg/m3 ] Separator Heat Capacity [J kg−1 ·K−1 ] Separator Density [kg/m3 ] No. Axial Nodes Anode Inlet Temperature [K] Anode Inlet Pressure [Pa] H2 Anode Inlet Mole Fraction H2 O Anode Inlet Mole Fraction Cathode Inlet Temperature [K] Cathode Inlet Pressure [Pa] O2 Cathode Inlet Mole Fraction N2 Cathode Inlet Mole Fraction Contact + Separator Resistance Limiting Current Density [A/m2 ] Exchange Current Density[A/m2 ]

8.00E+02 1.50E+03 4.00E+02 8.00E+03 20 1073 1.01E+05 9.70E-01 3.00E-02 1073 1.01E+05 2.10E-01 7.90E-01 1.0E-01 4.0E+03 5.5E+03

served for the 5 cell and 20 cell stacks respectively. Of interest however is the nearly linear variation observed for the 5 cell stack whereas for the larger 20 cell stack an asymmetric profile was obtained where the cell voltage was influenced by the top and bottom cells. This trend was also observed by Lin et al. (2003). It is normally assumed that interior cells will have nearly uniform performance which is an appropriate assumption for this case given the relatively small variation observed in cell voltage. However, it should be noted that in these simulations the same reactant flow rate is prescribed for all the cells in the stack whereas in reality the inlet flow rates will be different for each cell which will amplify the cell to cell variations. Figure 5.10 depicts cell to cell voltage variations observed in a 20 cell stack with non-uniform fuel flow distribution. A 7% variation in cell voltage was observed when 20% of the fuel flow was taken from the bottom cell (cell 0) and added to the neighboring cell (cell 1). This non-uniform distribution of fuel flow resulted in negligible changes in the temperature profile. Thermal stress calculations in the five cell stack for the temperature distribution presented above were performed by Valluru (2005) using the solid modeling software ANSYSTM . The stack is modeled to be consisting of five cells with one air channel and gas channel in each cell. Two dimensional stress modeling was performed at six different cross-sections of the cell. The temperature in each layer obtained from the above model of Burt et al. (2005) is used as the nodal value at a single point in the corresponding layer of the model developed in ANSYSTM and steady state thermal analysis is done in ANSYSTM to re-construct a two-dimensional temperature distribution in each of the cross-sections. The reconstructed two dimensional temperature is then used for thermal stress analysis. The boundary conditions applied for calculations presented here are: the bottom of the cell is fixed in y-direction (stack direction), the node on the bottom left is fixed in x-direction (cross flow direction) and y-direction and the top part is left free to

150


Fig. 5.8 Temperature profile along the thickness of the stack at x/L = 0.55 (node 10) for an average current density of 667 mA/cm2 .

expand. This condition is more or less realistic since the SOFC stacks are usually fixed at the bottom and allowed to expand freely towards the top of the stack.


151

Fig. 5.9 Cell voltage variation a stack, normalized with the highest cell voltage of 0.70 V for an average current density of 667 mA/cm2 .

Contours of maximum principal stress in the first slice (near the gas inlets) and the sixth slice (near the gas outlet) are shown in Figures 5.11 and 5.12 respectively. It can be seen that the stack is partially under compression and partially under tension due to the mismatch in the thermal expansion coefficient of the materials and nonuniform temperature. In each cross-section, the stresses are higher near the top of the stack than near the bottom. Also, the stresses are higher near the gas outlet than near the gas inlets. Maximum tensile and compressive stresses in all the slices are found to be 60 MPa and 57.2 MPa respectively which are in the electrolyte layer of the last slice. The maximum stresses in all the layers are found to be well within the failure limits of their respective materials and hence thermal stress failure is not predicted for this stack.

152


Fig. 5.10 Cell voltage variation within a 20 cell stack due to non-uniform fuel inflow normalized with the highest cell voltage of 0.71 V for an average current density of 667 mA/cm2 .

Fig. 5.11 Maximum principle stress contours in the slice near the gas inlets of the five cell SOFC stack predicted using the temperature from pseudo 2-D stack model and ANSYSTM (after Valluru, 2005).

5.6.2 Pseudo 3-D Model Example The same approach as used above in the pseudo 2-D approach for stack modeling can also be used in the case of multi-dimensional modeling. The next set of results


153

Fig. 5.12 Maximum principle stress contours in the slice near the gas outlets of the five cell SOFC stack predicted using the temperature from pseudo 2-D stack model and ANSYSTM (after Valluru, 2005). Table 5.3 Geometrical data for individual SOFC used for stack simulations with DREAM-SOFC. Parameter

Value

No of air/fuel channels Anode thickness (µm) Electrolyte thickness (µm) Cathode thickness (µm) Active area (mm × mm) Total interconnect thickness (mm) Height of fuel and air channels (mm) Width of fuel and air channels (mm) Length of air and fuel channels (mm) Width of the current collectors (mm)

18 50 150 50 100 × 100 2.5 1 3 100 2.42

is obtained using the pseudo 3-D model presented in Section 5.2.5 and the parallelization approach shown in Figure 5.7. The results are presented for a five cell SOFC stacks in co-flow and counter-flow configurations. The geometry, material properties and the operating conditions for the individual stacks are identical and are given in Tables 5.3 and 5.4. The inlet air and fuel temperature are prescribed to be 1173 K and the fuel is 90% hydrogen and 10% water vapor by volume. The co-flow stack is operated at 20 A with fuel and air utilizations of 56.7% and 8.3% respectively. The counter-flow stack is operated at 30 A with fuel and air utilizations of 85% and 5.5% respectively.

154

I.B. Celik and S.R. Pakalapati Table 5.4 Property data for stack simulations with DREAM-SOFC. Parameter

Value

Density (kg/m3 ) Anode Cathode Electrolyte Interconnect

6600 6600 6600 6600

Heat Capacity (J kg−1 ·K−1 ) Anode Cathode Electrolyte Interconnect

400 400 400 400

Thermal conductivity (W m−1 ·K−1 ) Anode Cathode Electrolyte Interconnect

2 2 2 2

Electrical Conductivity (−1 m−1 ) Anode Cathode Electrolyte Interconnect

exp − 1150 T 6 exp − 1200 σa = 42×10 T T 4 10300 exp − σa = 3.34×10 T T 6 1100 exp − σa = 9.3×10 T T

σa =

95×106 T

Figure 5.13a shows the distribution of the current density at the top and bottom faces of the stack. Here the thickness of the SOFC is aligned along the y-axis and hence the y-current is the primary current. Since the positive y-direction is from cathode to anode, the y-current is negative everywhere. It can be seen that there are only slight differences in current density distribution between these two surfaces. Further, the profiles of current density along the channel direction at the center of each cell plotted in Figure 5.13b show that the current density varies very little from cell to cell. This trend however may not be true for large stacks. The temperature contours at the anode/electrolyte interface of each cell are shown in Figure 5.14a and the profiles of temperature along the gas-flow direction at the center of each cell are shown in Figure 5.14b. Here the temperatures vary from cell to cell with a gradual increase from the bottom cell to the top cell as predicted by the pseudo 2-D model results presented earlier. Concentration of hydrogen distribution shown in Figure 5.15, however, shows very little variation from cell to cell. Similar trends are observed for the counter-flow stack (see Figures 5.16–5.18), even though the total current is higher in this case resulting in higher temperatures and lower concentrations. In the counter flow case, the air enters at z = 0 m and the fuel enters from z = 0.1 m. The overall stack voltage is 3.96 for co-flow stack and 3.16 for counter-flow stack.


155

Fig. 5.13 Prediction of y-current density (A/m2 ) distribution for five cell co-flow SOFC stack obtained using DREAM-SOFC. (a) Contours at the top and bottom surfaces of the stack. (b) Profiles along the direction of channels at the center of each cell.

Fig. 5.14 Prediction of Temperature (K) distribution for five cell co-flow SOFC stack obtained using DREAM-SOFC. (a) Contours at anode/electrolyte interface of each cell. (b) Profiles along the direction of channels at the center of each cell.

5.6.3 Truly Three-Dimensional Model Examples Fully 3-D modeling of a stack can be performed on a single processor for stacks with small number of cells with some simplifying modeling assumptions. Liu et al. (2006) for example simulated a six cell cross-flow PEMFC stack with 8 cm2 active area. The computational domain included the air and fuel manifolds as well. However, the flow through the gas channels in the interconnect is modeled as flow through porous media with straight pores (tortuosity = 1.0). With this assumption, grid requirement of the 3-D modeling was reduced enough that a six cell stack could fit on a single processor. Figure 5.19 shows the variation of electric potential and the

156


Fig. 5.15 Prediction of hydrogen mass fraction distribution for five cell co-flow SOFC stack obtained using DREAM-SOFC. (a) Contours at anode/electrolyte interface of each cell. (b) Profiles along the direction of channels at the center of each cell.

Fig. 5.16 Prediction of y-current density (A/m2 ) distribution for five cell counter-flow SOFC stack obtained using DREAM-SOFC. (a) Contours at the top and bottom surfaces of the stack. (b) Profiles along the direction of channels at the center of each cell.

overpotential along the thickness of the mini stack simulated by Liu et al. (2006) The potential at the anode side endplate (z = 0) is set to a reference potential of 0 V and the potential in the rest of the domain is given with respect to the reference potential. The six locations where there is a jump in the potential correspond to the six MEAs in the stack where electrochemical processes occur producing a net voltage. Figure 5.19 also shows the distribution of overpotential along the thickness of the stack from which it can be seen that the activation overpotentials in the catalysts are the major contributors to the total voltage loss in the stack. The temperature distribution in a plane cutting across the cells along the channel length is shown in Figure 5.20 for three different air flow rates. It can be seen that the high temperatures and tem-


157

Fig. 5.17 Prediction of Temperature (K) distribution for five cell counter-flow SOFC stack obtained using DREAM-SOFC. (a) Contours at anode/electrolyte interface of each cell. (b) Profiles along the direction of channels at the center of each cell.

Fig. 5.18 Prediction of hydrogen mass fraction distribution for five cell counter-flow SOFC stack obtained using DREAM. (a) Contours at anode/electrolyte interface of each cell. (b) Profiles along the direction of channels at the center of each cell.

perature gradients occur near the MEAs where the reactions occur. Also the overall temperature is higher for the case with lower air flow rates. The temperature for the case with air inlet velocity of 1 m/s is too high for steady operation of the stack. Variation of maximum temperature in the stack and the stack voltage as a function of air inlet velocity, plotted in Figure 5.21, shows that the voltage increases with increased air flow rate. However, this gain could be offset by the higher pumping work needed for high flow rates of air. Thus a complete system model is needed to determine an optimum air flow rate for the optimum power density of a given PEM stack. As a second example, results from fully three dimensional simulations of a 5 cell fuel cell stack performed using FLUENT’s SOFC module are presented. The geo-

158


Fig. 5.19 Potential distribution across the six cell cross-flow PEMFC stack at 4A discharge (after Liu et al., 2006). (Reprinted from Journal of Power Sources, Vol. 160, Liu, Z., Mao, Z., Wang, C., Zhuge, W., and Zhang, Y., “Numerical simulation of min PEMFC stack”, pp. 1111–1122, Copyright 2006, with permission from Elsevier.)

metry, properties and operating conditions are the same as those used in DREAM SOFC simulations (Tables 5.3 and 5.4) except for one difference. Here, the electric conductivities are constant (not a function of temperature) and their values are calculated using equations in Table 5.4 at a temperature of 1300 K. For electric potential solution, the top surface of the stack is prescribed to be a constant potential boundary and a total current constraint is used for the bottom surface of the stack. The total current is 20 Ampères in both the cases and simulations are performed in parallel on four processors. Figures 5.22 and 5.23 show the y-current density and temperature distribution respectively in the central x-section and various z-sections of the co-flow stack. The air and fuel in this case are both entering from left (z = 0 m). It can be seen from Figure 5.22 that the current densities are the highest in the z-planes near the gas inlets and they gradually decrease in the subsequent planes. The cell to cell variations in current density distribution are not obvious in Figure 5.22 but the profiles of y-current density along the z-direction taken at the center of each cell, plotted in Figure 5.24a show that there is maximum variation between the first two cells with the rest more or less being the same. It is not clear why such a large difference is observed between the first two cells. Perhaps this might be an artifact of the type of boundary conditions imposed at the bottom and the top surfaces. Temperature distribution in Figure 5.23 shows the gradual increase in the fuel cell temperature from the gas inlets to the gas exits. It can also be seen from Figure 5.23 that the temperature increases from the bottom of the stack to the top of the stack, especially near the inlet. This trend is even clearer in the temperature profiles shown


159

Fig. 5.20 Temperature (K) distribution of the six cell cross-flow PEMFC stack with different air inlet velocities in xy section plane: (a) 5 m/s; (b) 3 m/s; (c) 1 m/s (after Liu et al., 2006). (Reprinted from Journal of Power Sources, Vol. 160, Liu, Z., Mao, Z., Wang, C., Zhuge, W., and Zhang, Y., “Numerical simulation of min PEMFC stack”, pp. 1111–1122, Copyright 2006, with permission from Elsevier.)

160


Fig. 5.21 Voltage and highest temperature in cross-flow PEMFC stack at different air inlet velocities at 4 A discharge (after Liu et al., 2006). (Reprinted from Journal of Power Sources Vol. 160, Liu, Z., Mao, Z., Wang, C., Zhuge, W., and Zhang, Y., “Numerical simulation of min PEMFC stack”, pp. 1111–1122, Copyright 2006, with permission from Elsevier.)

Fig. 5.22 y-current density (A/m2 ) distribution inside the five cell co-flow SOFC stack calculated using FLUENT SOFC module.


161

Fig. 5.23 Temperature (K) distribution inside the five cell co-flow SOFC stack calculated using FLUENT SOFC module discharge.

(a)

(b)

Fig. 5.24 Profiles along the direction of channels at the center of each cell of the co-flow SOFC stack: (a) y-current density (A/m2 ), (b) temperature (K).

in Figure 5.24b. In these simulations, the three-dimensional distributions of velocity, temperature and specie concentrations inside the gas channels and porous regions are resolved thus eliminating the need for reduced order empirical modeling. Figures 5.25–5.27 show the similar plots for counter-flow stack. Here, fuel is entering from left (z = 0 m) and air from the right (z = 0.1 m). It can be seen from Figures 5.25 and 5.27a that the current densities are decreasing from the fuel inlet to fuel exit and there is, again, a big variation in the current distribution between

162


Fig. 5.25 y-current density (A/m2 ) distribution inside the five cell counter-flow SOFC stack calculated using FLUENT SOFC module.

Fig. 5.26 Temperature (K) distribution inside the five cell counter-flow SOFC stack calculated using FLUENT SOFC module discharge.


(a)

163

(b)

Fig. 5.27 Profiles along the direction of channels at the center of each cell of the counter-flow SOFC stack: (a) y-current density (A/m2 ), (b) temperature (K).

the first and second cells. The temperatures in Figures 5.26 and 5.27b are increasing from the air inlet to the air exit and the effect of the cold fuel gas entering the stack could be seen at the air exit.

5.6.4 Lumped Model Example As the last example we present some results from the 1-D stack model introduced in Section 5.5. These are for a 40 cell stack with cells of same geometry and properties as used in FLUENT simulations above. Each cell in the stack is assumed to produce 30 A current at 0.71 V and with an efficiency of 55%. The inlet temperature of gases, air and typical coal syn-gas, is prescribed to be 1173 K. Figure 5.28 shows the outlet temperatures of air and fuel streams when convection heat transfer is prescribed from the external surfaces of the stack. The outlet temperatures are colder close to the top cell (cell one) and higher around the middle of the stack. The high outlet temperature of the middle cells can be explained by the fact that these cells are only transferring energy by convection through the side surfaces which are small (5.1 mm × 100 mm) compared with the top and bottom surfaces (100 mm × 100 mm). The distribution is not symmetric because the prescribed convective heat transfer coefficient is higher for top surface than that for bottom surface (see Incropera and DeWitt, 2002). To demonstrate the capability of the code to handle different mass flow rates among the cells, hypothetical cases are simulated where the flow of gases is redistributed such that a few randomly selected cells receive lesser flow rates than the rest of the cells in the stack. The total flow rate to the stack, however, is same as in the base line presented above. These cases are representative of some of the cells in a stack being partially blocked due to say clogging. The results shown in Figure 5.29 correspond to the case where three cells are partially blocked, allowing only 70% of

164


Fig. 5.28 Outlet temperature of fuel and air streams across the cells of the stack.

the mass flow rate of base line case with the difference equally distributed among the other cells. Since the mass flow rates are different for the cells, the utilization in each cell is not the same. Cell number two is always kept partially blocked and the other two cells are selected in such a way that their distance from cell two increases. It can be seen from Figure 5.29 that the blocked cells are slightly hotter than their neighboring cells. This is due to the relatively lower cooling from the channels for these cells. Also, it can be seen from Figures 5.29a and 5.29b that as the distance between cell two and the other blocked cells increases, the maximum cell temperature increases slightly. Increasing the distance between the two blocked cells (Figure 5.29c), the maximum temperature decreased and the difference between the temperatures of the two blocked cells is reduced. As the distance between the partially blocked cells increases the top cell cools down while the bottom cell heats up slightly. If all three partially blocked cells are close to the bottom cell, both maximum stack temperature and the bottom cell temperature are at their highest. On the other hand, the temperature of the top cell is the lowest in this case. From these results it could be deduced that possible mal-distributions of reactants among the cells of a stack could have an adverse effect on the thermal stress behavior of stack through localized steep thermal gradients. Simple models like the one presented here could be very effective in control systems for real time prediction of say mechanical failure given the current operating conditions, so that corrective action may be taken in time. From the 1-D stack model, it is also possible to obtain a two pseudo 2-D temperature distribution inside the stack. The two dimensions are along the length and height of the cell stack. In order to obtain a temperature distribution, two additional temperatures are calculated for solid regions in each cell using the following equa-


165

Fig. 5.29 Cell temperature with manifold mal-distribution when some channels are partially blocked; (a) channels 2, 5 and 8, (b) channels 2, 8 and 14, (c) channels 2, 14 and 26, (d) channels 33, 36, 39 partially blocked.

tions: near the inlet Tcin = Tain +

Q˙ a ; ha Aa

near the outlet Tcout = Taout +

Q˙ a . ha Aa

(5.73)

(5.74)

Equations (5.73) and (5.74) are based on Equation (5.68). In the middle of the cell, the temperature is the cell temperature (Tc ) calculated from the steady state model. The 2-D temperature distribution thus obtained is as shown in Figure 5.30. Though only three temperatures are calculated along x-direction for each layer the interpolation done by the plotting software gave detailed contours that show the distinction

166


Fig. 5.30 Temperature contours from steady state model for 40 cell stack with manifold maldistribution: fuel flow in the cells is varying linearly from top to bottom of the stack.

between the solid and fluid region temperatures. Given the simplicity of the model used, the detail in the predicted temperature distribution is quite impressive.

5.7 Conclusions and Recommendations In this chapter we presented all the necessary steps for simulation of fuel cell stacks starting from single cell models and extending to relatively simple stack arrangements. The major ingredients in successful stack simulations are calculation verification, model validation (see Appendix to this chapter) and parallel processing. Verification is necessary to assure that the equations that are involved in the mathematical model are solved correctly, and validation can be defined in short as making sure that the right equations are solved (Roache, 1998). These steps are necessary at the cell-level modeling as well as at the stack-level modeling. Fuel cell stacks are amenable to easy application of parallel processing due to the fact that each cell is connected to the neighbors in series and the physical communication between the two neighbors is primarily via current flow and heat flux. In order to make computer simulations a viable tool for stack design analysis parallel computations are necessary.


167

Although there are extensive academic studies in the area of cell and stack level modeling of fuel cells, unfortunately, it can not be said that this effort has found its proper place in industrial applications. In other words, the technology transfer step is not being pursued as a follow up to the extensive academic studies published in the literature. The main reasons for this undesirable situation seem to be (i) lack of proper validation of the calculations, (ii) unreasonably long executions times still necessary for stack calculations. For example, completion of a 5-cell planar stack (made of 10 × 10 × 0.525 cm cells, with 18 fuel and air channels) simulation on a relatively coarse mesh (e.g. 2.5 million computational nodes) on five parallel processors using the commercial code FLUENT takes approximately 13 hours. Five cell stack with half the cell size (9 channels, 1.3 million nodes) takes 10 hours on five processors. If the same scaling applies when number of cells is increased, it may take a few days for simulation of more realistic stacks with tens of cell. This is too long of a turn-around time for the purpose of design improvement. In the light of the above, we recommend that while verification is applied, validation of the cell and stack calculations in comparison to carefully designed experiments must take priority in the fuel cell modeling community. Only by proper validation of the 3-D calculations using, at least, the spatial distribution of temperature and current measurements, the computer simulations can take its proper role in design analysis and improvement with relevance to industrial application. Needless to say, the computation time must be reduced for practicality purposes. Another issue that needs to be addressed is the accurate calculations of the transients of stack operations under variable loading due to changes in power utilization demand and/or under start-up and shut-down conditions. Tracking fast transients, especially during the start-up process, requires at least second order accurate temporal resolution which will impose additional computational cost on stack simulations. It seems that in the near future the best alternative would be to use reduced order physics based models such as those presented in Section 5.2 with appropriate empirical input and experimental validation to get the most benefit out of computational studies.

Appendix: Verification and Validation Issues Mathematical modeling of physical processes in fuel cells inevitably involves some assumptions that may or may not be valid under all circumstances. Furthermore approximations have to be introduced to make the computational models robust and tractable. These approximations in the mathematical models lead to the so called “modeling errors”. That is if the equations posed are solved exactly, the difference between this exact solution and the corresponding “true” but usually unknown physical reality is known as the modeling error. However, it is rarely the situation that the solution to the mathematical models is exact due to the inherent numerical errors such as round off errors, iteration convergence and discretization errors, among oth-

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ers. In what follows, we define some of these concepts and present some theoretical background along with examples pertaining to fuel cell modeling

Validation Validation is usually perceived as comparison of computed results to those of the corresponding experiments. This is a simplistic understanding of validation process in computational modeling. A broader interpretation of validation is the assessment of many calculations which are designed to emulate a set of physical experiments for the same physical phenomenon being computed or simulated. Since there is always some degree of uncertainty in experiments and also in calculations it is more precise to state the problem of validation as a comparative statistical analysis of computational results against experimental results. The major sources of uncertainties in calculations are those arising from discretization errors, δdis , iteration convergence errors, δiter , and modeling assumptions, δmod . Stated symbolically, we write 2 2 2 2 δcalc = δiter + δdis + δmod 2 2 = δnum + δmod .

(A.1)

Only in situations where numerical uncertainty is small compared to modeling uncertainty we can successfully validate a calculation. After minimizing numerical errors there will still be other uncertainties in calculations due to for example variations in inlet conditions or due to inherent uncertainty in tabulated material properties, etc. These can be best handled by repeating the calculations with appropriate variations in the uncertain input quantities, thus resulting in say nc calculations with seemingly nc random outcomes the mean and variance of which are donated by X¯ c and Sc2 . Similarly there would be ne repeated experiments of the same phenomenon with ne random outcomes with the corresponding mean and variance, X¯ e and Se2 , respectively. The estimated modeling error is by definition the difference between the experimental mean and calculation mean, i.e. Em = X¯ e − X¯ c .

(A.2)

Using classical statistical analysis (see for example Mendenhall and Sincich, 2007), the (1 − α)% confidence interval for the true error, Eµ = µe − µc , is given by C.I. of Eµ = (Em − k, Em + k),

(A.3)

k = S ∗ ta/2

(A.4)

where and ∗

S =

S2 Sc2 + e nc ne

1/2 ,

(A.5)


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and ta/2 is determined from the t-distribution such that P (T > ta/2 ) = α/2 with ν degrees of freedom where ν is given by ν=

(S ∗ )4 (Sc2 /nc )2 (nc −1)

+

(Se2 /ne )2 (ne −1)

.

(A.6)

Note that ν should be rounded to nearest integer value. Example: Suppose nine repeated calculations of a certain SOFC resulted in mean maximum solid temperature of 1167◦C with a sample standard deviation of 12◦ C (see Table A.2 for an example of such a calculation), whereas a set of five repeated experiments for a similar cell resulted in mean max solid temperature of 1140◦C with a sample standard deviation of 15◦ C. Calculate the 95% confidence interval for the modeling error assuming that the numerical errors in the calculations are negligible. Solution: Given data nc = 9,

Tc = 1167◦C,

Sc = 12◦C,

Te = 1140◦C, Se = 15◦C, 1/2 2 152 12 ∗ + = 7.8, S = 9 5

ne = 5,

ν=

(7.8)4 (16)2 (8)

+

(45)2 (4)

= 6.9 ∼ = 7.

From t-distribution tables with ν = 7 degrees of freedom and α = 0.05 (for 95% confidence), we obtain ta/s = 2.365. Hence k = (7.8)(2.365) = 18.45◦C, Em = Te − Tc = −27◦ C, 95% C.I of Eµ = (−45.45, −8.55). In other words, the true experimental value, θe will be bounded by (θc − 45.45 < θe < θc − 8.55). Note that, because of the modeling errors even the exactly calculated values θc may not match the true experimental value θe .

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Fig. A.1 Scaled residuals for a solution of 2-D Navier–Stokes Equations (after Celik and ElizaldeBlancas, 2006).

In the above discussion we have assumed that the numerical errors are negligibly small. In what follows we outline some procedures by which these errors can be quantified. This quantification process is known as calculation verification.

Iteration Convergence Iteration convergence errors are those due to incomplete convergence of iterative solutions. Iterations are necessary for several reasons such as Gauss–Seidel iteration for solution of linear system of equations, iteration to account for the coupling among different partial differential equations (PDEs), and iterations to account for linearization of non-linear terms. Usually, all these are lumped together and called iteration convergence errors. Incomplete convergence in any aspect of calculations would have to be quantified to determine the magnitude of such errors. One way of estimating the iteration convergence error is done by way of monitoring the normalized residual, Riter , i.e. some norm (such as the L2 norm) over the computational cells of the remainder after the numerical solution is substituted into the discretized counter part of the PDE. An example of residual monitoring is depicted in Figure A.1. It is seen in this case that the residual of each equation reduces to machine accuracy. However this may not always be possible and then the residuals by themselves may not be a good indicator of the magnitude of iteration convergence errors in the variables being solved for. In such situations, the difference in the solution between


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Fig. A.2 Behavior of x-velocity component as function of the iteration number (after Celik and Elizalde-Blancas, 2006).

successive iterations should also be monitored; this is defined as the approximate iteration error, (A.7) Ea,iter = φ n+1 − φ n . Consider for example the variation of the axial velocity as a function of iteration number shown in Figure A.2. If the calculations were stopped at iteration number n = 50, there would be an iteration error of about 200%. The fully converged solution in this case is given by u = 0.160495 m/s. Since in cases of incomplete iterative convergence the fully converged solution is not known an estimate for this error is necessary. The approximate relative iteration error and the relative true iteration error are defined respectively by new φ − φ old ea,iter = (A.8) , φ new converged φ − φ n et,iter = . (A.9) φ converged According to Ferziger and Peric (1996) the iteration convergence error should be approximated by ea,iter ∗ ea,iter , (A.10) = |λn − 1|

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where ‘lambda’ is the principal eigen value of the solution matrix and it can be approximated by |φ n+1 − φ n | λ∼ . (A.11) = n |φ − φ n−1 | In order to avoid fluctuations in ‘lambda’ it is further recommended (Celik and Elizalde-Blancas, 2006) to estimate the modified approximate relative iteration error from ea,iter ∗ . (A.12) ea,iter = |λaver − 1| Here λaver represents the average value of λ over sufficient number of iterations. The approximate relative iteration error, ea,iter, the relative true iteration error, et,iter, and ∗ , are shown in Figure A.3 for a sample calthe modified relative iteration error, ea,iter culation using the FLUENT software (see Celik and Elizalde-Blancas, 2006, for more details). It is seen that the approximate relative iteration error is smaller than the relative true iteration error at the beginning of the iterations since the change of φ between consecutive iterations is much smaller than the difference of the converged value and the current value. From Figure A.3 it is also seen that the modified approximate relative iteration error bounds the true relative error which is the objective of ea∗ . This characteristic behavior of the approximate relative iteration error, may not always be observed in CFD applications. Hence it is recommended that the iterative residuals should be reduced to machine accuracy whenever possible.

Grid Convergence All numerical errors that arise from or are related to the discretization, i.e. representing conservation equations in discrete form using for example finite elements or finite differences, can be expressed by a Taylor series (Richarson, 1910) Et = φ(0) − φ(h) = ci hi , i = 1, 2, 3, . . . (A.13) Here Et is the true discretization error, and h represents the nominal size of a computational cell. It is preferable to view φ and h as non-dimensional quantities such that φ h φ→ , h→ , (A.14) φref Lmax where φref is a reference value of the dependent variable being computed on a physical domain with a maximum dimension of Lmax so that h is always less than 1.0 and it is relatively small. In the asymptotic range where only the first few terms are dominant, Equation (A.13) can be approximated by φext − φ(h) = c1 h2 + c2 h2 + c3 h3 .

(A.15)


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Fig. A.3 Relative iteration errors at a point considering the last 20% of iterations.

This approximation should suffice for first, second and third order methods, however, at least four numerical solutions are required on substantially different meshes to determine the coefficients and the extrapolated value, φext . Three sets of calculations can be used to reduce computational cost by taking one of two following path ways: (i) φext − φ(h) = c1 h + c2 h2 ,

(A.16)

(ii) φext − φ(h) = chp .

(A.17)

If the theoretical order of the numerical scheme, p, is known to be less than or equal to 2 use (i), otherwise use (ii). After determining, c1 , c2 , h (Equation (A.16)) or c, p (Equation (A.17)) and φext , a fourth calculation can be used to confirm the apparent order (i.e. the effective order that is exercised during the actual calculations which may differ from the theoretical order for various reasons) by recalculating from Equation (A.17). Let us denote the numerical solutions φ1 , φ2 , and φ3 on grids represented by h1 < h2 < h3 . Assuming that the coefficients in the Taylor series expansion (Equation (A.15)) will not change with grid size, the following linear system of equations should be solved [A]{X} = {R}, (A.18)

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⎡

⎤ 1 −h1 −h21 [A] = ⎣ 1 −h2 −h22 ⎦ , 1 −h3 −h23

⎡

⎤

⎡

⎤ φ1 {R} = ⎣ φ2 ⎦ . φ3

φext {X} = ⎣ c1 ⎦ , c2

(A.19)

Similarly if option (ii) is used the following non-linear equation should be solved to determine p φ23 p , (A.20) r21 = f (r, p) φ12 φ12 = φ1 − φ2 ,

(A.21a)

φ23 = φ2 − φ3 ,

(A.21b)

r21 =

h2 , h1

(A.22a)

r32 =

h3 , h2

(A.22b)

p

f (r, p) =

(r21 − 1) p

(r32 − 1)

φext = φ1 +

,

φ12 . p (r21 − 1)

(A.23) (A.24)

It is usually recommended to test the results of 3-grid calculations with an additional fourth calculation to confirm the order of accuracy p and the extrapolated value, φext . A relative discretization uncertainty can then be calculated from (φext − φ1 ) . δdis = abs (A.25) φext

Verification Application The following grid study is performed on a 1-D gas channel model with axial diffusion and conduction. The fuel channel geometry used is the same as the one used for 3-D simulations of Pakalapati et al. (2006). It is 10 cm long and has a cross-section of 3 mm × 1 mm. Other input conditions to the model are the inlet concentration and temperature of fuel and the current density distribution along the flow direction. The current distribution prescribed here was obtained from the 3-D simulations of Pakalapati et al. (2006) for the case with the same inlet conditions as used here. The specie and heat sources inside the fuel channel are calculated using the prescribed current density distribution. Figure A.4 shows the hydrogen mass fraction and temperature profiles along the flow direction obtained using four different grid sizes. It can be seen from these figures that the solution is somewhat grid independent after 40 nodes. However, a comparison of grid convergence error in estimating the exit


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Fig. A.4 Profiles along the direction of fuel flow obtained using different grids.

hydrogen concentration and temperature, plotted against the grid size in Figure A.5, shows that about 200 grid points are needed for fairly grid independent solution. The error is calculated using the estimates from the finest grid (640 nodes) as the true solution. It can be seen from Figure A.5 that for acceptable level of accuracy in the prediction (e.g. 0, where 2O= → O2 ) and cathodic (i < 0, where O2 → O= ) driven conditions are given. As is evident from the figure, a simple multi-first-order exponential response is shown for both transients. The timescales for the cathodic transient are an order of magnitude greater than that for the anodic transient. Both of these transients result from a ‘conditioning’ effect when a cell is operated at specific current/voltage conditions. These results are representative of all types of cell designs using these materials, and can be expected to occur in other material systems as well. While such behavior is clearly non-negligible in the prediction of cell operation, the exact cause for the conditioning remains the subject of active research. In present day modeling, it is generally assumed that this behavior is significant only at the start of cell operation. Once the cell is conditioned, it is assumed that its performance will thereafter

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Fig. 9.2 Transient voltage (a) and current (b) behavior of an SOFC cell due to changes in load resistance (Qi et al., 2005).

be stable. Hence, while it is important to recognize the existence of this transient behavior, at this time such behavior is ignored in most transient modeling work. Figure 9.2 shows the short-term transient behavior of a fuel cell as obtained from a dynamic model derived from experimental electrochemical impedance studies (Qi et al., 2005). Figure 9.2a shows the cell voltage versus time due to two different resistive load changes (a resistance increase and decrease). The inset shows the existence of three distinct process timescales. The first, VRo , is an immediate response in the cell voltage which results from pure resistive elements within the cell. The second, VRct , is also relatively fast (circa sub-millisecond), that results from the time it takes a charge transfer process at the electrode-electrolyte interface to

9 Dynamic Modeling of Fuel Cells

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respond to the change in load. Finally, there is a relatively slow response, noted as Vp , that results from the time it takes local reactant concentrations to change. Figure 9.2b shows the corresponding changes in current through the cell due to the change in load resistance. These three processes are understood to exist at all times during the operation of a fuel cell, and hence need to be considered when developing models for fuel cell dynamic analysis. The model equation derivation of VRct is given in Section 9.3.3.

9.2.2 System Performance The analysis of system stack transients is complicated given that there are numerous causes for current/voltage changes. Not only is the main external load important in determining stack conditions, but also internal loads used to support ‘balance-ofplant’ control hardware. The system controller continually adjusts various active process control elements in its goal to manage temperatures, pressures, oxidationreduction potentials, and the rates of chemical reactions (e.g., consumption of fuel) so as to meet the instantaneous load demands while protecting the unit against undesirable operating conditions (e.g., carbon deposition, corrosion of materials, thermal stress, component oxidation, etc.) Common parameters used for control of the system are the fuel and air flow rates (both primary flow loops as well as secondary and possibly tertiary flow loops may be used), anode and/or cathode recycle flow rates, and any water used to support fuel reforming. In the early development of a new system, a backup purge gas (e.g., 4% hydrogen in nitrogen) may also be used for certain transient events. Pumps, blowers and flow control valves are the main control elements used. Finally, to manage the safe operation of the unit, it is common to employ multiple control loops. The main point to be understood here is that from the view point of an external observer, such systems make analyzing the externally accessible system data for cause and effect relations affecting the stack normally impossible. The conclusion is that the only way to properly model and predict stack level behavior is to fully resolve and accurately predict the dynamic characteristics of all system subcomponents. The development of models capable of such complete system level analysis is given in Chapter 8. It is the goal of this chapter to support such analysis through the development of dynamic models specific to the fuel cell stack. An example of a system transient is shown in Figure 9.3. The figure shows a 20 amp load increase on a nominal 3 kW SOFC system using a stream-reformed methane fuel. Stack load current, stack voltage, and input fuel flow rate are shown. Here, the system is pre-warmed to the conditions shown (ca. 20 amps), following which the controller permits exporting more power to a load at the ramp rate shown. As the transition occurs, numerous other system variables also adjust, some in direct response to the increase in load, and others imposed by the control system in order to keep all system components within their design limits.

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R.S. Gemmen

Fig. 9.3 Load increase transient of a three kilowatt SOFC system. Data points are shown at 1minute intervals.

Figures 9.4a and 9.4b show transients of the same unit undergoing two different step changes in load. Figure 9.4a shows the unit undergoing a load fault. The stack voltage quickly rises upon loss of current as would be expected from a standard cell voltage–current characteristic curve (e.g., see Figure 1.1 or Figure 4.6). At the moment of load fault, the fuel flow rate is immediately dropped. The magnitude of fuel reduction is clearly not in proportion to the magnitude of load loss as might be conjectured from an overall energy utilization analysis. A minimum level of fuel flow shown is used, in part, to maintain the thermal conditions within the system while at idle state. Figure 9.4b shows the unit undergoing a simultaneous loss of fuel and a removal of external load. Because of the loss of fuel, the stack voltage begins to drop. The stack voltage recovers shortly after the fault, however, do to the addition of a purge fuel having a minimum hydrogen content of about 4% in nitrogen (4% hydrogen is used due to its near non-flammable concentration.) These figures highlight the statement made previously that complete information of the internal state of all components and methods of control are needed in order to accurately predict system stack behavior, and therefore, the performance of the system itself.


275

Fig. 9.4a Transient of a three kilowatt SOFC system undergoing a load fault. Data points are shown at 1-minute intervals. Following the fault, a small amount of current remains on the stack to service ‘balance-of-plant’ loads.

9.2.3 Issues for Cell Operability As with all materials and structures, solid oxide fuel cells have limits in their operability. Various failure modes are possible. Structural failure modes are frequently seen during fuel cell development as designers work to understand the stress limitations of a particular cell design. Structural failures can occur at the interface between electrodes and electrolytes, within electrodes and electrolytes, and at the interface of seals and the cell or interconnect. The forces that lead to these structural failures can be thermal based (e.g., high temperature gradients in the cell, or from high differential thermal expansion of laminated components), or pressure based (e.g., gas phase loadings which can be a concern for direct hybrid systems). The phase stability of materials depends on local reactant concentrations and temperatures. Nickel anode supported designs normally must avoid reduction/oxidation cycles that can also result in structural failure. While failure modes within SOFC’s remain the subject of active research, already significant gains to our understanding are being achieved through modeling (Aguiar et al., 2005). Dynamic modeling can provide the detailed information needed to help understand how each of these methods of failure can arise during the normal operation of a fuel cell. Many more gains to our understanding are yet to be had through continued dynamic modeling analysis (Benhaddad and Protkova, 2005).

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Fig. 9.4b Transient of a three kilowatt SOFC system undergoing a simultaneous loss of fuel and removal of load (emergency stop). Data points are shown at 1-minute intervals. The higher potential achieved here versus that in Figure 9.4a at zero load results from the relatively dry purge gas used. The slightly negative current shown here at zero load is due to a small DC-bias in the experimental measurement.

9.3 Model Development 9.3.1 Defining Model Requirements As already noted, the present study of dynamic fuel cell behavior involves the analysis of systems with capacitive elements. These elements control the rate at which process parameters change due to net ‘forces’ imposed by other coupled process parameters. A general dynamic equation showing capacitance behavior is: d (cP )d∀ = Gn , (9.1) dt where P is some intrinsic process parameter associated with a conserved physical quantity Z. That is, P = Z/c, where c is the so called capacitance parameter. In general, P , Z and c can vary in space, and so Equation (9.1) shows an integration over space, ∀, to arrive at the total amount of the conserved quantity Z within the given control space. The control space is either a volume or an area depending on whether the type of fundamental phenomena being considered is volume based or surface based (examples follow below). The net addition of Z to the control space is described by Gn . The addition of Z can occur by either flux at the boundary of the control space, or by the conversion of other conserved quantities within the control


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space. To show the generality of Equation (9.1), we examine the case of conservation of momentum of a body under the acceleration of gravity, where Z is the said momentum. Here, Gn is the rate of momentum addition to the body (also known as the net force, F [N], on the body), c is the momentum capacitance of the body (also known as the mass, M [kg], of the body), P is the velocity (V [m·s−1 ]) of the body, and ∀ is the volume. Equation (9.1) then becomes the conservation of momentum equation. Finally, an example of the use of Equation (9.1) for surface based phenomena is the conservation of charge on the surface of a dielectric material, where Z is the said charge density on the surface. Here, Gn is the rate of charge addition (I [Coul·s−1 ]) to the surface of a dielectric material, c is the charge capacitance (C [Farad·m−2]) of the dielectric material, P is the voltage (V [V]) on the dielectric material, and ∀ is the surface area. We now use Equation (9.1) in a first-order analysis that provides estimates for the various timescales occurring in SOFC systems (Gemmen and Johnson, 2005). Such an analysis will be helpful later when we need to determine the requirements of the various submodels so as to provide an efficient calculation approach to a given transient problem. To determine the first-order timescale of a particular transient, we rearrange Equation (9.1) as: t = τ =

∀ CP [cP ∀] = cP = , Gn Gn Pr

(9.2)

where τ is the characteristic time for the transient under study, and P is the firstorder estimate for the change in the process variable. The grouping cP is referred to herein as the capacity parameter, CP , since it describes the amount of change in the fundamental physical parameter, and Gn /∀ is referred to as the Process Rate Parameter, Pr . The ratio, CP /Pr gives the estimate for the timescale of the transient. Several timescales are provided in Table 9.1 by employing Equation (9.2) on a representative 10 cm × 10 cm anode supported SOFC fuel cell operating at typical SOFC conditions (see Table 9.3 for reference). The name of a particular process timescale is given in the first column of Table 9.1. Associated with each process is its rate, Pr , and capacity, CP , parameters, and equations are provided in the table for each value. In Table 9.1, I is current flow, ρ = density (mass or mole), Ac = active area, Ax = flow passage cross-section area, ∀ = flow passage volume or gas volume flow rate, Vcell = cell voltage, D = molecular diffusion coefficient, Cp = gas specific heat, C = electric capacity or solid thermal capacity, t = electrode thickness, L = cell length, h = gas channel height, and U = gas velocity. Because some transport processes depend on a characteristic length scale (e.g., diffusion), and because there are several distinct characteristic length scales within a fuel cell (electrode thickness, cell length, etc.), numerous transient timescales are present for the same fundamental transport mechanism. As an example, consider line ‘A’ of Table 9.1 for the timescale of the cell charging time due to so called double-layer capacitance. As discussed later in this chapter, effective capacitance values are on the order of 1 Farad·m−2. If current densities of a cell are on the order of 1.0 A·cm−2 , and representative over-potentials at the triple-

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R.S. Gemmen

phase boundary is on the order of 0.1 V, then we have cP /(Gn/∀) = CP /Pr = Cηact /I = 0.1 Coul·m−2 /104 A/m2 = 10−5 s. As a final example, consider line ‘D’ of Table 9.1. We represent this problem as a body of density ρ, and heat capacity cp and whose surface is in contact with another medium of temperature Ts . Assume the initial body temperature is the same as the temperature of the other medium at Tb1 = Ts1 . From the fundamental equation we can write, ρcp LdTb /dt = λ(Ts − Tb )/L, where L is the characteristic conductive length, and λ is the thermal conductivity. We now scale this problem over the entire time of the thermal transient. Once the entire time of the transient passes (t2 − t1 ), the body will have reached the new temperature of Ts2 . For the overall transient, the temperature rate of change is (Tb2 − Tb1 )/(t2 − t1 ), and the average driving potential for the thermal conduction will be (Ts2 − Ts1 )/2 = (Tb2 − Tb1 )/2. We now define the first-order relationship between the parameter as ρcp L(Tb2 − Tb1 )/(t2 − t1 ) ∼ = λ(Tb2 − Tb1 )/L. Solving for the time constant, we have τ∼ = (ρcp L)/(λ/L). We therefore write Pr = λ/L and CP = ρcp L, as shown in Table 9.1.

9.3.1.1 Fuel Cell Timescale Analysis The results of Table 9.1 show a wide range of timescales (10−5 s to 104 s) spanning over nine orders of magnitude. These results can be helpful to guide engineering analysis by showing how transport models can be simplified, and yet still accurately predict a given fuel cell performance parameter. More specifically, to provide an efficient use of computational resources, it is customary in dynamic modeling to only consider the details of a given transport process if its characteristic timescale is within one or two orders of magnitude from the principle effect being examined. Hence, if one is investigating the characteristics of a fuel cell having a timescale of on the order of 102 s, then any transport process having a characteristic time greater than 104 s can be assumed constant; hence, no detailed model equation is necessary to relate the parameters of that transport process to the rest of the model. On the other hand, if there is a transport process having a characteristic timescale less than 100 s, then the related physical parameters can be assumed to behave quasi-steady – for this latter condition, a steady state equation is required to relate the parameters of that transport process to the rest of the model. All other transport processes will require the use of their respective fundamental dynamic equation to relate their parameters to the rest of the process. The choice for the separation of timescales (102 s used here) will affect the temporal accuracy of the solution. A wider time span will result in more phenomena being predicted using their respective dynamic equation as well as improved accuracy, albeit at the cost of greater computational effort. This span also sets the bounds over which computations will be performed,


279

Table 9.1 First-order calculation for characteristic timescales (Gemmen and Johnson, 2005).

as well as the time step – computing results beyond the assumed steady state limit would be inconsistent with such an assumption, while using time steps less than the quasi-steady limit may be computationally unnecessary (unless computational stability dictates otherwise).

9.3.1.2 Example Timescale Analysis Assume a principle timescale, T , has been identified for study dictated by the principle phenomenological effect of interest (e.g., a thermal response study). Once the value of this timescale is known, the integration time step, t, is then chosen so as to provide enough information (solution data) to allow an accurate picture of the transient, e.g., 1/100th of T . Once the value for t is known, then the type of formulation (quasi-steady formula versus the fundamental dynamic formula) of each individual physical phenomena can be selected (see Table 9.2).

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R.S. Gemmen

Table 9.2 Quasi-steady versus constant versus dynamic solution domains. The darker middle boxes denote the timescale, τ , for the given transient phenomena. Phenomena with timescales to left of the computational time step, t, can be considered quasi-steady. Phenomena with timescales to the right of the integration time, T , can be considered constant.

For the present example, consider a transient thermal analysis over a duration T = 600 s with a resolution of t = 5 s. Such an analysis is common for analyzing the thermal transient of an SOFC. Table 9.2 shows the phenomena with characteristic timescales 100 s and less (lines A to G) that can be considered to be quasi-steady. Additionally, phenomena with timescales 104 s and greater (line N) can be neglected (their behavior will appear to be constant.) This analysis quickly showed us how to treat many of the transport issues for this given problem. Further comparisons of timescales can be made to discover how to treat other transport mechanisms. For example, examination of the streamwise convection (line E) and diffusion (lines H and K) in Table 9.2 (two independent transport mechanisms that control streamwise species distribution), shows that the timescale for streamwise convective transport is 10−1 s while the timescale for streamwise diffusion is 101 s. Because the convective response is faster than diffusion, it will control the reactant conditions within the cell (note however, that because there is only 1-order of magnitude difference in timescales, very detailed studies may require the consideration of streamwise diffusion in addition to convection). Ignoring streamwise diffusion for the type of modeling we wish to pursue within this chapter, and given that the convective timescale is much faster than the time step, the gas flow will also be considered quasi-steady. Finally, the timescales for diffusion transverse to the streamwise direction, i.e., normal to the plane of the cell, are less than 10−3 s (see lines B and C in Table 9.2). Because this is much faster than the convective timescale, the transverse distribution of reactants in a channel can be considered quasi-steady (in fact, given the fast transverse diffusion rates, the concentration profiles transverse to the flow direction can be considered uniform). Additionally, because changes in the streamwise direction are often larger and more significant, the specie distribu-


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tion can be considered uniform in the transverse direction – i.e., one-dimensional. The ability to ignore these (and other) transport mechanisms, and to select quasisteady or constant models for individual phenomena, helps to speed computational predictions making engineering analysis more efficient.

9.3.2 Solution Method Selection In this section we take a moment to highlight several types of tools that are available for solving transient fuel cell problems. Some are designed to support large scale plant design and engineering such as Aspen Plus or the ProTRAX software package, while others are more general and provide primarily a mathematical solution framework such as the Matlab Simulink software package (Figure A.9.1). The former often come prepackaged with numerous basic submodels (e.g., transient heat exchanger models, and transient gas-mixer models), while the latter often require that the user derive and develop the necessary mathematical equations which will then be solved by the framework solver. Alternatively, one can develop one’s own numerical solution method, which offers improved control over the solution strategy and numerical solution methods used. The selection of the solution method will depend on the scale of the problem (whether being a focused analysis on the transient diffusion through an electrode or a study of the overall transient analysis of a plant undergoing a load transient), and the time available to arrive at a solution. Several examples of different solution approaches will be presented in later sections of this chapter, and others are given in Chapter 8. Regardless of the solution method taken, it is wise for the user of these models to be closely familiar with the limitations/assumptions of the model used. As now understood from Section 9.3.1, some prepackaged models may allow for a steady (quasi-steady) solution of a particular phenomenon, but such may not be feasible if the timescale of interest is not compatible with such an approach.

9.3.3 Transient Phenomena and Their Model Equations In this section we examine the primary transient phenomena that are of interest to SOFC analysis, and provide the fundamental model equations for each one. Examples for the use of these models are given in later sections. While the focus is on reduced-order models (lumped and one-dimensional), depending on the needs of the fuel cell designer, this may, or may not be justifiable. Each fuel cell model developer needs to ensure that the solution approach taken will provide the information needed for the problem at hand. For the goal of calculating overall cell performance, however, it is often that one-dimensional methods such as outlined below will be viable.

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Fig. 9.5 Equivalent circuit of inherent cell impedance. Cdla = double layer capacitance of anode; Cdlc = double layer capacitance of cathode; Rcta = charge transfer resistance of anode; Rctc = charge transfer resistance of cathode (Qi et al., 2003).

9.3.3.1 Electrochemical Conversion While much remains to be understood regarding the details of the chemical kinetics at the triple-phase boundary, what is clear is the existence of a tight coupling between cell load current and cell electrochemical loss. This tight coupling is normally expressed in the form of a Butler–Volmer equation given by Equation (3.37), Chapter 3. The Butler–Volmer equation expresses the steady state relationship between net current flow across a triple-phase boundary, i, and the overvoltage across the triple-phase boundary, η. On closer examination, however, it is seen that in fact there is a noticeable decoupling between voltage and current. While a range of behavior has been shown to exist, it has been common in the literature to describe the dominant decoupling as arising from the presence of a capacitive-type element at the triple phase boundary – the actual physical source of this capacitance is the subject of current research (Jørgensen and Mogensen, 2001). This element is also sometimes referred to as the ‘double-layer capacitance’ which becomes charged/discharged on changes in current loading (noted as ‘Cell Charging Time’ in Tables 9.1 and 9.2). This charge buildup is proportional to the voltage drop across the triple phase boundary (Hendriks et al., 2001). To represent this layer (phenomenologically), a variety of resistive and capacitive models have been proposed (Qi et al., 2003). One of the simplest models is a parallel resistance and capacitance (R-C) circuit in series with electrode and electrolyte resistances, see Figure 9.5. For this model, the current through the capacitor is given by (Adler et al., 2003): d(Cdl η) ic , dt

(9.3)

where Cdl is the capacitance of the electrode double-layer. A generally recognized value for Cdl has not been established, but values as high as 1 Farad·m−2 have been suggested (Hendriks et al., 2002; Adler et al., 2003; Qi et al., 2005). With the assumption of quasi-steady electrochemical loss, the current through the charge transfer resistor, iRct , is given by the Butler–Volmer equation. Figure 9.5 shows one charge transfer resistor-capacitor for each electrode, Rcta and Rctc . Because of the


283

parallel R–C circuit, the net instantaneous current is the sum of current through both R and C elements. Hence, i = ic + iRct . (9.4) As shown in Table 9.1, the typical timescale for the electrochemistry (‘Cell Charging Time’) is on the order of 10−5 s. As a result, it is often that this transient is ignored in cell performance calculations, and the quasi-steady Butler–Volmer relationship is used alone (Qi et al., 2005). An example model for this particular type of dynamic cell behavior is given in Section 9.5.

9.3.3.2 Reactant Electrode Transport Cell voltage is governed, in part, by the reactant concentrations at the electrolyte interface (triple phase boundary). Hence, reactant electrode transport must be considered when there are high consumption rates of reactants which result in a significant deviation between the reactant concentrations at the triple-phase boundary and those in the freestream. As seen in Section 3.3.2, Chapter 3, there are a variety of approaches used in the analysis of reactant electrode transport. A simple empirical quasi-steady model that describes the loss in voltage due to hydrogen diffusing though the anode is: i RT ln 1 − , (9.5) ηd,H2 = − 2F id where id is the limiting diffusion current which must be determined either empirically or by a higher level model. A similar form of model can be derived for the loss in voltage due to oxygen diffusion through the cathode (see Appendix A3.2). While the above diffusion model is widely used, especially for large system level modeling where the speed of the solution is critical, other more accurate models may be needed such as when improved reforming kinetics within the anode are needed (Gemmen and Trembly, 2006). Depending on the cell design, multi-dimensional transport may also need to be considered within the electrode (see Figure 9.6). For the present analysis, however, we focus on one-dimensional transport. Such an approach for the electrode transport is often justified as readily inferred from the ratio of the timescales shown in rows B (cathode electrode diffusion time) and K (cathode streamwise diffusion time) in Table 9.1. From this analysis we see that the diffusion transport in the direction through the thickness of the electrode is much faster (stronger) than the transport in the streamwise direction. Hence, we can ignore the diffusion in the streamwise direction and a local one-dimensional analysis through the thickness of the electrode is justified. Electrode transport is often represented by Fickian diffusion models. Such models, however, are applicable to bi-molecular transport, whereas for fuel cells it is often that more than two species exist, especially when considering the use of practical hydrocarbon fuels. For fuels such as natural gas, CH4 , H2 , H2 O, CO and CO2 are all present, and the diffusion of one is coupled to the diffusion of all others. To properly analyze the transport across the electrode in such cases, the Stephan–

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Fig. 9.6 Diffusion driven transport through the electrodes of an SOFC.

Maxwell relationship for multi-specie diffusion is needed (Gemmen and Trembly, 2005): J Xj Nid − Xi Njd Nid dXi −nt . (9.6) = ∗k + dx Dij∗ Di j =1

In Equation (9.6), x is the direction of flux, nt [mol m−3 s−1 ] is the total molar density, X [1] is the mole fraction, Nd [mol m−2 s−1 ] is the mole flux due to molecular diffusion, D ∗k [m2 s−1 ] is the effective Knudsen diffusion coefficient, D ∗ [m2 s−1 ] is the effective bimolecular diffusion coefficient (Dij∗ = Dij ε/τ ), ε is the porosity of the electrode, τ is the tortuosity of the electrode, and J is the total number of gas species. Here, a subscript denotes the index value to a specific specie. The first term on the right of Equation (9.6) accounts for Knudsen diffusion, and the following term accounts for multicomponent bulk molecular diffusion. Further, to account for the porous media, along with induced convection, the Dusty Gas Model is required (Mason and Malinauskas, 1983; Warren, 1969). This model modifies Equation (9.6) as: ⎡ J N ⎤ m J ∗k m=1 Dm Xj Ni − Xi Nj B0 Xi Ni dXi ⎣ ⎦ , (9.7) −nt − = ∗k + m dx Dij∗ Di Di∗k 1 + B0 Jm=1 X∗k j =1

Dm

where Nt [mol m−2 s−1 ] is the total mole flux, B0 [m2 s−1 ] is the effective permeation coefficient, and nt [mol m−3 ] is the total molar density which can be related to pressure through the ideal gas law. The three transport parameters, ε, τ , and B0 , need to be determined experimentally via correlation with the model. As Table 9.1 indicates, diffusion behavior can exist over a range of timescales depending on the domain of interest (e.g., cathode versus anode electrode). Hence, if one is interested in cell dynamics on the order of 10−5 s to 10−3 s, then the transient nature of this transport should also be considered using the following transport


285

equation: ε

dNi dni =− + Ri , dt dx

(9.8)

where ni [mol·m−3 ] is the concentration of specie i, Ni [mol·s·m−2 ] is the molar flux of specie i (calculated from Equation (9.7)), and Ri [mol m−3 s−1 ] is the rate of production of specie i due to chemical reactions such as internal reforming and water-gas shift. An example for the application of this model is given in Sections 9.4.4 and 9.5.5.

9.3.3.3 Gas Channel Energy and Reactant Transport The analysis of the conditions within a gas channel can also be assumed to be onedimensional given that the changes in properties in the direction transverse to the streamwise direction are relatively small in comparison to the changes in the streamwise direction. In this section, we examine the transport in a fixed cross-sectional area gas channel. The principle conserved quantities needed in fuel cell performance modeling are energy and mass. A dynamic equation for the conservation of momentum is not often of interest given the relatively low pressure drops seen in fuel cell operation, and the relatively slow fluid dynamics employed. Hence, momentum, if of interest, is normally given by a quasi-steady model, −dP Af d M˙ fi dAi = + , (9.9) dx dx dx while both gas channel thermal energy and reactant mass transport are given by a one-dimensional model, respectively, as: qi dAi dρCp T dρCp T V =− + + Gg , (9.10) dt dx Af dx si dAi dNi dηi =− + + Ri , (9.11) dt dx Af dx where x is now the coordinate in the streamwise direction, M˙ is the momentum flow at any streamwise location (M˙ = ρV 2 Af ); P is pressure; fi is the wall shear force per unit area at each bounding surface i; Cp is the gas mixture heat capacity; T is temperature; V is velocity; qi and si are the heat flux and mole flux, respectively, into the elemental control volume at each bounding surface i; dAi is the respective area of the bounding surface; Af is the flow cross-sectional area; and G g is the hi Ri internal thermal energy production due to any gas phase reactions (Gg = where hi [J·mol−1 ] is the absolute enthalpy of specie i), where Ri is the gas phase reaction for specie i. For the case of gas channel reactant transport, however, the species production, Ri , can normally be ignored (and therefore also the internal thermal energy production, Gg ) unless catalysts are present to support fast reactions (Gupta et al., 2005). Without some form of catalyst, reaction kinetics of H2 , H2 O,

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CO, and CO2 at the temperatures of an SOFC (e.g.,

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